Results and open questions

Claims

Browse definitions, theorems, conjectures, counterexamples, and open problems from across FPRD Lab. Each record includes its exact statement, supporting evidence, dependencies, limitations, and links to related papers.

902 indexed claims and results

Starting points

A suggested reading order

01
Automata and formal languages · proof
A direct scaffold for even palindromes
Conditional on the complete Draft 5 Section 11.1 sequence contract, persistent receipts and simultaneous records give a direct finite-scaffold and PEG route with symbolic resource bounds.
FPRD-T128 Read the publication
02
Automata and formal languages · mechanically checked
Strict real-time multitape machines compile to scaffolds
A mechanically checked construction turns each fixed-tape real-time machine step into one bounded persistent scaffold update.
FPRD-T89 Read the publication
03
Automata and formal languages · informal proof
Reversed real-time multitape languages sit properly inside PEG
The transfer theorem and classical palindrome recognition together place even palindromes in PEG.
FPRD-T90 Read the publication
04
Automata and formal languages · mechanically checked
The scaffold-to-PEG direction is mechanically checked
The full sufficient direction of the Loff–Moreira–Reis correspondence is formalized for finite scaffolding automata.
FPRD-T110 Read the publication
05
Logic, semantics, and rewriting · proof
Local moves generate final behavioral equivalence
In the unary distance-one setting, detours, absorptions, garbage moves, and carrier slides connect exactly the histories with the same final behavior.
FPRD-T135 Read the publication
06
Logic, semantics, and rewriting · proof
Unary scaffold histories admit a finite coherent presentation
A finite family of cubical and bounded critical cells generates all parallel paths inside a final behavioral fibre.
FPRD-T137 Read the publication
07
Algebra and discrete mathematics · proof
Explicit metric-group products on the half-line
A nested readout transports bitwise XOR to group laws on the nonnegative reals whose product is also a metric.
FPRD-T115
08
Algorithms and complexity · mechanically checked
The direct GapCVP reduction reaches 1/30-hardness
Independently checked parameter choices preserve the direct reduction while exposing a retained-architecture frontier at 1/28.
FPRD-T114 Read the publication
09
Models of computation · informal proof
Compact finite feasibility sews into one presentation
Under the stated compactness hypotheses, feasibility through every observation horizon yields one uniform exact presentation.
FPRD-T60
10
Models of computation · informal proof
Escape modulus detects failure of exact sewing
Bounded escape identifies membership in one compact resource core; unbounded escape records finite feasibility without uniform realization.
FPRD-T61

Browse by research area

Automata and formal languagesAutomata, grammars, parsing, language recognition, residuals, pushdown structure, and scaffolding automata.203 claims Models of computationFinite persistent structures, abelian networks, schedule-sensitive computation, geometric machines, and exact interfaces between models.196 claims Algorithms and complexityReductions, lower bounds, automata resources, streaming memory, linear recurrences, matrix powers, and arithmetic circuits.260 claims Logic, semantics, and rewritingOperational semantics, causal equivalence, rewriting histories, confluence, cubical coherence, proof dynamics, and verification.50 claims Algebra and discrete mathematicsTensor categories, representation theory, elliptic integrable systems, mathematical physics, formal series, semirings, metric groups, linear invariants, and discrete topology.138 claims Number theory and discrete dynamicsInteger complexity, arithmetic iteration, guarded-affine traces, modular seams, recurrence, orbit structure, recovery budgets, and contraction.55 claims

All claims and evidence

902 records

Search by statement, research area, result type, or contribution assessment. Each row links to its proof or evidence, sources, dependencies, limitations, and open work.

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902 of 902 entries

Stable ID and statement	Type and current status	Contribution assessment	Research area and topic	Proof or evidence	Dependencies	Source and review	Next action
FPRD-CK-T01 For the standard right-tetrahedron/left-edge Laurent presentation in Chen–Kricker Section 5, the six-root shape relation module is exactly the intersection of the ambient Panitch–Park relation module with the six-root shape torus: $S\cap\mathcal R_{\mathrm{ambient}}=J$ . Hence the Chen–Kricker comparison map is an isomorphism.	Resolution of Chen–Kricker Conjecture 5.27Proved and independently hostile-audited within FPRD Lab; no external review	Evidence and limits →	Models of computationQuantum topology	Full proof	Chen–Kricker Section 5 presentation and their surjective comparison map; Panitch–Park ambient face-cone presentation	FPRD Lab degree-zero descent research recordReviewed 2026-09-08No external or specialist review is recorded; the proof has passed an independent hostile internal audit.	Obtain external specialist review and monitor the source conjecture for revisions or an author-confirmed sidedness correction.
FPRD-CK-C01 Under Chen–Kricker's common orientation, labeling, product and state conventions, the comparison isomorphism intertwines their named extended Garoufalidis–Yu and normalized Panitch–Park traces.	Corollary · source-dependent trace comparisonProved from FPRD-CK-T01 and Chen–Kricker's named-trace theorem	Evidence and limits →	Models of computationQuantum topology	Corollary and convention boundary	FPRD-CK-T01; Chen–Kricker Theorems 4.9 and 5.34	FPRD Lab degree-zero descent research recordReviewed 2026-09-08No external or specialist review is recorded; the proof has passed an independent hostile internal audit.	Use the completed generic-parameter local-collar dictionary as supporting evidence; the surviving frontier is actual exterior edge relations, intrinsic disk/product structure, and unrestricted nonvanishing.
RA-01 The explicit CFG generates U unambiguously, including arbitrary ignored data tails; reversal preserves unambiguity.	PropositionProved in the manuscript; no external review	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	Grammar and direct traversal invariants	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Audit the grammar independently on arbitrary malformed data.
RA-02 Full residual counts at arbitrary update cuts and canonical query prefixes are powers of the corresponding binary matrix ranks; fixed update prefixes give affine images with equal-sized fibres.	PropositionProved in the manuscript; no external review	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	RA-01; rank-nullity	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Check the full-continuation interpretation independently.
RA-03 Terminal-marker hard and easy families have equal residual counts at every update cut and query prefix, including after any fixed initial update segment, and matching probability multisets under uniform completions. Their access-cost separation uses Ko's lower bound.	TheoremProfile identities proved; access exclusion depends on Ko	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	RA-02; Ko 2026, Theorem 1.1 for the access exclusion only	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Seek an independent review of the construction and its novelty.
RA-04 A fixed SCA for U would solve arbitrary n-squared-row Multiphase Inner Product with constant coordinate-update cost, logarithmic query cost, logarithmic words, and polynomial space.	PropositionReduction proved in the manuscript; no external review	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	RA-01; canonical encoding; fixed-degree, fixed-radius SCA definition	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Independently audit the external lower-bound model fit.
RA-05 Under Ko's Multiphase Inner Product exclusion and the Loff–Moreira–Reis reversal theorem, U reversed is an unambiguous CFL outside PEG.	CorollaryDependent on explicitly cited external theorems	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	RA-01; RA-04; Ko 2026; Loff–Moreira–Reis 2020, Theorem 16	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Obtain external review; retain Ko as an explicit dependency.
RA-06 Under the same external premises, U union U reversed is self-reversing and unambiguous, and belongs to neither PEG nor SCA.	CorollaryDependent on explicitly cited external theorems	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	RA-05; disjoint first-symbol alphabets; regular-intersection closure	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Review the closure and unambiguity arguments independently.
RA-07 Neither U nor its reversal is linear context-free, by a regular slice, a homomorphism, and the classical linear pumping lemma.	PropositionProved in the manuscript; classical proof method	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	RA-01; Horváth–Nagy 2010, Lemma 6	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Seek a different witness if linearity and unambiguity are both required.
RA-08 Both direct orderings of the CFG path rules fail, as do the corresponding outer end-guard repairs; explicit nested counterexamples are given. U's PEG status remains open.	Counterexample / failed approachSpecified conversions refuted; general PEG question open	Evidence and limits →	Models of computationFormal languages and automata	Manuscript proof and evidence	RA-01; ordinary PEG ordered-choice semantics	Remembering information and reaching it, research note, revision 4Reviewed 2026-09-05None documented for this work.	Find a different construction for U; the failed conversions do not prove non-membership.
FPRD-D38 Fix the accelerated map $T(n)=n/2$ for even $n$ and $T(n)=(3n+1)/2$ for odd $n$ . A guarded-affine trace records a chronological word $w\in\{0,1\}^k$ , its affine action $T^k(n)=(3^s n+c_w)/2^k$ , its guard $r_w\equiv-c_w3^{-s}\pmod{2^k}$ , and—at a fixed layer $(K,S)$ — $D=2^K-3^S$ and $\Pi(w)=c_w-Dr_w$ .	DefinitionFixed and self-contained; no Collatz-conjecture implication	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Guarded-affine presentation and worked trace	No recorded dependencies	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Compare the presentation vocabulary with the closest parity-vector and 2-adic literature before any novelty language.
FPRD-T138 If a prefix $u$ has weight $p$ , endpoint $y_u$ , multiplier $a=3^p$ , and guard $r_u$ , then its exact horizon- $\ell$ action on a suffix guard is the affine permutation $F_u^{(\ell)}(q)=a^{-1}(q-y_u)\pmod{2^\ell}$ .	TheoremProved by direct modular substitution; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Exact affine sewing proof	FPRD-D38	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Compare the formulation with the closest parity-vector and 2-adic conjugacy literature before any novelty language.
FPRD-T139 For every $\ell\ge3$ , the prefixes $u_s=1^s$ induce $2^{\ell-2}$ distinct exact horizon- $\ell$ seam maps. Thus an interface naming the complete map on every suffix guard needs at least $\ell-2$ bits, while slope and intercept use $O(\ell)$ bits; exact seam-profile demand is $\Theta(\ell)$ .	Theorem · exact resource boundProved for the explicitly defined exact-profile interface; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Exact seam-profile demand proof	FPRD-D38; FPRD-T138	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Keep this exact-interface bound separate from recognition lower bounds; seek a closer comparator for the profile-count formulation.
FPRD-T140 For a contracting layer $(K,S)$ and split $w=uz$ after $j$ symbols, $\Pi(uz)=2^j(B_u-\delta_z-D\mu_z)$ , where $\delta_z\ge0$ is loss from the best fixed-weight suffix and $\mu_z\ge0$ is the modular seam lift.	Theorem · exact decompositionProved algebraically and finitely corroborated; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Prefix recovery calculus and proof	FPRD-D38; FPRD-T138	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Use the exact decomposition only with joint guard–disorder information; marginal counts have already failed to give contraction.
FPRD-T141 Prefix budgets satisfy $2B_{ub}=B_u-\varepsilon_bD-bG_{L,q}$ . Sending a child slot $\nu$ to the parent slot $\varepsilon_b+2\nu$ gives disjoint parity images, so $N^{\mathrm{slot}}_{u0}+N^{\mathrm{slot}}_{u1}\le N^{\mathrm{slot}}_u$ and the level potential is nonincreasing.	Theorem · conservation lawProved; equality cases refute uniform one-step contraction; internally audited	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	One-bit recurrence and slot proof	FPRD-D38; FPRD-T140	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Do not seek a universal one-step factor loss; any improvement must use block structure or a stronger joint statistic.
FPRD-T142 If $t_1\ge\cdots\ge t_L\ge0$ are decreasing guard thresholds and $A_i\ge0$ are unconditional bounds for increasing disorder sublevels, then $M_L=t_L+\sum_{i=2}^L\min(t_{i-1}-t_i,A_i)+A_1 \ge\min_i(t_i+A_i).$	Theorem · no-gain obstructionProved by an elementary induction and internally audited; no external review	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Nested-threshold theorem and proof	Nonincreasing thresholds; Nonnegative unconditional sublevel bounds	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Do not add more unconditional Ferrers bands; seek a statistic that retains the joint guard–disorder correlation.
FPRD-T143 Let $N_u^{\mathrm{cap}}$ bound admissible guard extensions, $W_u=\binom{L}{q}$ count fixed-weight suffixes, $m_u=\min(N_u^{\mathrm{cap}},W_u)$ , and $P_h=\sum_{\|u\|=h}m_u$ . Then the accepted count is at most $P_{h+1}\le P_h$ , with equality at full depth, and the local loss obeys the exact bottleneck identity recorded below.	Theorem · capped potentialWritten all-length proof; internally audited; no independent checker for this theorem	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Lowering-capped potential proof	FPRD-D38; FPRD-T140	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Seek joint guard–disorder loss; the capped potential alone gives no uniform factor contraction.
FPRD-C03 Determine whether the forced continuation admits an all-length joint estimate for $r_w\le R,\qquad \delta_w\le B-Dr_w,$ strong enough to improve the leading candidate exponent, or instead construct an exact accepted-trace counterfamily showing that actual forced-suffix disorder can preserve exponential mass.	Open problemOpen; marginal routes are closed and no exponent improvement is claimed	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Surviving frontier and exact alternatives	FPRD-T140; FPRD-T142; FPRD-T143; FPRD-FAIL-COLLATZ-LF5	Guarded-affine trace dynamics proof packageReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Analyze the actual uniquely forced suffix, not another marginal capacity envelope or a larger finite enumeration.
FPRD-FAIL-COLLATZ-LF3 The exact monotonicity $P_{h+1}\le P_h$ and its additive bottleneck-loss identity do not imply a uniform bound $P_{h+1}\le\rho P_h$ with $\rho<1$ ; capacity-one paths can preserve their only slot.	Failed approach · negative boundaryClosed as a universal contraction route	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Why monotonicity supplies no universal rate	FPRD-T143; FPRD-FAIL-COLLATZ-LF5	Collatz post-freeze failure ledger · LF3Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Use the monotone potential for pruning and exact accounting only; require a stronger joint statistic for rate loss.
FPRD-FAIL-COLLATZ-LF4 At layer $(K,S)=(27,17)$ and depth $h=14$ , exact reconstruction gives $2{,}321$ live prefixes, all with capped multiplicity one, so $P_{14}=2{,}321$ and $F_{14}=0$ ; forced continuation therefore does not by itself mean that few traces survive.	Computational finding · failed scarcity proxyIndependently reproduced with an exact-integer checker	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Exact LF4 reconstruction and evidence boundary	FPRD-T143	Collatz post-freeze failure ledger · LF4Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Preserve the executable regression and avoid treating this finite row as an asymptotic contraction result.
FPRD-FAIL-COLLATZ-LF5 For every fixed $C>0$ , there are $2^{\gamma K-O(\log K)}$ genuine fixed-weight-feasible capacity-one prefixes, with $\gamma=(1-\log_3 2)(1-\log_2(3/2))=0.1531779921\ldots,$ that cross a $C\log_2K$ block with zero attrition under maximal-completion capacity.	Theorem · counterfamilyWritten all-length proof; critical-layer estimate has an external dependency	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Forced-family obstruction and proof boundary	Critical-layer estimate h=Z+O(log K); FPRD-T143	Collatz lowering-and-contraction supplement · Section 6Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Move to actual forced-suffix disorder; do not seek a universal capacity-only contraction over logarithmic blocks.
FPRD-FAIL-COLLATZ-LF7 The marginal cap $m_u=\min(N_u^{\mathrm{cap}},W_u)$ records how many guard slots and fixed-weight tails exist, but not which tail occupies which slot or its actual disorder; it is therefore insufficient by itself to resolve the forced-continuation frontier.	Failed abstraction · limitationSubstantiated as a limitation; the replacement joint statistic remains open	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Why the marginal cap loses the matching	FPRD-T143; FPRD-FAIL-COLLATZ-LF5; FPRD-C03	Collatz post-freeze failure ledger · LF7Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Track the joint assignment of exact guards to actual forced suffix deficits.
FPRD-FAIL-COLLATZ-LF8 The strongest preserved candidate estimate remains exponential, $2^{0.34416K+o(K)}$ ; none of the guarded-affine, marginal-capacity, or forced-family results proves finiteness of paradoxical sequences, Terras's conjecture, or the Collatz conjecture.	Nonclaim · scope boundaryExplicitly established as the program boundary	Evidence and limits →	Number theory and discrete dynamicsGuarded-affine trace dynamics	Explicit no-Collatz boundary	FPRD-T142; FPRD-T143; FPRD-FAIL-COLLATZ-LF5; FPRD-C03	Collatz post-freeze failure ledger · LF8Reviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Preserve dormancy unless a concrete joint guard–disorder lemma, exact counterfamily, or external audit reopens the route.
CRD-RW-1a In the occurrence-labelled rewriting system $R=\{aa\to a\}$ , the complete reductions $a^n\Rightarrow^*a$ are in bijection with the $(n-1)!$ orders in which the original gaps are deleted. After identifying consecutive contractions with disjoint residual supports, the equivalence classes are the $C_{n-1}$ planar full binary trees on $n$ leaves.	Theorem · exact enumerationProved in the stated occurrence-labelled model	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Enumeration and binary-tree proof	No recorded dependencies	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-1aReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the classification with standard trace-monoid and associahedral formulations.
CRD-RW-1b After adjoining the contextual overlap move $((AB)C)\leftrightarrow(A(BC))$ to the disjoint-square moves, every pair of complete occurrence-labelled reductions $a^n\Rightarrow^*a$ is connected.	Theorem · connectednessProved for complete reductions in the stated model	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Rotation connectedness proof	CRD-RW-1a	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-1bReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine which higher cells are required for coherence beyond connectedness.
CRD-RW-2a For complete occurrence-labelled reductions of $a^4$ , the graph generated by disjoint and overlap moves is the six-cycle $C_6$ . Contracting its unique disjoint-scheduling edge gives the five-cycle $C_5$ , the rotation graph of the five planar binary trees on four leaves.	Proposition · finite classificationProved by complete enumeration and edge classification	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Six-history enumeration	CRD-RW-1a; CRD-RW-1b	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-2aReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Retain the raw and schedule-quotient graphs as distinct presentations in later coherence arguments.
CRD-RW-2b The unique loop in the arity-four transformation graph survives backtrack cancellation and residual-interchange squares. In the fixed presentation, attaching one 3-cell along the raw hexagon—or equivalently along the quotient pentagon—is necessary and sufficient to fill it.	Proposition · coherence cellProved for the fixed arity-four component	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Loop and cell-attachment proof	CRD-RW-2a	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-2bReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Test whether the arity-four filler participates in a uniform all-arity presentation.
CRD-RW-3a The five-leaf binary-tree rotation complex $K_5$ has 14 vertices, 21 edges, six contextual pentagons, and three residual-interchange squares. Its nine faces are indexed by the proper contiguous clusters of the five leaves.	Theorem · cell-complex classificationProved for the schedule-quotient five-leaf complex	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Face enumeration and completeness proof	CRD-RW-2b	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-3aReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Generalize the cluster-indexing argument without assuming the five-leaf enumeration.
CRD-RW-3b Every edge of $K_5$ lies in exactly two native faces. Removing one pentagon leaves a collapsible complex; consequently $K_5\simeq S^2$ , with $H_1(K_5)=0$ and $H_2(K_5)\cong\mathbb Z$ .	Theorem · homotopy typeProved by an explicit elementary-collapse certificate	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Edge-incidence and collapse proof	CRD-RW-3a; CRD-RW-F5	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-3bReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Investigate whether comparable collapse certificates exist in higher associahedral dimensions.
CRD-RW-F1 In the terminating confluent system $aa\to a$ , swapping consecutive disjoint redex contractions does not connect all complete reductions with the same source and normal-form endpoint: the two reductions of $a^3$ begin at overlapping redexes and lie in different square classes.	Counterexample · failed completeness claimRefuted in the stated occurrence-labelled model; repaired by adding contextual overlap moves	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Self-contained counterexample and repair	CRD-RW-1a; CRD-RW-1b	Comparative Recognition Dynamics claim ledger · CRD-RW-F1; failure ledger · CRD-F-5Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Retain the counterexample as the gate against square-only history calculi; audit the all-arity overlap calculus before any coherence claim.
CRD-RW-F2 For $aa\to a$ , the sequence of underlying words is too coarse to recover redex independence or overlap: every complete occurrence-labelled reduction of $a^n$ projects to the same chain $a^n\to a^{n-1}\to\cdots\to a$ .	Representation obstructionRepresentation doctrine refuted for history classification	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Projection obstruction and replacement representation	CRD-RW-1a	Comparative Recognition Dynamics claim ledger · CRD-RW-F2; failure ledger · CRD-F-6Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Keep occurrence, redex-position, and residual-provenance data whenever the question concerns independence of histories.
CRD-RW-F3 For complete occurrence-labelled reductions of $a^4$ under $aa\to a$ , the graph generated by disjoint-square and contextual-overlap moves is connected but has $\pi_1\cong H_1\cong\mathbb Z$ ; connectedness therefore does not establish coherence.	Negative result · coherence obstructionConnectedness-to-coherence inference refuted; one higher cell repairs the fixed component	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Hexagon/pentagon loop and higher-cell repair	CRD-RW-2a; CRD-RW-2b	Comparative Recognition Dynamics claim ledger · CRD-RW-F3; failure ledger · CRD-F-7Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Require an explicit higher-dimensional coherence criterion and audit it beyond the fixed arity-four component.
CRD-RW-F4 Disjoint leaf intervals are not a complete test for independent contextual rotations: at five leaves, an inner rotation on $((AB)C)$ commutes with an outer rotation $((XD)E)\to X(DE)$ even though the inner leaf support is strictly contained in the outer support.	Counterexample · definition obstructionLeaf-support criterion refuted; active-constructor support survives	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Nested interchange witness and finite graph audit	CRD-RW-2b; CRD-RW-3a	Comparative Recognition Dynamics claim ledger · CRD-RW-F4; failure ledger · CRD-F-9Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use residual commutation or disjoint active-constructor support when enumerating interchange squares at five leaves and beyond.
CRD-RW-F5 For the five-leaf square–pentagon complex, the calculation $H_1=0$ cannot by itself establish $\pi_1=0$ , because first homology records only the abelianization of the fundamental group.	Failed proof methodHomology-only proof withdrawn; conclusion recovered by an explicit collapse certificate	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Proof-method boundary and collapse repair	CRD-RW-3a; CRD-RW-3b	Comparative Recognition Dynamics claim ledger · CRD-RW-F5; failure ledger · CRD-F-10Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use the explicit elementary-collapse certificate as the proof; retain the homology calculation only as corroboration.
CRD-SAT-10a If every clause in a strict $r$ -CNF has $r$ distinct literals, then every consistent seed assigning fewer than $r-1$ variables has trivial unit-propagation closure: no clause can yet be unit or false.	Lemma · interaction-depth boundProved for strict finite r-CNF databases	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Strict-width lemma	No recorded dependencies	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare this seed-depth formulation with standard notions of propagation hardness and absorption.
CRD-SAT-10b For $D=(x\vee\neg z\vee\neg p)\wedge(y\vee\neg z\vee p)$ and $D^+=D\wedge(x\vee y\vee\neg z)$ , the databases are semantically equivalent and have the same singleton unit-propagation signature, yet fixed false-first DPLL in the order $x<y<z<p$ uses three decisions on $D$ and two on $D^+$ , with no conflicts.	Theorem · exact separating exampleProved for the displayed databases and fixed DPLL policy	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Second-order separation proof	CRD-SAT-10a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Use the parameterized CRD-SAT-11 family for conclusions across clause widths.
CRD-SAT-10c A satisfiable strict-3CNF needs at least four variables and at least two source clauses to entail a distinct strict width-three addition; the databases in CRD-SAT-10b attain both lower bounds.	Proposition · analytic minimalityProved for distinct entailed strict-3 clauses	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Minimality proof	CRD-SAT-10b	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Retain the strict, satisfiable, same-width scope when citing the minimality claim.
CRD-SAT-11a For every $r\ge3$ , there are connected, semantically equivalent strict- $r$ -CNF databases $D_r$ and $D_r^+$ whose unit-propagation signatures agree through seed size $r-2$ and first differ at seed size $r-1$ . The added clause is a pure resolvent and a reverse-unit-propagating consequence.	Theorem · sharp parameterized familyProved for every clause width r at least three	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Arbitrary-width interaction-depth proof	CRD-SAT-10a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare the family with standard propagation-hardness and absorption examples before making a novelty claim.
CRD-SAT-11b Under false-first DPLL order $x<y<z_1<\cdots<z_{r-2}<p$ , exhaustive unit propagation, and stopping when every stored clause is satisfied, $D_r$ uses $r$ decisions and $D_r^+$ uses $r-1$ , with no conflicts on either successful trace.	Theorem · exact decision lawProved for the stated family and fixed DPLL interface	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Exact decision calculation	CRD-SAT-11a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Keep the fixed order, polarity, and syntactic stopping rule attached to the resource claim.
CRD-SAT-11c Across unbounded clause width, no fixed finite unit-propagation seed depth determines the fixed-policy DPLL decision count: for any $t\ge1$ , choosing $r=t+2$ gives semantically equivalent databases with equal signatures through depth $t$ but different decision counts.	Corollary · fixed-depth obstructionProved from the arbitrary-width family	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Fixed-depth obstruction	CRD-SAT-11a; CRD-SAT-11b	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Use CRD-SAT-12a for the complementary positive theorem at fixed added-clause width.
CRD-SAT-12a Let $G=F\cup A$ , where every added clause has width at most $w$ . Equality of the complete unit-propagation signatures through seed size $w-1$ , UP-absorption of every clause in $A$ by $F$ , and equality of UP closure and conflict outcomes on every consistent seed are equivalent.	Theorem · exact equivalenceProved for arbitrary CNF bases and finite bounded-width additions	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Fixed-width absorption theorem	No recorded dependencies	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare the definition and equivalence with standard SAT absorption terminology before making a novelty claim.
CRD-SAT-12b The exact unit-propagation completeness radius for additions of width at most $w$ is $w-1$ : depth $w-1$ always suffices, while for every $w\ge3$ there are semantically equivalent pairs agreeing through depth $w-2$ and separating at a seed of size $w-1$ .	Corollary · sharp boundProved from the absorption theorem and the preserved width-w family	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Sharp completeness-radius proof	CRD-SAT-12a; CRD-SAT-11a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare the exact radius with standard propagation-hardness and absorption formulations.
CRD-SAT-12c For a fixed variable order and polarity, exhaustive unit propagation, and stopping when every stored clause is syntactically satisfied, the equivalent conditions of CRD-SAT-12a imply that $F$ and $F\cup A$ have identical complete DPLL traces: the same closures, decisions, conflicts, and stopping points.	Corollary · trace preservationProved for the stated fixed-policy DPLL interface	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Fixed-policy DPLL corollary	CRD-SAT-12a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Keep the fixed-policy and syntax-insensitive scope explicit when comparing later reason-sensitive results.
CRD-SAT-13a For every $m\ge1$ , the connected strict-3CNF databases $F_m$ and $G_m=F_m\cup\{x\vee y\vee b\}$ are semantically equivalent and have identical unit-propagation closure and conflict outcomes on every consistent seed. The added clause is UP-absorbed and has the displayed $m$ -step pure-resolution derivation.	Theorem · closure equivalenceProved in the stated connected strict-3CNF family	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Absorption and complete UP-equivalence proof	CRD-SAT-12a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare the absorption formulation with standard SAT and proof-complexity terminology before making any novelty claim.
CRD-SAT-13b Under the fixed earliest-clause propagation policy, immutable first reasons, decisions $x=0$ then $y=0$ , and latest-trail first-UIP analysis, $F_m$ requires exactly $m+1$ resolution pivots while $G_m$ requires one. Both traces learn $x\vee y$ , make two decisions, and encounter one conflict.	Theorem · exact resource separationProved for the stated deterministic solver interface	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Fixed-policy first-UIP derivation	CRD-SAT-13a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Use CRD-SAT-14a and CRD-SAT-14b for priority-independent conclusions; retain this entry as the exact fixed-policy calculation.
CRD-SAT-F13 Semantic equality together with equality of complete all-seed unit-propagation closure does not determine first-UIP resolution work for a reason-sensitive solver interface.	Counterexample · representation obstructionRefuted by the CRD-SAT-13 family	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Closure-equivalent reason-compression counterfamily	CRD-SAT-13a; CRD-SAT-13b	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	When comparing proof work, retain reason provenance or state explicitly which canonical reason policy replaces it.
CRD-SAT-14a For every legal sequential unit-propagation schedule, reason choice, and detected conflict under $x=y=0$ , the first-UIP cost spectrum of $F_m$ is $\{m+1\}$ , while the spectrum of $G_m$ is $\{1,m+1\}$ . Every trace learns $x\vee y$ .	Theorem · all-schedule classificationProved for the stated sequential-UP/first-UIP model	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	All-schedule path-frontier proof	CRD-SAT-13a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Use the exact spectrum when comparing best available proof work with guaranteed online work.
CRD-SAT-14b The priority-independent minimum first-UIP costs are $c_{\min}(F_m)=m+1$ and $c_{\min}(G_m)=1$ , giving exact gap $m$ ; the maximum costs are equal: $c_{\max}(F_m)=c_{\max}(G_m)=m+1$ .	Corollary · exact extremal costsProved from the all-schedule classification	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Minimum and maximum cost corollary	CRD-SAT-14a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Test a different proof resource, such as minimum resolution width or clause space, rather than extending the classified path family.
CRD-SAT-14c Across all legal reason choices, $F_1$ has maximum analysis width three; for $m\ge2$ , the attainable maximum widths for $F_m$ are exactly $\{3,4\}$ ; every trace of $G_m$ has maximum width three.	Proposition · exact width spectrumProved in the stated all-schedule model	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Hostile width audit	CRD-SAT-14a	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Keep the width-four mixed-reason witness visible when comparing the fixed-policy and all-schedule statements.
CRD-SAT-F14 Retaining the absorbed shortcut $L$ does not guarantee that every legal propagation and reason schedule realizes the one-pivot conflict analysis.	Counterexample · scheduling obstructionRefuted by a legal long trace in the augmented family	Evidence and limits →	Algorithms and complexityComparative Recognition Dynamics · SAT and proof complexity	Delayed-shortcut countertrace	CRD-SAT-14a; CRD-SAT-14b	Comparative Recognition Dynamics cumulative proofs and claim recordReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Do not translate best available proof work into a guaranteed runtime improvement without a policy theorem.
AFD-11 The increasing binary ZILP function $f(x,y)=\lfloor x/3\rfloor+\lfloor y/3\rfloor+\mathbf1_{\{x\equiv y\equiv2\pmod3\}}$ belongs to $R_2\setminus R_1$ , so its exact floor depth is two.	Theorem · explicit separationProved internally; literature and priority audit pending	Evidence and limits →	Models of computationAbelian gates and chronology erasure	Binary depth-two theorem and proof structure	Holroyd–Levine–Winkler: binary ZILP upper bound; AFD-01 depth-one normal form; AFD-10 pulse-fiber lemma	Abelian Gates and Chronology Erasure · durable Run 5 and Run 8 artifactsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Conduct a focused primary-literature and priority audit before novelty or current-open language.
UPT-01 Every scalar function computed by a finite acyclic network of splitters, adders, and unprimed topplers satisfies $F(x+y)\ge F(x)+F(y)$ .	Lemma · universal invariantProved internally by structural induction	Evidence and limits →	Models of computationAbelian gates and chronology erasure	Superadditivity proof	Acyclic unprimed splitter/adder/toppler syntax	Abelian Gates and Chronology Erasure · durable Run 5 and Run 8 artifactsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit placement in the broader literature on floor functions and unprimed abelian gates.
UPT-02 The unary increasing ZILP function $h(n)=n-\mathbf1_{\{n\equiv2\pmod3\}}$ has periodic residual $P\le0$ but is not computable by any unprimed acyclic gate network.	CounterexampleProved internally; refutes a precise sufficiency conjecture	Evidence and limits →	Models of computationAbelian gates and chronology erasure	Sign-condition counterexample	UPT-01; Holroyd–Levine–Winkler: the sign condition is necessary	Abelian Gates and Chronology Erasure · durable Run 5 and Run 8 artifactsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Keep superadditivity in every future characterization attempt; do not revive sign-only sufficiency.
UPT-03 For a zero-at-zero unary ZILP function with residual period dividing three, unprimed-DAG computability, superadditivity, $2f(1)\le f(2)$ and $2f(2)\le f(3)+f(1)$ , and an explicit one-layer unprimed formula are equivalent.	Theorem · exact subclass classificationProved internally and finitely corroborated	Evidence and limits →	Models of computationAbelian gates and chronology erasure	Unary period-three theorem	UPT-01	Abelian Gates and Chronology Erasure · durable Run 5 and Run 8 artifactsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine whether superadditivity remains sufficient for every zero-at-zero unary ZILP function, or find the smallest later failing period.
UPT-04 For unary residual period dividing two, $P\le0$ is sufficient for an unprimed formula; therefore period three is the smallest unary coordinate period at which the sign condition alone fails.	Proposition · minimal counterexample periodProved internally and finitely corroborated	Evidence and limits →	Models of computationAbelian gates and chronology erasure	Period-two formula and minimality	UPT-02	Abelian Gates and Chronology Erasure · durable Run 5 and Run 8 artifactsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Retain period three as the exact first obstruction when studying larger unary periods.
FPRD-D29 A literal-local Fold-and-Look presentation uses a connected strip with finitely many input-controlled feature sites, an input-independent sequence of all-layers simple folds, no cutting, punching, or unfolding after the input is encoded, and one final through-hole observation. The standing geometric hypotheses are clean alignment, no crease through a punch, interiority, uniform overlay, and positive punch radius.	DefinitionPrecisely stated after correction of the zero-radius boundary case	Evidence and limits →	Models of computationFold-and-Look computation	Definition and standing hypotheses	No recorded dependencies	FPRD Fold-and-Look notes and verification recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add one worked example and one example excluded by each geometric hypothesis.
FPRD-T63 Under the hypotheses of FPRD-D29, the final observation is $\operatorname{Look}(w)=\bigvee_{y\in Y_{\mathrm{eff}}}\ \bigwedge_{p\in\Phi(y)}\ell_p(w)$ , where $Y_{\mathrm{eff}}$ contains the interior final locations with a nonempty fibre consisting entirely of feature sites.	TheoremRepaired after counterexample and rechecked within FPRD Lab	Evidence and limits →	Models of computationFold-and-Look computation	Incidence normal form and proof	FPRD-D29	FPRD Fold-and-Look notes and verification recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the fibre formulation directly with the source construction in An–Demaine–Demaine–Ku and add an exact bibliographic locator.
FPRD-T64 If a Fold-and-Look presentation realizes a Boolean function $g$ , then its maximum layer depth $D$ bounds the minimum DNF term width, and its number of potential feature sites $P_{\mathrm{sites}}$ bounds the minimum total literal occurrences. If $g$ is nonconstant, the number $S$ of effective variable-bearing fibres also bounds the minimum DNF term count.	TheoremProved and rechecked within FPRD Lab	Evidence and limits →	Models of computationFold-and-Look computation	Formula-resource inequalities	FPRD-D29; FPRD-T63	FPRD Fold-and-Look notes and verification recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Clarify which inequalities remain valid for some-layers folds and for nonuniform punch images.
FPRD-T65 For $n\ge1$ , the prime implicants of $\operatorname{MISMATCH}_{2n}$ are exactly the $2n$ essential width-two terms witnessing a disagreeing mirrored pair. Hence every literal-local Fold-and-Look presentation has $D\ge2$ , $S\ge2n$ , $P_{\mathrm{sites}}\ge4n$ , and $F\ge1$ ; a one-fold construction attains all four bounds.	TheoremProved; normal-form dependency rechecked within FPRD Lab	Evidence and limits →	Models of computationFold-and-Look computation	Mismatch prime implicants and optimal construction	FPRD-D29; FPRD-T64	FPRD Fold-and-Look notes and verification recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Give the one-fold construction a compact diagram and keep moved material separate from the four proved optimal coordinates.
FPRD-T66 For every permutation $\pi\in S_n$ , the matching language $L_\pi$ has exactly $2^n$ midpoint continuation rows. The former reflection-line/LIS fold lower bound is withdrawn: within FPRD-D29 a three-layer fibre may have same-orientation outer feature sites and an intervening TRUE or duplicate site, yet reduce to a width-two term. Whether equality of two halves requires $\Omega(\log n)$ folds in this model remains unresolved.	Partial theorem and failed lower-bound argumentExact row theorem proved; geometric separation withdrawn after counterexample	Evidence and limits →	Models of computationFold-and-Look computation	Matching rows and reflection-line counterexample	FPRD-D29; FPRD-T42; FPRD-T63	FPRD Fold-and-Look notes and verification recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Either prove a lower bound using complete reduced fibres, including constants and duplicate literals, or construct a sublogarithmic equality presentation.
FPRD-T67 Under FPRD-D29, $P_{\mathrm{sites}}\le 2^F(\Lambda_{\mathrm{fp}}/\delta+1)$ . Consequently a presentation of parity on $m$ bits satisfies $\Lambda_{\mathrm{fp}}/\delta+1\ge m2^{m-1-F}$ , which is exponential in $m$ whenever the fold budget is, for example, $F=O(\log m)$ .	Proposition and corollaryProved and rechecked within FPRD Lab	Evidence and limits →	Models of computationFold-and-Look computation	Fold, material, and precision coupling	FPRD-D29; FPRD-T64	FPRD Fold-and-Look notes and verification recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Extend the resource table to charge encoder work and the fan-in of the final observation.
FPRD-D22 A fixed-length recognition-cut datum consists of a finite nonempty alphabet, a total input length, a Boolean function on words of that length, the exact prefix-to-suffix future rows at every cut, strict reachable deterministic layered recognizers, deterministic one-way protocols across a cut, and fixed-length binary message cost.	DefinitionPrecisely stated with reachability and fixed-length conventions	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Recognition-cut definitions	FPRD-D06	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a two-cut worked example showing literal rows, the row quotient, the canonical layered states, and the corresponding messages.
FPRD-T42 For every fixed-length Boolean recognition function, the recognizer whose grade- $i$ states are the distinct exact future rows $R_{T,i}$ is correct, reachable, and reduced. Every reachable correct strict layered recognizer has a unique grade-preserving surjection onto it. Consequently $r_f(T,i)$ is the exact minimum state count at every grade, $\max_i r_f(T,i)$ is the exact minimum width, and every reachable reduced realization is uniquely grade-isomorphic to the canonical one.	TheoremProved and internally audited	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Canonical cut classes and layered width	FPRD-D22	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map the theorem explicitly to fixed-order branching programs and distinguish strict layered width from skip-permitting diagram size.
FPRD-T43 At every fixed cut $(T,m)$ , the distinct future rows are the minimum reachable message alphabet of an exact deterministic prefix-to-suffix one-way protocol. The exact fixed-length binary cost is $\lceil\log_2 r_f(T,m)\rceil$ . Every deterministic one-pass recognizer has at least $r_f(T,m)$ reachable complete configurations at that cut and at least the same ceiling-log number of mutable bits.	TheoremProved and internally audited	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Exact one-way cut cost and configuration transfer	FPRD-D22; FPRD-T39; FPRD-T41; FPRD-T42	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Separate the exact deterministic theorem from possible randomized, interactive, amortized, and uniform-streaming variants.
FPRD-T39 For generalized Dyck words of length $2N$ with $k$ bracket types and $\ell$ neutral symbols, let $A_{k,\ell}(N)=\sum_{h=0}^N k^h$ when $\ell\ge1$ , and $A_{k,0}(N)=\sum_{j=0}^{\lfloor N/2\rfloor}k^{N-2j}$ . The exact number of reachable midpoint future rows is $\rho_{k,\ell}(0)=1$ and $\rho_{k,\ell}(N)=A_{k,\ell}(N)+1$ for $N\ge1$ . Hence every exact deterministic one-pass recognizer needs at least $\lceil\log_2\rho_{k,\ell}(N)\rceil$ midpoint bits; the standard typed-stack recognizer matches this asymptotically.	TheoremProved, mechanically corroborated, and internally audited	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Common-midpoint rows and deterministic configurations	FPRD-D20; FPRD-T38	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a small table for one bracket type, two bracket types, and the neutral-free parity restriction.
FPRD-T41 For any finite controlled-Dyck system, total length $T$ , and cut $m$ , the reachable exact-length future signatures are exactly the distinct rows of the controlled Hankel cut. They form the unique reachable reduced deterministic layered recognizer up to layerwise isomorphism and have minimum cardinality at every layer. At $T=2N$ and $m=N$ , the minimum midpoint width is the number of distinct rows of the square controlled Hankel window.	TheoremProved, mechanically corroborated, and internally audited	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Exact rows and universal layered minimum	FPRD-D21; FPRD-T38; FPRD-T39	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Provide one nonsquare cut example and keep fixed-length layered minimality separate from global visibly-pushdown state minimization.
FPRD-D20 At level N, a generalized-Dyck Hankel profile records the residual rows visible to continuations of length at most N, the square prefix-versus-continuation window with both sides of length exactly N, their field and Boolean-semiring factor ranks, and the deterministic one-pass configuration interface at the midpoint of length-2N inputs.	DefinitionPrecisely stated with rank, row, and midpoint conventions separated	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Generalized-Dyck Hankel profile	FPRD-D18; FPRD-D19	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a level-two example showing the bounded window, exact midpoint window, zero row, and identity minor.
FPRD-D21 A controlled-Dyck future-quotient datum combines a generalized-Dyck stack alphabet with a finite total controller, the start-reachable stack/controller dynamic and absorbing failure state, bounded-height carriers with a tail point, at-most-length and exact-length Boolean future signatures, reachable graded summaries preserving readout, bonding, and named actions, and the corresponding fixed-length Hankel cuts and layered recognizers.	DefinitionPrecisely stated with reachable and relative-minimality boundaries	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Controlled-Dyck quotient specification	FPRD-D15; FPRD-D18; FPRD-D19; FPRD-D20	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a two-state controller example in which distinct raw configurations merge under a bounded future signature.
FPRD-T38 For generalized Dyck language $D_{k,\ell}$ , the bounded residual window through horizon $N$ has field rank and Boolean factor rank $S_k(N)=\sum_{h=0}^N k^h$ . Its exact length-N midpoint window has both ranks equal to $A_{k,\ell}(N)$ , the number of typed stacks reachable at that exact length, with all heights available when a neutral symbol exists and only parity-compatible heights otherwise.	TheoremProved, mechanically corroborated, and internally audited	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Bounded and exact midpoint rank proofs	FPRD-D20; FPRD-T34; FPRD-T35; FPRD-T37	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add small literal matrices illustrating the identity minor and the zero-row correction to deterministic width.
FPRD-T40 For every finite controlled-Dyck system and horizon $N$ , finite balanced-word saturation computes all reachable configurations through stack height $N$ , including those reached by arbitrarily long balanced detours, and bounded future signatures compute a finite quotient $S_N$ preserving readout, bonding, and named degree-lowering letter actions. The family $(S_N)$ is the minimum reachable reduced graded summary in its declared category. With the exact semantic quotient given the topology of agreement on bounded futures, its inverse limit is action- and readout-preserving homeomorphic to that quotient.	TheoremProved, mechanically corroborated, and internally audited	Evidence and limits →	Algorithms and complexityRecognition cuts and finite-state memory	Bounded quotients, universal property, and inverse limit	FPRD-D21; FPRD-T28; FPRD-T34; FPRD-T35; FPRD-T36; FPRD-T37	FPRD recognition-cut and Dyck-language proof recordsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a compact computed example of the quotient tower and clarify how its topology compares with standard visibly-pushdown semantics.
FPRD-D18 For a generalized Dyck alphabet with k bracket types and any number of neutral symbols, the exact residual carrier consists of finite typed stacks together with an absorbing failure state. A live stack s has future language K_s of words that empty it without failure; failure has the empty future language.	DefinitionPrecisely stated and self-contained	Evidence and limits →	Automata and formal languagesGeneralized-Dyck residual dynamics	Model and exact residuals	No recorded dependencies	FPRD Dyck-residual and finite-observation proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Add a small worked residual calculation for two bracket types and one neutral symbol.
FPRD-T34 Every left residual of a generalized Dyck language is exactly $K_s$ for one live stack $s$ , or $K_\bot$ ; all are reachable and pairwise distinct. Distinct live stacks satisfy $\operatorname{sep}(K_s,K_t)=\min(\|s\|,\|t\|)$ , while $\operatorname{sep}(K_s,K_\bot)=\|s\|$ . The induced ultrametric makes openers half-contractions, neutrals isometries, and closers at most $2$ -Lipschitz.	TheoremProved and internally audited	Evidence and limits →	Automata and formal languagesGeneralized-Dyck residual dynamics	Exact residual theorem	FPRD-D18	FPRD Dyck-residual and finite-observation proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare the transition monoid and ultrametric formulation with standard polycyclic-monoid treatments.
FPRD-T35 The depth- $N$ quotient keeps every stack of height at most $N$ and places all deeper stacks and failure in one tail class. It has $1+\sum_{h=0}^{N}k^h$ states, and its inverse limit reconstructs exactly the live finite stacks and failure. A word of length $r$ induces a natural graded map from level $N+r$ to level $N$ ; a same-level action by a closer is impossible in general.	TheoremProved and internally audited	Evidence and limits →	Automata and formal languagesGeneralized-Dyck residual dynamics	Finite shadows and reconstruction	FPRD-D18; FPRD-T34	FPRD Dyck-residual and finite-observation proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Publish a diagram of the bonding maps and one explicit failure of same-level closer action.
FPRD-D19 A complete finite-observation tower is a compact Hausdorff state space $X$ equipped with separating continuous surjections $q_N:X\to X_N$ to finite discrete quotients and compatible bonding maps. An action $F:X\to X$ has exposure at most $e$ when equality at observation level $N+e$ always forces equality of the images at level $N$ .	DefinitionPrecisely stated and self-contained	Evidence and limits →	Models of computationExposure-filtered finite observation	Finite observations and exposure	No recorded dependencies	FPRD Dyck-residual and finite-observation proof recordsReviewed 2026-09-09No documented external or specialist review of this FPRD result is recorded.	Find a useful application of the exposure distinction; the worked binary-digit example now explains zero, finite, and unbounded exposure.
FPRD-T36 For a complete finite-observation tower, an exact action $F$ has exposure at most $e$ exactly when it is represented by a unique compatible family $F_N^{[e]}:X_{N+e}\to X_N$ . Conversely, every such shifted-natural family reconstructs a unique continuous exact action. Exposures are closed under padding and add under composition; in the shortest-observation ultrametric, exposure $e$ is equivalent to a $2^e$ -Lipschitz bound.	TheoremProved and internally audited	Evidence and limits →	Models of computationExposure-filtered finite observation	Exposure representation theorem	FPRD-D19	FPRD Dyck-residual and finite-observation proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare the shifted-natural formulation with standard pro-object and profinite dynamical-system language.
FPRD-T37 On the full language carrier $\mathcal P(\Sigma^*)$ , the left derivative $\delta_u(K)=u^{-1}K$ has exposure at most $\|u\|$ , and this universal bound is sharp. The completed derivative orbit of any language has the same finite images as its literal derivative orbit and is reconstructed exactly from them; for regular languages the tower stabilizes to the usual finite residual automaton.	TheoremProved and internally audited	Evidence and limits →	Models of computationExposure-filtered finite observation	Derivative calibration	FPRD-D19; FPRD-T36	FPRD Dyck-residual and finite-observation proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Determine exact exposure on selected invariant language closures rather than relying on the full-carrier universal bound.
FPRD-D23 A compatible uniform cut field over a nonempty finite alphabet contains every literal Boolean table indexed by an exact prefix length and exact future length, the diagonal law that moves one named symbol across the cut, combined bounded-future signatures, prefix-budget rectangles, global residual quotients, horizon bonds, Boolean output, and exposure-one letter actions.	DefinitionPrecisely stated and self-contained	Evidence and limits →	Algorithms and complexityUniform residual quotients and growth	Uniform cut field	FPRD-D02; FPRD-D12; FPRD-D19; FPRD-D22	FPRD uniform-cut and residual-growth proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Add one complete two-letter example showing exact cut cells, a combined row, a horizon bond, and a letter action.
FPRD-T44 Diagonally compatible literal cut fields are exactly the cut fields of unique languages. Their combined bounded-future quotients are the finite observations of the reachable residual orbit, their section maps are exposure-one derivatives, and the two iterated prefix-reach and future-observation constructions recover respectively the residual orbit and its bounded-observation completion, with exact actions and readout.	TheoremProved and internally audited	Evidence and limits →	Algorithms and complexityUniform residual quotients and growth	Assembly theorem	FPRD-D23; FPRD-T22; FPRD-T36; FPRD-T37	FPRD uniform-cut and residual-growth proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Compare the two iterated constructions with standard profinite and residual-automaton presentations.
FPRD-T45 A language is regular exactly when the global quotient counts $q_L(N)=\|Q_N(L)\|$ are bounded, equivalently when one adjacent pair satisfies $q_L(N)=q_L(N+1)$ . At the first global plateau, the shifted letter actions descend to same-level actions and the quotient is the unique minimal residual DFA. Uniformly small individual fixed-length cuts do not suffice: nonregular length-only languages can have one row and zero one-way bits at every fixed cut.	TheoremProved and internally audited	Evidence and limits →	Algorithms and complexityUniform residual quotients and growth	Global plateau theorem	FPRD-D23; FPRD-T22; FPRD-T36; FPRD-T43; FPRD-T44	FPRD uniform-cut and residual-growth proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Relate the global quotient sequence to established state-complexity and follower-set invariants without transferring incompatible constants.
FPRD-T46 Over the ordered binary alphabet, $q_L(N)\le 2^{2^{N+1}-1}$ for every language. One decidable language built from prefix-free Elias-gamma headers and table payloads attains this ceiling at every horizon simultaneously. For any language, simultaneous attainment is equivalent to density of its residual orbit in the full Boolean behavior shift and to equality of its bounded-observation completion with that full shift.	TheoremProved and internally audited	Evidence and limits →	Algorithms and complexityUniform residual quotients and growth	Maximum residual growth	FPRD-D23; FPRD-T22; FPRD-T36; FPRD-T37; FPRD-T44	FPRD uniform-cut and residual-growth proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Classify attainable growth and access profiles inside named finite recognition-presentation classes.
FPRD-T47 If a binary language realizes all $2^{V_N}$ horizon-N rows, its access radius is at least $V_N=2^{N+1}-1$ . The gamma-coded simultaneous maximizer realizes every row by radius at most $V_N+2\lfloor\log_2(N+1)\rfloor+1$ , and therefore has access $2^{N+1}+O(\log N)$ .	TheoremProved and internally audited	Evidence and limits →	Algorithms and complexityUniform residual quotients and growth	Access bounds	FPRD-T46	FPRD uniform-cut and residual-growth proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Determine whether a simultaneous maximizer can attain the volume lower bound with smaller additive overhead.
FPRD-T48 For every strictly increasing $f:\mathbb N\to\mathbb N$ , the exact residuals of $L_f=\{a^n b a^{f(n)}:n\ge0\}$ are precisely the pre-marker families $A_n$ , singleton completions $R_r=\{a^r\}$ , and the empty residual. At horizon $N$ , $q_f(N)=N+2+\|\{n:f(n)\le N-1\}\|$ .	TheoremProved and internally audited	Evidence and limits →	Algorithms and complexityUniform residual quotients and growth	Sparse-marker profile	FPRD-D23; FPRD-T22; FPRD-T44; FPRD-T45	FPRD uniform-cut and residual-growth proof recordsReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Identify finite presentation classes that constrain the jump terms hidden by the scalar quotient sequence.
FPRD-T49 For $L_f=\{a^n b a^{f(n)}:n\ge0\}$ , where $f:\mathbb N\to\mathbb N$ is strictly increasing, the first appearances of every active singleton row, pre-marker row, and empty row give exact formulas for the prefix/future rectangle (G_f(M,N)) and access radius (a_f(N)). A fixed scalar quotient history through horizon (N) can coexist with arbitrarily large access at that same horizon.	TheoremProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Sparse-marker access	FPRD-T48; FPRD-T47	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Classify which finite-presentation families constrain the jump term that controls access.
FPRD-T50 For a finite controller (M) intersected with a generalized-Dyck language, shortest balanced-controller distances determine exact raw-coordinate arrivals. Minimizing those arrivals inside each bounded semantic row and then maximizing computes (a_{L_M}(N)). If (B_M) is the largest finite balanced distance from an underlying-DFA-reachable source, then $a_{L_M}(N)\le\max\{1,(N+1)B_M+N\}$ ; the class-wide bound is pointwise optimal.	Theorem and algorithmProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Controlled-Dyck access calculus	FPRD-D24; FPRD-T40; FPRD-T44; FPRD-T49	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Compare the presentation certificate with established visibly-pushdown and weighted shortest-witness formulations.
FPRD-T51 For generalized-Dyck words whose stack height reaches at least $r\ge1$ , $q_{k,r}^{\ge}(N)=1+\sum_{h=0}^{N}k^h+\sum_{h=\max(0,2r-N)}^{r-1}k^h$ and $a_{k,r}^{\ge}(N)=\max(2r,N)$ . For words staying below (r), $q_{k,r}^{<}(N)=1+\sum_{h=0}^{\min(N,r-1)}k^h$ and $a_{k,r}^{<}(N)=\max(1,\min(N,r-1))$ .	TheoremProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Depth-threshold profiles	FPRD-D24; FPRD-T40; FPRD-T50	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Use the threshold families as calibrations when comparing access certificates across equivalent controllers.
FPRD-T52 Only controller states that actually begin balanced segments of legal prefixes are needed in the replacement proof. Their entry balanced diameter cannot increase under a surjective reachable DFA morphism. Minimizing this diameter over regular controllers with the same trace on a fixed Dyck language defines $\beta_D(L)$ and gives $a_L(N)\le\max\{1,(N+1)\beta_D(L)+N\}$ .	TheoremProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Dyck-trace minimax	FPRD-D25; FPRD-T40; FPRD-T50	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Investigate computable subclasses of the trace-class minimization problem and dependence on the chosen Dyck envelope.
FPRD-T53 For a controlled-Dyck language, entry-context residual access $\eta_D$ , top-level balanced residual access $\gamma_D$ , and first balanced acceptance flip $\tau_D$ satisfy $\beta_D\ge\eta_D\ge\gamma_D\ge\tau_D$ . The value $\eta_D$ is exactly computable by finite completion-summary saturation. Together with any minimized presentation it yields a finite minimax sandwich; endpoint equality certifies optimality. The defect $\sigma_D=\beta_D-\eta_D$ is nonnegative and limit-computable from above, and zero is semidecidable.	Theorem and exact finite procedureProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Semantic access hierarchy	FPRD-D25; FPRD-T51; FPRD-T52	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Find either a positive sewing-defect example or a proof that every controlled-Dyck trace has zero defect.
FPRD-T54 Let (K_3) contain epsilon and the nonempty one-bracket Dyck words whose terminal run of closing brackets has length $2\bmod 3$ . Its semantic entry access and trace-minimax entry cost agree: $\eta_D(K_3)=\beta_D(K_3)=4$ . Hence its sewing defect is zero.	Theorem and exact exampleProved; finite-state lower bound is computer-assisted and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Terminal-descent tradeoff	FPRD-D26; FPRD-T53	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine whether any controlled-Dyck trace has positive sewing defect.
FPRD-T55 The minimum controller size for $K_3$ is three, while the minimum size among presentations with legal-entry diameter at most four is six. Its exact nondominated state/access pairs are therefore $(3,6)$ and $(6,4)$ .	Theorem and exact finite-state classificationProved; the five-state exclusion is computer-assisted and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Modulus-three frontier	FPRD-D26; FPRD-T54	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Produce a portable unsatisfiability certificate if the finite-state encoding is revisited.
FPRD-T56 The modulus-two trace $K_2$ satisfies $\eta_D(K_2)=\beta_D(K_2)=4$ . Its natural two-state controller attains that cost, so the single nondominated state/access pair is $(2,4)$ .	Theorem and exact finite-state classificationProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Modulus-two frontier	FPRD-D27; FPRD-T53	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Use the exceptional top-level residual at modulus two as a boundary test for broader terminal-descent constructions.
FPRD-T57 The modulus-four trace $K_4$ satisfies $\eta_D(K_4)=\beta_D(K_4)=6$ . Its natural state-minimal four-state controller has cost eight; an explicit six-state controller has cost six; and no controller with at most five states attains cost six. The exact frontier is $(4,8)$ and $(6,6)$ .	Theorem and exact finite-state classificationProved; the five-state exclusion is computer-assisted and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Modulus-four frontier	FPRD-D27; FPRD-T55	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek a checkable certificate for the five-state exclusion and avoid extrapolating an unproved minimum-state law.
FPRD-T58 For terminal-descent traces, $\eta_D(K_2)=\beta_D(K_2)=4$ , while for every $m\ge3$ , $\eta_D(K_m)=\beta_D(K_m)=2(m-1)$ . A uniform $2m$ -state height-residue/obligation controller attains the semantic lower bound for every modulus.	Theorem and uniform constructionProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	All-modulus theorem	FPRD-D27; FPRD-T53	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine the minimum number of states needed to attain optimal access beyond moduli two, three, and four.
FPRD-T59 For every even modulus $m$ , an $(m+2)$ -state positional controller has Dyck trace $K_m$ and entry cost $\beta_D(K_m)$ . At $m=4$ it is isomorphic to the explicit six-state controller in FPRD-T57.	Theorem and uniform constructionProved and internally audited	Evidence and limits →	Automata and formal languagesFormal languages and automata	Even-modulus construction	FPRD-D27; FPRD-T58	FPRD proofs on residual access and controlled-Dyck presentationsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine whether m+2 is minimal at optimal cost for any even modulus beyond four.
FPRD-D28 A presentation demand field records, for each target behavior $x$ and observation horizon $N$ , the upward-closed set $F_x(N)$ of target-independently bounded resources that realize the same horizon- $N$ observation as $x$ . Its exact feasible set is $F_x(\infty)$ ; first-obstruction depths and compact-core escape moduli record how finite fits fail to become exact presentations.	DefinitionSelf-contained definition	Evidence and limits →	Models of computationModels of computation	Definitions	FPRD-D02; FPRD-D23; FPRD-D25	FPRD presentation-demand and max-plus calibration proofsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Compare the field and its Pareto boundary with established approximation, identification-in-the-limit, and presentation-complexity terminology.
FPRD-T60 If matching observation cylinders form a neighborhood basis, then a target $x$ is feasible at resource $r$ through every finite horizon exactly when $x\in\overline{S_r}$ . Consequently a compact semantic stratum in a Hausdorff behavior space sews all uniformly bounded finite-horizon fits into one exact fit. For nested compact natural-number strata, exact demand is the supremum of finite-horizon demand.	TheoremProved and internally audited	Evidence and limits →	Models of computationModels of computation	Closure and compact sewing	FPRD-D28; FPRD-T03; FPRD-T44	FPRD presentation-demand and max-plus calibration proofsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Identify useful presentation classes whose semantic resource strata are closed or compact, and document their topology explicitly.
FPRD-T61 For an increasing compact exhaustion $S_{r,t}\uparrow S_r$ , exact membership in a core is equivalent to infinite escape modulus, while feasibility through every horizon is equivalent to unbounded escape moduli across the cores. Thus finite-horizon fitting fails to sew exactly at resource $r$ precisely when every core is excluded at some finite horizon but those exclusion horizons are unbounded.	TheoremProved and internally audited	Evidence and limits →	Models of computationModels of computation	Escape characterization	FPRD-D28; FPRD-T03; FPRD-T60	FPRD presentation-demand and max-plus calibration proofsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Find natural presentation classes with a genuine fixed-resource sewing defect, rather than demand that merely diverges across resource bounds.
FPRD-T62 For $F(w)=\max(\|w\|_a,\|w\|_b)$ , two one-state integer max-plus sequential components are exact, while no single finite sequential component is exact. Through horizon $N$ , the minimum width-one fit has $2\lfloor N/2\rfloor+1$ states and minimum canonical height $\lfloor N/2\rfloor$ . Hence width-one state and canonical-height demand grow linearly, whereas an exact width-two cover has constant resource $(2,2,1)$ .	Theorem and exact calibrationProved and internally audited	Evidence and limits →	Models of computationWeighted automata	Max-count calibration	FPRD-D28; FPRD-T60; FPRD-T61	FPRD presentation-demand and max-plus calibration proofsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Compare the finite-horizon state and canonical-height frontiers with standard weighted-automata approximation and sequentiality measures.
FPRD-D30 A pre-query causal-track presentation fixes a one-dimensional address line and finitely many persistent tracks before a future selector is known. At each input step a track holds or moves one address toward smaller coordinates; a demand is served only when a track occupies the demanded address at the required time.	Definition and model boundarySelf-contained model; classical multi-head context identified	Evidence and limits →	Algorithms and complexityPre-query causal tracks and fixed-cut factorization	Model and conventions	No recorded dependencies	FPRD causal-track and fixed-cut factorization proofsReviewed 2026-09-02No documented external or specialist review of this FPRD result is recorded.	Compare the pinned-coordinate convention with additional multi-head and streaming models before extending any lower bound beyond this exact presentation.
FPRD-T68 For the explicit even-palindrome survivor recurrence, virtual axes are stationary and query events form a two-dimensional dominance order, so offline persistent-track demand equals poset width. Width two can force three tracks online. With (r) late selector bits, no earlier demand, and targets (q(z)=(r+1)operatorname{value}_2(z)), every completed path has width one while exact pre-query demand is (2^r).	Theorem and exact separationProved and independently rechecked	Evidence and limits →	Algorithms and complexityPre-query causal tracks and fixed-cut factorization	Causal width theorem	FPRD-D30; Dilworth's decomposition theorem	FPRD causal-track and fixed-cut factorization proofsReviewed 2026-09-02No documented external or specialist review of this FPRD result is recorded.	Determine which selector encodings retain an exponential gap when target spacing or one-way movement is relaxed.
FPRD-T69 At one selected cut, the minimum number of complete snapshots in a (k)-track factorization equals the minimum number of cells partitioning the past set such that each cell is resolving for the positions readable from some (k)-track layout. For (d=2^r) independent bits spaced (r+1) apart and an (r)-bit selector, the exact number is (2^{max(d-k,0)}).	Theorem and exact frontierProved and independently rechecked	Evidence and limits →	Algorithms and complexityPre-query causal tracks and fixed-cut factorization	Fixed-cut factorization	FPRD-D30; FPRD-T68	FPRD causal-track and fixed-cut factorization proofsReviewed 2026-09-02No documented external or specialist review of this FPRD result is recorded.	Find compatibility conditions that sew optimal factorizations across multiple cuts or prove that no bounded compatible family exists.
FPRD-T70 For a trim deterministic sequential integer-weighted machine with $m$ states, raw height $h$ , pinned BFS access tree $T$ , and canonical height $H$ , one has $H\le\max\{D+2,\max_{e\notin T}\ell_T(e)\}h\le(m+1)h$ . After minimizing within a fixed state budget $n$ , $h_{f,\le n}\le H_{f,\le n}\le(n+1)h_{f,\le n}$ , and the coefficient $n+1$ is attained.	Theorem and sharp boundProved and independently rechecked	Evidence and limits →	Algorithms and complexityRaw and canonical height in weighted automata	Height coupling	FPRD-D28	FPRD weighted-automata height proofsReviewed 2026-09-02No documented external or specialist review of this FPRD result is recorded.	Compare the coupling envelope with standard weight-pushing bounds and isolate useful graph classes with a smaller coefficient.
FPRD-T71 On the augmented presentation graph, the minimum sup-norm over real state gauges equals the maximum absolute mean circulation over oriented closed walks, and the minimum over integer gauges is its ceiling. Initial and final weights are represented by edges to a boundary vertex, so there is no further integrality loss.	TheoremProved and independently rechecked; classical duality application	Evidence and limits →	Algorithms and complexityRaw and canonical height in weighted automata	Cycle-mean gauge duality	No recorded dependencies	FPRD weighted-automata height proofsReviewed 2026-09-02No documented external or specialist review of this FPRD result is recorded.	Seek specialist review and compare the fixed-graph formulation with standard weighted-transducer normalization algorithms.
FPRD-T72 There is one partial integer-valued target over $\{a,b,c,d\}$ and, for every $k\ge3$ , a horizon $N_k=k+1+2^k$ whose exact minimum state budget is $n_k=k+3\cdot2^{k-1}+2$ . Even after minimizing over every global length-and-constant shift, repaired raw height is one while repaired canonical height is $3\cdot2^{k-2}=(1/2-o(1))n_k$ .	Theorem and explicit separationProved and internally audited	Evidence and limits →	Algorithms and complexityRaw and canonical height in weighted automata	Parity-comb separation	FPRD-D28	FPRD weighted-automata height proofsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine whether the incompatible-carrier mechanism persists for a natural total rational series or under richer admissible baselines.
FPRD-T73 If an integer max-plus series has a trim unambiguous realization, then it is either sequential, with bounded finite-horizon state, raw-height, and canonical-height demand, or it is nonsequential, in which case both height demands grow as $\Theta(N)$ . No intermediate raw- or canonical-height rate occurs inside this unambiguous class.	Theorem and dichotomyProved and internally audited	Evidence and limits →	Algorithms and complexityRaw and canonical height in weighted automata	Unambiguous height dichotomy	FPRD-D28	FPRD weighted-automata height proofsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the quantitative finite-horizon formulation with the determinization and degree-of-sequentiality literature before making any positioning claim.
FPRD-T74 For $f_{\min}(w)=-\min(\|w\|_a,\|w\|_b)$ , intrinsic max-plus ambiguity degree and exact sequential cover width are both two. For every $N\ge2$ , the minimum width-one state demand is $2\lfloor N/2\rfloor+1$ , while both minimum raw height and minimum canonical height equal one. Thus the unambiguous height dichotomy cannot extend to ambiguity degree two.	Theorem and exact counterexampleProved and internally audited	Evidence and limits →	Algorithms and complexityRaw and canonical height in weighted automata	Min-count counterexample	FPRD-D28; FPRD-T73	FPRD weighted-automata height proofsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Seek a rational series with a genuinely sloped state-versus-canonical-height frontier rather than another flat corner example.
FPRD-T75 Let $f:\Sigma^\rightharpoonup G$ have nonempty domain $D$ , let $\Lambda_f(N)$ be its finite-horizon prefix-metric variation, and let $\beta_N$ be the largest shortest completion length among prefixes of $D\cap\Sigma^{\le N}$ . Then $\Lambda_f(N)\le h_f^1(N)\le\max\{\|f(w_)\|,(2\beta_N+3)\Lambda_f(N)\}$ , and for every domain $h_f^1(N)\le \|f(w_)\|+24(\|w_\|+1)\Lambda_f(N)+1$ . Moreover $\beta_N\le N$ ; if a trim $q$ -state NFA recognizes $D$ , its uniform completion radius is at most $q-1$ .	Theorem and constructive upper boundProved and independently reconstructed	Evidence and limits →	Algorithms and complexityHeight correction and ambiguity in weighted automata	Finite-horizon variation bound	FPRD-D28	FPRD proof using the Basso–Sidler Lipschitz extension theoremReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine whether either construction coefficient is optimal after minimizing over all unrestricted-state realizations.
FPRD-T76 For the exact partial domain $a^\#\cup a^\$$ , with $f(a^m\#)=x^m$ and $f(a^m\$)=y^m$ , set $d_m=x^{-m}y^m$ , $D(N)=\max_{m<N}\|d_m\|$ , and $E(N)=\max_{m<N}\max\{\|x^m\|,\|y^m\|\}$ . For $N\ge1$ , if $d_0,\ldots,d_{N-1}$ are distinct, then $A_f^1(N)=N+1$ , $D(N)/4\le h_f^1(N)\le\max\{\min(\|x\|,\|y\|),D(N)\}$ , and $E(N)/2\le H_f^1(N)\le E(N)$ . The raw-height window does not require delay distinctness.	Theorem and exact state countProved and independently audited	Evidence and limits →	Algorithms and complexityHeight correction and ambiguity in weighted automata	Two-arm transfer theorem	FPRD-D28	FPRD weighted-automata resource proofsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Complete the targeted comparison with quantitative delay, twinning, and cyclic-subgroup-distortion results.
FPRD-T77 On the exact four-branch support $\{0a^m\#0,0a^m\$0,1a^m\#1,1a^m\$1:m\ge0\}$ , there is a fixed trim unambiguous $H_3(\mathbb Z)$ -valued target with $A_f^1(N)=2N$ for $N\ge2$ . After minimization over every single global gauge $L_{r,c}f(w)=r^{\|w\|}f(w)c$ , where $(r,c)$ may depend on the horizon but acts on all four branches, optimized raw height is $\Theta(\sqrt N)$ and optimized canonical height is $\Theta(N)$ .	Theorem and asymptotic separationProved and independently audited	Evidence and limits →	Algorithms and complexityHeight correction and ambiguity in weighted automata	Incompatible-drift separation	FPRD-D28; FPRD-T76	FPRD weighted-automata resource proofsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine the correct restricted family of presentation-aware corrections before proposing any broader gauge hierarchy.
FPRD-T78 There is a fixed partial integer target on the regular support $b^+ca^*\#\mathrm{Code}\,\mathrm{Code}$ for which $h_f^1(N)\le1$ and $H_f^1(N)=\Theta(N)$ . For every fixed $k\ge1$ , let $R^h_{f,k}$ and $R^H_{f,k}$ take separate infima over all horizon-dependent $k$ -state deterministic sequential baselines with unrestricted integer weights and all exact residual presentations. Then $R^h_{f,k}(N)\le1$ and $R^H_{f,k}(N)=\Theta_k(N)$ .	Theorem and robust separationProved and independently audited	Evidence and limits →	Algorithms and complexityHeight correction and ambiguity in weighted automata	Fixed-state correction theorem	FPRD-D28	FPRD proof with classical unary max-plus and twins-theorem boundariesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Settle whether bounded raw height can coexist with unbounded corrected canonical height inside the finitely ambiguous rational class.
FPRD-T79 Restrict a $k$ -ambiguous max-plus automaton to $a_1^\cdots a_d^$ . On every sufficiently fine eventual product-congruence cell, its value is the maximum of at most $k$ affine functions of the block lengths. Therefore a cell containing $m$ full-dimensional regions with $m$ distinct affine gradients forces ambiguity degree at least $m$ .	Theorem and reusable lower-bound criterionProved and independently audited	Evidence and limits →	Algorithms and complexityHeight correction and ambiguity in weighted automata	Letter-bounded gradient criterion	No recorded dependencies	FPRD deduction from Gaubert's unary rational-series theoremReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the gradient lower bound with broader tropical and bounded-language ambiguity criteria before making any novelty claim.
FPRD-T80 The integer max-plus series $f_{\triangle}(w)=\max\{-3\|w\|_a,-\|w\|,-3\|w\|_b\}$ has intrinsic ambiguity degree exactly three.	Theorem and exact lower boundProved and independently audited	Evidence and limits →	Algorithms and complexityHeight correction and ambiguity in weighted automata	Three-gradient example	FPRD-T79	FPRD deduction using the letter-bounded gradient criterionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use the gradient criterion on less symmetric examples and compare the resulting lower bounds with existing ambiguity hierarchies.
MMR-001 RPR-S1 is a precise one-shot scalar-register model with read-only free inputs, last-use overwrite, retained outputs, and no recomputation.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-002 Naive $2\times2$ multiplication has $(M,A,D,S)=(8,4,2,5)$ in RPR-S1.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-003 The standard Strassen formulas use 7 multiplications and 18 additions/subtractions.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-004 The standard products admit a one-shot schedule with $(M,A,D,S)=(7,18,5,5)$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-005 Across all 225 per-output binary associations of the fixed products, minimum storage is five for 104 associations and six for 121.	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-006 The standard Strassen product formulas have arithmetic depth at least four, and balanced output trees attain four.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-007 Every one of the nine balanced depth-four per-output trees for (C_{11}) and (C_{22}) needs six RPR-S1 registers.	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-008 A depth-four, six-register schedule exists for the standard balanced pairing.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-009 The same rank-seven certificate and the same $7/18$ operation counts permit distinct executable $(D,S)$ points.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-010 Memory-efficient scheduling and pebble-game search for Strassen-Winograd are established topics.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-011 Recursive Strassen CDAGs have established communication/I/O lower bounds, including no-recomputation and recomputation-aware results.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-012 A future arithmetic receipt should contain an explicit linear-circuit realization or a certified Pareto envelope, not merely tensor rank.	Provisional claim or open boundaryProvisional; not promoted beyond the source ledger	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the stated proof or model obligation before increasing its evidence level.
MMR-013 A rank- $r$ tensor decomposition is exactly an exact bilinear algorithm using $r$ nonscalar products; tensor rank minimizes that count.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-014 In a sealed exact extension context, an explicit receipt $(L_U,L_V,L_W)$ preserves semantics and adds exactly $r$ nonscalar products and the three SLPs' declared addition and fixed-scalar counts.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-013	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-015 After cross-boundary circuit optimization, the sealed receipt counts remain executable upper bounds but need not equal optimized counts.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-016 A coefficient matrix $B$ has row-wise compilation bounds $A(B)\le\operatorname{nnz}(B)-z(B)$ and $C(B)\le\operatorname{nnz}(B)$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-017 Coefficient matrices do not by themselves witness exact chosen-circuit addition counts or dependency order.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-018 Explicit linear SLPs compose in depth by an exact max-plus arrival map under sealed substitution.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-019 Fixed-base literal recursion obeys $M_k=r^k$ , $A_k=rA_{k-1}+ab^{2(k-1)}$ , and $C_k=rC_{k-1}+cb^{2(k-1)}$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-020 The arithmetic exponent of that fixed recursive construction is $\max(2,\log_b r)$ , with an $N^2\log N$ factor when $r=b^2$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-021 A current fixed-rank algorithm can improve addition count solely by changing the explicit linear SLP.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-022 Theorem R2.1 does not realize border-rank degenerations, asymptotic-sum constructions, or laser-method extractions.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-023 Two exact realizations of $(P,Q,R)$ have the same $(N,A,C,D,S)=(3,2,0,3,3)$ , labelled arrivals $(1,3,1)$ , and unlabelled fan-out multiset $\{0,0,1,1,2\}$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-024 For the context $w=b+r$ , $z=p+w$ , transparent composition gives $S(E[G_R])=2$ and $S(E[G_P])=3$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-025 Semantics, $(N,A,C,D,S)$ , labelled output arrivals, and an unlabelled fan-out multiset do not suffice for optimal peak-storage composition.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-023; MMR-024	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-026 The diagonal three-coordinate bilinear map in Run 3 has tensor rank three.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-027 Exhaustive order-ideal search independently returns standalone peaks $(3,3)$ and composed peaks $(2,3)$ for the Run 3 pair.	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-028 The Run 3 separation depends on transparent rather than atomic module composition.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-029 Executed prefixes of a downstream sewn DAG are in bijection with compatible pairs of gadget and context order ideals.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-030 The boundary-ideal state weight $w(I,J)$ equals the exact number of live scalar values after that prefix.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-031 The complete boundary-ideal receipt Π(G) is sufficient to compute exact optimal peak storage in every finite downstream transparent RPR-S1T context.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-029; MMR-030	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-032 A finite acyclic network of modules has an exact compatible-tuple ideal recurrence, canonically independent of parenthesization.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-033 The Run 4 receipt is finite but may have $2^{\|V_G\|}$ states and is not yet a succinct transfer theorem.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-034 Product-state and monolithic ideal DPs independently return peaks 2 and 3 on the Run 3 pair.	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-035 Peak-memory DAG scheduling is the weighted one-shot black-pebbling problem, and one-shot pebbling has strong hardness results.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-036 Strong bisimulation preserving the port-storage label is a congruence for Run 4 sewing and preserves exact optimal peak in every downstream RPR-S1T context.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-031	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-037 Bottom-up signatures $\sigma(I)=(L(I),\{\sigma(I'):I\to I'\})$ compute the coarsest strong-bisimulation quotient of the finite ideal DAG.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-038 The naive $2\times2$ ideal receipt has exactly 625 ideals and 256 strong-bisimulation classes.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-039 The current label $(R,\mu,\lambda,\mathrm{terminal})$ alone is insufficient for downstream optimal storage.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-040 Finite computation confirms the naive $625\to256$ quotient, a quotient/monolithic sewn peak of 4, the same-label peaks 2 and 3, and Run 3 quotient sizes $12\to12$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-041 Strong bisimulation and coarsest-partition refinement are classical transition-system theory.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-042 Equality of completed storage-observation trace sets after contracting adjacent repeated labels is sufficient for exact peak storage in every downstream RPR-S1T context.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-031	FPRD storage-sewing proof chain; one-shot pebbling and behavioural-equivalence literature calibrationsReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the trimmed-DAG strictness question and test whether a smaller exact contextual quotient replaces the completed trace language.
MMR-043 Run 5 strong bisimulation is strictly finer than exact storage-context equivalence in the current untrimmed fixed-DAG model.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-044 A three-gate witness has equal completed stutter-trace languages and equal exact peaks over all 118 enumerated productive one-port contexts with at most four context gates.	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-045 Whether strong bisimulation is strictly finer than storage-context equivalence for trimmed gadgets remains open.	Provisional claim or open boundaryProvisional; not promoted beyond the source ledger	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the stated proof or model obligation before increasing its evidence level.
MMR-046 Stuttering, trace, and bisimulation equivalences are classical concurrency-theory distinctions.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-047 A fixed bilinear base DAG plus a macro schedule $\pi$ has exact pinned depth-first storage recurrence $S_k^\pi=\max\{\sigma_\pi b^{2(k-1)},\alpha_\pi b^{2(k-1)}+S_{k-1}^\pi\}$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-048 With $\kappa_\pi=\max(\alpha_\pi,\sigma_\pi-1)$ , the construction satisfies $S_k^\pi\le1+\kappa_\pi(b^{2k}-1)/(b^2-1)$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-049 Across all 225 fixed-product Strassen output associations, exact recursive-frontier Pareto signatures are: 104 $(5,5)$ , 8 $(5,6)$ , and 113 $(6,6)$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-050 No declared standard-Strassen association has recursive-call frontier four; the global nondominated pair is $(5,5)$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-051 Every one of the nine depth-four balanced standard-Strassen associations has recursive-frontier pair $(6,6)$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-052 The storage-oriented standard-Strassen recursive realization has $D(N)=1+4\log_2N$ and $S(N)=(5N^2-2)/3$ , while a balanced depth-oriented realization has $D(N)=1+3\log_2N$ and $S(N)=2N^2-1$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-047; MMR-049	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-053 Both Run 7 Strassen realizations have $M(N)=N^{\log_2 7}$ and $A(N)=6N^{\log_2 7}-6N^2$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-054 Literal standard-Strassen recursion has exact maximum wire fan-out $F(N)=3\log_2N+1$ for $N=2^k$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-055 Recursive Strassen-Winograd schedules with extra storage below $2N^2/3$ for read-only standard multiplication are established.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-056 In a sealed copy-only graph transformation, a value with $d$ original consumers needs at least $c_\Delta(d)=\max\{0,\lceil(d-\Delta)/(\Delta-1)\rceil\}$ identity buffers under fan-out cap $\Delta\ge2$ ; for $\Delta=2$ , a caterpillar attains $d-2$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-057 The strict pinned-reset fan-out-two Strassen compiler has $C^{\rm pin}(N)=\frac{16}{3}(N^{\log_2 7}-N^2)$ , $D^{\rm pin}(N)=1+6\log_2N$ , and $S^{\rm pin}(N)=4N^2-3$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-058 Raw-versus-owned provenance is sufficient to avoid resetting every generated Strassen child input: $P_1,P_6,P_7$ have two owned operands, while $P_2$ -- $P_5$ have one owned and one borrowed operand.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-059 For $N=2^k\ge2$ , the ownership-aware fan-out-two Strassen compiler uses $C_2(N)=\frac{24}{7}N^{\log_2 7}-4N^2$ identity buffers.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-056; MMR-058	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-060 The count in MMR-059 is minimal among sealed copy-only fan-out-two compilations of the fixed literal expanded Strassen DAG.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-061 The ownership-aware fan-out-two realization has exact dependency depth $D(N)=1+5\log_2N$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-059	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-062 The same realization has constructive peak live storage $S(N)\le3N^2$ , with output counted and immutable root inputs excluded.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-059	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-063 Exact expanded-DAG checks for $N=2,4,8,16,32$ verify MMR-059 and MMR-061 and return storage values $10,46,190,766,3070=3N^2-2$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-064 Charging multiplications, additions, and fan-out-two buffers as unit gates gives $\frac{73}{7}N^{\log_2 7}-10N^2$ total gates, so the fixed Strassen realization retains exponent $\log_2 7$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-065 Constant-factor bounded-fanout transformations preserving constant-factor size and depth for bounded-fanin logical networks are classical.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-066 The ownership-aware schedule may satisfy the exact identity $S(N)=3N^2-2$ for all powers of two.	Provisional claim or open boundaryProvisional; not promoted beyond the source ledger	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Resolve the stated proof or model obligation before increasing its evidence level.
MMR-067 An explicit length- $r$ polynomial degeneration with normalized error degree $e$ compiles over any field with $e+1$ distinct nonzero points to an exact bilinear DAG with $(e+1)r$ nonscalar products.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-068 Explicit evaluated input/output SLPs make the interpolation DAG's additions and fixed-scalar products exactly additive under sealed substitution; depth follows their max-plus arrival maps, and a serial accumulator schedule gives a declared storage upper bound.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-069 Truncating each factor polynomial after degeneration order $q$ gives normalized error degree $e\le2q$ and hence exact rank at most $(2q+1)r$ when the field has enough points.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-070 The $k$ th tensor power of a receipt with normalized error degree $e$ has an exact interpolation realization using $(ke+1)r^k\le(2qk+1)r^k$ nonscalar products.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-071 Evaluated tensor-power SLPs obey the literal addition recurrence (9.14); interpolation adds at most $ke\,(\dim C)^k$ vector additions, preserves $O(k)$ dependency depth when each evaluated recursion has $O(k)$ depth, and preserves $O((\dim C)^k)$ serial storage when each evaluated recursion has such a schedule.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-072 Over a finite base field lacking $ke+1$ nonzero points, interpolation over an extension of degree $h_k=\lceil\log_{\|K\|}(ke+2)\rceil$ and naive extension multiplication gives at most $(ke+1)r^kh_k^2$ base-field nonscalar products.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-073 For rational factor coefficients of bit height $H$ and degree at most $q$ , integer-node interpolation gives fixed coefficients of height $O(k(q+1)H+kq\log(ke+1)+ke)$ at tensor-power level $k$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-074 Under the direct APA error envelope $A\varepsilon+Bu\varepsilon^{-q}$ , the exact minimizing parameter is $(qBu/A)^{1/(q+1)}$ and the minimum is $(q+1)q^{-q/(q+1)}A^{q/(q+1)}(Bu)^{1/(q+1)}$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-075 Border length, degeneration order, and error degree do not determine numerical precision for a supplied DAG.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-076 The rank-two $W$ -tensor degeneration with $(q,D,e)=(1,3,2)$ is recovered exactly by three evaluations and six pure tensors with multipliers $(3,-3/2,1/3)$ on $P(1),P(2),P(3)$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-077 Error degree and degeneration order are active quantitative debordering parameters; a current result states rank $\le(e+1)$ times border rank and $e\le2\,\mathrm{ord}$ in the general interpolation step.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-078 Smirnov's length-46 APA $4\times4$ matrix-multiplication scheme has polynomial order three and reports that 16-digit arithmetic yields about four accurate digits after one recursion at its preferred parameter.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-079 The proof-consistent five-term BCLR curve for the partial $2\times2$ matrix-multiplication tensor has $(r,q,D,e)=(5,1,2,1)$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-080 The BCLR tensor has the literal exact ten-product realization $T_{\rm BCLR}=2P(1)-\frac12P(2)$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-081 The displayed one-level BCLR evaluation admits a shared-output SLP with five nonscalar products, eleven additions/subtractions total, three or four input fixed-scalar products depending on sharing, and two output scalings by $t^{-1}$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-082 The direct BCLR truncation error is componentwise linear in $t$ , with error vector $(t x^2_2y^2_2,-t x^2_2y^2_2,-t x^2_2(y^1_1-y^1_2),0)$ in the source basis. Under componentwise unit input bounds, the sharp output constants are $A_\infty=2$ and $A_2=\sqrt6$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-084 Tensor powers of the fixed BCLR curve have normalized error degree $k$ and literal exact interpolation product count $(k+1)5^k$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-085 Tensor powers of the five-term BCLR partial tensor do not by themselves realize ordinary square matrix multiplication.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-086 Two complementary relabelled BCLR curves sum to $P_{322}(t)=tM_{\langle3,2,2\rangle}+t^2E_{322}$ with length ten.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-087 The cyclic tensor product of the $322$ , $223$ , and $232$ curves is a length-1000 degeneration of $M_{\langle12,12,12\rangle}$ with $(q,D,e)=(3,6,3)$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-086	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-088 Literal exact interpolation of the square BCLR curve uses 4000 products at one base level and $(3k+1)1000^k$ at tensor level $k$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-089 The arithmetic exponent transferred by the square BCLR certificate is $\log_{12}1000=2.779885224087\ldots$ .	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-090 The normalized square BCLR curve has exact output-row coefficient maxima $(12,48,60,24)$ in degrees zero through three.	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-091 The relative row-coefficient truncation of $k$ square BCLR tensor levels is at most $[(1+\|t\|)^2(1+2\|t\|)]^k-1$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-092 Conditional on roundoff envelope $B_ku\|t\|^{-3k}$ with $B_k/12^k$ at most exponential, choosing $\|t\|=\Theta(1/k)$ gives sufficient mantissa precision $3k\log_2k+O(k+\log(1/\tau))$ .	Conditional resultConditional on the stated hypothesis	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-093 A declared separable one-level length-1000 square BCLR SLP uses $(A_U,A_V,A_W)=(1232,1232,2040)$ , totaling 4504 additions/subtractions.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-094 Cyclic tensor dualization preserves rank but does not preserve a chosen addition SLP.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-095 The canonical coefficient-row length-1000 BCLR DAG has 5184 nonzeros in each of $U,V,W$ and base receipt $(M,A,C,D,S,F)=(1000,13408,15552,14,2001,60)$ .	Theorem with computational certificateSupported by an analytic argument and retained computation	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-087	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-096 Literal recursion of MMR-095 gives $A_k=13408(1000^k-144^k)/856$ , $C_k=15552(1000^k-144^k)/856$ , and exact depth $D_k=1+13k$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-097 The declared eager row-wise depth-first schedule has exact peak storage $S_k=1+2000(144^k-1)/143<14N^2$ for $N=12^k$ .	Theorem with computational certificateSupported by an analytic argument and retained computation	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-098 Explicit coefficient gates keep raw fan-out at 60 at every positive recursion level; sealed fan-out-two compilation uses exactly $13040(1000^k-144^k)/856$ buffers.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-099 For the row-wise BCLR matrices, $e_{\max}=768$ , the rounded-operation path increment is 17, and $(\\|U_t\\|_{\max},\\|V_t\\|_{\max},\\|W_t\\|_{\max})\le(1,1,\|t\|^{-3})$ for $0<\|t\|\le1$ .	Theorem with computational certificateSupported by an analytic argument and retained computation	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-100 In the standard multiplicative rounding model, RPR-D11 has finite- $u$ forward error at most $768^k\|t\|^{-3k}((1+u)^{1+17k}-1)\\|A\\|_{\max}\\|B\\|_{\max}$ .	Specialization of a published theoremSpecialized from a cited published theorem	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-101 Relative to the $12^k$ natural output scale, combining MMR-100 with Run 10 truncation gives a sufficient mantissa law $p=3k\log_2k+O(k+\log k+\log(1/\tau))$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-102 Exact $(3k+1)$ -point interpolation of the same row DAG has $M=(3k+1)1000^k$ , $A=(3k+1)A_k+3k\,144^k$ , $C=(3k+1)C_k+(3k+1)144^k$ , parallel depth $D_k+1+\lceil\log_2(3k+1)\rceil$ , and serial storage at most $S_k+144^k+1$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-103 The 4504-addition separable compiler and the 13408-addition common row DAG are distinct executable resource points.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-104 The declared separable square-BCLR DAG has exact base receipt $(M,A,C,D,S,F)=(1000,4504,8128,19,2002,5)$ and product-call frontier $\alpha=2000$ .	Theorem with computational certificateSupported by an analytic argument and retained computation	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-087; MMR-093	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-105 Literal recursion of MMR-104 gives $A_k=4504(1000^k-144^k)/856$ , $C_k=8128(1000^k-144^k)/856$ , $D_k=1+18k$ , and declared peak $S_k=2+2000(144^k-1)/143$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-106 Raw fan-out of RPR-D12 stays five; sealed fan-out-two compilation needs exactly $R_k=2808(1000^k-144^k)/856$ buffers.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-107 One explicit eager balanced fan-out-two compilation of RPR-D12 has $D_k^{(2)}=1+23k$ and the same declared peak $S_k^{(2)}=2+2000(144^k-1)/143$ .	Theorem with computational certificateSupported by an analytic argument and retained computation	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-108 Exact Laurent-majorant propagation through RPR-D12 gives $K_r(s)\le768s^{-3}$ for every output and rounded-path increment $\nu=23$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-109 In the declared floating-point model, RPR-D12 has finite- $u$ forward error at most $768^k\|t\|^{-3k}((1+u)^{1+23k}-1)\\|A\\|_{\max}\\|B\\|_{\max}$ .	Specialization of a published theoremSpecialized from a cited published theorem	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-110 RPR-D11 and RPR-D12 are Pareto-incomparable: RPR-D12 improves $A,C,F,R$ and keeps amplification base 768, while worsening raw depth, declared storage by one scalar, and rounded-path increment.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-095; MMR-104	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-111 Current primary work explicitly co-optimizes operation count and numerical growth for recursive bilinear algorithms, so Run 12 is not new general arithmetic--accuracy optimization theory.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-112 If $D=\bigoplus_{i=1}^K\langle n_i,m_i,p_i\rangle$ , then $D^{\otimes N}$ is the labelled direct sum over words $\mathbf i\in[K]^N$ of $\langle\prod_j n_{i_j},\prod_j m_{i_j},\prod_j p_{i_j}\rangle$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-113 A length- $r$ , order- $q$ , normalized-error- $e$ explicit degeneration to $D$ powers to length $r^N$ , order $Nq$ , error degree $Ne$ ; literal exact interpolation uses $(Ne+1)r^N$ nonscalar products.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-114 Direct-sum type classes have multinomial multiplicity and satisfy $\sum_{\mathbf i}\operatorname{Vol}(D_{\mathbf i})^{\rho/3}=(\sum_i v_i^{\rho/3})^N$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-115 Tensor-powering a complete direct sum is neutral under the asymptotic sum inequality: the powered inequality is exactly the $N$ th power of the one-level inequality.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-114	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-116 A strongly disjoint survivor set of a partitioned tensor power, together with explicit constituent isomorphisms, compiles by coordinate zeroing into a labelled direct-sum batch with zero arithmetic gates for the masks in the hardwired model.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-117 Equal-size zeroings of $\Delta_M=\bigoplus_{i=1}^M\langle1,1,1\rangle$ have identical aggregate ASI/runtime statistics but a labelled selector receipt needs $\lceil\log_2\binom Mh\rceil$ bits in the worst case.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-118 The ASI application is not literal scheduling or block-diagonal embedding of the supplied batch DAG; it is a separate asymptotic tensor transformation whose theorem statement omits non-multiplicative executable resources.	Specialization of a published theoremSpecialized from a cited published theorem	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-119 Schönhage's classical $\underline R(\langle4,1,4\rangle\oplus\langle1,9,1\rangle)\le17$ example yields $\omega\le2.547992912204408\ldots$ .	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-120 Given an exact square base $(n,R)$ , a type- $\mathbf c$ of an $N$ th direct-sum power, and explicit normalized error degree $e$ , the three-cyclic common-curve compiler produces $s=n^{3\lfloor\log_R\binom N{\mathbf c}\rfloor}\prod_i v_i^{c_i}$ with exact rank at most $(3Ne+1)r^{3N}$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-112; MMR-113	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-121 For Gibbs weights $p_i=v_i^\tau/r$ , the ideal ASI bootstrap map is $\Phi(\beta)=3\beta\log r/(3H+\beta L)$ and satisfies $1/\Phi(\beta)=\alpha/\beta+(1-\alpha)/(3\tau)$ with $\alpha=H/\log r<1$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-122 An explicit polynomial direct-sum degeneration and every fixed rational $\varepsilon>0$ algorithmically yield a finite exact square base $(s_\varepsilon,R_\varepsilon)$ with $\log_{s_\varepsilon}R_\varepsilon\le3\tau+\varepsilon$ , starting from naive multiplication rather than an oracle for $\omega$ .	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-120; MMR-121	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-123 The canonical coefficient-row realization of the final ASI base has $A_0<3Rs^2$ , $C_0\le3Rs^2$ , fewer than $3Rs^2$ fan-out-two buffers, logarithmic base depth, and peak below $10Rs^2$ ; sealed recursion preserves the multiplication exponent, has $O(\log n)$ depth, $O(n^2)$ pinned scalar storage, and fan-out two.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-122	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-124 For an explicit rational ASI certificate, coefficient height is finitely computable through the Run 14 common-denominator recurrence; for the final exact base, standard stationary bilinear analysis gives sufficient precision $O_\varepsilon(\log n+\lambda)$ for absolute error $2^{-\lambda}$ .	Analytic result with published dependencySupported analytically with a cited published dependency	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-122	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MMR-125 The constructive ASI selector is succinctly generable from a type vector and combinatorial unranking, so Run 13's arbitrary-mask receipt lower bound does not apply to this structured family.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-120	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-126 A coarse effective analysis gives $\log s_\varepsilon\le\exp(O((\log(1/\varepsilon))^2))$ and hence $s_\varepsilon\le\exp(\exp(O((\log(1/\varepsilon))^2)))$ for a fixed nontrivial certificate.	Theorem, construction, or exact resource formulaProved in the FPRD research record	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	MMR-122	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Independently re-audit the complete derivation before any further promotion.
MMR-127 For volume data $(16,9)$ and $r=17$ , the reciprocal contraction is $0.234421037097059\ldots$ and the ideal exponents begin $3,2.6412830017,2.5692657873,2.5529480498,\ldots$ toward $2.5479929122\ldots$ .	Computational findingVerified by retained finite computation; analytic proof is not implied	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Obtain an independent implementation or analytic proof before any further promotion.
MMR-128 Schönhage's ASI and its type/compression core are classical; modern Strassen-calculus proofs can instead pass through the asymptotic spectrum.	Literature calibrationEstablished in cited literature and retained as calibration	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Mathematical overview and evidence boundary	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26The cited result appears in external literature; that publication is not external review of the FPRD synthesis.	Confirm the primary citation, hypotheses, and exact scope through an independent source audit.
MRF-001 Treating the tensor/rank certificate as a complete executable receipt. the fixed seven Strassen products and fixed (7/18) arithmetic counts admit different depth/storage points. Rank alone does not select a schedule or association.	Failed approach · scope boundaryrejected in the strong resource-preserving sense.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-002 Expecting depth-four Strassen to fit the five-register schedule unchanged. all nine pairings of the two four-term outputs fail with five registers and succeed with six.	Failed approach · scope boundaryrefuted for the declared balanced per-output family by exact finite search.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-003 Calling the Run 1 schedule novel. memory-efficient and in-place Strassen-Winograd scheduling, including exhaustive pebble-game searches, is established literature.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-004 Reading scalar live-register storage as an I/O or communication bound. RPR-S1 has no memory hierarchy, transfer operation, processor partition, latency, or bandwidth coordinate. Established Strassen communication bounds use recursive CDAG structure and memory models absent here.	Failed approach · scope boundaryinvalid transfer.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-005 Treating 18 additions as optimal for every rank-seven $2\times2$ scheme. Run 1 fixes the standard Strassen formulas. The Winograd variant and other linear circuits have different additive counts.	Failed approach · scope boundaryfalse framing; excluded.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-006 Inferring numerical stability from coefficients $0,\pm1$ . small formal coefficients do not by themselves bound intermediate magnitudes, rounding amplification, recursive error, or cancellation.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-007 The disjoint-output-frontier proof of the five-register bound. before fan-out has been consumed, one live Strassen product can serve two output trees. Four logical output partials therefore need not be four distinct live values.	Failed approach · scope boundaryrejected during hostile audit.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-008 Treating coefficient-matrix sparsity counts as exact. row-wise evaluation of two identical rows $(1,1)$ uses two additions, while a shared SLP computes one sum and fans it to both outputs, using one addition in the unbounded-fan-out model.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-009 Preserving exact gate-count equality through an optimizing seam. cross-boundary common-subexpression elimination, folding, reassociation, or cancellation may delete or merge gates from the literal composition.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-010 Equating tensor rank with total scalar arithmetic. classical bilinear complexity counts products of two input-dependent quantities and discounts additions and multiplication by fixed coefficients.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-011 Inferring a storage recurrence from the arithmetic receipt. peak live storage depends on scheduling across the seam, output readiness, boundary liveness, fan-out, and possible interleaving. The disjoint gate-count partition does not determine those data.	Failed approach · scope boundaryblocked, not proved.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-012 Promoting fixed exact recursion to asymptotic tensor machinery. the Run 2 recurrence repeats one exact finite rank decomposition. Border rank needs a limiting/degeneration transfer; asymptotic rank may exploit strict submultiplicativity; Schönhage's inequality amortizes direct sums; and the laser method uses partitioning and zeroing in high tensor powers. None is present in Theorem R2.1.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-013 Using only different arrival times on independent outputs. with free inputs and a global topological scheduler, an independent output can usually be postponed until the exterior needs it. Merely swapping which independent branch is shallow need not change optimal peak storage.	Failed approach · scope boundaryrejected as an easy separation strategy.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-014 Repairing aggregate storage with labelled output arrivals. the Run 3 pair has the same arrival vector $(1,3,1)$ but composed peaks two and three.	Failed approach · scope boundaryrefuted.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-015 Silently treating transparent and atomic boundaries as equivalent. transparent sewing permits exterior and interior gates to interleave; atomic execution first materializes all gadget outputs. The Run 3 separation exists only in the transparent model.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-016 Treating the cancellative witness as a tensor invariant. global algebraic resynthesis simplifies both gadgets to the same three independent products and can erase the storage difference.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-017 Declaring port-labelled live subsets alone sufficient. a live subset does not say which internal progress states are reachable next or what nonboundary frontier they carry. Run 4 retains the entire order-ideal transition system to make sufficiency provable.	Failed approach · scope boundarynot licensed; replaced by an overcomplete sufficient receipt.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-018 Calling the complete boundary-ideal receipt minimal. Theorem R4.1 proves sufficiency only. Distinct ideals may be indistinguishable by every downstream context and therefore quotientable.	Failed approach · scope boundaryrejected as an unsupported promotion.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-019 Calling the Run 4 receipt efficient or scalable. a gadget can have $2^{\|V_G\|}$ ideals, and network product states can multiply. The theorem is an exact finite specification, not a polynomial algorithm or useful recursive asymptotic bound.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-020 Treating one-shot live storage as red-blue pebble I/O. Run 4 has one memory level, free reads, no transfer gates, and no cache capacity parameter. Red-blue composition results price data movement in a different model.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-021 Quotienting ideals by the current port-storage label alone. in the naive DAG, one live product on unused $C_{01}$ and one live product on demanded $C_{00}$ have identical exposed ports, pending bits, frontier one, and terminal status. The context $z=C_{00}+u$ gives exact remaining peaks two and three.	Failed approach · scope boundaryrefuted.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-022 Calling strong bisimulation the minimal contextual quotient. strong bisimulation preserves every scheduling branch, including branches never used by an optimal min-max schedule. Peak-context equivalence may therefore be coarser.	Failed approach · scope boundaryrejected as unproved.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-023 Treating the quotient as a polynomial compilation. the construction currently enumerates the ideal graph, and even the compressed naive family has $4^k$ classes for $k$ labelled independent output components.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-024 Using the storage quotient as the whole resource receipt. Run 5 transitions are unlabelled. They do not by themselves preserve gate types, arithmetic counts, depth, coefficients, copying, or precision.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-025 Treating strong bisimulation as storage-context minimal. an independent dead gate creates a transition between equal storage observations. The states before and after that gate are not strongly bisimilar, but gate insertion/deletion preserves every composite schedule weight in every downstream context.	Failed approach · scope boundaryrefuted in the untrimmed model.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Storage-sewing strictness boundary	No recorded dependencies	FPRD storage-sewing proof chain; classical branching/stuttering equivalence calibrationReviewed 2026-09-01No documented external or specialist review of the FPRD synthesis is recorded.	Determine whether strictness survives after imposing the natural trimmed-DAG condition.
MRF-026 Dropping completion together with the Run 5 terminal bit. stutter-trace compression may omit repeated terminal-looking storage labels only if completion is still marked explicitly. Otherwise a nonterminal zero-live state can be confused with a completed state.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-027 Promoting the dead-gate witness to productive DAGs. the separating pair uses a vertex with no path to any boundary output. Dead-code elimination or a trimmed-gadget convention removes it.	Failed approach · scope boundaryrejected as unsupported.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-028 Treating storage-observational stutters as resource-free. the dead gate still consumes an arithmetic operation and may affect depth, latency, energy, or bit cost even though its result is never live.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-029 Recursing the scalar five-register schedule unchanged. a scalar multiplication may overwrite a last-use operand, while a recursive child must retain its input blocks and create a fresh output. Exact search finds minimum recursive-call frontier five, so the $2\times2$ recursive base uses six stored scalars rather than the Run 1 scalar peak five.	Failed approach · scope boundaryrefuted under the Run 7 pinned read-only-call contract.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-030 Assuming a fixed base DAG preserves constant fan-out. raw operand passes identify a parent input wire directly with a child input wire. Along nested diagonal blocks, the same scalar feeds three new local linear forms at every level, yielding fan-out $3k+1$ .	Failed approach · scope boundaryrefuted for literal standard-Strassen recursion.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-031 Calling the Run 7 storage formulas globally optimal. Theorem R7.1 is exact for a declared sequential depth-first macro schedule with pinned read-only child inputs. Transparent interleaving, streamed block formation, mutable inputs, accumulation, cross-output SLPs, other rank-seven schemes, and recomputation may change the optimum.	Failed approach · scope boundaryrejected as unsupported.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-032 Treating Run 7 live storage as communication or I/O. the model has one storage level, free reads, no cache capacity, and no transfer edges. Recursive CDAG expansion alone does not identify red-blue pebbling cost.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-033 Assuming a globally chained copy-optimal relay preserves logarithmic depth. applying a per-wire fan-out-two caterpillar to the fully expanded Strassen DAG and ordering relay leaves by the scalar schedule attains the minimum buffer count, but exact expansions through five levels have critical depth $1+\frac32k(k+3).$ A raw value traverses longer relay chains at successive recursion levels.	Failed approach · scope boundaryrefuted for the tested consumer-order construction.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-034 Copying every recursive child input. a generated linear form at last use has no parent-side consumer to protect and can transfer ownership into the child. Only raw pass-through operands need an entry copy.	Failed approach · scope boundaryrejected as sufficient but unnecessarily expensive.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-035 Reading the sealed copy lower bound as an arithmetic lower bound. Lemma R8.1 retains every original arithmetic gate once and permits only identity relays. Gate duplication, recomputation, equal-value sharing, different output SLPs, or a different bilinear algorithm can change the graph and invalidate the per-wire consumer count.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-036 Importing ownership-aware formulas into the Run 7 pinned contract. Run 7 pins every child input and requires a fresh output. Run 8's sharp compiler permits a last-use generated temporary to transfer its register to the child; raw values and root inputs remain protected.	Failed approach · scope boundaryrejected as a contract mismatch.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-037 Promoting the checked storage pattern to a theorem. the exact checker returns $S(N)=3N^2-2$ only for $N=2,4,8,16,32$ . The induction currently proves only $S(N)\le3N^2$ .	Failed approach · scope boundaryrejected pending an analytic solution.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-038 Compiling bare border-rank membership. membership in the closure of rank- $r$ tensors does not supply a polynomial curve, degeneration order, error degree, coefficient field, or linear SLPs. None of the executable resources can be recovered from the integer $r$ alone.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-039 Interpolating the unnormalized polynomial at $D+1$ points. the known factor $\varepsilon^q$ can be divided out at every nonzero evaluation point. The normalized curve has degree $e=D-q$ , so only $e+1$ evaluations are needed, not $D+1$ .	Failed approach · scope boundaryrejected as valid but unnecessarily expensive.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-040 Assuming the base field contains enough evaluation points. a small finite field may contain fewer than $ke+1$ nonzero elements at tensor-power level $k$ . Ordinary Lagrange interpolation then cannot use the declared nodes.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-041 Treating exact interpolation as numerically benign. integer nodes $1,\ldots,m$ give exact weights $(-1)^{j-1}\binom mj$ whose absolute sum is $2^m-1$ . They are convenient for rational height but can amplify floating-point error. Roots of unity improve the interpolation weights but change the field and do not control within-decomposition cancellation.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-042 Deriving working precision from $(r,q,e)$ . adjoining $MR-MR=0$ as two rank-one terms preserves the target, order, and error degree while making coefficient norms and adverse roundoff arbitrarily large.	Failed approach · scope boundaryrefuted for sealed supplied DAGs.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-043 Treating decimal samples as a verified APA certificate. Smirnov's published length-46 table is displayed mainly as values at $x=0.1$ , with one non-monomial exception and Laurent powers represented numerically. Those samples are useful source data but are not yet an exact machine-verified polynomial identity.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-044 Treating source OCR as a faithful tensor certificate. the public 1979 scan exposes the decisive formulas as images, and the page itself became access-restricted during inspection. OCR text was insufficient for an exact sign-sensitive identity. A later primary arXiv source, Landsberg--Ryder 2015, supplies the complete BCLR curve.	Failed approach · scope boundaryrejected and worked around.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-046 Calling the ten-product BCLR interpolation rank-optimal. two-node interpolation literally creates ten rank-one terms, but global algebraic simplification or a different exact algorithm can use fewer. The checker proves identity, not minimality.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-047 Recursing the five-term partial BCLR tensor as square multiplication. the verified Run 9 tensor has mode dimensions $(3,4,4)$ , so its $k$ th tensor power has dimensions $(3^k,4^k,4^k)$ rather than the three equal mode dimensions of square matrix multiplication.	Failed approach · scope boundaryrefuted by dimension audit.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-048 Assuming cyclic tensor duality preserves a chosen addition SLP. a cyclic mode permutation preserves the rank-ten tensor decomposition, but it moves the output recombination map into an input-form role and transposes another input map into the output role. The displayed linear SLP costs therefore change across the $(3,2,2)$ , $(2,2,3)$ , and $(2,3,2)$ factors.	Failed approach · scope boundaryrefuted for the declared BCLR realization.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-049 Promoting the $3k\log k$ precision estimate to an unconditional theorem. the estimate assumes a floating-point roundoff envelope $B_k u s^{-3k}$ and at most exponential growth of $B_k/12^k$ . Run 10 proves the truncation envelope but has not yet derived $B_k$ for the expanded length-1000 DAG. Staged normalization or rescaling may also improve the literal $s^{-3k}$ amplification.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-050 Combining resource coordinates from incompatible BCLR realizations. the exact $4504$ -addition count belongs to one separable Kronecker compilation, while no matching storage schedule, raw-fan-out profile, buffer placement, or floating-point error proof has yet been constructed for that same graph.	Failed approach · scope boundaryrejected pending a common expanded DAG.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-051 Assigning the 4504-addition count to the row-wise stability DAG. the Demmel-style common DAG independently evaluates every coefficient row and uses 13408 additions. The 4504 construction shares Kronecker-stage intermediates and has a different error-path and liveness graph.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-052 Calling the Run 11 roundoff theorem new stability theory. the finite- $u$ estimate is the classical stationary-partition $\mathbf\Delta$ -set argument specialized to the verified BCLR coefficient matrices.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-053 Eliding unit coefficient gates while retaining fan-out 60. explicit $\pm1$ gates break raw identities at recursive seams. Removing them lowers the fixed-scalar count from 15552 to 11952 at the base, but raw wires may then persist and accumulate consumers across levels.	Failed approach · scope boundaryrejected as a graph mismatch.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-054 Promoting the Run 11 storage and copied bounds to optima. $S_k$ is exact only for one eager materialized-row schedule. The copy count is minimal only under sealed copy-only transformation, while the copied depth and storage bounds deliberately keep coarse worst-case splitter overheads.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-055 Treating power-of-two scaling as a complete numerical repair. it removes coefficient representation error in an unbounded-exponent binary model but does not prevent amplification of earlier roundoff by later $\|t\|^{-3}$ scaling, nor does it address finite exponent range, overflow, or underflow.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-056 Promoting the Run 11 precision upper bound to every BCLR SLP. $e_{\max}=768$ and the path exponent 17 belong to independent balanced coefficient rows. Shared separable SLPs have different rounding paths and intermediate magnitudes.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-057 Expecting Kronecker sharing to worsen the exponential error base. exact gate-level Laurent propagation returns the same uniform majorant $768\|t\|^{-3}$ as the independently evaluated row graph. Sharing changes the rounded-path increment from 17 to 23, but not the amplification base in the proved envelope.	Failed approach · scope boundaryrefuted for RPR-D12.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-058 Treating RPR-D11 as the simultaneous BCLR baseline after Run 12. RPR-D12 has a verified common-DAG receipt and substantially lowers additions, scalar gates, fan-out, and copy count. RPR-D11 remains preferable in raw dependency depth, one scalar of declared storage, and rounded-path length.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-059 Continuing BCLR by tuning local constants indefinitely. Runs 9--12 now cover square closure, debordering, truncation, two recursive DAGs, storage, bounded copying, and graph-specific roundoff. Further local scheduling may be useful engineering, but no remaining BCLR obligation currently outranks asymptotic-sum extraction for the excursion's north star.	Failed approach · scope boundaryrejected as the current campaign direction.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-060 Treating a direct-sum batch schedule as the asymptotic sum inequality. the explicit batch DAG computes independent products. A block-diagonal embedding still omits dense cross-block products. The multiplicity-to- $\omega$ inference is supplied by Schönhage's separate asymptotic tensor theorem, not by an ordinary topological schedule of the batch.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-061 Expecting tensor powers of a complete direct sum to improve ASI value. the powered volume sum factorizes exactly as $(\sum_i v_i^{\rho/3})^N$ , while the border-rank upper bound becomes $r^N$ . Taking roots recovers the original inequality.	Failed approach · scope boundaryrefuted analytically.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-062 Calling coordinate zeroing free without stating the model. hardwired algebraic circuits can omit zeroed sources and outputs at zero arithmetic cost, but the mask, address map, strong-disjointness proof, and constituent isomorphisms still need description or generation.	Failed approach · scope boundaryrejected in uniform or physical compilation models.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-063 Calling ASI nonconstructive in general. classical proofs give asymptotic tensor derivations, and later speedup theorems can be constructive. The asymptotic-spectrum route may package existence nonconstructively, but that does not erase constructive proofs.	Failed approach · scope boundaryrejected as an overstatement.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-064 Using the unknown exponent $\omega$ as the outer combiner. the usual limiting derivation can say that an outer square algorithm of exponent $\omega+o(1)$ exists, but that is circular as a finite compiler. Run 14 instead begins with naive $(2,8)$ multiplication and iterates an explicit finite exponent-improvement map.	Failed approach · scope boundaryrejected as an executable proof.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-065 Debordering three cyclic factors separately. separate coefficient extraction gives $(Ne+1)^3r^{3N}$ products. Keeping the polynomial curves through cyclic tensoring and extracting once gives $(3Ne+1)r^{3N}$ .	Failed approach · scope boundaryrejected as avoidable overhead.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-066 Treating Run 13's arbitrary-mask lower bound as an ASI obstruction. the selected mask is the complete class of words with a given type. It has a succinct count-vector description and a deterministic combinatorial unranking generator, despite an exponentially large expanded port list.	Failed approach · scope boundaryrefuted for the constructive type-class proof.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-067 Compiling from a bare numerical border-rank upper bound. interpolation, field choice, coefficient height, and numerical conditioning require the actual polynomial degeneration. Membership in a border-rank variety is not itself an executable witness.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-068 Expecting ASI's hidden DAG resources to change its arithmetic exponent. canonical linear-map expansion, fan-out-two buffering, and sealed recursion give additions/copies with the same exponent, logarithmic depth, and quadratic pinned scalar storage. Coefficient and precision costs remain finite constants after $\varepsilon$ is fixed.	Failed approach · scope boundaryrefuted in the declared fixed-$\varepsilon$ model.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
MRF-069 Interpreting effective ASI as a practical algorithm. the sufficient finite powers compound, and even a loose historical calibration produces a base dimension with billions of decimal digits. The coefficient table and height can be comparably extreme.	Failed approach · scope boundaryrejected.	Evidence and limits →	Algorithms and complexityResource-preserving matrix multiplication	Rejected transfers and surviving boundaries	No recorded dependencies	Resource-preserving matrix realization research recordReviewed 2026-08-26No documented external or specialist review of the FPRD synthesis is recorded.	Keep this exclusion attached to any future reuse of the corresponding construction.
FPRD-D32 For $X_n=\mathbb Z/n\mathbb Z$ , the two reflections (h_0(i)=-i) and (h_1(i)=2-i), and append-permutation updates $T_{(p,A)}(S)=A\cup p(S)$ , the observable is completion of a hinge-group orbit. Complete future rows identify causal prefixes, and $[d]\preceq_R[v]$ means that some temporal continuation carries the future-row class of (d) to that of (v).	Definition and finite semantic modelDefined self-containedly and used by the proved obstruction	Evidence and limits →	Models of computationModels of computation	Carrier and causal order	No recorded dependencies	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine which parts of the carrier survive in a typed irreversible incidence category with physically legal compositions.
FPRD-T81 Complete future-row equality in the cyclic Boolean two-hinge carrier is a computable right congruence with $1+\prod_j(2^{\|F_j\|}-1)$ classes for orbit fibres (F_j). Raw updates and fixed-frame row classes have meet GCDs, but on five sites the targets (C_1=\{0,1,2\}) and (C_2=\{0,1,3\}) have no greatest common causal semantic factor: the edge ({0,1}) and diagonal ({0,2}) are nonassociated maximal common factors.	Theorem, exact count, and finite counterexampleProved and internally audited	Evidence and limits →	Models of computationModels of computation	Five-site obstruction	FPRD-D32	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Compile the certificate into the smallest typed irreversible incidence category and test whether legal composition forces a common alignment.
FPRD-T82 For a finite nonempty alphabet (Gamma) and $B\ge1$ , the joint syntactic congruence of $operatorname{Per}_p=\{w:p\text{ is a period of }w\}$ , $1\le p\le B$ , is the weakest finite monoid congruence deciding every bounded-period predicate in arbitrary two-sided contexts. Its quotient (M_B) is computable and has a compositional cover by capped length, length-(B) prefix and suffix, and the period vector. If $\|Gamma\|\ge2$ , then $\|Gamma\|^B\le\|M_B\|\le(B+1)(\sum_{i=0}^B\|\Gamma\|^i)^2 2^B$ , so exact generic storage is (Theta(B)) bits; for unary (Gamma), the quotient is trivial.	Theorem and information boundProved and internally audited	Evidence and limits →	Models of computationAutomata theory	Canonical bounded-period quotient	No recorded dependencies	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Study smaller quotients for genuinely restricted typed contexts rather than unrestricted word contexts.
FPRD-T83 For every finite nonempty output alphabet there is one fixed deterministic three-sweep transducer and one fixed legal idempotent crossing-effect/component type ((E,C)) whose legal same-type two-sided output-context equivalence is exactly the generic congruence defining (M_B), for every (B). The witness uses silent outer sweeps and a copying reverse sweep, so every output word and every context (x[-]y) occurs within that one rooted type.	Theorem and saturation witnessProved and internally audited	Evidence and limits →	Models of computationAutomata theory	Rooted-type saturation	FPRD-T82	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Replace bare rooted typing by a carrier that marks two output-minimal anchor occurrences and their legal intervening factor.
FPRD-T84 A fixed deterministic three-sweep reverse-copy transducer over ({0,1,\#}) has two named output-minimal one-cell anchors of one fixed effect/component type and trace output (#). For every $K\ge4$ , legal right insertion into the inter-anchor slot gives a contextual quotient for $operatorname{SmallPer}_K=\bigcup_{p=1}^K\operatorname{Per}_p$ with at least (2^{K-3}) and at most (\|M_K\|) classes. Thus the exact Boolean small-period archive still requires (Theta(K)) bits.	Theorem and contextual lower boundProved and internally audited	Evidence and limits →	Models of computationAutomata theory	Output-minimal anchor archive	FPRD-T82; FPRD-T83	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine the complexity of exposing the relevant marked record from a successful run, rather than compressing the record after exposure.
FPRD-T85 Let (K=n+3), (W_x=\#\#x\#), and $D_n=\{W_xW_y:x,y\in\{0,1\}^n\}$ . Relative to (D_n), (operatorname{SmallPer}_K) is payload equality and its complement is inequality. Equality needs at least (2^n) NFA states; either deterministic polarity needs at least (2^n) states; inequality has an NFA with (2n+2) states and every such NFA has at least the least (r) satisfying (inom r{\lfloor r/2\rfloor}\ge2^n).	Theorem and promise-complexity separationProved and internally audited	Evidence and limits →	Models of computationAutomata theory	Promise-state separation	FPRD-T84	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Identify a certificate-cover measure for the full language where run, anchor, and mismatch obligations must be guessed together.
FPRD-T86 Let $\Gamma$ be finite and nonempty and $B\ge2$ . At every length at least $2B-2$ , each nonempty profile of periods through $B$ consists exactly of the positive multiples through $B$ of its least period. One next letter preserves the entire profile and every other letter destroys it. Hence $\operatorname{SmallPer}_B$ and its complement share a complete DFA transition graph with at most $1+\sum_{i=0}^{2B-3}\|\Gamma\|^i+\sum_{d=1}^B\|\Gamma\|^d$ states. Over a non-unary alphabet the delay $2B-2$ is exact.	Theorem, exact delay, and DFA constructionProved and internally audited	Evidence and limits →	Models of computationCombinatorics on words	Principal period saturation	FPRD-T82	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine whether the exposed-word normal form yields any useful bound after run and inversion exposure are incorporated.
FPRD-T87 Let $\Sigma_3=\{0,1,2\}$ , $B\ge2$ , $n=\lfloor\log_2((B+2)/2)\rfloor$ , and (m=2^n). There are words (U_x,V_y), indexed by $x,y\in\{0,1\}^n$ , with (\|U_xV_y\|=B+1) and $U_xV_y\in\operatorname{SmallPer}_B$ exactly when $x\ne y$ . Hence $\operatorname{nsc}(\operatorname{Bad}_B)\ge m>(B+2)/4$ , and every DFA for either polarity has at least (m) states. For an NFA with state set (Q) recognizing $\operatorname{SmallPer}_B$ , this cut proves only $m\le\binom{\|Q\|}{\lfloor\|Q\|/2\rfloor}$ .	Theorem and automata lower boundsProved and internally audited	Evidence and limits →	Models of computationAutomata theory	All-cap period selectors	No recorded dependencies	FPRD finite-carrier and bounded-period proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine whether a legal successful-run or inversion construction can expose this selector family without destroying the state lower bound.
FPRD-T165 For the Kim-Park witness pair $L=C^R$ , there are deterministic visibly pushdown languages $A_C,A_L$ whose forgotten-action projections satisfy $\pi(A_C)=C\in\mathsf{LIN}\cap\mathsf{PEG}\setminus\mathsf{SCA}$ and $\pi(A_L)=L\in\mathsf{LIN}\cap\mathsf{SCA}\setminus\mathsf{PEG}$ . Thus one projected frontier is stationary-transport hard while the other is PEG hard, and the two obstructions are exchanged by reversal.	Corollary · two-sided visible-lift separationSource-attributed composition of FPRD-T91 with Kim-Park Theorem 4.19	Evidence and limits →	Models of computationAutomata and formal languages	Two-sided visible-lift theorem	FPRD-T91; FPRD-T161; Kim-Park 2026, Theorem 4.19	Negative resolution of the forgotten-action questionReviewed 2026-09-24The separation is imported from Kim and Park, arXiv:2608.29592v1. FPRD Lab has checked the class composition and reversal orientation, not independently rerun their Lean artifact.	Use a visible lift of C, rather than of C^R, as the hard calibration for intrinsic stationary-frontier lower bounds.
FPRD-T164 Forgotten-action projections of deterministic visibly pushdown languages are exactly the context-free languages. Since Kim and Park exhibit $L=C^R\in\mathsf{LIN}\setminus\mathsf{PEG}$ , there exists a deterministic visibly pushdown language whose forgotten-action projection is not a PEG language. Therefore FPRD-C02 is resolved negatively.	Theorem · negative resolution by exact class compositionSource-attributed proof from FPRD-T91 and the Kim-Park witness theorem	Evidence and limits →	Models of computationAutomata and formal languages	Negative resolution of FPRD-C02	FPRD-T91; Kim-Park 2026, Theorem 4.19 and Corollary 4.20	Negative resolution of the forgotten-action questionReviewed 2026-09-24FPRD-T91 is source-verified from Alur and Madhusudan; the PEG separation is imported from Kim and Park, arXiv:2608.29592v1. The short composition has not received specialist review.	Replace the universal inclusion target with a characterization of the projected frontier subclasses lying in SCA and PEG.
FPRD-D33 A forgotten-action frontier machine has a finite transition list $(q,a,X,q',\gamma)$ , an explicit initial configuration, and final-state acceptance. Each transition consumes one input symbol and replaces the leftmost stack symbol (X) by $\gamma$ ; there are no epsilon or endmarker steps. Its frontier component (S_u(q)) contains exactly the stacks reachable in control state (q) after the observed prefix (u).	Definition and executable machine modelDefined self-containedly and used by the checked recurrence	Evidence and limits →	Automata and formal languagesPushdown frontiers and visible projections	Machine and frontier	No recorded dependencies	FPRD pushdown-frontier proof and formalization recordsReviewed 2026-08-26No documented external or specialist review of this FPRD exposition is recorded.	Use this fixed interface when comparing stationary frontier representations; do not conflate it with a PEG or scaffold.
FPRD-T91 For every finite alphabet $\Sigma$ , a language $L\subseteq\Sigma^$ is context-free exactly when $L=\pi(A)$ for a visibly pushdown language $A\subseteq(\Sigma\times\{\mathsf c,\mathsf r,\mathsf l\})^$ , where $\pi$ forgets the action tag. The projection is letter-to-letter and length preserving, and (A) may be recognized by a deterministic visibly pushdown automaton.	Source theorem and elementary converseSource-verified and reconstructed; not mechanically checked	Evidence and limits →	Automata and formal languagesPushdown frontiers and visible projections	Visible-projection theorem	Alur–Madhusudan Proposition 1 and Theorem 2; Closure of context-free languages under homomorphism	FPRD pushdown-frontier proof and formalization recordsReviewed 2026-08-26The forward construction and determinization are published results of Alur and Madhusudan; the FPRD application has no documented specialist review.	Add a direct theorem-level primary-source locator to the public bibliography and seek a specialist check of the normalization conventions.
FPRD-T92 For a forgotten-action frontier machine, (S_{ua}(q')) is the union of $\gamma(X^{-1}S_u(q))$ over transitions $(q,a,X,q',\gamma)$ . A word is accepted exactly when some final control state has a nonempty frontier component. Both identities hold from the initial frontier, including at the empty word.	Theorem with Lean verificationMechanically checked and internally audited	Evidence and limits →	Automata and formal languagesPushdown frontiers and visible projections	Frontier recurrence and readout	FPRD-D33	FPRD pushdown-frontier proof and formalization recordsReviewed 2026-08-26No documented external or specialist review of this FPRD exposition is recorded.	Use the recurrence as the exact semantic target for proposed finite or persistent frontier representations.
FPRD-T93 Let $\mathcal K_0\subseteq\mathcal K_1\subseteq\cdots$ be finite families of canonically normalized scaffolding automata whose union contains every finite scaffold source, with the index bounding all finite source data. If (e_L(N)) is the least (k) for which a member of $\mathcal K_k$ agrees with (L^R) through length (N), then $L\in\mathsf{PEL}$ exactly when $\sup_N e_L(N)<\infty$ .	Theorem and finite pigeonhole argumentProved and internally audited	Evidence and limits →	Automata and formal languagesPushdown frontiers and visible projections	Finite-core sewing	Loff–Moreira–Reis Theorem 16	FPRD pushdown-frontier proof and formalization recordsReviewed 2026-08-26No documented external or specialist review of this FPRD exposition is recorded.	Apply the criterion only to finite normalized strata; test whether projected visible frontiers escape every fixed stratum.
FPRD-T94 Suppose a fixed synchronous multistack executor represents a forgotten-action frontier through a fixed derivative basis and fixed slots, with exact initialization, a preserved validity invariant, exact one-letter updates, and exact final-state readout. Then after every word its represented relation equals the pushdown frontier. The executor and its compiled scaffold therefore recognize exactly the source language, including at the empty word.	Conditional compilation theorem with Lean verificationMechanically checked and internally audited	Evidence and limits →	Automata and formal languagesPushdown frontiers and visible projections	Stationary-cover transfer	FPRD-T89; FPRD-T92	FPRD pushdown-frontier proof and formalization recordsReviewed 2026-08-26No documented external or specialist review of this FPRD exposition is recorded.	Construct a stationary bounded-slot cover for broader projected frontiers, or prove a representation-independent resource obstruction.
FPRD-T89 Every finite deterministic strict real-time (m)-tape Turing machine with ordinary single-head work tapes, no input-length advice, one global transition per arriving input symbol, and no post-input computation has an equivalent Loff--Moreira--Reis scaffolding automaton of degree (2m) and distance two. Each tape is represented by a two-stack zipper, and one appended scaffold node updates every persistent stack root after each input letter.	Compiler theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Strict real-time transfer	No recorded dependencies	FPRD persistent-stack and root-sharing proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek specialist review of the machine-interface correspondence and the finite-carrier boundary used in the PEG composition.
FPRD-T90 The reversal of the strict real-time multitape Turing-machine languages is a proper subclass of $\mathsf{PEG}$ . Combining this containment with the classical real-time palindrome theorem and its ordinary single-head multitape implementation yields $\mathsf{EPAL}_2=\{ww^R:w\in\{0,1\}^*\}\in\mathsf{PEG}$ , so Conjecture 7 of Loff--Moreira--Reis is false.	Proper-containment theorem and language corollarySource-verified informal proof with mechanized compiler composition; internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Recognition-class consequence	FPRD-T89; FPRD-T111	FPRD persistent-stack and root-sharing proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Obtain external formal-languages review of the classical recognizer interface, ordinary-machine realization, and properness-source composition.
FPRD-T88 In the tuple-valued fixed-port decoder used by the original persistent-stack compiler, redirecting destination port (1) to source port (0) does not clone the source stack. A backward two-stack history with decoded stacks ([2,1]) and ([9]) becomes ([2,1]) and ([2,9]) under the naive redirection, not two copies of ([2,1]).	Counterexample and failed approachMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Fixed-port clone counterexample	FPRD-T89	FPRD persistent-stack and root-sharing proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek external review of the coordinate correspondence and test whether other natural fixed-port clone operations fail for the same reason.
FPRD-T100 Let a scaffold cell carry a symbol together with the coordinate to use after following its current port. For every backward (s)-port scaffold satisfying clean missing edges and every simultaneous family in which each output pushes onto, keeps, or pops any selected source stack, one appended self-free node with paths of length at most two decodes each output as exactly that action applied to its source. Backwardness and clean missing edges are preserved.	Representation theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Coordinate-carrying root sharing	FPRD-T88; FPRD-T89	FPRD persistent-stack and root-sharing proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek external review of the coordinate-carrying decoder and investigate stationary allocation beyond fixed causal refinements.
FPRD-T101 Every finite deterministic letter-synchronous fixed-slot source-selecting stack machine compiles to an exact Loff--Moreira--Reis scaffolding automaton. Each output slot may simultaneously push onto, keep, or pop any source slot from the preceding round. The compiler appends one self-free node per input symbol, has degree (s) and distance two, and preserves finite control, every decoded stack, acceptance, and the empty word.	Compiler theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Source-selecting stack compiler	FPRD-T89; FPRD-T100	FPRD persistent-stack and root-sharing proof recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek external review of the all-prefix simulation and continue beyond the known finite causal allocator tiers.
FPRD-T102 Every partial deterministic visible machine equipped with one stationary fixed-slot frontier allocator compiles to an exact Loff–Moreira–Reis scaffolding automaton for its forgotten call, return, and local-action projection. The allocator is letter-synchronous and source-selecting, and the compiled scaffold recognizes exactly the successful hidden-action runs after the same prefix, including epsilon.	Conditional compiler theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Exact allocator interface	FPRD-T91; FPRD-T101	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek external review of the allocator laws and their correspondence with forgotten-action visibly pushdown semantics.
FPRD-T103 A total deterministic visible machine with a designated absorbing rejecting sink has exactly the same forgotten-action projection as the partial machine obtained by discarding every command whose target is that sink. Equality holds after the same prefix, including epsilon, and any allocator for the pruned frontier compiles through FPRD-T102 and FPRD-T101.	Normalization theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Rejecting-sink pruning	FPRD-T96; FPRD-T102	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Check the pruning theorem against alternative totalization conventions used in the visibly pushdown literature.
FPRD-T104 If every reachable visible control carries at most one live concrete stack, then one fixed slot per control is a canonical stationary allocator. Its transition depends only on the active-control mask, the current input, and current stack tops, and it compiles through FPRD-T102 and FPRD-T101.	Allocator theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Control-functional allocation	FPRD-T102; FPRD-T103	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Determine useful structural tests or sufficient conditions for control functionality in natural projected machines.
FPRD-T105 If a partial deterministic visible machine has a finite control refinement whose refined frontier is control-functional, then a stationary scaffold decides the original forgotten-action projection. The refinement preserves the same input, hidden action, and concrete stack in both directions after every prefix.	Refinement transfer theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Finite causal refinement	FPRD-T103; FPRD-T104	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Characterize when a bounded frontier admits a finite causal refinement, rather than merely assuming one is supplied.
FPRD-T106 A named control-functional refinement with a supplied equivalence from its refined controls to a finite slot type can be reindexed and compiled without changing any input, hidden action, concrete stack, frontier, or accepting answer. Refinements compose, including at epsilon.	Reindexing and composition theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Finite-control reindexing	FPRD-T104; FPRD-T105	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Package standard finite enumerations for common control carriers while retaining the supplied-equivalence boundary.
FPRD-T107 If successful histories reaching a fixed target control with the same last hidden action always carry the same stack, then remembering none at epsilon and the last call, return, or local action afterward gives a finite control-functional refinement and hence a stationary scaffold.	Canonical refinement theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Last-action refinement	FPRD-T105; FPRD-T106	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Identify natural visibly pushdown subclasses in which last-action separation follows from syntax or determinism conditions.
FPRD-T108 For any fixed natural number k, if the newest-first window of the last at most k successful hidden actions separates stacks at each observed word and target control, then pairing control with that window gives a finite control-functional refinement and a stationary scaffold. The theorem includes k=0 and epsilon.	Uniform bounded-window theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Bounded action-window refinement	FPRD-T106; FPRD-T107	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Find structural conditions that bound a sufficient action-window depth, or move to a different stationary coordinate system when no fixed depth works.
FPRD-T109 There is a one-letter, one-control partial deterministic visible machine whose forgotten-action projection is the universal unary language, yet for every fixed finite action-window depth two successful traces over the same word and control have the same retained window and different stacks, [7] and [].	Counterexample and limitation theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Action-window escape	FPRD-T108	FPRD visible-frontier allocator formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Search for another stationary representation that succeeds on the escape machine and separates representation failure from language complexity.
FPRD-D34 A finite ordinary complete-match parsing expression grammar has a finite input alphabet and a finite nonterminal carrier transported to Fin n; expressions use ordinary ordered-choice relational semantics, evaluation is total, and a word is recognized exactly when evaluation succeeds with empty remainder.	Definition and semantic contractDefined self-containedly and mechanically exercised	Evidence and limits →	Models of computationAutomata and formal languages	PEG semantic target	No recorded dependencies	FPRD scaffold-to-PEG formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek specialist review of the semantic correspondence with ordinary complete-match PEG conventions.
FPRD-T110 Every finite Loff–Moreira–Reis scaffolding automaton satisfying the published distance constraint has a finite total ordinary complete-match parsing expression grammar recognizing the reversal of its language, including automata whose transitions use SELF.	Mechanized scaffold-to-PEG theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Full sufficient direction	FPRD-D34	FPRD scaffold-to-PEG formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Obtain external formal-languages review of the LMR-model transcription, generated grammar, and SELF invariant.
FPRD-T111 The full scaffold-to-PEG theorem composes unchanged with the strict real-time multitape compiler and the source-selecting root-sharing compiler. Each finite source machine therefore receives an ordinary PEG for its reverse language; the even-palindrome implications remain explicitly conditional on an exact source recognizer.	Mechanized composition theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Compiler corollaries	FPRD-T89; FPRD-T101; FPRD-T110	FPRD scaffold-to-PEG composition recordsReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek external review of both interface compositions while keeping the classical palindrome premise separately sourced.
FPRD-T112 If every one-symbol derivative of each basis language is supplied as a finite list-valued principal cover, then every one-symbol derivative of a finite principal cover has an exact finite cover. Moreover, every finite listed stack frontier has an exact cover over the epsilon basis with one term per listed stack.	Mechanized normalization theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Cover normalization	FPRD-T92; FPRD-T94	FPRD principal-cover normalization formalizationReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Investigate the joint stationary tradeoff among width, basis complexity, fresh allocation, and access rather than treating derivative depth alone as an obstruction.
FPRD-T113 For the retained exponent inequalities of the direct deterministic 3SAT-to-Euclidean-CVP reduction, every real c with 0 <= c < 1/28 admits natural-number field, support, and moment exponents with strict slack, while no retained tuple with nonnegative field exponent exists for c >= 1/28.	Mechanized parameter-frontier theoremMechanically checked and internally audited	Evidence and limits →	Models of computationAlgorithms and complexity	Frontier derivation	No recorded dependencies	FPRD GapCVP parameter-retuning excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Compare alternative reduction architectures and seek specialist review before interpreting the retained frontier more broadly.
FPRD-T114 With c=1/30, Q=242, kappa=34, and tau=172, the formalized construction has output dimension at most 40 N^485 and yields Euclidean GapCVP hardness exponent 1/30 together with binary nearest-codeword and syndrome-decoding hardness exponent 1/15.	Mechanized reduction and executable parameter certificateMechanically checked and internally audited	Evidence and limits →	Models of computationAlgorithms and complexity	Concrete certificate	FPRD-T113	FPRD GapCVP parameter-retuning excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek independent lattice-complexity review of the retuned proof interfaces and the complete Lean specialization.
FPRD-D35 A metric-group product is one binary operation star on the nonnegative reals that both makes the set a group and makes d(x,y)=x star y a metric.	Definition and structural reductionSelf-contained definition	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Definition and forced structure	No recorded dependencies	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Add broader examples and compare this same-operation requirement with ordinary invariant metrics on abstract groups.
FPRD-T115 For every c>=1, the nested readout generated by h_c(t)=t/(c+t) is a continuous bijection from digit streams N^N onto the nonnegative reals, is subadditive under coordinatewise nim-XOR, and therefore transports XOR to an explicit metric-group product; for 0<c<1 the readout is discontinuous and misses (0,1-c].	Construction theorem with sharp family thresholdProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Nested construction	FPRD-D35	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek specialist review of the infinite nested-readout argument and its sharp endpoint treatment.
FPRD-T116 For c=2 the transported product is Baire class 1 and Riemann integrable on compact rectangles, its translations are cadlag, and it is discontinuous exactly when one coordinate is a positive rational and the other is positive; nevertheless no metric-group product can be separately continuous in the ordinary topology.	Exact regularity and impossibility theoremProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Ordinary regularity	FPRD-D35; FPRD-T115	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Obtain descriptive-set-theory review of the exact discontinuity set and decoder complexity distinction.
FPRD-T117 At c=2 the intrinsic metric is complete, separable, zero-dimensional, homogeneous, isosceles-free, and has full distance spectrum [0,infinity); the product is computable from digit names but no metric-group product is total Type-2 computable from ordinary Cauchy names or o-minimal definable.	Geometry and computability theoremProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Intrinsic and ordinary representations	FPRD-T115; FPRD-T116	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Formalize the representation interfaces and seek review of the Type-2 and o-minimal no-go arguments.
FPRD-T118 For every metric-group product and a>0, each closed interval [r,r+a] contains a discontinuity of the translation T_a(x)=a star x, and the bound is sharp; moreover sup_b \|T_b(x)-T_b(y)\|=x star y.	Universal sharpness theoremProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Universal translation bounds	FPRD-D35	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Clarify the classical profile identity's provenance and seek review of the sharp interval construction.
FPRD-T119 For every c>=1 in the nested family, all translations are cadlag and share the countable dense discontinuity set B_c^+=E_c(Fin)\{0}; infinite-support parameters jump at all of B_c^+, while a finite nonzero parameter has an exact closed scattered jump set of Cantor-Bendixson height s+1 and largest gap h_c^s(2^r).	Simultaneous regularity classificationProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Translation classification	FPRD-T115; FPRD-T118	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek specialist review of the finite-support Cantor-Bendixson calculation and simultaneous-family regularity terminology.
FPRD-T120 Bitwise XOR is the unique Boolean group law circle on the nonnegative integers with identity zero satisfying p circle q <= p+q for every p and q; equivalently, it is the unique metric-group product on the nonnegative integers.	Uniqueness theoremProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Integer rigidity	FPRD-D35	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Seek specialist review and complete the literature comparison with Elliott's Boolean-algebra uniqueness theorem.
FPRD-T121 For every c>=1, each finite-support translation in the nested family is locally of bounded variation, but some sparse infinite-support translation has infinite total variation already on [0,1].	Sharp family obstructionProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Bounded-variation boundary	FPRD-D35; FPRD-T115; FPRD-T119	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Classify more infinite digit streams and compare the nested-family failure with non-coordinate actions.
FPRD-T122 For every metric-group product and ordinary-open interval U, the parameters whose translation discontinuity set meets U form an intrinsic-open dense set; under intrinsic Baireness, whole-ray-dense sections are generic, and unconditionally every requested scale contains a smaller initial interval on which one translation has a dense discontinuity set.	Universal density theoremProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Universal dense-section theorem	FPRD-D35; FPRD-T118	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Resolve whether every product has one whole-ray-dense section without assuming intrinsic Baireness.
FPRD-T123 A positive additive score on a Boolean support subgroup containing every singleton, with finite score on every allowed support, cannot be a bijection onto the full nonnegative half-line.	Representation no-go theoremProved and internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Atomic-score obstruction	FPRD-D35	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Investigate interaction-dependent, signed, and non-coordinate presentations outside the theorem's hypotheses.
FPRD-T124 A metric-group product with all translations locally BV exists exactly when the half-line admits a regular elementary abelian 2-action by locally BV permutations satisfying \|h_a(x)-x\|<=a; locally finite interval-exchange actions cannot do this, and every dyadic-only repair of binary labels gives infinite variation to infinitely alternating translators.	Equivalence theorem and restricted no-go resultsProved frontier; general existence remains open	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Simultaneous local-BV frontier	FPRD-D35; FPRD-T122	FPRD metric-group regularity excursionReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Construct or obstruct a regular locally-BV Boolean action beyond coordinate readouts and locally finite interval exchanges.
FPRD-D36 A bounded simultaneous record transducer chooses, from a fixed-radius observation of old records, one bounded labelled graph of fresh records whose fields may target old addresses or any active fresh slot, including self, forward, and mutually cyclic references; no fresh record is inspected while the plan is chosen.	Definition and machine modelSelf-contained repaired definition	Evidence and limits →	Models of computationAutomata and formal languages	Machine model	No recorded dependencies	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Draft 5Reviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Compare the model with established cyclic term-graph and simultaneous-allocation formalisms without assuming equivalence.
FPRD-T125 Binary even-palindrome dynamics factor exactly into one longest even-palindromic suffix occurrence and one reversed left-context rope. In the abstract persistent-sequence interface, typed mirror receipts, bridge-shadow transport, and common direct/reset rope recurrences close every online pop, prepend, reset, fresh-record, and retained-tail update with a fixed number of sequence operations. This is receipt semantics, not a certificate that a concrete sequence implementation meets the native per-symbol resource contract. A raw historical endpoint, immutable type-birth table, raw anchor-independent chunk, or bounded bridge chase does not supply the same interface.	Representation and closure theoremProved in Draft 2 and internally audited	Evidence and limits →	Models of computationAutomata and formal languages	Receipt dynamics and proof	FPRD-D36	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Draft 5Reviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Obtain an external check of the bridge-shadow, mirror-prefix, and selector-receipt arguments.
FPRD-T126 Every bounded simultaneous record transducer with s roots, at most A fresh q-field records per letter, and old-address bound r compiles exactly and letter-synchronously into a native scaffolding automaton of degree max{1,s+Aq} and observation/update distance at most r+1, provided its initial data is one fixed finite immutable pointer-closed record graph, its plans factor through finite control, the input letter and bounded labelled rooted unfoldings, and its roots and fields target missing, bounded old addresses, or current fresh slots. Physical pointer identity is not an observable test. Initial records are virtual finite decoder data; dynamic fresh targets use SELF plus a finite slot tag, including self-loops, forward references and mutual cycles.	Exact compiler theoremCorrected compiler proof; broader application audit remains	Evidence and limits →	Models of computationAutomata and formal languages	Tuple-packed compiler	FPRD-D36	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Draft 5Reviewed 2026-09-10No documented external or specialist review of this FPRD result is recorded.	Audit the remaining palindrome dependency chain and seek an external specialist check of the corrected compiler; compare it with existing graph-machine encodings.
FPRD-T127 Assume a persistent sequence implementation satisfies the direct Draft 5 Section 11.1 contract: list semantics, persistence, opaque elements, a finite source-record signature and inspected nonpointer data, uniformly bounded strict evaluation and field selection, explicit old/fresh pointer origins, and no escaping temporary control values. Its bounded old-dependent structural allocation plan can stage a new receipt as an opaque atom and tie that atom to its fresh owner in one D36 simultaneous batch, preserving sequence semantics. This conditional staging theorem does not certify the concrete OCaml implementation or recover a withdrawn numerical radius from Kaplan--Tarjan's abstract worst-case operation bound.	Conditional staging theorem under the complete sequence contractConditional theorem proved; numerical radius certificate withdrawn	Evidence and limits →	Models of computationAutomata and formal languages	Opaque-atom staging	FPRD-D36; FPRD-T126; Kaplan–Tarjan 1999, Theorem 5.1; Viennot et al. 2026, v3	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Draft 5Reviewed 2026-09-10No documented external or specialist review of this FPRD result is recorded.	Extract canonical symbolic type schemes for the 474 context-sensitive cat source sites, preserving arbitrary Pair^k(alpha) depth, then run branch-correlated callback substitution while failing closed on both assertions.
FPRD-T128 Conditional on a persistent sequence implementation satisfying the complete direct Draft 5 Section 11.1 contract, the receipt construction, whole-client normalization, opaque-atom staging and simultaneous record compiler yield a fixed finite letter-synchronous native scaffolding automaton for EPAL_2={ww^R:w in {0,1}*}, with symbolic finite resource bounds. Loff--Moreira--Reis then gives an ordinary complete-match PEG for reversal(EPAL_2)=EPAL_2. This conditional direct route uses neither a strict real-time multitape machine nor Slisenko's palindrome premise; the separate T89/T90 transfer route remains independent.	Conditional direct recognition consequenceConditional construction with an explicit sequence contract; symbolic resource bounds	Evidence and limits →	Models of computationAutomata and formal languages	Direct theorem and dependency chain	FPRD-T125; FPRD-T126; FPRD-T127; FPRD-T110	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Draft 5Reviewed 2026-09-10No documented external or specialist review of this FPRD result is recorded.	Extract and check symbolic type schemes for the 474 contextual cat source sites, preserving arbitrary Pair^k(alpha) depth, finish branch-correlated callback substitution, then extend the fail-closed interpreter separately to uncons and seek external review.
FPRD-D37 For fixed finite scaffold degree, observation distance, and working alphabet, a scaffold construction history is a word over a finite row alphabet. Its chronological physical history, complete strong trace of minimal rooted behavioral figures, final behavioral figure, and acceptance readout are distinct observations. A presentation move must state which of those observations it preserves.	DefinitionFixed and self-contained; no completeness or novelty claim	Evidence and limits →	Models of computationTopology, sewing, and coherence	Histories, observations, and two congruences	FPRD-D01; FPRD-D03	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Use the observation ladder consistently when classifying later scaffold moves and lower bounds.
FPRD-T129 Equality of rooted scaffold unfoldings is equivalent to equality of all finite neighborhoods and deterministic labelled bisimilarity, is preserved by every common finite scaffold continuation, and is separated when unequal by a finite distance-two unary continuation. Every finite rooted scaffold has a canonical minimal residual figure, unique up to rooted labelled port-graph isomorphism.	Theorem · full abstraction and minimizationProved, independently rederived, and finitely stress-tested	Evidence and limits →	Models of computationTopology, sewing, and coherence	Future-context full abstraction	FPRD-D37	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Compare the scaffold-specific continuation and separator statements with the closest coalgebraic testing and context-equivalence formulations.
FPRD-T130 Every legally generated rooted scaffold and its minimal behavioral figure are weakly acyclic reachability chains. Conversely every finite behaviorally reduced labelled rooted reachability-chain port graph has a legal scaffold presentation at some finite distance. After adjoining a missing sink, every port action is regressive and the finite port-word navigation monoid is R-trivial.	Theorem · structural characterizationProved, independently rederived, and finitely stress-tested	Evidence and limits →	Models of computationTopology, sewing, and coherence	Generated-chain theorem	FPRD-D37; FPRD-T129	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Position the reachability-chain characterization against standard ordered transformation monoids without transferring it to the streamed language.
FPRD-T131 On every finite weakly acyclic deterministic labelled partial port graph, elementary guarded pair folds generate exactly the greatest bisimulation equivalence. Orienting those folds toward their quotients terminates and is confluent up to rooted labelled port-graph isomorphism, with the canonical minimal residual figure as normal form.	Theorem · static normalizationProved, independently rederived, and exhaustively tested on the declared finite domain	Evidence and limits →	Models of computationTopology, sewing, and coherence	Guarded folds generate bisimilarity	FPRD-T129	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Keep object-level fold normalization separate from a presentation of all paths between fold sequences.
FPRD-T132 For every fixed finite scaffold degree d, distance, and working alphabet, least physical-endpoint detours and literal guarded SELF unrolls form a terminating confluent graph-local presentation of complete strong-trace equivalence. Two histories have the same behavioral figure after every prefix exactly when they normalize to the same SELF-eager history, and a length-n history takes at most d*n whole-row moves.	Theorem · strong-trace normal formProved, independently audited, and finitely stress-tested	Evidence and limits →	Models of computationTopology, sewing, and coherence	Convergent strong-trace presentation	FPRD-D37; FPRD-T129; FPRD-T131	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Investigate path-level coherence for the strong calculus without implying that object confluence already supplies a finite two-cell presentation.
FPRD-T133 The least replay-closed causal endpoint support of a scaffold history is stable: preserving its supported descriptors and final-reachable labels preserves the final rooted concrete scaffold and the support itself. Every resulting causal-garbage fibre is a finite Cartesian product; single-coordinate triangles and distinct-coordinate commuting squares present its path groupoid and make its filled two-complex simply connected.	Theorem · replay invariance and path coherenceProved and independently audited; no dedicated standalone checker	Evidence and limits →	Models of computationTopology, sewing, and coherence	Replay invariance	FPRD-D37	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-26No documented external or specialist review of this FPRD result is recorded.	Study how causal support changes under strong and temporal moves; do not extend the garbage-cube theorem beyond one stable fibre without a transport proof.
FPRD-T134 Fix finite scaffold degree and distance and one behaviorally reduced old prefix. Two unexposed acyclic router/consumer extensions with the same consumer label, literal agreement on MISSING and SELF coordinates, and corresponding valid path demands that reach the same old residual behavior are connected by endpoint detours, causal-garbage changes, continuation-aware injective routing, and dedicated relation exchange. Every common suffix is preserved.	Theorem · restricted two-row sewingProved, independently audited, and computationally stress-tested	Evidence and limits →	Models of computationTopology, sewing, and coherence	Unexposed two-row sewing	FPRD-T129; FPRD-T133	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Determine whether dedicated relation exchange can be derived by refactoring older rows instead of citing the pointed endpoint congruence of the fixed prefix.
FPRD-T135 For every fixed history length, nonempty finite working alphabet, scaffold degree one, and scaffold distance one, endpoint detours, literal guarded absorptions, causal-garbage changes, and temporal carrier slides generate exactly final behavioral equivalence. Every final fibre has one literal right-packed normal form determined by its final figure and history length.	Theorem · final-equivalence normal formProved, independently audited, and exhaustively tested on the declared finite domain	Evidence and limits →	Models of computationTopology, sewing, and coherence	Unary final presentation	FPRD-T132; FPRD-T133	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Investigate which part of right-packing survives at degree two or distance two without assuming a chain of unique relays.
FPRD-T136 After quotienting a fixed-length degree-one distance-one final fibre by its internal strong and causal-garbage chambers, temporal carrier slides form the undirected Hasse graph of an order-ideal rectangle. The rectangle is [m-1] x [n-m] for an ordinary terminal and [m-1] x [n-m-1] for a recursive terminal with m positive; the pure recursive case is a singleton. Filling independent slides gives a rooted CAT(0) cube complex whose hyperplanes are delay boxes, dimension is the rectangle width, edge distance is total carrier displacement, and medians are coordinatewise.	Theorem · quotient geometry and path coherenceProved, corrected, independently audited, and computationally stress-tested	Evidence and limits →	Models of computationTopology, sewing, and coherence	Carrier-surgery CAT(0) geometry	FPRD-T133; FPRD-T135	Persistent Scaffold Computation: Causal Presentations, Sewing, and FeedbackReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Extend the cubical base through, rather than only below, the strong and garbage presentation fibres.
FPRD-T137 For every fixed history length, nonempty finite working alphabet, degree one, and distance one, all parallel paths inside one final behavioral fibre generated by endpoint detours, literal absorptions, causal-garbage changes, and temporal slides are related by a length-independent guarded family of garbage triangles and squares, collapse cells, strong and temporal Peiffer or carrier-cube cells, strict mixed naturality squares, and one bounded absorption-transport hexagon. The three support captures reduce by garbage cancellation and detour-transport is derived.	Theorem · guarded path presentationProved, repaired after independent internal review, and computationally stress-tested	Evidence and limits →	Models of computationTopology, sewing, and coherence	Uniform guarded path presentation	FPRD-T132; FPRD-T133; FPRD-T135; FPRD-T136	A Coherent Cubical Algebra for Unary Scaffold HistoriesReviewed 2026-08-27No documented external or specialist review of this FPRD result is recorded.	Test whether a comparable guarded coherence theorem exists at higher degree without claiming a finite context-closed polygraph or finite derivation type.
FPRD-D40 A finite adaptive work history consists of independent source cells, well-founded derived cells computed by terminating deterministic oracle procedures over earlier cells, and a terminating deterministic observer. Each execution records an ordered finite query transcript, including repeated queries; its queried-cell set is the support of that transcript. Replay support is the least set containing the observer's queried cells and closed under the queried-cell sets of every supported derived cell.	Definition · execution-specific dependency modelDefined self-containedly and exercised by an exhaustive finite model	Evidence and limits →	Logic, semantics, and rewritingAdaptive replay slices	Adaptive work-history model	No recorded dependencies	Adaptive replay-slice theoremReviewed 2026-10-05No documented external or specialist review of this FPRD result is recorded.	Compare the model more closely with dynamic slicing, provenance traces, and self-adjusting computation before making any priority claim.
FPRD-T150 In every finite adaptive work history, changing only source cells outside one execution's replay support preserves every supported derived value and ordered query transcript, the observer result and ordered query transcript, and the replay support itself. The support is the least trace-closed certificate. At a downward-closed history cut, later demands pull back through the earlier dependency graph of the same execution. With independent source domains, fixing supported source coordinates gives a product cylinder contained in a replay-equivalence class; equality with the whole class is not asserted.	Theorem · replay stability, composition, and support cylindersSelf-contained proof and exhaustive small-model falsification audit	Evidence and limits →	Logic, semantics, and rewritingAdaptive replay slices	Adaptive replay-slice theorem	FPRD-D40	Adaptive replay-slice theoremReviewed 2026-10-05No documented external or specialist review of this FPRD result is recorded.	Formalize the well-founded replay induction and determine which additional conditions turn per-execution slices into a stationary bounded-memory representation.
FPRD-T175 For a letter projection from a call-return-local alphabet, define the clock diameter of an observed word as the largest final stack-height difference among compatible hidden lifts. If no calls exist, the stack is never populated. Otherwise, the diameter is uniformly bounded exactly when each visible-letter fiber has one action type, in which case it is identically zero. If any fiber mixes action types, then for every n some observed word of length at most 2n has clock diameter at least n.	Theorem · exact zero-or-unbounded stack-clock dichotomySelf-contained elementary proof with exhaustive action-fiber audit	Evidence and limits →	Models of computationAutomata and formal languages	Stack-clock dichotomy	FPRD-T174; visibly pushdown action partitions; letter projections	The stack-clock dichotomy for letter projectionsReviewed 2026-09-24The visibly pushdown action partition is classical. No external or specialist review of the projection-level dichotomy formulation is recorded.	Develop source-conditioned clock coherence, since no nontrivial bounded-skew tier exists at the unrestricted letter-projection level.
FPRD-T174 Let A be deterministic visibly pushdown and let pi preserve the call, return, and local partition. For every fixed finite family of synchronous causal selectors, the language of accepted observations missed by the family is visibly pushdown: all source and selected stacks zip into one product stack. Coverage is therefore decidable. For every fixed selector budget (m,s), a computable finite horizon B_(A,pi)(m,s) decides global feasibility and bounds obstruction radius.	Theorem · effective selector compactness under synchronized stack actionsSelf-contained product-stack proof with classical VPL calibration and bounded exhaustive audit	Evidence and limits →	Models of computationAutomata and formal languages	Synchronized stack-clock theorem	FPRD-T169; FPRD-T171; FPRD-T173; visibly pushdown automata emptiness	Synchronized stack clocks make causal selection effectiveReviewed 2026-09-24The synchronized-stack closure mechanism and visibly pushdown decision properties are established by Alur and Madhusudan. No specialist review of the FPRD selector application is recorded.	Use T175's clock dichotomy to study intermediate tiers restricted by source language, selector outputs, bounded phase structure, or contextual maps.
FPRD-T173 Let A be accepted by an r-state DFA, let pi be a letter projection, and fix a family of m synchronous selectors with at most s states each. The family covers pi(A) globally if and only if it covers every accepted observed word through length r^(m+1)s^m-1. Consequently, fixed-budget selector feasibility is decidable for regular hidden sources, and every infeasible budget has an obstruction suite whose radius obeys the same explicit bound.	Theorem · effective finite-horizon collapse for regular sourcesSelf-contained finite-automaton proof with exhaustive bounded product audit	Evidence and limits →	Models of computationAutomata and formal languages	Regular-source selection horizon	FPRD-T169; FPRD-T171; FPRD-T172; finite automata emptiness	Effective selection horizons for regular hidden sourcesReviewed 2026-09-24The proof is an elementary finite-product and shortest-accepting-run argument. No external or specialist review of the FPRD formulation is recorded.	Use T174's effective common-stack-clock tier and T175's projection-clock dichotomy; study quantitative escape for action-erasing or structurally restricted sources.
FPRD-T172 For a regular source language A, finite causal selector covers are equivalent, up to explicit state overhead, to domain-covering synchronous multi-sequential subrelations of the accepting-lift relation R_(A,pi). For arbitrary nonregular sources the notions separate: under the identity projection every language has a one-state causal selector, while a standard sequential uniformizer can exist only when its required domain is regular.	Boundary theorem · exact regular bridge and nonregular separationSelf-contained reduction proof with literature calibration and exhaustive finite audit	Evidence and limits →	Models of computationAutomata and formal languages	Acceptance-blind uniformization bridge	finite automata; synchronous sequential transducers; multi-sequential relations	Acceptance-blind selection and sequential uniformizationReviewed 2026-09-24The note explicitly calibrates against Jecker-Filiot multi-sequential relations and the sequential-uniformization literature. No specialist review of the FPRD acceptance-blind formulation is recorded.	Determine which structural properties of nonregular source recognizers let acceptance-blind selectors be synthesized or ruled out effectively.
FPRD-T171 For every tagged language A, letter projection pi, and infeasible selector budget (m,s), there is a finite set W of accepted observed words that defeats every family of at most m selectors with at most s states. Its cardinality is at most the number N_(m,s)(pi) of candidate labeled selector families. The minimum obstruction cardinality is an exact finite set-cover number, and the T169 escape length equals the minimum possible maximum word length of an obstruction suite.	Theorem · finite obstruction duality with explicit certificate boundSelf-contained finite-universe proof with exhaustive bounded duality audit	Evidence and limits →	Models of computationAutomata and formal languages	Finite obstruction duality	FPRD-T169; finite Mealy machine enumeration; finite set-cover duality	Finite obstruction duality for causal selectionReviewed 2026-09-24The proof is a finite candidate-universe argument. No external or specialist review of the obstruction formulation is recorded. The optional hard-lift application retains T168's imported premise.	Seek structural bounds on obstruction cardinality and radius for positive allocator classes, then compare them with the hard visible lift.
FPRD-T170 Let Gamma -> Sigma -> Delta be finite letter projections, A a language over Gamma, and B=pi(A). If (m,s) lies in Sel(A,pi) and (n,t) lies in Sel(B,rho), then (mn,st) lies in Sel(A,rho composed with pi). The law holds at every finite horizon. Hence an empty composite selection spectrum forces an empty spectrum at some constituent stage, and escape lengths satisfy the corresponding minimum lower bound.	Theorem · multiplicative chain rule and obstruction localizationSelf-contained product-state proof with exhaustive bounded composition audit	Evidence and limits →	Models of computationAutomata and formal languages	Causal selection chain rule	FPRD-T167; FPRD-T169; finite-state transducer composition	The causal selection chain ruleReviewed 2026-09-24The proof is a direct finite-state product and cover-composition argument. No external or specialist review of the selection-spectrum formulation is recorded.	Use the chain rule to factor complicated forgotten-coordinate maps and isolate the first projection stage whose selection spectrum becomes empty.
FPRD-T169 For a tagged language A and letter projection pi, let Sel_n(A,pi) contain the selector-count and selector-state budgets that cover all accepted observations through length n. Then Sel(A,pi)=intersection_n Sel_n(A,pi). Each set is upward closed, the horizon sequence decreases, and every infeasible finite budget has a least finite escape length. Applied to the hard visible lift A_C, every fixed finite selector budget fails on some finite input horizon.	Theorem · exact finite-horizon compactness and escape modulusSelf-contained finite-candidate compactness proof with exhaustive bounded audit	Evidence and limits →	Models of computationAutomata and formal languages	Causal selection spectrum theorem	FPRD-T167; FPRD-T168; finite Mealy machines; Dickson's lemma	Causal selection spectra and finite escapeReviewed 2026-09-24The compactness argument is self-contained and uses a finite candidate universe. No external or specialist review of the selection-spectrum formulation is recorded. The hard-lift application retains T168's imported Kim-Park premise.	Bound or estimate the escape modulus on positive allocator tiers and seek an intrinsic growth law for the Kim-Park visible lift.
FPRD-T168 There is a deterministic visibly pushdown language A_C with a degree-one, distance-two scaffold such that its forgotten-action projection C has empty stationary transport spectrum. Moreover, no fixed finite family of synchronous finite-state tag selectors covers all accepting observed words: csel(A_C,pi)=infinity.	Corollary · exact hidden-choice blowup in a deterministic visible liftSource-attributed corollary with explicit transport and selector-cover proof	Evidence and limits →	Models of computationAutomata and formal languages	Hard visible-lift selector theorem	FPRD-T91; FPRD-T101; FPRD-T163; FPRD-T167; Kim-Park 2026, Theorem 4.19	Finite causal selector coversReviewed 2026-09-24The SCA non-membership of C is imported from Kim and Park, arXiv:2608.29592v1. FPRD Lab has not independently reverified their lower bound or Lean artifact.	Find an intrinsic growth measure for accepting selector families that can be bounded on the positive allocator tiers and forced to diverge on the C lift.
FPRD-T167 Let A have transport tuple (d,k,g,q), and let m synchronous finite-state selectors T_i with s_i states satisfy pi(A)=union_i T_i^{-1}(A). Then pi(A) has transport tuple (m*d,k,g^m,q^m product_i s_i). Thus every fixed finite online cover of the accepting hidden histories preserves bounded stationary transport.	Theorem · finite hidden-choice closure with explicit transport boundSelf-contained composition proof with exhaustive finite product audit	Evidence and limits →	Models of computationAutomata and formal languages	Finite causal selector-cover theorem	FPRD-T157; FPRD-T166	Finite causal selector coversReviewed 2026-09-24The proof composes two local FPRD product constructions. No external or specialist review of the selector-cover formulation is recorded.	Replace the finite selector count by a horizon-dependent selection profile and compare its growth with scaffold access capacity.
FPRD-T166 Let T be a deterministic synchronous Mealy transducer with s states. If (d,k,g,q) lies in Tr(L), then (d,k,g,sq) lies in Tr(T^{-1}(L)). The compiled presentation builds exactly the source persistent graph and adds only the transducer state to finite control. Hence nonempty transport spectrum is invariant under finite-state synchronous causal coordinate changes with finite-state inverses.	Theorem · exact inverse-image closure and unchanged-graph resource boundSelf-contained product proof with exhaustive finite-state audit	Evidence and limits →	Models of computationAutomata and formal languages	Finite-state causal preimage theorem	FPRD-D45; Loff--Moreira--Reis scaffolding automata; synchronous Mealy transducers	Finite-state causal preimages preserve stationary transportReviewed 2026-09-24The construction is a finite-state product argument. No external or specialist review of the transport-spectrum formulation is recorded.	Use the theorem as the deterministic baseline when classifying which existential hidden-history operations destroy bounded stationary transport.
FPRD-T163 For the Kim-Park record-first witness C, delta_str(C)=infinity while delta_str(C^R)<infinity, and both C and C^R are linear context-free. Thus reversal can change an empty stationary transport spectrum into a nonempty one even within LIN.	Corollary · exact directional asymmetry of stationary transportSource-attributed corollary of Kim-Park Theorem 4.19 and FPRD-T161	Evidence and limits →	Models of computationAutomata and formal languages	Reversal-asymmetry corollary	FPRD-T161; Kim-Park 2026, Theorem 4.19	Cell-probe obstructions to stationary transportReviewed 2026-09-23The language-class separation is imported from Kim and Park, arXiv:2608.29592v1. FPRD Lab has not independently rerun their Lean artifact or rechecked the underlying cell-probe lower bound.	Use a deterministic visible lift of C as the hard calibration for intrinsic stationary-frontier invariants.
FPRD-T162 Let K be an (a,b)-SCA encoding of a three-stage Boolean problem in the sense of Kim and Park. If no exact deterministic cell-probe data structure has word size Theta(log n), update time O(a(n)), query time O(b(n)), and polynomial space, then Tr(K) is empty and delta_str(K)=infinity.	Transfer corollary · encoding-robust obstruction to every finite scaffold presentationSource-attributed corollary with explicit dependency and verification boundary	Evidence and limits →	Models of computationAutomata and formal languages	Cell-probe transport obstruction	FPRD-T161; Kim-Park 2026, Definition 3.2 and Theorem 3.5	Cell-probe obstructions to stationary transportReviewed 2026-09-23The SCA-to-cell-probe transfer is imported from Kim and Park, arXiv:2608.29592v1. FPRD Lab has checked only the translation into its transport invariant.	Translate the known hard language C into the FPRD frontier model and seek an intrinsic obstruction matching its cell-probe lower bound.
FPRD-T161 Let a scaffolding automaton have tuple (d,k,g,q) with k>=1, put D=sum_(i=1)^k d^i, and m=ceil(log_2(1+gq)). It has an equivalent tuple (D+m,2,1,2). Shortcut ports expose every source endpoint in one hop, and marker ports encode each ordinary source label-control pair as local SELF/missing geometry. The zero marker code is reserved for the distinguished base node. Hence Tr(L) is nonempty exactly when it contains some tuple (D,2,1,2).	Theorem · structural normal form and one-dimensional existence criterionSelf-contained constructive proof with combined compiler and structural-code audits	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Structural scaffold normal-form theorem	FPRD-T160; FPRD-T157; Loff--Moreira--Reis scaffolding automata	Structural normal form for scaffolding automataReviewed 2026-09-23A targeted literature search found the original scaffold model but no published unary-label, two-control structural normal form; no specialist review or priority claim is recorded.	Study the minimum canonical structural degree delta_str(L), including composition laws and lower bounds that survive every finite label and control encoding.
FPRD-T160 If a scaffolding automaton has tuple (d,k,g,q) with k>=2, then it has an equivalent tuple (sum_(i=1)^k d^i,2,g,q). The compiler gives each physical node one shortcut port for every original port word of length at most k. The compiled radius-two view reconstructs the original radius-k view, distinguishing base from missing at shortcut endpoints; at most two hops install every shortcut on the next node. Hence SA_<=2=SA. Combined with FPRD-T159, SA_<=1=REG is a strict subset of SA_<=2=SA.	Theorem · complete distance normalization for scaffolding automataSelf-contained constructive proof with machine, symbolic-descriptor, and strictness audits	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Distance-two normal-form theorem	FPRD-T159; Loff--Moreira--Reis scaffolding automata	Distance-two normal form for scaffolding automataReviewed 2026-09-23A targeted literature search found the original scaffold model but no published distance-two normalization; no specialist review or priority claim is recorded.	Replace distance-based expressiveness questions with degree-versus-distance succinctness bounds and seek lower bounds against the entire distance-two normal form.
FPRD-T159 Every distance-one scaffolding automaton of degree d, working-alphabet size g, and q controls compiles to a DFA with at most N=q[1+(g+1)^(d+1)] states, so its residual tower stabilizes by horizon N-1. Conversely, every regular language has a distance-one scaffold presentation. Hence SA_<=1=REG. The explicit degree-two, distance-two persistent selector is nonregular, so REG=SA_<=1 is a strict subset of SA_<=2.	Theorem · exact base tier and strict first step of the scaffold distance hierarchySelf-contained constructive proof with compiler and strictness audits	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Distance-one collapse theorem	FPRD-T153; FPRD-T151; Loff--Moreira--Reis scaffolding automata	Distance-one scaffolds collapse to finite automataReviewed 2026-09-23A targeted literature search found the original scaffold model but no published statement of this distance-one characterization; no specialist review or priority claim is recorded.	Use T160's complete distance normalization to study degree-versus-distance succinctness and optimal degree rather than further expressiveness levels.
FPRD-T158 Let (d,k,g,q) lie in Tr(L), let U be a nonempty finite set of fixed suffixes, and let b be any Boolean function of their membership answers. With R_(k,h)=0 for h=0 or k=0 and R_(k,h)=k+(h-1)(k-1) otherwise, put D_U=max over u in U of R_(k,\|u\|+1). Then the observer language {p : b((1_L(pu))_(u in U))=1} has transport tuple (d,D_U,g,q 2^\|U\|). The compiled machine uses exactly the original persistent graph. In particular, every fixed right quotient L/u preserves nonempty stationary transport spectrum.	Theorem · finite-observer closure with explicit transport boundsSelf-contained proof with compiled-control, locality, and sharpness audits	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Finite future-observer theorem	FPRD-T153; FPRD-D45	Finite future observers internalize into stationary transportReviewed 2026-09-23The proof uses the locally established readable-cone theorem FPRD-T153; no external or specialist review of this resource-bound formulation is recorded.	Define and test observer families whose suffix sets grow with the cut, since every fixed finite right-context observer is now known to compile.
FPRD-D45 The stationary transport spectrum Tr(L) is the set of resource ceilings (degree, descriptor distance, working-alphabet size, control-state count) under which the canonical residual dynamics of L has a one-node-per-symbol bounded-locality persistent presentation. It minimizes over all presentations within the scaffolding model.	Definition · implementation-minimized stationary transport invariantDefined exactly at the finite scaffolding-automaton boundary	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Stationary transport spectrum	FPRD-T156; Loff--Moreira--Reis scaffolding automata	Stationary transport spectraReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Place positive forgotten-action tiers inside explicit spectrum cones and search for presentation-independent emptiness criteria.
FPRD-T157 For every language L, Tr(L) is upward closed in the componentwise order and therefore, by Dickson's lemma, is generated by a unique finite antichain of Pareto-minimal resource tuples whenever nonempty. Complement preserves Tr exactly. If tuples (d_i,k_i,g_i,q_i) present L_i, then every binary Boolean combination has a presentation bounded by (d_1+d_2, max(k_1,k_2), g_1g_2, q_1q_2). Nonempty spectrum is equivalent to recognition by a finite scaffolding automaton and implies a PEG for the reverse language through FPRD-T110.	Theorem · finite basis and Boolean calculus for transport resourcesSelf-contained proof with finite-basis and paired-machine audits	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Finite Pareto basis theorem	FPRD-D45; Dickson's lemma; FPRD-T110	Stationary transport spectraReviewed 2026-09-23The finite-basis step is classical Dickson's lemma and the model is due to Loff, Moreira, and Reis; no external review of this FPRD spectrum formulation is recorded.	Develop lower bounds that exclude all Pareto tuples simultaneously, or compile projected frontier representations into an explicit feasible tuple.
FPRD-D44 For a language L, Q_h is the finite set of response rows of prefixes on continuations of length at most h. Restriction gives surjections Q_(h+1) to Q_h, forming an inverse system. Reading one input symbol maps Q_(h+1) to Q_h by selecting the corresponding response subtree.	Definition · canonical finite-horizon quotient systemDefined self-containedly and connected to residual derivatives	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Residual tower	FPRD-D42; left quotients; inverse systems of finite sets	Residual towers and canonical transportReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Use the inverse-limit shift as the semantic object for an encoding-independent theory of stationary transport presentations.
FPRD-T156 The inverse limit of the finite-horizon residual tower is a compact totally disconnected deterministic response machine with continuous input transitions. The full Myhill--Nerode residual space embeds equivariantly and densely. A language is regular exactly when the tower stabilizes, equivalently when its quotient sizes are bounded or one restriction map Q_(h+1) to Q_h is bijective.	Theorem · canonical residual completion and stabilization criterionSelf-contained proof with exhaustive finite audit	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Residual completion theorem	FPRD-D44; Myhill--Nerode theorem; inverse limits of finite sets	Residual towers and canonical transportReviewed 2026-09-23The theorem is aligned with established profinite and topological recognition theory; no external review of this FPRD formulation is recorded.	Use the stationary framework D45/T157 and later compilation results; the remaining frontier is structural lower bounds, spectrum emptiness, or explicit compilation for consequential presentations.
FPRD-D43 For a demanded map on a finite product source, operational support records coordinates touched by a replay trace, semantic support records coordinates capable of changing the demanded answer, and quotient memory identifies sources with equal complete answers. These are distinct resources.	Definition · three-layer replayable-memory modelDefined self-containedly with exact comparison maps	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Three memory layers	FPRD-T150; FPRD-D42	The support-quotient theorem for replayable memoryReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Define stationary transport cost for updating and naming quotient classes as the cut advances.
FPRD-T155 For a map F on a finite product source, the essential-coordinate set E(F) is the unique least sufficient coordinate support; the minimum exact deterministic summary has \|F(X)\| states; and \|F(X)\| is at most the product of the alphabets on E(F), with equality exactly when the induced map is injective. For a family of futures, essential support is the union of component supports. Every stable adaptive replay envelope contains E(F).	Theorem · exact support-quotient factorizationSelf-contained proof with exhaustive finite audit	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Support-quotient theorem	FPRD-D43; FPRD-T150; FPRD-T152	The support-quotient theorem for replayable memoryReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Use D45/T157 and the later normal-form and selector results to study the surviving structural cost of stationary transport across cuts.
FPRD-D42 A finite cut-response system records the answer beta(p,c) for each past p and possible continuation c. Its response dimension is log2 of the number of distinct response rows, separating the information needed for readiness across a continuation family from the replay support used by any one completed continuation.	Definition · finite-cut communication invariantDefined self-containedly and connected exactly to residual rows	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Cut-response definition	FPRD-D41; deterministic one-way communication	Cut-response factorization and linear observer dimensionReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Extend the finite-horizon scaffold capacity bound to richer stationary frontier presentations.
FPRD-T152 For every finite cut-response system, the minimum number of exact deterministic summary states equals the number of distinct response rows. Hence the fixed-length one-way cost is ceil(log2 R) bits for R rows. For linear observers a·x over F_q^n whose span has dimension r, the exact minimum is q^r states, or ceil(r log2 q) bits. Any resource class with at most B(s) cut presentations can realize the system only if R<=B(s).	Theorem · exact cut factorization and capacity transferSelf-contained proof and exhaustive finite falsification audit	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Cut-response factorization	FPRD-D42; deterministic one-way communication; rank-nullity	Cut-response factorization and linear observer dimensionReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Use FPRD-T154 to replace row cardinality by a structural response invariant or a tighter reachable-presentation capacity theorem.
FPRD-T153 Fix a Loff--Moreira--Reis scaffolding automaton of degree d, distance k, working alphabet size g, and control set Q. For continuation horizon h, every response row is determined by the cut control state and the radius R=k+(h-1)(k-1) unfolded neighbourhood when k,h>=1, with radius zero when k=0 or h=0. If V_0=g+1 and V_(r+1)=1+(g+1)V_r^d, then every horizon-h cut has at most \|Q\|V_R distinct response rows. The radius R is uniformly sharp.	Theorem · readable-cone capacity and sharp horizon radiusSelf-contained proof with exhaustive finite structural audit	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Finite-horizon scaffold capacity	FPRD-T152; FPRD-T133; Loff--Moreira--Reis scaffolding automata	Finite-horizon response capacity of scaffolding automataReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Use FPRD-T154 to replace raw row cardinality by a structural invariant that survives larger fixed scaffold parameters.
FPRD-T154 For every finite input alphabet of size s, one fixed generic T153 parameter tuple, degree max(2,s), distance 2, one working label, and one control state, has a numerical abstract-view envelope at least as large as the maximum possible number of Boolean response rows over every continuation horizon h. Therefore a lower-bound proof using only finite-horizon response-row cardinality cannot exceed every T153 envelope and cannot by itself separate a language from all scaffolding automata. Any PEG transfer must account for reversal.	Theorem · parameter-uniform barrier for a cardinality methodSelf-contained elementary proof with exact and symbolic finite audit	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Universal row-count barrier	FPRD-T152; FPRD-T153; finite Boolean response tables	The universal row-count barrierReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Define a cut invariant that records advance compatibility, horizon coherence, or realizability structure rather than only the number of rows.
FPRD-D41 For a fixed family of continuations, a prefix's continuation-exposed replay profile records the accept/reject response to each continuation. Pointwise replay width is the largest source slice used by one completed continuation, while the future-exposed union contains every old source coordinate selected across the family.	Definition · bridge from execution slices to residual observationsDefined self-containedly and calibrated on an exact selector family	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Replay-profile definition	FPRD-D40; FPRD-T150	Replay fan-out theoremReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Define transport orbits for persistent roots and test them against a visible lift of the Kim-Park SCA-hard language C.
FPRD-T151 Let M contain the nonempty binary words whose middle symbol, at one-based position ceil(n/2), is 1. Every completed word has a one-source-cell replay certificate when length is public shape data. Nevertheless, for every r there are 2^r prefixes of length 2r and r continuations whose response profiles are all distinct. Thus M has at least 2^r residuals at that cut, and every exact deterministic one-pass recognizer needs at least 2^r configurations there.	Theorem · replay-width/residual-fan-out separationSelf-contained elementary proof and exhaustive finite audit	Evidence and limits →	Automata and formal languagesReplayable memory and cut responses	Replay fan-out separation	FPRD-D41; FPRD-T150; Myhill--Nerode residual separation	Replay fan-out theoremReviewed 2026-09-23No documented external or specialist review of this FPRD result is recorded.	Use the explicit degree-two unary-query selector as a positive calibration, then study branching and merging demand families rather than a single backwards cursor.
FPRD-IC-O01 Selfridge asked whether $\lVert 2^n\rVert=2n$ for every $n$ , where $\lVert m\rVert$ is the least number of ones needed to build $m$ using addition, multiplication, and parentheses.	Recognized open problemOpen; the FPRD track proves only restricted expression-family results	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Problem, scope, and current frontier	No recorded dependencies	Selfridge's integer-complexity question; current context in Konyagin--Oganesyan (2026)Reviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Keep the restricted track paused pending a new normal-form or effective-height theorem; more residue tables are not the current frontier, and unrestricted Selfridge remains open.
FPRD-IC-D01 The active restricted grammar has value $M[A(B+C)(D+E)+F]$ , where displayed operands are positive $(2,3)$ -smooth constructions and all three displayed additions are paid. After cost-preserving normalization, $\gcd(B,C)=\gcd(D,E)=1$ , the inner equation is primitive, and a fork is proper when both inner sums are nonsmooth.	Definition and cost-preserving normal formPrecisely stated and proved within the restricted grammar	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Grammar and normalization	No recorded dependencies	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Preserve the exact paid-atom cost and unit charges in every subsequent reduction.
FPRD-IC-T01 Every normalized primitive proper fork $3^a(B+C)(D+E)+2^f3^g=2^N$ satisfies $f\le2$ . Moreover the odd parts of the two nonsmooth inner sums have opposite values of the quadratic character $\chi(x)=+1$ for $x\equiv1,3\pmod8$ and $-1$ for $x\equiv5,7\pmod8$ .	Computer-assisted restricted theoremProved and independently reconstructed within FPRD Lab	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Thin-branch theorem	FPRD-IC-D01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use the restriction to keep all later attacks inside the surviving f=0,1,2 branches.
FPRD-IC-T02 In the normalized $g=0$ , positive-type-I branch, the ladder $3^aP=(2^{2rx}-1)/(2^x+1)$ has no surviving canonical negative-character solution with $r\ge2$ . The remaining $r=1$ controls cost at least $2N+2$ .	Exact computer-assisted restricted theoremProved with complete periodic certificates and an independent evaluator	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Cyclotomic ladder closure	FPRD-IC-D01; Michael Bennett, Theorem 1.6, for inherited boundary equations	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Keep this branch closed unless a defect is found in a displayed periodic certificate.
FPRD-IC-T03 The complete odd- $N$ , $g=0$ , positive-type-I branch of the normalized proper fork has no displayed expression of cost at most $2N-1$ . Its only solution in the two nonsmooth canonical shapes is $7\cdot73+1=2^9$ , whose two presentations cost $2N+2$ and $2N+3$ .	Exact restricted branch theoremProved with congruence and primitive-divisor arguments; independently audited	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Odd-exponent branch	FPRD-IC-D01; FPRD-IC-T02; Zsigmondy's theorem on one boundary	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not reopen the odd-N branch; concentrate on the surviving even-N families.
FPRD-IC-T04 At the first even layer $a=1$ , all but two large-x residue families and one x=4 control lift are excluded. In the x=4 family, every positive solution of $3(4+3^v)(1+16\cdot3^y)+1=2^N$ must satisfy $(v,y,N)\equiv(0,0,8)\pmod{(576,144,360)}$ , $v_3(2^N-13)=\min(v,y)+1$ , and $v+y>10^{70}$ .	Exact restricted reduction and lower boundProved and independently audited as an intermediate reduction; the lift is closed by FPRD-IC-T05	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Even-layer reduction	FPRD-IC-D01; FPRD-IC-T01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Retain the Hensel class as the input to FPRD-IC-T05; do not treat the x=4 lift as an active survivor.
FPRD-IC-T05 The inherited $a=1$ , II2, $x=4$ equation $3(4+3^v)(1+16\cdot3^y)+1=2^N$ has no positive solution. Its first 3-adic Hensel digit forces $N\equiv3248\pmod{9720}$ ; modulo $19441$ , the right side is then $15812$ , while the nine possible left-side residues omit $15812$ .	Exact computer-assisted restricted theoremProved by a finite nine-residue certificate and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Exceptional-lift closure	FPRD-IC-T04	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T07 for the type-II2 continuation and concentrate on the remaining large-x type-II1 family.
FPRD-IC-T06 Every remaining $a=1$ large-x solution would already have displayed cost at most $2N-1$ . The 2-adic gates force $v\ge55313581$ in II1 and $v\ge60254788$ in II2. In the II2 subbranch $y<v$ , $y\ge5$ , an exact modulus- $19441$ quotient eliminates 17 of the 28 inherited classes for $x\pmod{540}$ , leaving 11 necessary classes.	Exact restricted reduction and partial branch theoremProved and independently audited as an intermediate reduction; II2 is closed by FPRD-IC-T07	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Large-x frontier	FPRD-IC-D01; FPRD-IC-T05	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Retain the 17/11 split as the input to FPRD-IC-T07; do not treat the eleven classes as active survivors.
FPRD-IC-T07 The large-x equation $3(4+3^v)(1+2^x3^y)+1=2^N$ has no solution under the inherited $a=1$ , type-II2 conditions. Modulo 7 forces $3\mid y$ ; the cases $y=2,4$ fall immediately and $y=3$ falls modulo 73. For $y\ge5$ , exact quotients modulo $19441$ , $6481$ , and $13$ eliminate all 28 inherited classes for $x\pmod{540}$ .	Exact computer-assisted restricted branch theoremProved with displayed finite certificates and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete type-II2 closure	FPRD-IC-D01; FPRD-IC-T05; FPRD-IC-T06	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T08 for the type-II1 continuation; do not treat this family as an active survivor.
FPRD-IC-T08 The last $a=1$ type-II1 family $3(2+3^v)(1+2^x3^y)+1=2^N$ , with $v$ odd, $N\equiv16\pmod{18}$ , and $x$ in the sixteen inherited classes modulo $270$ , has no solution. Modulo $7$ forces $(v,y)\equiv(3,0)\pmod6$ ; modulo $13$ , the left residues are then $\{0,7\}$ and the right residues are $\{3,10\}$ . Together with the preceding canonical closures, this completes the full $a=1$ layer of the active normalized branch.	Exact elementary restricted layer theoremProved by two displayed residue quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete first-layer closure	FPRD-IC-D01; FPRD-IC-T05; FPRD-IC-T07	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-B01 and FPRD-IC-T09 for the first a=2 family; do not reopen a=1.
FPRD-IC-B01 For the $a=2$ type-I2 equation $9(1+4\cdot3^v)(1+2\cdot3^y)+1=2^N$ , the coarse necessary classes $v\equiv5\pmod6$ , $y\equiv0\pmod6$ , and $N\equiv6\pmod{36}$ cannot be excluded by any finite collection of moduli coprime to $6$ : the formal boundary $(v,y,N)=(-1,0,6)$ supplies compatible residues. The sharper 3-adic condition $N\equiv42$ or $78\pmod{108}$ breaks this obstruction and enables FPRD-IC-T09.	Exact coarse-sieve method-boundary lemmaProved algebraically and checked on independent finite quotients	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Next-layer boundary	FPRD-IC-D01; FPRD-IC-T08	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Apply the 3-adic refinement before coprime order moduli; FPRD-IC-T09 shows that this closes the family.
FPRD-IC-T09 The equation $9(1+4\cdot3^v)(1+2\cdot3^y)+1=2^N$ has no positive solution in the inherited $a=2$ type-I2 branch. Modulo $7$ and $13$ force $v\equiv5\pmod6$ , $y\equiv0\pmod6$ , and $N\equiv6\pmod{36}$ . An exact 3-adic comparison at $2^6$ sharpens this to $N\equiv42$ or $78\pmod{108}$ . Modulo $271$ , the resulting 25 left residues and 10 right residues are disjoint.	Exact computer-assisted restricted branch theoremProved by displayed valuation and finite residue certificates; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	First a=2 family closure	FPRD-IC-D01; FPRD-IC-T08; FPRD-IC-B01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T10 for the final a=2 family; do not treat a=2 as an active survivor.
FPRD-IC-T10 The remaining odd-exponent type-II2 equation $9(4+3^v)(1+2^x3^y)+1=2^N$ , with $v,y\ge1$ and $x=v_2(37+3^{v+2})$ , has no solution. Modulo $73$ and the primary 2-adic gate force $v\equiv5\pmod{12}$ and $(x,y)\equiv(1,3),(4,11),(7,7)\pmod{(9,12)}$ . Modulo $577$ , where $3=2^{105}$ and $\operatorname{ord}_{577}(2)=144$ , the required logarithm classes $\{1,4,7\}\pmod9$ are disjoint from the available classes $\{2,3,5,6\}\pmod9$ . Together with the preceding canonical closures, this completes the $a=2$ layer of the active normalized branch.	Exact computer-assisted restricted layer theoremProved by displayed finite quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete second-layer closure	FPRD-IC-D01; FPRD-IC-T09	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T11 for the a=3 entry table; do not reopen any entry with x at most 20.
FPRD-IC-T11 In the active normalized branch $27P(1+2^x3^y)+1=2^N$ , where $P$ is one of the four negative-character canonical factors, the two $x=1$ families and every entry with $3\le x\le20$ have no solution. Modulo $81$ gives $N\equiv18$ or $36\pmod{54}$ . The exact gate $27P+1\equiv2^x3^y\pmod{2^{2x}}$ , followed by a complete quotient modulo $73$ , forces $(x,y)\equiv(0,6),(3,2),(6,10)\pmod{(9,12)}$ . Exact finite covers eliminate $x=3,6,9,12,15,18$ . Hence every survivor has $x\ge21$ and $3\mid x$ ; FPRD-IC-T12 closes this remote tail.	Exact computer-assisted entry theorem and finite range closureProved by exact residue quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Third-layer entry theorem	FPRD-IC-D01; FPRD-IC-T10	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T12 for the remote tail; do not reopen the a=3 positive-type-I entry table.
FPRD-IC-T12 The remote branch $27P(1+2^x3^y)+1=2^N$ , with $x\ge21$ , $3\mid x$ , and $P$ one of the three surviving negative-character shapes, has no solution. Modulo $262657$ , where $\operatorname{ord}(2)=27$ and $\operatorname{ord}(3)=14592$ , the complete quotient leaves eight residue rows. Six fail modulo $7$ and two fail modulo $13$ . Together with FPRD-IC-T11, the complete $a=3$ positive-type-I layer is empty.	Exact computer-assisted restricted layer theoremProved by complete residue quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Remote third-layer closure	FPRD-IC-D01; FPRD-IC-T11	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Treat positive type I as closed and move to the three low positive-type-II layers.
FPRD-IC-T13 For every normalized primitive proper equation $3^aPQ+1=2^N$ with $f=g=0$ and $a\ge4$ , both canonical odd types $Q=1+2^x3^y$ and $Q=2^x+3^y$ are impossible. The valuation identity gives $27\mid N$ , so modulo $73$ and $262657$ both equations force $Q=0$ . Every canonical negative factor $P$ is nonzero at both primes, while the complete exponent classes making $Q=0$ at the two primes have no simultaneous lift.	Uniform exact computer-assisted restricted theoremProved by displayed subgroup and incompatible-lift certificates	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Uniform high-layer obstruction	FPRD-IC-D01; FPRD-IC-T01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not resume exterior-layer induction above a=3; only the low positive-type-II layers remain in the g=0 branch.
FPRD-IC-T14 Every normalized primitive proper $f=g=0$ fork whose positive-character factor is $1+2^x3^y$ has displayed cost at least $2N$ . The cyclotomic and odd- $N$ theorems close $a=0$ ; FPRD-IC-T08 and T10 close $a=1,2$ ; FPRD-IC-T11 and T12 close $a=3$ ; and FPRD-IC-T13 excludes every $a\ge4$ .	Restricted branch-closure corollaryProved by the cited exhaustive branch decomposition	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complete positive-type-I branch	FPRD-IC-T02; FPRD-IC-T03; FPRD-IC-T08; FPRD-IC-T10; FPRD-IC-T12; FPRD-IC-T13	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Classify positive type II at a=1,2,3; the uniform theorem already supplies its high-layer boundary.
FPRD-IC-T15 The normalized complementary family $27P(v)(2^x+3^y)+1=2^N$ , with $P$ any canonical negative-character factor, has no solution. Its exact gate quotient contains 524288 states; relaxed but solution-preserving filters modulo $6481,257,73,7,193$ leave respectively $16512,1268,390,124,0$ states.	Exact computer-assisted restricted layer theoremProved by a complete finite quotient and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary third-layer closure	FPRD-IC-D01; FPRD-IC-T13	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Keep a=3 closed and concentrate on the exact residual quotient at a=1,2.
FPRD-IC-T16 Every canonical solution of $3^aP(v)(2^x+3^y)+1=2^N$ with $a\in\{1,2,3\}$ satisfies $x\le10$ . A coupled quotient at the aligned primes $17,193,257,12289$ leaves two high-gate rows modulo $(512,512,6144,6144)$ ; all 162 simultaneous lifts to periods $(1536,1536,18432,18432)$ fail modulo $7,13,73,97,577$ .	Uniform exact computer-assisted restricted theoremProved by coupled exponent quotients and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Uniform high-x obstruction	FPRD-IC-D01; FPRD-IC-T13	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not resume a high-x search; the low-x quotient is now classified by FPRD-IC-T21.
FPRD-IC-T17 Every remaining canonical solution of $3^aP(v)(2^x+3^y)+1=2^N$ , $a\in\{1,2,3\}$ , lies in one of 21 displayed periodic rows modulo $(v,y,N)=(1536,1536,18432)$ , all at $a=1,2$ , with $x\in\{1,3,4,5,6,8\}$ . These are necessary lift classes, not a classification.	Exact residual-quotient theoremProved as a complete necessary quotient; every row is now classified by FPRD-IC-T21	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Twenty-one residual lift classes	FPRD-IC-T15; FPRD-IC-T16	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use this finite quotient only as the input to the completed second-gate classification FPRD-IC-T21.
FPRD-IC-T18 For the five residual rows of $9P(v)(64+3^y)+1=2^N$ inherited from FPRD-IC-T17, every solution satisfies $v\equiv v_0\pmod{36288000}$ , $y\equiv y_0\pmod{36288000}$ , and $N\equiv12\pmod{48384000}$ , where $(v_0,y_0)$ is its displayed boundary residue. Twelve simultaneous multiplicative-order quotients preserve exactly one lift of each row.	Exact computer-assisted lift theoremProved and independently reconstructed; superseded as a frontier by FPRD-IC-T21	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Shared five-row boundary lock	FPRD-IC-T17	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Retain as an exact intermediate theorem; do not extend this lock after FPRD-IC-T21 closes every nonboundary lift.
FPRD-IC-T19 Every non-boundary solution in the five-row cluster of FPRD-IC-T18 satisfies $v+y>26774635116096000$ and $N\ge42436792629504012$ . Equivalently, after writing $v=v_0+36288000r$ , $y=y_0+36288000s$ , every solution has $r+s>737837167$ .	Certified computer-assisted height theoremProved, but superseded by the complete nonboundary exclusion FPRD-IC-T21	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	One-sided approximation barrier	FPRD-IC-T18	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not extend the height sweep; the full second gate now excludes every lift to which this conditional bound applied.
FPRD-IC-T20 Let $F(v,y)=3^aP(v)(2^x+3^y)+1$ . Eleven rows of FPRD-IC-T17 have fixed valuation $v_2(F)=N_0$ . In each of the other ten rows, the solutions of $F(v,y)\equiv0\pmod{2^e}$ form a one-dimensional Hensel graph: modulo $L_e=3\cdot2^{e-2}$ , there are exactly $2^{e-11}$ states and projection to the $v$ -coordinate is bijective.	Exact 2-adic lifting theoremProved by LTE and complete finite base checks; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Full valuation gate and Hensel graph	FPRD-IC-T17	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Reuse the full-side valuation gate before applying odd-prime quotients in the remaining fork orientations.
FPRD-IC-T21 The normalized positive-type-II family $3^aP(v)(2^x+3^y)+1=2^N$ with $a\ge1$ has exactly ten canonical proper solutions, representing $3\cdot5\cdot17+1=2^8$ , $3\cdot31\cdot11+1=2^{10}$ , $9\cdot5\cdot91+1=2^{12}$ , and $9\cdot13\cdot35+1=2^{12}$ . Every presentation costs at least $2N+4$ .	Exact computer-assisted restricted branch classificationProved from the complete residual quotient, full 2-adic gate, and one aligned prime; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Positive-type-II classification	FPRD-IC-T13; FPRD-IC-T15; FPRD-IC-T16; FPRD-IC-T17; FPRD-IC-T20	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Move to the complementary a=0<g orientation and final 2-adic exponents f=1,2; do not reopen the classified tower.
FPRD-IC-T22 Let $RS+3^g=2^K$ be the reduced $a=0<g$ equation, with $R$ the negative-character canonical odd factor and $S$ the positive-character factor. Of the twelve possible canonical pairs modulo 24, the equation and the two Jacobi identities leave exactly $(R,S;K,g)\equiv(5,1;1,1),(5,11;0,0),(7,1;0,0),(7,17;1,0),(13,1;0,1),(13,19;0,0)\pmod{(24,24;2,2)}$ .	Exact quadratic-residue restrictionProved elementarily and independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary quadratic gate	FPRD-IC-D01; FPRD-IC-T01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use the six-row gate before any 3-adic lifting or odd-prime quotient in the complementary orientation.
FPRD-IC-T23 No normalized primitive proper solution with $a=0<g$ and $f=2$ has $v_2(B+C)=v_2(D+E)=1$ . Indeed, the two odd parts must have opposite character and hence residues $5$ and $17\pmod{24}$ , a pair excluded by FPRD-IC-T22.	Exact restricted branch exclusionProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Double-even branch exclusion	FPRD-IC-T01; FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Classify the three surviving f=1 residue rows, beginning with their unique-minimum and tied 3-adic strata.
FPRD-IC-T24 In every normalized primitive proper solution in the three complementary $f=1$ residue rows, let $(1+3^b)/2$ be the odd part of the even factor and write the other factor as $W=2^h+2^q3^y$ . Then $\min(b,y,g)=1+v_3(K+1-h)$ . All six tied-minimum strata whose leading 3-adic digit could cancel are impossible modulo 7 and 73.	Exact valuation theorem and tied-stratum exclusionProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=1 valuation law	FPRD-IC-T01; FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Combine this law with the f=2 valuation theorem and classify the first two minimum layers m=1,2.
FPRD-IC-T25 In every normalized primitive proper solution in the complementary $f=2$ , split- $(2,0)$ branch, let $V_b=(1+3^b)/4$ , where $b\ge3$ is odd, and write the other odd factor as $W=2^h+2^q3^y$ . Then $\min(b,y,g)=1+v_3(K+2-h)$ . All ten tied-minimum strata whose leading 3-adic digit could cancel are impossible: nine by complete periods modulo 7 and 73, and the sole residual class by incompatible complete periods modulo 487 and 2593.	Exact valuation theorem and tied-stratum exclusionProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=2 valuation law	FPRD-IC-T01; FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Combine the f=1 and f=2 laws at minimum layers m=1,2, seeking a finite order-modulus cover before opening the f=0 rows.
FPRD-IC-T26 In the normalized complementary $f=1$ and split- $(2,0)$ $f=2$ branches, impose $m=\min(b,y,g)\in\{1,2\}$ . Of the 32 canonical minimum strata, nine are empty. Every state in the other 23 lies on one of 40 exact periodic lift graphs with common coordinate period $P=20{,}652{,}025{,}680$ . Twenty-nine graphs are rooted at seventeen proper harmless presentations and eleven at nine improper rational boundary identities. Every noncontrol proper solution therefore has output exponent $N\ge P$ .	Exact modular lift classification and height barrierProved without exponent bounds; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary minimum-layer lift lock	FPRD-IC-T22; FPRD-IC-T24; FPRD-IC-T25	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use the next 3-adic digit and an archimedean descent on the eleven boundary-rooted lift graphs; do not merely enlarge the order-modulus period.
FPRD-IC-T27 In the normalized complementary $f=1$ and split- $(2,0)$ $f=2$ branches, every solution with $m=\min(b,y,g)\in\{1,2\}$ is one of exactly seventeen explicit proper presentations. All seventeen cost at least $2N+6$ . The eleven lift graphs rooted at improper rational identities are empty modulo $3^6$ ; 26 control-rooted graphs freeze completely; and the remaining three reduce to $2^{K+1}=11\cdot3^b+173$ or $2^K=3^g+295$ , whose nonzero lifts are excluded by Matveev's explicit lower bound and exact finite modular descent.	Exact computer-assisted classificationProved with one classical linear-forms dependency; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary minimum-layer classification	FPRD-IC-T26; FPRD-IC-T24; FPRD-IC-T25	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Test whether the valuation period and an explicit linear-forms cutoff cross uniformly in m; otherwise open the complementary f=0 quotient.
FPRD-IC-T28 The normalized complementary $f=0,1,2$ family has only finitely many presentations. More precisely, its eight fixed factor-shape patterns contain at most $8\exp(7\cdot30^{15})$ presentations in total. This is a direct corollary of the Evertse--Schlickewei--Schmidt bound for nondegenerate S-unit equations: after expansion and division by $2^{K+f}$ , five positive $\{2,3\}$ -units sum to one in a multiplicative group of rank at most six.	Classical S-unit corollaryProved from an established theorem; not an effective height bound	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Uniform complementary finiteness	FPRD-IC-D01	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Do not treat the solution-count bound as a height cutoff; obtain finite congruence quotients inside the remaining f=0 strata.
FPRD-IC-T29 In the normalized complementary $f=0$ branch $RS+3^g=2^K$ , the quadratic gate leaves forty canonical factor placements and exactly twenty possible tied-minimum 3-adic cancellations. Only two can have $v=y=g$ . One has no solutions; the other has the unique solution $(2+3)(1+8\cdot3)+3=2^7$ , whose displayed construction costs $18=2K+4$ ones.	Exact elementary classification with finite residue certificateProved; independently reconstructed	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=0 triple-minimum classification	FPRD-IC-T22	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Use FPRD-IC-T30 for the pair-minimum continuation; do not treat the eighteen strata as active survivors.
FPRD-IC-T30 In the normalized complementary $f=0$ branch $RS+3^g=2^K$ , the eighteen pair-minimum strata whose first nonzero 3-adic digit can cancel contain exactly one solution, $(4+3)(1+8\cdot3)+3^4=2^8$ , of displayed cost $29=2K+13$ . Together with FPRD-IC-T29, the entire exceptional cancellation locus consists of this control and $(2+3)(1+8\cdot3)+3=2^7$ . Every other solution satisfies $\min(v,y,g)=1+v_3(K-r-u)$ .	Exact computer-assisted restricted theoremProved with complete finite quotients and independent reconstruction	Evidence and limits →	Number theory and discrete dynamicsSelfridge integer complexity	Complementary f=0 cancellation closure	FPRD-IC-T22; FPRD-IC-T29; LTE	FPRD Selfridge integer-complexity research trackReviewed 2026-09-06No documented external or specialist review of these FPRD results is recorded.	Pause layer-by-layer enumeration; reopen this branch only with an effective six-term reduction, a uniform freezing theorem, or another structural bridge.
FPRD-ST-L01 Let $I$ be a minimal nonzero connected-supergroup-invariant two-sided ideal of a finite algebra $A$ , with $I^2=I$ . If $J\triangleleft I$ is nonzero and invariant under every superderivation of $I$ , then $AJA=I$ and $I=IJI\subseteq J$ ; hence $I$ is differentiably simple.	Structural lemmaProved and hostile-audited within FPRD Lab	Evidence and limits →	Models of computationTensor categories	Ambient saturation proof	Victor Kac, Proposition 5.1.1, is used after this lemma	FPRD Lab supertannakian semisimplicity research recordReviewed 2026-09-06No external or specialist review is recorded; the proof has passed a hostile internal audit.	Seek specialist review of the connected-integration and saturation step.
FPRD-ST-L02 Let $C\to B$ be finite superalgebras in $\operatorname{Rep}^fH$ . If $B\otimes_C B\to B$ has an even bimodule section after forgetting $H$ , and $C$ is internally absolutely flat, then $B$ is internally absolutely flat.	Categorical transfer lemmaProved and hostile-audited within FPRD Lab	Evidence and limits →	Models of computationTensor categories	Faithful-forgetful retract proof	FPRD-ST-L01	FPRD Lab supertannakian semisimplicity research recordReviewed 2026-09-06No external or specialist review is recorded; the proof has passed a hostile internal audit.	Seek tensor-category review of the arbitrary-ind-module scope.
FPRD-ST-T01 For a finite associative algebra object $A$ in a supertannakian category over an algebraically closed field of characteristic zero, absolute flatness, internal $\operatorname{Ext}^1$ -vanishing for finitely presented modules, product decomposition into simple algebra objects, internal semisimplicity, absence of nonzero nilpotent two-sided ideal objects, and rigidity of finitely presented bimodules are equivalent.	Resolution of Coulembier--Sherman Question 5.16Proved and hostile-audited within FPRD Lab; no external review	Evidence and limits →	Models of computationTensor categories	Full proof	FPRD-ST-L01; FPRD-ST-L02; Coulembier--Sherman, Theorem 3.17 and Corollary 6.6; Kac, Proposition 5.1.1; Wall's matrix/queer classification	FPRD Lab supertannakian semisimplicity research recordReviewed 2026-09-06No external or specialist review is recorded; the proof has passed a hostile internal audit.	Obtain external specialist review and monitor the source question for revisions.
QA-RA-S6-D15 A two-reservoir receipt amplifier transports a latent distinction through one observation-null region, uses a seam to manufacture a second presentation with a fresh null region, and exposes the distinction only at the second seam.	Definition · presentation dynamicsPrecisely stated and realized by an exact reverse-and-add family	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Definition and exact realization	FPRD-D39; FPRD-T144	FPRD Quotient Arsenal Stage 6 two-reservoir attackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Formulate a doctrine-neutral composition theorem for seams that convert exhausted resources into fresh null-receipt coordinates.
QA-RA-S6-T15 For every $q\ge4$ , the equal-length Hamming-adjacent integers $a_q=2^{9q-1}+154$ and $b_q=2^{9q-1}+1178$ have identical binary digit-growth observations for exactly $12q-12=4L/3-12$ steps, where $L=9q$ , and differ at the next observation.	Exact theorem · latent quotient failureProved by an all-parameter symbolic run induction and hostilely audited	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Theorem, mechanism, and proof	QA-RA-S6-D15; FPRD-T144	FPRD Quotient Arsenal Stage 6 two-reservoir attackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the serial-reservoir mechanism with the Stage 2 obstruction-and-receipt calculus without promoting the unproved complete-future shift.
QA-RA-S6-C15.1 The family satisfies $\tau_g(a_{q+1},b_{q+1})=\tau_g(a_q,b_q)+12$ : nine added source bits create exactly twelve additional null observations. Consequently every sufficient complete-response horizon on all $9q$ -bit inputs obeys $D_{9q}\ge12q-11$ .	Corollary · quantitative receipt lower boundExact consequence of QA-RA-S6-T15	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Receipt-growth corollary	QA-RA-S6-T15	FPRD Quotient Arsenal Stage 6 two-reservoir attackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine whether serial reservoir composition has an additive or multiplicative distortion law in a doctrine-neutral setting.
QA-RA-S6-N15 The exact twelve-step increase in first-divergence latency does not establish equality of complete shifted futures such as $\gamma(b_q)_{[21,\infty)}=\gamma(b_{q-1})_{[9,\infty)}$ .	Negative boundary · failed strengtheningStronger coinductive inference rejected; finite-latency theorem retained	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Latency versus bisimulation boundary	QA-RA-S6-T15	FPRD Quotient Arsenal Stage 6 hostile auditReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Prove the shifted complete-future identity coinductively or keep it permanently separate from the latency theorem.
FPRD-D39 Fix $b\ge2$ , $A_b=\{0,\ldots,b-1\}$ , and $n\in\mathbb N_0$ . Write the canonical expansion $n=\sum_{i=0}^{L-1}a_i b^i$ least-significant first, with $L=1,a_0=0$ for zero and $a_{L-1}\ne0$ otherwise. Define $\operatorname{rev}_b(n)=\sum_i a_i b^{L-1-i}$ , $T_b(n)=n+\operatorname{rev}_b(n)$ , $c_i=a_i+a_{L-1-i}$ , $\gamma_0=0$ , $s_i=(c_i+\gamma_i)\bmod b\in A_b$ , and $\gamma_{i+1}=\lfloor(c_i+\gamma_i)/b\rfloor$ , retaining $\gamma_L$ . Reversal-created leading zeroes do not change the integer. For canonical digits $d_i$ of $m$ , $D_b(m)=\#\{i:d_i\ne d_{M-1-i}\}$ counts ordered positions.	DefinitionExact conventions reviewed; no theorem is embedded in this record	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Definitions and conventions	No recorded dependencies	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Find the smallest update-closed receipt that carries this one-step presentation across iteration.
FPRD-T144 For each fixed $b\ge2$ and $n\in\mathbb N_0$ , the quantities in FPRD-D39 satisfy $\gamma_i\in\{0,1\}$ and $T_b(n)=\sum_{i=0}^{L-1}s_i b^i+\gamma_L b^L$ . After the symmetric fold supplies $c_0,\ldots,c_{L-1}$ in increasing digit index, the carry stage is a deterministic subsequential transducer over $C_b=\{0,\ldots,2b-2\}$ , with states $\{0,1\}$ , initial state 0, transition $\lfloor(c+\gamma)/b\rfloor$ , output $(c+\gamma)\bmod b$ , and terminal output 1 exactly when the final carry is 1. Within this folded one-step presentation, carry is the only orientation-dependent sequential state.	Written theorem · fixed-base transducer familyProved on the public page; finite replay is corroboration, not proof	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Folded carry theorem and proof	FPRD-D39	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare this elementary decomposition with prior reverse-and-add and finite-transducer literature before making any novelty claim.
FPRD-T145 Under FPRD-T144, if $\gamma_L=0$ (equivalently, the step does not increase digit length), then for $j=L-1-i$ , $s_i-s_j\equiv\gamma_i-\gamma_j\pmod b$ and the integer $\gamma_i-\gamma_j$ lies in $\{-1,0,1\}$ . This is a residue statement, not an absolute-distance bound; its residue-set corollary is nonrestrictive in bases 2 and 3.	Written corollary · modular digit structureProved on the public page; finite replay is corroboration, not proof	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Mirrored carry law and proof	FPRD-D39; FPRD-T144	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit prior art, especially the documented observation that a palindromic output requires a palindromic carry pattern.
FPRD-T146 A positive decimal input has a palindromic one-step reverse-and-add output exactly in one of two cases: every symmetric column sum is at most $9$ , or every sum lies in $\{0,11\}$ and the outer sum is $11$ . On $1\le n\le1{,}000{,}000$ , the two classes contain $151{,}250$ and $808$ inputs.	Digit-structure theorem · reproduced finite censusGeneral classification proved; complete bounded census independently reproduced	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Classification proof and bounded census	FPRD-D39; FPRD-T144	Vaughn Suite, reverse-and-add carry note (16 September 2003); FPRD independent proof and censusReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Translate the decimal carry proof to general bases, treating the exceptional base-two growth pattern separately.
FPRD-T147 For the outputs $T^k(196)$ , $1\le k\le4{,}000$ , the reflection distance has minimum $2$ , its minimum normalized value is $4/33$ at step $68$ , and its mean normalized value over the $3{,}771$ outputs longer than $100$ digits is $0.659803985542$ .	Reproduced bounded computational findingExact finite window independently replayed by two implementations	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Reflection-distance observation	FPRD-D39	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the same pinned statistic with convergent controls; do not extrapolate the finite lower bound to the infinite orbit.
FPRD-T148 For the first $3{,}000$ transitions from $196$ , the mean normalized shifted reflection distance is $0.496558797244$ at shift $0$ on the $1{,}735$ non-growing outputs and $0.497066641824$ at shift $+1$ on the $1{,}265$ growing outputs; the mismatched shifts tested from $-2$ through $+2$ lie between $0.884$ and $0.897$ .	Reproduced bounded computational findingExact finite table independently replayed; universal shifted-axis identity unproved	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Axis-shift correction	FPRD-D39; FPRD-T145	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Derive or refute an exact shifted-index identity; retain the present result as a finite table until then.
FPRD-T149 Across the states $T^k(196)$ , $0\le k<4{,}000$ , every within-state decimal digit block through length $4$ and every incoming-carry block through length $12$ occurs. The pinned replay gives carry-in density $0.499366103113$ , mirrored-digit mutual information $0.563591475633$ bits, and mirrored carry-in mutual information $0.000013795714$ bits.	Reproduced bounded computational findingExact finite window and estimators independently replayed	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Raw-complexity observation	FPRD-D39; FPRD-T144	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add explicit control sequences under the same estimators before making comparative complexity claims.
FPRD-FAIL-LYCHREL-01 The aperiodic-monotile substitution analogy was stopped because the sampled raw digit and carry streams showed none of the low-complexity hierarchy that route intended to exploit. This failure does not rule out conditional, finite-state, backward-preimage, or anti-concentration approaches.	Important failed approachScope-corrected negative research record	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Failed-route boundary	FPRD-T149	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Preserve the fold-and-carry representation while avoiding claims that raw block complexity excludes richer invariants.
FPRD-C04 Does there exist a positive decimal integer whose reverse-and-add orbit never reaches a palindrome? In particular, does the orbit of $196$ avoid palindromes forever?	Open problemOpen; no FPRD resolution	Evidence and limits →	Number theory and discrete dynamicsNumber theory and discrete dynamics	Open boundary	FPRD-D39	FPRD Lab, Folded reverse-and-add and carry analysis (6 June 2026)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Pursue finite structural results and exact interfaces without treating bounded orbit searches or random-like statistics as an infinite proof.
QA-BK-D01 For a disk with n marked points and b labeled interior boundary components, capping the interior boundaries defines a quotient π, while Buckman’s vector D records the b integral boundary-twist coordinates forgotten by π.	Definition · quotient interfacePrecisely stated and used in the exact splitting theorem	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	Capping setup	Buckman thesis, §§3.1–3.3	Buckman thesis; FPRD quotient-receipt normalizationReviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Compare the coordinate convention with framed-braid conventions when boundary labels are allowed to permute.
QA-BK-T01 Under the labeled-disk hypotheses, (π,D) is an isomorphism from Mod(D_n^b) to Mod(D_{n+b},Q)×Z^b; equivalently, every reduced mapping class has a unique zero-D lift.	Theorem · exact quotient reconstructionProved from the capping kernel and D normalization; internally audited	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	Splitting theorem and proof	QA-BK-D01; Classical capping-kernel theorem	Deduction from Buckman §§3.1–3.3; framed-braid splitting backgroundReviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Seek a mapping-class specialist check of the exact correspondence between Buckman’s D convention and the standard framed-braid splitting.
QA-BK-T02 For any normalized set-section σ of π, its factor-set cocycle satisfies c_σ=δ(D∘σ); correcting σ by that 1-cochain produces the unique zero-D homomorphic section.	Theorem · explicit cocycle primitiveProved by direct substitution; internally audited	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	Cocycle calculation	QA-BK-T01	FPRD deduction from Buckman’s D invariantReviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Extend the calculation to the semidirect coefficient system arising from permuted boundary labels.
QA-BK-T03 If two lifts U,V have the same capped image and Δ=D(V)−D(U), all two-sided positive boundary corrections are x=Δ₊+c and y=Δ₋+c with c∈N^b; the unique minimum total cost is \|\|Δ\|\|₁.	Theorem · exact resource minimumProved coordinatewise; internally audited	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	Positive balancing theorem	QA-BK-T01	FPRD deduction applied to Buckman’s balancing stepReviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Measure how this boundary-twist cost interacts with Hurwitz distance and lantern substitutions in the Chapter 4 examples.
QA-BK-T04 For a connected double cover, a cover-preserving mapping class has a boundary-pointwise lift exactly when its relative phase ε in reduced H⁰ of the boundary vanishes; for a planar twist, ε(t_c^m)=(m mod 2)·ω(c)·χ_c.	Theorem · lifting obstructionProved by path lifting and mod-two intersection; internally audited	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	Boundary-phase lift theorem	Cover character ω and path-lifting criterion	FPRD refinement of the Birman–Hilden lifting step used in BuckmanReviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Label the actual Chapter 4 factor curves by their cut vectors and compare the resulting zero-receipt blocks with the thesis’s liftable packets.
QA-BK-N01 When ε(t_c) is nonzero, the element t_c² is liftable although the displayed factorization (t_c)(t_c) has two individually unliftable factors; regrouping changes the admissible atom grammar without being a Hurwitz move.	Counterexample · presentation boundaryExact minimal witness proved	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	Packetization boundary	QA-BK-T04	FPRD formulation of the squared-twist convention used across Buckman §§2.4, 3.1, and 3.3Reviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Treat packetization as an explicit allowed operation in any algorithm comparing liftable positive factorizations.
QA-BK-T05 In B₃ there are two length-three factorizations of the same element, all factors positive half-twists with identical unary conjugacy data, whose generated subgroups have permutation images of orders 2 and 6; hence they are not Hurwitz equivalent, even up to simultaneous conjugation.	Theorem · finite-quotient obstructionExact witness proved and independently replayed	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	B₃ obstruction and certificate	Braid relation xyx=yxy; Hurwitz invariance of the generated subgroup	FPRD construction; B₃ algorithmic context compared with OrevkovReviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Search the fixed boundary-multitwist and Buckman factor types for the first pair not decided by this finite quotient.
QA-BK-N02 The attempted search for a nonzero H² obstruction to the labeled-disk capping extension fails: Buckman’s D supplies an explicit primitive, so the extension class vanishes; nontriviality can reappear only after changing hypotheses such as boundary labeling or ambient surface.	Failed approach · exact negative resultRefuted under the stated hypotheses; boundary of refutation recorded	Evidence and limits →	Algebra and discrete mathematicsPlanar factorization quotients and Hurwitz receipts	Why the first obstruction vanishes	QA-BK-T02	FPRD hostile hypothesis test; closed-surface contrast from Bellingeri–GervaisReviewed 2026-08-28No documented external or specialist review of this FPRD deduction is recorded.	Test the permuted-boundary semidirect extension rather than reopening the refuted labeled direct-product case.
FPRD-QA-MX-D01 The presentation card fixes $M_n(\mathbb C)$ , conjugation, the characteristic-polynomial quotient, the balanced normal locus, Jordan receipts, stabilizers, and future-observation interfaces.	Definition · quotient interfaceExact definition; internally audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the card as the normalization template for new quotient doctrines.
FPRD-QA-MX-T01 The characteristic polynomial is the affine quotient for one matrix; every fibre has one closed semisimple orbit, and its normal representatives form one unitary orbit.	Theorem · classical import and exact adapterProved and internally checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Compare simultaneous-conjugation fibres where traces of words replace one characteristic polynomial.
FPRD-QA-MX-T02 The conjugacy-orbit Frobenius infimum is $\sum_i\|\lambda_i\|^2$ , attained exactly for semisimple matrices; for $0<t\le1$ , the displayed diagonal scaling of $J_m(\lambda)$ has excess $(m-1)t^2$ and condition number $t^{-(m-1)}$ .	Theorem · resource distortionSelf-contained proof and exact checks	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Determine optimal conditioning for prescribed distance above the orbit floor.
FPRD-QA-MX-T03 For every eigenvalue $\lambda$ , the increments of $\dim\ker(A-\lambda I)^k$ recover the conjugate Jordan partition; the finite tower recovers the full similarity class inside an invariant fibre.	Theorem · exact receiptProved and internally checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Study which truncated towers suffice for restricted matrix classes.
FPRD-QA-MX-T04 The centralizer dimension is $\sum_{\lambda,k}(\pi'_{\lambda,k})^2$ , and shifted-power ranks can only drop in orbit closure.	Theorem · stabilizer and degenerationDirect proof with classical converse boundary	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Extend the receipt-order comparison to several commuting matrices.
FPRD-QA-MX-T05 The infinite scalar quotient $A\mapsto(\operatorname{tr}A^k)_{k\ge1}$ equals the characteristic-polynomial quotient and therefore forgets all Jordan-block data.	Theorem · exact lossProved from Newton identities and Cayley–Hamilton	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Classify minimal observation families that recover selected Jordan data.
FPRD-QA-MX-T06 For $V\cong\mathbb C^n$ and outputs $CA^kv$ , the universal state quotient is $V/\bigcap_{k=0}^{n-1}\ker(CA^k)$ ; horizon $n-1$ suffices and can be sharp.	Theorem · universal quotientSelf-contained proof and exact checks	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Compare nonlinear response quotients with their linearized towers.
FPRD-QA-MX-X01 $2I_2$ and $J_2(2)$ have the same invariant quotient but different Jordan receipts, stabilizers, and Frobenius-infimum attainment.	Counterexample · quotient lossExact and reproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use as the mandatory smallest hostile test for matrix quotients.
FPRD-QA-MX-X02 Repeated-eigenvalue semisimple and nonsemisimple 3-by-3 matrices collide in polynomial invariants while their receipts and stabilizers differ.	Counterexample · boundary caseExact and reproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Retain as the repeated-eigenvalue regression fixture.
FPRD-QA-MX-C01 A public dependency-free census checks 44 partitions through dimension seven, 233 dominance comparisons, centralizer formulas, six Jordan-scaling rows, the two-dimensional collision, and sharp observation chains.	Computational findingReproduced independently	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Preserve the certificate as a regression suite.
FPRD-QA-MX-F1 Balancing selects a unitary orbit, not a distinguished matrix; treating it as a canonical point silently discards residual symmetry.	Failed approach · stabilizer boundaryRefuted exactly	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Carry the residual unitary stabilizer as part of the receipt.
FPRD-QA-MX-F2 A nonsemisimple conjugacy orbit approaches but never reaches its balanced Frobenius floor.	Negative resultProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Measure approximation cost together with conditioning.
FPRD-QA-MX-F3 Extending scalar trace observations to every positive power still leaves the Jordan receipt invisible.	Failed approach · observation lossRefuted by theorem and examples	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Change the observation type rather than merely adding more scalar powers.
FPRD-QA-MX-F4 Restricting attention to the balanced normal locus records the closed semisimple orbit in each invariant fibre, but it does not retain the Jordan data of nonsemisimple orbits in that fibre.	Negative comparisonExact scope boundary	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained matrix calibration and public audit	No recorded dependencies	FPRD one-matrix calibration; Kempf–Ness, Procesi, Gerstenhaber, and Kalman foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Combine balanced-locus data with a separate fibre-degeneration receipt.
QA-SEW-T1 An edge receipt on a finite connected graph is killed by vertex corrections exactly when its circulation vanishes on a cycle-space basis.	Theorem · finite sewingProved and verifier-backed	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	No recorded dependencies	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use this as the degree-one control case for every new sewing doctrine.
QA-PROBE-T1 Given cochain maps $i^:C^\bullet(P)\to C^\bullet(A)$ and $q^:C^\bullet(Q)\to C^\bullet(A)$ , let $D^k=\{a:i^a\in\operatorname{im}q^\}$ and $N^k=\ker i^$ . Then $D$ and $N$ are subcomplexes, and restriction induces $D^\bullet/N^\bullet\cong\operatorname{im}i^\cap\operatorname{im}q^*\subseteq C^\bullet(A)$ .	Theorem · first-isomorphism adapterProved and exactly audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	No recorded dependencies	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Specify admissible probe categories before exporting the construction.
FPRD-QA-T06 For a short exact sequence $0\to N\xrightarrow{i}D\to Q\to0$ , lift a quotient cocycle $q\in Z^k(Q)$ to $x\in D^k$ and write $d_Dx=i(c)$ . A closed lift exists exactly when $[c]\in H^{k+1}(N)$ vanishes; if $c=d_Na$ , then $x-i(a)$ is a closed lift.	Classical connecting-homomorphism criterion · FPRD receipt adapterProved and internally checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	No recorded dependencies	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Track minimal or resource-weighted primitives in applications.
FPRD-QA-T07 For a fibrewise short exact sequence of abelian local systems on a finite graph, the edge-overlap disagreement of local lifts is the cochain connecting cocycle. Its class vanishes exactly when the quotient section has a globally transport-compatible lift.	Theorem · graph-local-system specializationProved and exhaustively audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	FPRD-QA-T06	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Extend to higher-dimensional nerves through totalization.
FPRD-QA-T08 For nested subcomplexes $N\subset M\subset D$ , a class in $H^k(D/M)$ lifts to $H^k(D)$ exactly when its primary class in $H^{k+1}(M/N)$ vanishes and the resulting secondary coset in $H^{k+1}(N)/\operatorname{im}\delta_{N,M}$ vanishes.	Theorem · staged obstructionProved and exhaustively audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	FPRD-QA-T06	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Identify higher-stage indeterminacy when more layers are present.
QA-CECH-A1 For a finite cover and a short exact sequence of abelian cochain presheaves, local lifts $a$ of a global closed quotient cochain determine the closed total cochain $O(a)=da+(-1)^k\check\delta a$ . A strict global closed lift exists exactly when $O(a)=-D_{\mathrm{tot}}n$ for a null receipt $n$ of Čech degree zero.	Theorem · local-to-global adapterProved and independently audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Local-to-global and reduction-in-stages proof and evidence	No recorded dependencies	FPRD local-to-global and stage receipts; classical double-complex and Eilenberg–Zilber foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Keep strict and homotopy-coherent lift targets distinct.
QA-XR-T1 In a commuting bicomplex, if $[\partial x]=0$ in $\delta$ -cohomology, choose $h$ with $\delta h=\partial x$ . The corrected total cochain has next defect $\partial h$ ; the choice-independent secondary obstruction is the class of $[\partial h]$ modulo the image induced by $\partial$ on $H_\delta$ , equivalently the corresponding $E_2$ -class.	Theorem · obstruction staircaseProved, sign-audited, and scope-corrected	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Local-to-global and reduction-in-stages proof and evidence	No recorded dependencies	FPRD local-to-global and stage receipts; classical double-complex and Eilenberg–Zilber foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	State the secondary receipt in its natural quotient after pinning both axes and signs.
QA-STAGE-T1 For commuting Hamiltonian actions of a direct product $N\times H$ with split moment map, the Koszul and curvature receipts split and direct versus two-stage reduction are compatible. At the simplicial level, diagonal and iterated totalizations are naturally chain-homotopy equivalent rather than literally identical.	Theorem · split calibrationProved and computationally checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Local-to-global and reduction-in-stages proof and evidence	No recorded dependencies	FPRD local-to-global and stage receipts; classical double-complex and Eilenberg–Zilber foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Carry action and curvature receipts before applying reduction in stages beyond direct products.
QA-STAGE-F2 The real three-dimensional Heisenberg extension has nonzero central splitting curvature: for any linear lifts of the two quotient basis vectors, their bracket is the central basis vector. Therefore a naive zero-curvature split-stage model cannot represent this extension.	Counterexample · missing hypothesisExact, checked, and domain-corrected	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Local-to-global and reduction-in-stages proof and evidence	No recorded dependencies	FPRD local-to-global and stage receipts; classical double-complex and Eilenberg–Zilber foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Carry the extension curvature as a stage receipt.
FPRD-QA-FAIL06 If the null family is not closed under the differential, the proposed quotient differential is not well-defined.	Counterexample · necessary hypothesisProved by finite example	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	No recorded dependencies	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Audit differential closure before defining any probe quotient.
FPRD-QA-FAIL07 A loop that is trivial after quotienting can still produce nontrivial stabilizer displacement upstairs.	Counterexample · hidden groupoid receiptExact	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	No recorded dependencies	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Retain the stabilizer or action-groupoid cocycle.
FPRD-QA-FAIL08 Different first-stage corrections can change the raw secondary receipt; only its coset modulo induced correction images is canonical.	Counterexample · indeterminacyProved and checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Self-contained cohomological sewing proof and evidence	No recorded dependencies	FPRD cohomological sewing proofs and finite controls; classical connecting-homomorphism and local-coefficient foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Publish staged receipts as cosets, not chosen representatives.
QA-CECH-F2 Overlap compatibility alone does not imply global extendability: on a two-set cover, the presheaf with global group $0$ , both local and overlap groups $\mathbb Z/2$ , identity local restrictions, and zero global restrictions has the compatible local pair $(1,1)$ , but no global section maps to it.	Counterexample · local/global boundaryExact and independently enumerated	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Local-to-global and reduction-in-stages proof and evidence	No recorded dependencies	FPRD local-to-global and stage receipts; classical double-complex and Eilenberg–Zilber foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	State the relevant descent or global-surjectivity hypothesis when using compatible local data.
QA-XR-F2 If two primary primitives differ by a $\delta$ -closed cochain $z$ , their raw secondary defects differ by $\partial z$ . Thus the unquotiented secondary class is not canonical; only its coset modulo the induced image of $\partial$ on $H_\delta$ is.	Negative result · choice dependenceExact and independently checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Local-to-global and reduction-in-stages proof and evidence	No recorded dependencies	FPRD local-to-global and stage receipts; classical double-complex and Eilenberg–Zilber foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Record the secondary obstruction in its natural quotient or spectral-sequence term.
FPRD-QA-COL-D01 A chronological parity word $w$ determines the exact affine block $T^k(n)=(3^{s(w)}n+a_w)/2^k$ and its unique residue cylinder $n\equiv-3^{-s(w)}a_w\pmod{2^k}$ .	Definition · exact chartClassical formula, precisely normalized	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the pinned chronology and residue convention in every derived claim.
FPRD-QA-COL-T01 For $w=uv$ , $a_{uv}=3^{s(v)}a_u+2^{\|u\|}a_v$ , and a unique tail coordinate gives $r_{uv}=r_u+2^{\|u\|}c_{u\mid v}$ .	Theorem · exact sewingProved and checked	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Treat the seam coordinate as part of the chart, not informal compatibility.
FPRD-QA-COL-T02 Computing $T^k(n)\bmod2^h$ requires and is determined by $n\bmod2^{h+k}$ ; no modulus $2^{h+k-1}$ suffices.	Theorem · future-observation costProved and exhaustively checked in bounded ranges	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Compare acceptance-specific quotients that discard most response values.
FPRD-QA-COL-T03 A word $w$ has an integral periodic lift exactly when $[a_w]=0$ in $\mathbb Z/(2^k-3^{s(w)})$ ; every nonempty word has a unique 2-adic lift because the coefficient $2^k-3^{s(w)}$ is odd and nonzero.	Theorem · lifting obstructionProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Keep integral, positive, and 2-adic acceptance as separate quotients.
FPRD-QA-COL-T04 Every deterministic depth- $h$ selector on the binary chamber graph has one equivariant 2-adic section; an integral section additionally requires integral cycle values and integral inverse-tree lifts.	Theorem · section criterionProved and exhaustively checked through depth four	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Analyze compatibility across selector depths rather than isolated finite sections.
FPRD-QA-COL-T05 Every coherent parity tower sews uniquely in $\mathbb Z_2$ ; its least residues represent an element of $\mathbb N_0$ exactly when they are bounded, equivalently eventually constant.	Theorem · admissible-locus criterionProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Audit admissible-locus membership in every inverse-limit construction.
FPRD-QA-COL-T06 Cyclic rotation transports the integral cokernel receipt by a unit, while repetition injects the original cokernel and preserves integrality and positivity.	Theorem · receipt transportProved and extensively checked	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use primitive cyclic classes when counting distinct periodic presentations.
FPRD-QA-COL-T07 An adjacent rewrite $x01y\to x10y$ changes the least residue by a term of exact 2-adic valuation equal to the swap position; the depth- $h$ receipt count is a truncated binomial sum.	Theorem · minimal static receiptProved and exhaustively checked through length fourteen	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the receipt count as a loss bound only for the pinned fixed-weight quotient.
FPRD-QA-COL-X01 For $w=001$ , the integral receipt is $4\bmod5$ , so no integral lift exists, while the unique 2-adic lift is $4/5$ .	Counterexample · completion mismatchExact	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Retain as the smallest completion-versus-integrality fixture.
FPRD-QA-COL-X02 Every finite prefix of $1^\infty$ has positive integer realizations, but the coherent least residues $2^k-1$ sew to $-1\in\mathbb Z_2\setminus\mathbb N_0$ .	Counterexample · ghost limitExact	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Require bounded least representatives when the target doctrine is the nonnegative integers.
FPRD-QA-COL-X03 The words $1010$ and $1100$ have multiplier $9/16$ but translations 7 and 5; only the first has an integral fixed lift.	Counterexample · slope insufficiencyExact	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Carry the ordered translation or cokernel class with the slope.
FPRD-QA-COL-C01 Among 8,800 primitive parity necklaces through period sixteen, exactly five have integral fixed lifts; the positive graph induced on $\{1,\ldots,100000\}$ contains only the $1\leftrightarrow2$ cycle.	Computational findingReproduced	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Do not enlarge the search without a new structural criterion.
FPRD-QA-COL-C02 At depths one through four, all selectors have 2-adic sections; integral counts are $3,5,6,1$ and positive counts are $1,0,0,0$ .	Computational findingReproduced	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Study cross-depth compatibility rather than another marginal count.
FPRD-QA-COL-C03 The fixed-weight checker verifies every valuation, receipt-count, and fine-cycle exactness formula through length fourteen.	Computational findingReproduced	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Preserve as a regression certificate for quotient-loss claims.
FPRD-QA-COL-FAIL01 Linear-part holonomy may be nontrivial on a realized cycle even though exact state differences telescope; the affine translation and its cokernel are indispensable.	Failed approach · incomplete invariantRefuted exactly	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the full affine block rather than its multiplier alone.
FPRD-QA-COL-FAIL02 Collapsing a fixed-slope fibre creates labelled comparison loops, but those loops record discarded vertical data rather than cycles of the Collatz dynamics.	Failed inferenceCorrected	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Collatz guard-chamber review and independent audit	No recorded dependencies	FPRD guard-chamber proofs; Terras, Böhm–Sontacchi, Lagarias, and Bernstein–Lagarias foundationsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Name the comparison graph and its semantics before interpreting circulation.
FPRD-QA-MG-D01 The quotient retains vertex and dart orbits and separately records stabilizer cosets, monodromy, and fibre occupancy.	Definition · quotient interfaceExact definition	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Keep parallel dart orbits until monodromy is computed.
FPRD-QA-MG-T01 For a dart orbit represented by $a$ leaving $u$ , its lifts from a fixed lift of $u$ are indexed by the coset set $\Gamma_u/\Gamma_a$ ; the lift is unique exactly when these stabilizers agree.	Theorem · minimal local receiptProved and checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Attach coset labels only where stabilizers differ.
FPRD-QA-MG-T02 The simple orbit quotient preserves orbit reachability, while equitable quotient matrices preserve class-level walk counts; neither statement implies unique lifting.	Theorem · interface separationProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Declare the intended observation before choosing a graph quotient.
FPRD-QA-MG-T03 A free action without edge inversion gives a regular cover; the product of edge voltages is the monodromy that determines the lifted endpoint and component structure.	Theorem · classical adapterProved and calibrated	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use dart or edge-orbit quotients when parallel edges carry different voltages.
FPRD-QA-MG-T04 A Hamiltonian cycle in the base lifts to one Hamiltonian cycle upstairs exactly when its fibre monodromy is transitive; for a cyclic regular cover, its voltage must generate the deck group.	Theorem · exact lifting criterionProved and checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Search quotient cycles together with monodromy, not before it.
FPRD-QA-MG-T05 The projection of an upstairs Hamiltonian cycle visits each quotient vertex exactly as many times as the size of its fibre.	Theorem · occupancy obligationProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Combine occupancy with local cosets and monodromy in tour searches.
FPRD-QA-MG-X01 Under half-turn symmetry, each of four outgoing dart orbits from the fixed center has two lifts, so one bit is required per chosen outgoing orbit.	Example · sharp local receiptReproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use as the local nonfree-action regression case.
FPRD-QA-MG-X02 The half-turn quotient of $C_4$ has two parallel edge orbits with different voltages; collapsing them to a simple $K_2$ erases whether the lift is one 4-cycle or two 2-cycles.	Counterexample · quotient lossProved and reproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Retain darts or parallel edge orbits whenever cycle lifting matters.
FPRD-QA-MG-C01 Exact half-turn quotients of the 4-by-4, 5-by-5, and 6-by-6 knight graphs verify local lift counts and exhibit 6-by-6 quotient cycles of voltages zero and one.	Computational findingReproduced independently	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the certificate as a covering/monodromy regression suite.
FPRD-QA-MG-FAIL01 Merging parallel quotient edges can preserve reachability while destroying the receipt needed for closed-walk lifting.	Failed approachRefuted by C4	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use a multigraph or dart quotient.
FPRD-QA-MG-FAIL02 A group action free on vertices may still invert an undirected edge, so regular-cover conclusions require a no-inversion hypothesis.	Negative result · missing hypothesisExact	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Check darts or barycentrically subdivide before invoking cover theory.
FPRD-QA-MG-FAIL03 A Hamiltonian quotient cycle need not lift to a Hamiltonian cycle without the occupancy and monodromy receipts.	Failed inferenceRefuted exactly	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Move-graph lifting and monodromy review	No recorded dependencies	FPRD move-graph proofs and finite audit; Gross–Tucker voltage covers and Schwenk's knight-tour classificationReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Apply FPRD-QA-MG-T04 and T05 before any tour claim.
FPRD-QA-RA-D01 The admissible presentation is the disjoint union of fixed-length mirror-profile charts, quotiented by equality of the evaluated one-step output.	Definition · quotient atlasExact definition	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Reverse-and-add cross-length atlas review and independent audit	No recorded dependencies	FPRD fixed-length mirror profiles and Stage 5 cross-length seam proofsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the atlas as the base for cross-length receipt calculations.
FPRD-QA-RA-T01 Every cross-length profile fibre has at most two points and lies in the two adjacent charts determined by the output length.	Theorem · cross-length sewingProved and exhaustively regressed	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Reverse-and-add cross-length atlas review and independent audit	No recorded dependencies	FPRD fixed-length mirror profiles and Stage 5 cross-length seam proofsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Carry only the chart choice across the seam.
FPRD-QA-RA-T02 The bit $\varepsilon=M-m$ suffices to recover the source chart from the post-step value, and one bit is necessary in every base.	Theorem · sharp receiptProved in every base	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Reverse-and-add cross-length atlas review and independent audit	No recorded dependencies	FPRD fixed-length mirror profiles and Stage 5 cross-length seam proofsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Combine it with the within-chart source-fibre receipt when raw reconstruction is required.
FPRD-QA-RA-T03 If $f_m(y)$ counts raw $m$ -digit sources over output $y$ , then the full fibre has size $f_{M-1}(y)+f_M(y)$ ; cross-length gluing adds at most one bit to the larger within-chart source-fibre budget.	Theorem · fibre budgetProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Reverse-and-add cross-length atlas review and independent audit	No recorded dependencies	FPRD fixed-length mirror profiles and Stage 5 cross-length seam proofsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Compare the budget with finite-horizon response quotients.
FPRD-QA-RA-C01 A dependency-free checker verifies 1,208,130 admissible profiles over bases two through twelve and lengths one through eight, including 310 overlap outputs.	Computational findingReproduced	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Reverse-and-add cross-length atlas review and independent audit	No recorded dependencies	FPRD fixed-length mirror profiles and Stage 5 cross-length seam proofsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Preserve as a regression fixture.
FPRD-QA-RA-FAIL01 Cross-length one-step fibres do not exhibit unbounded isotropy; they have size at most two.	Failed conjectural routeRefuted by FPRD-QA-RA-T01	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Reverse-and-add cross-length atlas review and independent audit	No recorded dependencies	FPRD fixed-length mirror profiles and Stage 5 cross-length seam proofsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Look for complexity in response depth, not one-step fibre size.
FPRD-QA-RA-FAIL02 The cross-length seam graph is a matching, so it carries no nontrivial cycle monodromy at one step.	Negative resultProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Reverse-and-add cross-length atlas review and independent audit	No recorded dependencies	FPRD fixed-length mirror profiles and Stage 5 cross-length seam proofsReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Move the obstruction search to repeated responses.
QA-RA-S6-T04 While two nonzero flanks remain separated by zeros, their reverse-and-add state follows an exact four-parameter recurrence and the gap changes by $G'=G+g-2$ .	Theorem · causal presentationProved and independently checked	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the gap as an explicit resource rather than extrapolating past collision.
QA-RA-S6-C04.1 An initial separated gap $G\ge2h$ guarantees $h$ exact causal updates.	Corollary · resource boundProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Determine sharp survival times for special flank families.
QA-RA-S6-T05 The outward aggregate together with current growth does not determine next growth; binary states 129 and 194 are an exact collision.	Counterexample · insufficient receiptProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Search for the smallest sufficient quotient between the aggregate and full flanks.
QA-RA-S6-T06 The separated-flank receipt inverse limit contains a compatible point whose every finite prefix is realized but which has no single finite separated-flank realization.	Theorem · ghost limitProved; finite prefixes checked	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Specify the completion doctrine before using inverse-limit points as states.
QA-RA-S6-T11 For $n_q=2^{9q-1}+3$ , the binary growth trace begins $(101010)^q1011$ ; $n_q$ and $n_{q+r}$ agree for $6q+3$ bits and then differ.	Theorem · unbounded response depthProved and independently checked	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Seek upper bounds or structural descriptions for the infinite response quotient.
QA-RA-S6-C11.1 The equivalence relations induced by finite growth-response horizons are not eventually constant.	CorollaryProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Measure the realized class-growth rate.
QA-RA-S6-C11.2 Binary reverse-and-add realizes infinitely many distinct complete growth-response sequences.	CorollaryProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Identify a natural generator or grammar for a substantial subfamily.
QA-RA-S6-C11.3 Every quotient from which the full binary growth response can be decoded has infinitely many states.	Corollary · state lower boundProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Separate cardinality lower bounds from description complexity.
QA-RA-S6-C11.4 Nonstabilization persists even when the current digit-length parity is observed.	Corollary · strengthened negative resultProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Test other finite side observations structurally, not by enumeration alone.
QA-RA-S6-C11.5 The horizon- $h$ growth quotient has at least $\lfloor(h+2)/6\rfloor-1$ realized classes; a fixed-parity fibre retains an order- $h/12$ bound.	Corollary · quantitative lower boundProved	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Search for a sharper family or an upper-bound presentation.
QA-RA-S6-FAIL11 Finite response censuses cannot establish nonstabilization at all horizons; an explicit delayed-divergence family is required.	Failed approach · evidence boundaryClosed by theorem	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Use enumeration only for discovery and regression.
FPRD-QA-RA-T04 The master ledger records a proved synchronization/lumpability theorem, but its proof payload was not durably available in the imported archive.	Provisional archival recordIndexed; proof payload unavailable	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Recover stage6-working/synchronizing-receipt-theorem.md and its verifier.
FPRD-QA-RA-D02 The master ledger records a reversal-compatible residue quotient family, but the exact definition packet is missing.	Provisional archival recordIndexed; source unavailable	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Recover the stage6-residue-working packet.
FPRD-QA-RA-T05-T07-ARCHIVE Three proved results on reversal-compatible residue and active-parity receipts are recorded as a family, but their exact statements and verifiers are missing.	Provisional theorem-family recordIndexed; exact payload unavailable	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Recover the theorem and split the range into exact records.
FPRD-QA-RA-X01 A counterexample in the reversal-compatible residue family is listed, but its witness is absent from durable storage.	Provisional counterexample recordIndexed; witness unavailable	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Recover and independently replay the missing witness.
FPRD-QA-RA-D03 The master ledger records a carry-frontier presentation, but its defining source file is missing.	Provisional archival recordIndexed; source unavailable	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Recover stage6-carry-frontier/carry-frontier-theorem.md.
FPRD-QA-RA-T08-T10-ARCHIVE Three carry-frontier, minimal predictive receipt, and residue-sewing theorems are recorded as proved, but their exact proof packet is unavailable.	Provisional theorem-family recordIndexed; exact payload unavailable	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Recover, split, and audit the three claims.
FPRD-QA-RA-CF01 A bounded realized parity-response census is recorded, but its certificate did not survive durable transfer.	Provisional computational recordIndexed; certificate unavailable	Evidence and limits →	Models of computationNumber theory and discrete dynamics	Quotient Arsenal synthesis	No recorded dependencies	QA-RA-STAGE5 revision 1 and durable Stage 6 working packetsReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Recover or rerun the original census.
QA-BUCK-HUR-009 A positive half-twist factorization of a positive full-twist power has connected endpoint graph, so its factor permutations generate the full symmetric group.	Theorem · quotient boundaryProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use cross-component linking rather than larger symmetric quotients.
QA-BUCK-LINK-010 Every pair of endpoint components in the pinned positive factorization problem satisfies an exact cross-component linking budget.	Theorem · abelian sewingProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Combine it with diagonal and nonabelian receipts.
QA-BUCK-MIN-011 For the pinned (3,1,1) endpoint partition, target $\Delta_5^4$ , and squared twists about proper curves, the three-packet cross-linking problem has 55 labelled support multisets in three occupancy types, and three packets are minimal.	Computational theorem · finite classificationExact proof and computation	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Apply stronger receipts before attempting geometric lifts.
QA-BUCK-DIAG-012 Each endpoint component obeys an exact internal crossing equation, completing the colored abelian receipt omitted by cross-linking alone.	Theorem · sewing receiptProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use it to remove false positive occupancy types.
QA-BUCK-SELECT-013 The three minimal occupancy types force half-twist counts $-12,4,8$ ; positivity eliminates the first and the pinned four-half inventory selects the second.	Theorem · resource obstructionProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Test its singleton-threading lift fibre.
QA-BUCK-SHADOW-014 The eight-half branch has the rigid standard-alphabet shadow $x^4(yx^2y)^2=(xy)^6$ after forgetting two singleton strands.	Theorem · necessary shadowProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore singleton threading through a relative Nielsen or nilpotent receipt.
QA-BUCK-CAN-015 All 27 labelled eight-half patterns fail the mod-three singleton-vector receipt in the canonical BKL section.	Computer-assisted obstructionExact but section-relative	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Analyze the complete winding fibre rather than another fixed section.
QA-BUCK-COLLAPSE-016 With arbitrary winding, the three atom grammars become one 40-element mod-three conjugacy class, and its eighth power fills $\operatorname{Sp}_4(\mathbb F_3)$ .	Computational theorem · quotient collapseExact certificate; imported conservatively	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore the missing helper and rerun the full closure.
QA-BUCK-MOVING-017 Canonical rejection, arbitrary-winding permission, one candidate rejection, and alternate-lift permission can coexist; rejecting a chosen lift does not obstruct the whole fibre.	Counterexample · quantifier correctionExact finite realization; replay partly blocked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Preserve the existential quantifier over complete lift fibres in every obstruction claim.
QA-HUR-FIB-T01 Through an abelian extension, allowed factor lifts sew to a target exactly when the prefix/suffix-transported affine factor fibres meet the target fibre.	Theorem · lifting criterionProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Apply the criterion layer by layer in a central filtration.
QA-HUR-FIB-T02 A one-step $p$ -power congruence kernel is an $\mathbb F_p$ -vector space, and multiplication linearizes in its top layer.	Theorem · linearizationProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Track the first nonvanishing cross term explicitly.
QA-HUR-FIB-T03 For corrections $1+2x_i+4y_i$ , a fixed first-layer word lifts by affine equations in $y_i$ , while variation across the complete $x_i$ -solution fibre is quadratic and order-sensitive.	Theorem · two-layer receiptProved and algebraically replayed	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the polar form to detect when the quadratic obstruction collapses.
QA-HUR-FIB-X02 Two noncommuting first-layer corrections can have the same affine sum but different ordered second-layer products.	Counterexample · noncommutative receiptProved and reproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Never replace ordered cross terms by an unordered quadratic count.
QA-BUCK-FIB-C01 The mod-four colored atom orbit has 3,072 states, exactly 32 above each of 96 mod-two states.	Computational findingStored certificate; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore audit_winding_mitm_hits.py and rerun.
QA-BUCK-FIB-C02 Each of nine residual classes has a certified selected eight-atom mod-four lift.	Computational findingStored certificate; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore the dependency and independently replay all witnesses.
QA-BUCK-FIB-X01 A selected nonexact word agrees with its target through degree three of the integral Magnus expansion and first differs in degree four.	Counterexample · finite observation lossExact certificate; imported conservatively	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Compute the full fibre product image before drawing an obstruction conclusion.
QA-BUCK-FIB-T05 For the selected residual-zero word, the vector-valued quadratic obstruction vanishes on all $2^{32}$ first-layer solutions, so every one lifts modulo eight in the pinned Magnus model.	Finite-model theoremCertificate independently audited	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore full recomputation before checked status.
QA-HUR-NI-T01 Appending the inverse target identifies a fixed-target factorization space with a relative product-one tuple slice, equivariantly for Hurwitz moves and quotient maps.	Theorem · comparisonProved and independently checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use relative Nielsen stabilizers to organize complete lift fibres.
QA-HUR-NI-T02 A compatible central extension supplies a lifted-product receipt invariant under Hurwitz moves and simultaneous conjugation; its action-groupoid cocycle is the central quotient-path receipt.	Theorem · obstruction receiptProved and independently checked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Compute the endpoint-stabilizer action in the first noncentral layer.
QA-HUR-NI-C01 The relative Nielsen comparison passes 1,625 exact identities on all 125 length-three nonidentity $S_3$ tuples.	Computational findingReproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Retain as a regression test for the general theorem.
QA-HUR-NI-T03 The colored degree-three quotient has order 393,216, atom stabilizer order 4,096, and degree-four atom fibres of size 128 over the 96-state degree-three atom orbit.	Finite-model theoremStored exact closures; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore the missing helper and rerun the group closures.
QA-HUR-NI-C02 Exact closures contain 393,216 degree-three group states and 12,288 degree-four atom states.	Computational findingCertificate available; local replay incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Repair the archive dependency and promote only after replay.
QA-HUR-FIB-T06 In a square-zero top layer, local lifts sew exactly when the target defect lies in the sum of their prefix/suffix-transported direction spaces; the residual class in the cokernel is the obstruction.	Theorem · exact sewing criterionProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use the cokernel before escalating to a higher quotient layer.
QA-HUR-FIB-D01 The colored atom stabilizer generates local atom lifts; the braid stabilizer of a complete quotient tuple governs path equivalence and must not be quotient out again in product existence.	Definition · quantifier correctionProved distinction	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Name both stabilizers explicitly in future quotient-path theorems.
QA-BUCK-FIB-T07 Each degree-four atom fibre above a degree-three colored atom is an affine $\mathbb F_2$ -space of rank seven with 128 states.	Finite-model theoremStored certificate; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore full replay and retain transported directions for each position.
QA-BUCK-FIB-X03 For one selected residual-two quotient word, the rank-seven transported product image misses the target defect, so none of its $2^{56}$ local lift tuples reaches the degree-four target.	Finite counterexample · nonzero obstructionExact stored certificate; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Repair the checker and retain this as a product-cokernel witness.
QA-BUCK-FIB-C06 Eight other selected quotient words have rank-ten product images containing the target, with explicit degree-four witnesses.	Computational findingStored witnesses; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore direct witness replay.
QA-HUR-FIB-T08 The complete attainable-product set of an admissible lift fibre is invariant under quotient Hurwitz moves; each move bijects Cartesian lift fibres while preserving total product.	Theorem · orbit invariantProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use product-image rank and target membership as quotient-orbit receipts.
QA-HUR-FIB-C08 In a square-zero layer, product-image dimension, target membership, and the product-cokernel class are invariants of the quotient Hurwitz orbit.	Corollary · orbit obstructionProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz fibre theorem core and independent audit	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz action and Fried–Völklein's Nielsen-class frameworkReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Compare its discriminatory power with relative Nielsen stabilizers.
QA-BUCK-FIB-X04 Two residual-two quotient words with the same degree-three product lie in different quotient Hurwitz orbits because their degree-four product receipts have ranks seven and ten and opposite target-lift decisions.	Finite counterexample · orbit separationStored exact certificates; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Repair the verifier before checked status.
QA-BUCK-FIB-C07 The obstructed word transports eight identical rank-seven spaces, while the lifting word transports three classes whose distinct pairs meet in dimension four and together span rank ten.	Computational findingCertificate available; replay dependency incomplete	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Find a structural explanation of the three transported classes.
QA-BUCK-FIB-C08.1 The alternative residual-two word has a sparse-polynomial degree-four target lift recorded by an independent witness.	Computational findingStored witness; import replay blocked	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Restore and replay the sparse checker.
F-HUR-02 Equal endpoint, length, atom type, factor conjugacy classes, and abelian receipts can still leave positive factorizations in distinct Hurwitz orbits.	Failed approach · relational obstructionRefuted by exact B3 witness	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Quotient Arsenal synthesis	No recorded dependencies	FPRD Hurwitz-fibre proofs; Lewis's Hurwitz-action convention; Fried–Völklein's Nielsen-class framework; Buckman's planar settingReviewed 2026-08-30No documented external or specialist review of this Quotient Arsenal result is recorded.	Use relational or subgroup receipts only within their proven scope.
F-HUR-05 For positive full-twist factorizations, connected endpoint support forces the full symmetric group, so subgroup order cannot separate the intended boundary completion.	Failed approach · quotient collapseRefuted by QA-BUCK-HUR-009	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use linking and singleton-threading receipts instead.
F-HUR-07 The 55 cross-linking support solutions are necessary quotient solutions, not realized braid factorizations, because internal braid and label-order data are lost.	Failed inferenceCorrected	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Treat support enumeration as a first-stage filter only.
F-HUR-09 The eight-half branch survives both the colored abelian budget and a positive three-strand shadow.	Negative result · surviving branchProved	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Analyze the nonabelian singleton-threading fibre.
F-HUR-10 A positive factorization after forgetting singleton strands is only a necessary shadow and does not provide a section back to the full braid problem.	Failed inferenceCorrected	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Supply a lift receipt or keep the result at shadow level.
F-HUR-11 Rejecting all patterns in one canonical BKL section does not reject the complete arbitrary-winding fibre.	Failed inference · section dependenceRefuted	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Quantify over the entire allowed fibre.
F-HUR-13 Enlarging the winding fibre saturates the mod-three image rather than sharpening it.	Negative result · quotient saturationStored exact computation	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Change the receipt doctrine instead of enlarging the same quotient.
F-HUR-14 A Nielsen receipt may reject a displayed lift while another lift in the same support fibre reaches the target.	Failed inference · quantifier errorRefuted by full-fibre certificate	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Compute complete attainable-product sets.
F-HUR-15 Both squared packet types are invisible modulo four, and the length-eight active product set contains the identity.	Failed finite-quotient approachExact stored computation	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Do not continue the same low-layer quotient without a structural reason.
F-HUR-16 Further low-prime Burau searches saturated large finite images without exposing the missing geometric coordinate, so the route was stopped for diminishing returns.	Important failed approachDocumented and stopped	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Hurwitz support and positivity review	No recorded dependencies	FPRD colored-crossing and threading proofs; Buckman, Orevkov, A'Campo, Bloomquist–Patzt–Scherich, and Birman–Ko–LeeReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use point-pushing, nilpotent threading, or the product-cokernel receipt.
QA-GR-D03 A type-slot synchronization assigns anonymous double-deletion types to the pairwise slots among vertex-deleted card copies, respecting each card's local type multiset.	Definition · sewing presentationExact	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Use it as the baseline quotient for enriched receipts.
QA-GR-T02 The deck of $2P_3$ has 18 type-slot synchronizations in two symmetry orbits; precisely the six-element $P=Q$ orbit sews, while the twelve-element $P\ne Q$ orbit does not.	Finite theorem · nonzero sewing obstructionProved and independently reproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Test receipts stronger than anonymous overlap types.
QA-GR-C02 All unlabeled graphs through order five have one type-slot chamber; sixteen order-six graphs have multiple chambers.	Computational findingReproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Classify the order-six chamber defects before extending the census.
QA-GR-X02 Complete per-card multisets of anonymous double-deletion types do not suffice to sew a deck coherently.	Counterexample · quotient lossExact finite	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Retain relational synchronization data rather than adding more marginal counts.
QA-GR-D04 A slot $(C,v)$ is refined by the $\operatorname{Aut}(C)$ -orbit of the marked vertex $v$ .	Definition · candidate receiptExact	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Test whether higher marked tuples refine the obstruction.
QA-GR-C03 For all 52 unlabeled graphs of orders one through five, deletion-type classes equal marked-vertex automorphism orbits.	Computational findingReproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Seek the first graph where marked orbits strictly refine deletion types.
QA-GR-X03 The marked-extension refinement adds no information to any tested deck through order six and does not remove the false $2P_3$ chamber.	Negative computational resultReproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Move from one-card automorphism orbits to cross-card matching data.
QA-GR-T03 For the $2P_3$ chamber, two perfect matchings $P,Q$ define an equivariant defect in $\mathbb F_2^2$ , and the synchronization sews exactly when that defect vanishes.	Finite theorem · exact obstructionProved and independently reproduced	Evidence and limits →	Algebra and discrete mathematicsAlgebra and discrete mathematics	Graph-deck synchronization proof and independent audit	No recorded dependencies	FPRD graph-deck synchronization proofs; Bondy–Hemminger, McKay, and smaller-card reconstruction literatureReviewed 2026-09-01No documented external or specialist review of this Quotient Arsenal result is recorded.	Generalize the matching module to the other fifteen order-six multi-chamber graphs.
FPRD-SH-D01 For a letter-labelled NFA $A$ and state $q$ , the loop spectrum is $\Lambda_A(q)=\{\sigma:q\xrightarrow{\sigma}q\}$ .	Definition · local quotient invariantPrecisely stated and used by the obstruction theorem	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Loop spectrum and quotient convention	No recorded dependencies	FPRD Lab definition applied to Broda et al., “Location based automata for expressions with shuffle and intersection” (2023)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use the stronger incoming-spectrum invariant FPRD-SH-T04 for non-recurrent shuffle failures.
FPRD-SH-L01 For every regular expression with shuffle, every noninitial state of its prefix automaton has one incoming transition label; hence no prefix state has self-loops on two distinct letters.	Lemma · structural propertySelf-contained proof at the stated construction	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Homogeneity proof	Broda et al., “Location based automata for expressions with shuffle and intersection” (2023), Section 6	Broda, Machiavelo, Moreira, and Reis, “Location based automata for expressions with shuffle and intersection” (2023); FPRD reconstructionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain homogeneity as the target-side invariant in generated quotient tests.
FPRD-SH-T01 For every existential state quotient $\pi:A\twoheadrightarrow B$ , $\Lambda_A(q)\subseteq\Lambda_B(\pi(q))$ for every state $q$ .	Theorem · elementary quotient invariantSelf-contained proof	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Loop-spectrum monotonicity proof	FPRD-SH-D01	Elementary labelled-graph fact; FPRD application to the shuffle/prefix seamReviewed 2026-08-31No documented external or specialist review of these FPRD results is recorded.	Use the invariant as a constant-time rejection test before exhaustive quotient search.
FPRD-SH-T02 Let $\alpha=\beta\amalg\gamma$ . In the full asynchronous-product location automaton, if the two components have recurrent locations with self-loops on distinct letters $a\ne b$ , then $A_{\mathrm{Pre}}(\alpha)$ is not an existential state quotient of $A_{\mathrm{POS}}(\alpha)$ . For a construction that trims unreachable locations, the paired product location must be reachable.	Theorem · infinite obstruction familySelf-contained proof; exact small-case verification	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Independent-recurrence theorem and proof	FPRD-SH-L01; FPRD-SH-T01; Broda et al., “Location based automata for expressions with shuffle and intersection” (2023)	FPRD Lab analysis of the shuffle location construction in Broda et al. (2023)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Treat this loop theorem as a special case of the exact incoming-spectrum classification FPRD-SH-T05.
FPRD-SH-X01 For distinct letters $a\ne b$ , the three-state prefix automaton of $a^\amalg b^$ is not a quotient of its four-state location automaton. The expression has exactly two letter occurrences, the minimum among shuffle expressions whose two operands both contain a letter.	Counterexample · sharp size boundSelf-contained proof and exact verifier reproduction	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Minimal counterexample and transition tables	FPRD-SH-T02	FPRD deduction; Broda et al. (2023), Example 13, supplies the four-occurrence comparisonReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as the sharp occurrence-count boundary; make no literature-priority claim.
FPRD-SH-X02 For $a^\amalg a^$ , merging the three noninitial location states gives the two-state prefix automaton as an ordinary existential quotient, but no such quotient partition is left-invariant.	Counterexample · sharp boundaryExact proof and exhaustive partition check	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Unary control and predecessor defect	FPRD-SH-T03	FPRD Lab deduction using the 2023 shuffle location/prefix constructionsReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain this as the smallest exceptional case in the all-length proof FPRD-SH-T06.
FPRD-SH-T03 A state partition $\pi$ is left-invariant exactly when initiality is saturated and every pair $q,q'$ in one block has identical predecessor-block sets $\{\pi(r):r\xrightarrow{\sigma}q\}$ for every letter $\sigma$ .	Theorem · exact criterionSelf-contained reformulation of the standard definition	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Predecessor criterion	Left invariance on the reversed automaton	2019 left-quotient architecture; FPRD predecessor-profile reformulationReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Combine this criterion with Boolean block equations for larger SAT-checkable quotient certificates.
FPRD-SH-C01 For $((ab)^)\amalg((bc)^)$ , exact reconstruction gives a 9-state, 18-transition location automaton, a 9-state marked prefix automaton, and an 8-state, 16-transition unmarked prefix automaton; exhaustive partition search finds no quotient.	Computational finding · source reproductionIndependently reproduced with an exact structural checker	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Published example reconstruction	Broda et al., “Location based automata for expressions with shuffle and intersection” (2023), Section 6	Broda, Machiavelo, Moreira, and Reis, “Location based automata for expressions with shuffle and intersection” (2023), Example 13Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use the incoming-spectrum witness from FPRD-SH-T04 as the direct explanation of this example.
FPRD-SH-C02 Among all 306 expressions $x^\amalg y^$ over $\{a,b,c\}$ with $\|x\|+\|y\|\le4$ , 288 have no ordinary prefix quotient; the 18 ordinary-quotient cases are all unary same-letter controls, and none has a left-invariant quotient.	Computational finding · bounded censusExhaustive within the stated finite domain; its pattern is now proved by FPRD-SH-T05	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Census table and scope	Exact automaton constructor; Exhaustive set-partition search	FPRD Lab bounded starred-word shuffle censusReviewed 2026-08-31No documented external or specialist review of these FPRD results is recorded.	Retain this census as the discovery record and finite cross-check for FPRD-SH-T05.
FPRD-SH-T04 For a letter-labelled NFA $A$ , let $I_A(q)=\{\sigma:\exists p\;(p\xrightarrow{\sigma}q)\}$ . If $\pi:A\twoheadrightarrow B$ is an existential state quotient, then $I_A(q)\subseteq I_B(\pi(q))$ for every state $q$ .	Theorem · local quotient invariantSelf-contained proof; strictly strengthens the loop-spectrum test	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Incoming-spectrum theorem	Existential block-transition convention	Elementary labelled-graph fact; FPRD application to shuffle locationsReviewed 2026-08-31No documented external or specialist review of these FPRD results is recorded.	Test the invariant on broader non-starred shuffle syntax.
FPRD-SH-T05 For nonempty words $x,y$ , $A_{\mathrm{Pre}}(x^\amalg y^)$ is an existential state quotient of $A_{\mathrm{POS}}(x^\amalg y^)$ if and only if $x=a^m$ and $y=a^n$ for one letter $a$ and integers $m,n\ge1$ .	Theorem · exact classificationSelf-contained proof with a closed-form quotient certificate and executable audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Exact dichotomy and constructive proof	FPRD-SH-L01; FPRD-SH-T04; Broda et al., “Location based automata for expressions with shuffle and intersection” (2023)	FPRD Lab classification using the shuffle location/prefix constructions of Broda et al. (2023)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-T06 for the stronger left-invariant classification.
FPRD-SH-T06 For all nonempty words $x,y$ , $A_{\mathrm{Pre}}(x^\amalg y^)$ is not a left-invariant quotient of $A_{\mathrm{POS}}(x^\amalg y^)$ .	Theorem · exact classificationSelf-contained all-length proof with exact boundary and parameter audits	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Complete left-quotient obstruction	FPRD-SH-T03; FPRD-SH-T05; Broda et al., “Location based automata for expressions with shuffle and intersection” (2023)	FPRD Lab deduction using the shuffle location/prefix constructions of Broda et al. (2023)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Test whether an analogous predecessor obstruction classifies broader non-starred shuffle syntax.
FPRD-SH-C03 For $1\le m,n\le16$ at the audited parameter pairs, the closed-form projection for $(a^m)^\amalg(a^n)^$ produces an NFA exactly equal to the recursively constructed prefix automaton; the largest audit maps 289 states to 257.	Computational finding · theorem auditExact equality checks pass at all nine selected parameter pairs	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Executable audit	FPRD-SH-T05; Exact automaton constructor	FPRD Lab parameterized certificate verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Keep the finite audit synchronized with any proof or implementation revision.
FPRD-SH-D02 For strong synchronization, solo terminal events are allowed only outside $\Gamma$ and paired terminal events only on equal letters in $\Gamma$ ; for arbitrary synchronization the same paired clause is added without suppressing solo events. Applying the usual backward $R_\varepsilon$ -recursion to these clauses defines the natural prefix automata studied here.	Definition · FPRD extension of the cited location frameworkPrecisely stated for strong and arbitrary synchronization; the weak operator uses a separate backward-memory definition	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Synchronized prefix convention	Broda et al., “The Prefix Automaton” (2021); Broda et al., “Location automata for synchronised shuffle expressions” (2023)	FPRD prefix extension of Broda et al., “The Prefix Automaton” (2021), and Broda et al., “Location automata for synchronised shuffle expressions” (2023)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Keep the strong/arbitrary convention separate from the proved weak backward receipt.
FPRD-SH-T07 For $m,n\ge1$ , under strong synchronization on $\Gamma=\{a\}$ , the accessible location automaton and the natural prefix automaton of $(a^m)^\mathbin{\amalg^{\Gamma}_{s}}(a^n)^$ are isomorphic cycles of length $\operatorname{lcm}(m,n)$ , with their initial state attached. In particular the prefix automaton is a left-invariant quotient of the accessible location automaton.	Theorem · exact all-parameter classificationSelf-contained orbit proof with parameterized executable audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Mandatory-synchronization theorem	FPRD-SH-D02; Broda et al., “Location automata for synchronised shuffle expressions” (2023), strong synchronized-shuffle construction	FPRD deduction from the Broda–Machiavelo–Moreira–Reis strong synchronized-shuffle construction (2023)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Determine which non-unary strong synchronized families retain the same accessible-orbit quotient.
FPRD-SH-T08 For $m,n\ge1$ , under arbitrary synchronization on $\Gamma=\{a\}$ , the natural prefix automaton of $(a^m)^\mathbin{\amalg^{\Gamma}_{a}}(a^n)^$ is an ordinary existential quotient of its location automaton exactly when $\min(m,n)=1$ , and it is a left-invariant quotient exactly when $m=n=1$ .	Theorem · exact all-parameter classificationSelf-contained necessity and constructive sufficiency proofs with exhaustive boundary audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Arbitrary-synchronization classification	FPRD-SH-D02; FPRD-SH-T03; Broda et al., “Location automata for synchronised shuffle expressions” (2023), arbitrary synchronized-shuffle construction	FPRD deduction from the Broda–Machiavelo–Moreira–Reis arbitrary synchronized-shuffle construction (2023)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Test whether the forced-initial-block obstruction extends to non-unary arbitrary synchronization.
FPRD-SH-B01 The published weak synchronized location construction records forward memory $\Delta,\Lambda$ for solo synchronized letters since the last joint event. A prefix construction based on last events instead needs a dual receipt describing allowable solo synchronized letters until the next joint event; reusing the forward memory does not by itself define that automaton.	Boundary · resolved definition gateRequirement discharged by the proved reversal-dual construction	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Weak-synchronization boundary	Broda et al., “Location automata for synchronised shuffle expressions” (2023), weak synchronized-shuffle construction; Broda et al., “The Prefix Automaton” (2021)	FPRD analysis of the directionality mismatch in the cited constructionsReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain the directionality distinction when extending the receipt beyond the unary family.
FPRD-SH-D03 Define $u\overleftarrow{\bowtie}_{\Gamma,D,E}v=\{z^R:z\in u^R\bowtie_{\Gamma,D,E}v^R\}$ , where the published auxiliary operator on the right records solo synchronized letters since the previous joint event. The reversal-dual parameters record solo letters before the next joint event and give a correct final-event $R_\varepsilon$ -recursion for weak synchronized shuffle.	Definition theorem · backward-memory constructionSelf-contained reversal proof with exact unary graph audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Backward-receipt definition and proof	FPRD-SH-B01; Broda et al., “The Prefix Automaton” (2021); Broda et al., “Location automata for synchronised shuffle expressions” (2023), weak synchronized-shuffle operator	FPRD reversal-dual extension of the cited weak synchronized frameworkReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Test whether the receipt remains finite and useful for non-unary or nested weak shuffles.
FPRD-SH-T09 For every $m,n\ge1$ , the backward-receipt prefix automaton of $(a^m)^\mathbin{\amalg^w_{\{a\}}}(a^n)^$ is an existential state quotient of its accessible weak location automaton.	Theorem · exact constructive quotientAll-length proof with closed-form quotient and exact parameter audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Weak ordinary-quotient theorem	FPRD-SH-D03; Broda et al., “Location automata for synchronised shuffle expressions” (2023), weak synchronized-shuffle construction	FPRD deduction from the Broda–Machiavelo–Moreira–Reis weak synchronized-shuffle construction (2023) and the FPRD reversal-dual receiptReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Search for a non-unary family in which the shifted receipt projection remains exact.
FPRD-SH-T10 For every $m,n\ge1$ , the backward-receipt prefix automaton of $(a^m)^\mathbin{\amalg^w_{\{a\}}}(a^n)^$ is not a left-invariant quotient of its weak location automaton.	Theorem · exact impossibility classificationAll-length determinization-orbit proof with exceptional-case audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Weak left-quotient obstruction	FPRD-SH-D03; Determinization preservation under left quotients	FPRD determinization-orbit consequence for the Broda–Machiavelo–Moreira–Reis weak location construction and the FPRD prefix receiptReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Test eventual subset-period mismatch as an obstruction for non-unary synchronized products.
FPRD-SH-C05 Exact construction verifies the weak ordinary quotient for 1,024 pairs with $1\le m,n\le32$ , verifies the APOS/APre subset-orbit formulas for 256 pairs with $1\le m,n\le16$ , and exhausts 11,825 compatible partitions across $(1,1),(1,2),(2,1)$ .	Computational finding · theorem auditAll exact structural, orbit, and boundary checks pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Weak synchronized audit	FPRD-SH-T09; FPRD-SH-T10; Exact automaton constructor	FPRD Lab weak-synchronization verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Keep the verifier synchronized with any extension beyond the unary family.
FPRD-SH-C04 Exact construction checks 576 strong and 576 arbitrary synchronized pairs for $1\le m,n\le24$ ; exhaustive search over 10,534 compatible partitions independently verifies the ordinary- and left-quotient boundary in the six smallest arbitrary-synchronization instances.	Computational finding · theorem auditExact structural and exhaustive boundary checks pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Synchronized threshold audit	FPRD-SH-T07; FPRD-SH-T08; Exact automaton constructor	FPRD Lab synchronized-shuffle verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Extend the exact checker to a non-unary synchronized family only after its prefix receipt is stated precisely.
FPRD-SH-T11 Let $\Vert_\psi$ be an arbitrary-arity memoryless Boolean product of NFAs, and let $E_i$ be a left-invariant equivalence on component $A_i$ . The coordinatewise product equivalence $E=\prod_iE_i$ is left-invariant on $\Vert_\psi(A_1,\ldots,A_n)$ , and $\Vert_\psi(A_1,\ldots,A_n)/E\cong\Vert_\psi(A_1/E_1,\ldots,A_n/E_n)$ .	Theorem · arbitrary-arity quotient transferSelf-contained predecessor proof with strengthened independent boundary audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Boolean-product transfer theorem	FPRD-SH-T03; The 2026 Boolean-product automaton construction	Broda–Machiavelo–Moreira–Reis (2026) product construction; FPRD left-handed consequence using the Broda–Holzer–Maia–Moreira–Reis (2019) definition of left invarianceReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Compare the canonical product of component prefix quotients with the natural prefix automaton of one Boolean-product expression.
FPRD-SH-B02 For ordinary component expressions, FPRD-SH-T11 gives a canonical left quotient from the Boolean product of their position automata to the Boolean product of their prefix automata. This target need not equal the natural prefix automaton of the combined Boolean-product expression: pure shuffle $a^\amalg b^$ is already a counterexample to that identification.	Boundary · exact construction separationExact boundary, now explained by the event-signature expansion theorem	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Two prefix constructions	FPRD-SH-T11; FPRD-SH-X01; The classical position-to-prefix left quotient	FPRD comparison of the 2026 Boolean-product automaton with the 2021/2023 prefix constructionsReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-T12 through FPRD-SH-T14 for the exact construction and policy criteria.
FPRD-SH-T12 For a flat Boolean-product expression with marked ordinary components, the natural backward prefix automaton is isomorphic to the useful canonical product of the component prefix automata after splitting every noninitial product state $q$ into one copy for each incoming event signature $(a,S)\in\operatorname{Sig}(q)$ . Hence $\|Q_{\mathrm{Pre}}\|=1+\sum_{q\ne q_0}\|\operatorname{Sig}(q)\|$ .	Theorem · exact construction identificationSelf-contained structural proof with exact projection audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Event-signature expansion theorem	FPRD-SH-T11; The 2021 prefix recursion; The 2026 Boolean-product support policy	FPRD extension of the 2021 prefix and 2026 Boolean-product constructionsReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Compare the event-signature expansion directly with the 2026 paper's Glushkov and follow constructions.
FPRD-SH-T13 For a fixed flat Boolean-product expression with marked ordinary components, the natural prefix automaton is isomorphic to the useful canonical component-prefix product if and only if every noninitial useful product state $q$ has exactly one incoming event signature. If some $q$ has two signatures, the natural marked automaton is strictly larger and cannot be a quotient of the product.	Corollary · expression-specific iff criterionImmediate exact consequence of the event-signature expansion	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Unique-signature criterion	FPRD-SH-T12	FPRD event-signature analysisReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use the signature count as the exact state-overhead statistic for algorithmic bounds.
FPRD-SH-T14 A memoryless Boolean policy makes the natural prefix automaton isomorphic to the useful canonical component-prefix product for every tuple of ordinary component expressions if and only if every active letter $a$ has one allowed support $S_a$ and $S_a\cap S_b\ne\varnothing$ for every two distinct active letters. In the positive case the quotient is the identity-block quotient; every policy violation has a starred-expression witness whose natural automaton is strictly larger than the product.	Theorem · sharp universal policy classificationSelf-contained necessity and sufficiency proof with independent complete small-policy audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Signature-rigidity classification	FPRD-SH-T12; FPRD-SH-T13; Homogeneity of component prefix automata	FPRD deduction from Broda–Maia–Moreira–Reis (2021) and Broda–Machiavelo–Moreira–Reis (2026)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-T15 and FPRD-SH-T16 for the exact local bound and its policy-only complexity boundary.
FPRD-SH-T15 Let $C_\psi$ have the policy signatures $(a,S)$ as vertices, with distinct vertices adjacent exactly when they share a letter or have disjoint supports. Over every tuple of ordinary component expressions, the largest attainable number of incoming signatures at one useful canonical product state is exactly $\omega(C_\psi)$ . Hence $\|\operatorname{Sig}(q)\|\le\omega(C_\psi)$ and $\Delta(E)\le(\omega(C_\psi)-1)(\|Q(P)\|-1)$ .	Theorem · exact support-graph state-complexity boundSelf-contained upper bound and exact starred realization; finite audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Compatibility-graph theorem	FPRD-SH-T12; Homogeneity of component prefix states; Graph clique number	FPRD support-graph consequence of the Boolean-product event-signature expansionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-T17 and FPRD-SH-T18 for the exact global collision receipt and disjoint-support count.
FPRD-SH-T16 Given an explicitly listed memoryless support policy $\psi$ and $k$ , deciding whether some tuple of ordinary component expressions can produce one useful canonical product state with at least $k$ incoming event signatures is NP-complete. Hardness holds when every letter has one support of size three.	Theorem · computational complexity classificationSelf-contained reduction from three-dimensional matching; exact finite reduction audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	NP-completeness proof	FPRD-SH-T15; Karp's NP-completeness of three-dimensional matching	FPRD reduction from the classical three-dimensional matching problemReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Identify tractable policy classes and parameterizations for the compatibility clique number.
FPRD-SH-T17 For a fixed flat memoryless Boolean-product expression, let $T_\sigma$ be the useful canonical product states entered by event signature $\sigma$ . Then $\Delta(E)=\sum_\sigma\|T_\sigma\|-\|\bigcup_\sigma T_\sigma\|$ . Inclusion–exclusion needs only compatibility cliques; with one support per letter these are exactly support matchings. The pair-intersection bound is exact iff no state receives three or more signatures.	Theorem · exact global state-count identitySelf-contained double-counting and inclusion–exclusion proof; complete finite audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Global collision calculus	FPRD-SH-T12; FPRD-SH-T15; Finite inclusion–exclusion	FPRD global consequence of the Boolean-product event-signature expansionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Optimize the matching-indexed target intersections for overlapping one-support policies.
FPRD-SH-T18 For a one-support-per-letter policy with pairwise-disjoint active supports, let $N_a$ be the number of useful states in the synchronous block advanced by letter $a$ , let $r$ be the number of active letters, and let $M=\prod_aN_a$ . Then $\Delta(E)=\sum_a(N_a-1)\prod_{b\ne a}N_b-(M-1)=(r-1)M-M\sum_aN_a^{-1}+1$ . Pure shuffle is the singleton-support specialization.	Theorem · exact prescribed-size global overheadSelf-contained Cartesian-factorization proof; strict-gap scope corrected and independently audited	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Disjoint-support overhead theorem	FPRD-SH-T17; Pairwise-disjoint event supports; Useful synchronous block products	FPRD extension of the shuffle and Boolean-product automata frameworksReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Determine the exact prescribed-size maximum when one-support events overlap.
FPRD-SH-T19 Create one primary selector item for every policy signature $\sigma=(a,S)$ , with an off option and an on option containing each participant in $S$ as a nonprimary item colored $a$ . XCC solutions are then in bijection with cliques of the signature-compatibility graph. With one support per letter, they are exactly support-hypergraph matchings.	Theorem · exact sparse constraint compilationSelf-contained bijection proof; exhaustive color-consistency audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Exact-cover-with-colors compilation	FPRD-SH-T15; Knuth's exact covering with colors	FPRD compilation into Knuth's XCC modelReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-T20 and FPRD-SH-T21 for the exact one-shot algorithm and its counting boundary.
FPRD-SH-T20 For a graph $G$ , assign one letter $a_e$ and support $e$ to each edge, and give vertex $v$ the component expression $\varepsilon+\sum_{e\ni v}a_e$ . Useful canonical product states are exactly graph matchings $M$ , the state for $M$ has $\|M\|$ incoming signatures, and $\Delta(G)=\sum_M(\|M\|-1)_+$ .	Theorem · exact automata-to-matching correspondenceSelf-contained all-graph proof; exact product exploration audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	One-shot matching model	FPRD-SH-T12; FPRD-SH-T19; Finite graph matchings	FPRD one-shot specialization of the Boolean-product frameworkReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use the XCC compilation to enumerate matching receipts without constructing the canonical product.
FPRD-SH-T21 Computing the one-shot overhead $\Delta(G)=\sum_M(\|M\|-1)_+$ is #P-complete under polynomial-time Turing reductions. If $Z(G)$ counts all graph matchings, then $Z(G)=1+\Delta(G\sqcup K_2)-2\Delta(G)$ . Hardness therefore holds with size-two supports and depth-one component expressions.	Theorem · exact counting-complexity classificationSelf-contained membership and isolated-edge reduction; exact finite audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Counting-hardness boundary	FPRD-SH-T20; Valiant's #P-completeness of counting matchings	FPRD reduction from Valiant's classical matching-count problemReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Study sparse, bounded-width, and structurally restricted policy classes where XCC enumeration is effective.
FPRD-SH-T22 For a one-support-per-letter policy with support family $\mathcal F\subseteq\binom{[n]}r$ , universal zero overhead is equivalent to $\mathcal F$ being independent in $KG_{n,r}$ . If $n<2r$ , every support family has zero overhead. If $n\geq2r$ , then $\|\mathcal F\|\leq\binom{n-1}{r-1}$ ; for $n>2r$ , equality requires a full star. When $r\geq2$ , $n>2r$ , and the core is empty, the sharp Hilton–Milner ceiling is $\binom{n-1}{r-1}-\binom{n-r-1}{r-1}+1$ . At $n=2r$ , maximum families choose one support from each complementary pair, and empty-core examples exist for $r\geq2$ .	Theorem · classical extremal transfer and sharp boundaryExact automata reduction with classical EKR and Hilton–Milner inputs; equality cases audited	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Uniform-support trichotomy	FPRD-SH-T14; Erdős–Ko–Rado theorem; Hilton–Milner theorem	FPRD automata transfer of the Erdős–Ko–Rado and Hilton–Milner theoremsReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-T23 for the quantitative receipt once the EKR ceiling is crossed.
FPRD-SH-T23 Assume $1\leq r\leq n/2$ . Let $N=\binom nr$ , $d=\binom{n-r}r$ , $\tau=\binom{n-r-1}{r-1}$ , and $\alpha=\binom{n-1}{r-1}$ . If an $r$ -uniform policy has $m>\alpha$ supports, then its number of disjoint pairs is at least $\left\lceil\frac{d+\tau}{2N}m(m-\alpha)\right\rceil$ . The depth-one one-shot overhead dominates this count.	Theorem · spectral collision and overhead lower boundSelf-contained spectral derivation from the known Kneser spectrum; exact finite audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Spectral collision receipt	FPRD-SH-T20; Kneser graph spectrum; Rayleigh quotient	FPRD composition of the classical Kneser spectrum with the one-shot matching theoremReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Replace the general spectral receipt with the exact diversity and supersaturation hierarchy in its valid parameter ranges.
FPRD-SH-T24 For an $r$ -uniform support family $\mathcal F$ , let $m=\|\mathcal F\|$ , let $d$ be its maximum participant degree, let $\gamma=m-d$ , and let $q(\mathcal F)$ count disjoint support pairs. Then $q(\mathcal F)\geq\gamma\max\{0,d-[{n-1\choose r-1}-{n-r-1\choose r-1}]\}$ . Every counted pair is a realizable two-signature prefix-splitting receipt.	Theorem · quantitative obstruction certificateSelf-contained all-length counting proof; complete small-family audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Diversity obstruction receipt	FPRD-SH-T14; FPRD-SH-T15; Uniform support counting	FPRD translation of an elementary extremal-set countReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Test sharpness at the first positive-overhead layer and determine how pair receipts aggregate into total prefix-state overhead.
FPRD-SH-T31 Let $x$ be a maximum-degree participant, let $\mathcal A$ be the supports through $x$ , and let $\mathcal B$ be the $\gamma$ exceptional supports avoiding $x$ . Every compatible signature family is either a matching $N\subseteq\mathcal B$ , or $N\cup\{A\}$ for exactly one compatible $A\in\mathcal A$ . The target-set collision identity therefore gives an exact algorithm using $O(\|\mathcal F\|2^\gamma)$ target-intersection oracle calls, without enumerating the component product.	Theorem · exact fixed-parameter target factorizationSelf-contained proof; exact policy-valid target audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Diversity factorization theorem	FPRD-SH-T17; FPRD-SH-T19; Support-family diversity	Kupavskii (2018) supplies the diversity parameter; the target factorization is an FPRD deductionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Test the factorization against an externally posed problem before extending the framework further.
FPRD-SH-T32 Group each hub support $A$ by $\mu(A)=\{j:A\cap B_j\ne\varnothing\}\subseteq[\gamma]$ . This is the coarsest quotient preserving compatibility with every exceptional matching. A subset-zeta transform computes exact one-shot overhead in $O(\|\mathcal F\|r\gamma+\gamma2^\gamma)$ time and $O(2^\gamma)$ auxiliary space.	Theorem · fixed-parameter algorithm and canonical quotientSelf-contained proof; complete small-domain and sharpness audits pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Conflict-mask quotient and zeta algorithm	FPRD-SH-T20; FPRD-SH-T31; Subset zeta transform	FPRD automata specialization using the classical subset-zeta transform of Björklund et al.Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Seek an independently motivated application; do not optimize the same kernel in isolation.
FPRD-SH-T33 Let $\mathcal C\subseteq\mathcal F$ be any pairwise-intersecting support family and $\mathcal B=\mathcal F\setminus\mathcal C$ , with $\|\mathcal B\|=k$ . Every compatible family is an exceptional matching in $\mathcal B$ , with at most one compatible member of $\mathcal C$ . Hence the target-oracle and one-shot factorizations hold with $k$ in place of diversity. Among decompositions obtained from one intersecting core, the smallest possible exception count is $\kappa(\mathcal F)=\tau(D_{\mathcal F})$ , and $\kappa\le\gamma$ .	Theorem · optimal intersecting-core decompositionSelf-contained proof; exact minimum-cover and target audits pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Intersecting-core factorization	FPRD-SH-T17; FPRD-SH-T31; Minimum vertex cover	FPRD correction using an elementary disjointness-graph decomposition; Harris–Narayanaswamy (2024) is the algorithmic comparatorReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use kappa only as the optimum of this supplied-core decomposition; do not infer a general arithmetic lower bound.
FPRD-SH-B10 For every prime power $q$ , the line supports of $PG(2,q)$ are pairwise intersecting, so $\kappa=0$ . There are $q^2+q+1$ supports and every participant lies on $q+1$ of them, so closest-star diversity is $\gamma=q^2$ . Thus $\gamma-\kappa$ is unbounded within the intersecting-core decomposition.	Boundary result · unbounded parameter gapSelf-contained finite-geometry proof; exact q=2,3,5 controls pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Unbounded diversity gap	FPRD-SH-T33; Dembowski, Finite Geometries	Dembowski (1968) supplies the projective-plane incidence facts; the kernel comparison is an FPRD boundary controlReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Do not describe diversity as a complete or optimal search kernel.
FPRD-SH-C18 Exact checks cover 1,088 complete support families, 31,377 matching states across complete and larger controls, 1,200 policy-valid target systems with 7,281 useful states, and projective planes of orders 2, 3, and 5. Direct enumeration and the minimum-cover kernel agree in every case.	Computational finding · corrected-kernel auditAll exact checks pass; deterministic rerun is byte-identical	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Intersecting-core audit	FPRD-SH-T33; FPRD-SH-B10; Exact enumeration	FPRD Lab intersecting-core verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as supporting evidence; arbitrary-size conclusions come from the proofs.
FPRD-SH-B11 With the slice length encoded in unary, exact counting of the Mazurkiewicz traces meeting a length slice of a DFA language is $\#\mathrm P$ -complete even for the fixed independence graph with the single edge $a\mathrel{\mathbb I}b$ . Thus the graph has vertex-cover number one, width two, maximum degree one, and matching number one. The hardness persists although every accepted reduction word has exactly one occurrence of the chosen cover letter. At cover number zero, traces are words and DFA length-slice counting is polynomial.	Boundary theorem · published-hardness parameter extractionPublished parsimonious reduction; parameter corollary checked directly	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Trace-cover hardness boundary	de Colnet–Meel–Mathur Theorem 3.2; Polynomial DFA word counting	de Colnet, Meel, and Mathur (POPL 2026), Theorem 3.2; FPRD parameter extractionReviewed 2026-09-01The underlying parsimonious reduction was peer reviewed at POPL 2026; the vertex-cover extraction has no separate external review.	Do not seek an FPT algorithm from an alphabet-only cover; also control the sequential presentation orbit exposed to the DFA.
FPRD-SH-T34 Given an $m$ -state DFA and a supplied vertex cover $B$ of the independence graph with $\|B\|=\kappa$ , the nonempty one-Foata-step traces touching the DFA language, including their size distribution, can be counted exactly in $O(\|\Sigma\|\kappa 2^\kappa m)$ time and $O(2^\kappa m)$ auxiliary space.	Theorem · exact fixed-parameter algorithmSelf-contained proof; exhaustive and randomized exact audits pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	One-step cover dynamic program	FPRD-SH-T33; DFA subset dynamic programming; Foata steps	FPRD trace specialization of the intersecting-core decompositionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use this as the exact scope boundary; any lift to full traces must add a parameter controlling sequential residual amplification.
FPRD-SH-C19 Direct trace-step enumeration and the cover dynamic program agree on all 64 four-letter independence graphs, all 65,536 complete two-state DFA instances and final sets, 2,500 larger deterministic controls, and 4,000 DNF reduction-pattern controls. Two complete runs are byte-identical.	Computational finding · algorithm and boundary auditAll exact checks pass; deterministic rerun is byte-identical	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Trace-cover exact audit	FPRD-SH-B11; FPRD-SH-T34; Exact enumeration	FPRD Lab trace-cover verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as supporting evidence; the arbitrary-size conclusions come from the proof and the published reduction.
FPRD-SH-B09 The canonical hub quotient can have all $2^\gamma$ conflict classes, and only $\gamma+1$ pairwise-disjoint supports already give $2^{\gamma+1}$ XCC solutions. Explicit quotient materialization and solution enumeration can require exponential size. Neither construction proves that arithmetic evaluation requires $2^{\Omega(\gamma)}$ time.	Boundary result · sharp representation and output costsExplicit constructions and exact formulas proved	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Sharpness and scope boundary	FPRD-SH-T31; FPRD-SH-T32; FPRD-SH-T21	FPRD explicit construction and scope analysisReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Do not claim an arithmetic lower bound without a separate parameterized reduction.
FPRD-SH-C17 Exact checks cover 33,856 complete support families, 738,063 matching states, 2,000 policy-valid target systems with 24,636 useful states, both small-diversity boundaries, and both sharpness constructions. All checks pass; a complete rerun is byte-identical.	Computational finding · exact algorithm and boundary auditAll exact checks pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Diversity-kernel audit	FPRD-SH-T31; FPRD-SH-T32; FPRD-SH-B09; Exact enumeration	FPRD Lab diversity-kernel verifier and independent result reviewReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as evidence; arbitrary-size conclusions come from the proofs.
FPRD-SH-C11 Exact enumeration checks 1,082,432 support families in four complete small domains, all 182 maximal intersecting families at $(n,r)=(7,3)$ , and 640 deterministic larger policies. The EKR, Hilton–Milner, complementary-pair, spectral, and one-shot matching receipts all agree; the 175 maximum empty-core $(7,3)$ families are exactly the classical equality constructions.	Computational finding · extremal and spectral auditSource replay and independent boundary audit pass; T22/T23 hypotheses corrected	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Uniform-support exact audit	FPRD-SH-T22; FPRD-SH-T23; FPRD-SH-T24; Exact enumeration	FPRD Lab uniform-support verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as implementation evidence; use the classical theorems and spectral proof for arbitrary parameters.
FPRD-SH-T25 Let $J_\psi$ join distinct signatures whose nonempty supports overlap. Define $\psi^{\otimes k}$ on the participant universe $U^k$ by $S_{(a_1,\ldots,a_k)}=S_{a_1}\times\cdots\times S_{a_k}$ . Then $J_{\psi^{\otimes k}}=J_\psi^{\boxtimes k}$ .	Theorem · exact policy-to-graph-product correspondenceSelf-contained all-length proof; exhaustive finite tensor audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Support-tensor theorem	FPRD-SH-T15; Strong graph product; Cartesian products of finite supports	FPRD deduction in the strong-product framework of Lovász (1979)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-T26 as a policy-level capacity dictionary; keep expression reachability as a separate proof obligation.
FPRD-SH-T26 With $\Omega(\psi)=\alpha(J_\psi)$ and $\Omega_\infty(\psi)=\sup_{k\ge1}\Omega(\psi^{\otimes k})^{1/k}$ , the support-tensor theorem gives $\Omega_\infty(\psi)=\Theta(J_\psi)$ . Every finite simple graph occurs as $J_\psi$ for a uniform support policy.	Theorem · exact asymptotic-capacity identitySelf-contained corollary and constructive graph-representation proof; exact audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Shannon-capacity identity	FPRD-SH-T25; FPRD-SH-T15; Definition of Shannon capacity	FPRD automata interpretation of classical Shannon capacityReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Compile structured graph codes into reachable expression automata without treating tensor vertices as already-reachable letters.
FPRD-SH-B03 For the complete $r$ -uniform policy, $J_\psi=\overline{KG_{n,r}}$ . If $r\mid n$ , its capacity is $n/r$ ; if $r\nmid n$ , the one-shot packing $\lfloor n/r\rfloor$ and fractional/Lovász ceiling $n/r$ diverge. The first complete two-support case is $T_5=\overline{KG_{5,2}}$ , with published values $\alpha(T_5^{\boxtimes d})=2,5,12,27$ for $d=1,2,3,4$ .	Boundary result · reduction to a published graph-capacity problemExact reduction to the published triangular-graph capacity problem; no new capacity bound claimed	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Complement-of-Kneser boundary	FPRD-SH-T26; Kneser graphs; Triangular-graph capacity results	Kizhakkepallathu, Östergård, and Popa (2013)Reviewed 2026-09-01The triangular-graph bounds are published; the FPRD policy translation has no external review.	Determine or tighten 60 <= alpha(T5 strong-power 5) <= 67 with symmetry-reduced XCC/DLX.
FPRD-SH-C12 Exact enumeration checks 39,693 tensor vertex pairs, uniform policy representations of all 1,099 graphs through five vertices, tensor identities for every graph through four vertices, the complete two-support realization of $T_5$ , and cyclic pair policies through $C_9$ . All graph, tensor, and cyclic identities pass.	Computational finding · Shannon bridge auditAll exact checks pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Support-tensor exact audit	FPRD-SH-T25; FPRD-SH-T26; FPRD-SH-B03; Exact enumeration	FPRD Lab support-tensor verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use the verified policy dictionary as a gate before claiming any expression-level realization.
FPRD-SH-T27 If the support-overlap graph $J_\psi$ is perfect, then $\Omega_\infty(\psi)=\Omega(\psi)=\alpha(J_\psi)$ . Thus the Cartesian support-tensor packing rate is already attained in one layer. Pure shuffle and full synchronization are the empty- and complete-graph boundary cases.	Corollary · exact no-amplification criterionSelf-contained sandwich proof from classical perfect-graph and Lovász-theta facts	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Perfect-policy criterion	FPRD-SH-T26; Perfect graph theorem; Lovász theta sandwich	FPRD corollary from Lovász's perfect-graph theorem (1972) and theta sandwich (1979)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use perfect-graph recognition as a policy compiler gate before finite-power XCC search.
FPRD-SH-X03 Give letter $a_i$ the adjacent-pair support $\{i,i+1\}$ on a five-participant cycle. Then $J_\psi=C_5$ , $\Omega(\psi)=2$ , and the tensor-square signatures $\{(i,2i\bmod5):i\in\mathbb Z_5\}$ are pairwise compatible, so $\Omega(\psi^{\otimes2})=5>4=\Omega(\psi)^2$ and $\Omega_\infty(\psi)=\sqrt5$ .	Exact example · strict support-tensor amplificationExplicit five-signature certificate; exact Lovász upper bound	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Pentagon policy split	FPRD-SH-T25; FPRD-SH-T26; Lovász's pentagon capacity theorem	FPRD cyclic-policy translation of Lovász's classical C5 codeReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use FPRD-SH-X04 for the expression-level realization; next compile the typed recursive C7 gadget rather than enumerating its output.
FPRD-SH-B04 For adjacent-pair supports on seven participants, $J_\psi=C_7$ , so $\Omega_\infty(\psi)=\Theta(C_7)$ . The current Lean-verified recursive certificate in $C_7^{\boxtimes200}$ gives $\Omega_\infty(\psi)\ge3.258805369885\ldots$ , while the Lovász upper bound is below $3.3177$ .	Boundary result · externally anchored support-policy rateExact policy reduction; current graph-capacity bounds imported from primary sources	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Seven-cycle policy boundary	FPRD-SH-T26; C7 Shannon-capacity bounds	Buys, Polak, and Zuiddam (2026), using Gao's product methodReviewed 2026-09-01The lower bound is a current preprint; the FPRD automata translation has no external review.	Use FPRD-SH-X06/T30 as the compiled five-dimensional base; next test whether residual quotients compose across Gao's typed product DAG without expanding its output.
FPRD-SH-T28 Let $G=J_\psi$ be the conflict graph of a one-support-per-letter policy, use optional-once component expressions, and let $\pi:G\to\overline G$ be a complementing permutation. The synchronized product with a $\pi$ -relabelled copy has exactly $1+\|V(G)\|$ reachable natural-prefix states, and its last-signature section $\{(v,\pi(v)):v\in V(G)\}$ is independent in $G^{\boxtimes2}$ .	Theorem · reachable expression compilerSelf-contained all-length proof; exhaustive complementing-permutation audit passes	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Twisted-section compiler	FPRD-SH-T12; FPRD-SH-T14; Self-complementary graphs	FPRD expression-level compiler for the classical self-complementary square codeReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use self-complementary Paley graphs as further square calibrations; use the separate path compiler FPRD-SH-X06 for C7.
FPRD-SH-X04 For the optional-once five-cycle expression, the useful component-prefix product has 11 states and the natural prefix automaton has 16. Synchronizing it with the copy relabelled by $i\mapsto2i\pmod5$ has only six reachable natural-prefix states; its five noninitial last-signature pairs are exactly $\{(i,2i):i\in\mathbb Z_5\}$ , a nonrectangular independent set of size five in $C_5^{\boxtimes2}$ .	Exact example · reachable expression-level twisted sectionExplicit expression, exact state/transition counts, and executable audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Reachable pentagon twist	FPRD-SH-T28; FPRD-SH-X03; FPRD-SH-T12	FPRD reachable realization of Lovász's classical pentagon codeReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use this as calibration while defining a nonvacuous succinct-generator resource.
FPRD-SH-T29 For $D\subseteq A^2$ , let $D_x=\{y:(x,y)\in D\}$ and index intermediate states by the distinct nonempty sets $R\in\{D_x\}$ . The transitions $r\xrightarrow{x}p_{D_x}$ and $p_R\xrightarrow{y}r$ recognize $D^*$ . This residual quotient is reversible iff its distinct successor sets are pairwise disjoint, and it is a graph-language for $G$ iff $D$ is independent in $G^{\boxtimes2}$ .	Compiler lemma · exact residual quotient, reversibility, and validity criteriaSelf-contained proof; exhaustive small-relation audit and corrected standard-alphabet C7 regression pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Reversible two-block compiler	FPRD-SH-X04; Partial reversible DFAs; Strong-product independence	FPRD compiler lemma calibrated against Meiburg's reversible-capacity constructionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Generalize the distinct-successor-set quotient to the typed five-block compiler and test compositional residual minimization across Gao's product nodes.
FPRD-SH-B05 If a classical graph-language $L\subseteq V(G)^*$ has growth $\rho$ , then every slice $L\cap V(G)^n$ is independent in $G^{\boxtimes n}$ , and for every $\varepsilon>0$ some finite slice has root rate above $\rho-\varepsilon$ . Reversible state splitting can compress or organize code families, but cannot yield a classical lower bound that does not factor through ordinary finite independent sets.	Calibration boundary · exact slice factorizationDefinition-level proof; consistent with the primary regular-language capacity framework	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	No-free-capacity boundary	Definition of graph-language growth; Definition of Shannon capacity	Direct consequence of Meiburg's graph-language formulationReviewed 2026-09-01Meiburg's regular-language framework is peer-reviewed; the FPRD methodological boundary has no documented specialist review.	Measure succinctness and verification cost separately from code size; call a capacity result new only when an emitted finite slice improves a published bound.
FPRD-SH-C13 The exact audit verifies the one-layer 11/16-state construction, the six-state reachable synchronized square, its five nonrectangular pentagon signatures, the six-state reversible lowering, both compiler criteria over all 530 two-block relations through three symbols, the corrected standard-alphabet six-state C7 residual quotient, 4,130 graph-language checks, and 265 complementing-permutation certificates through five vertices. All checks pass.	Computational finding · expression and reversible compiler auditAll exact checks pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Expression-twist exact audit	FPRD-SH-T28; FPRD-SH-X04; FPRD-SH-T29; Exact enumeration	FPRD Lab expression-twist verifier, corrected by the independent result reviewReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as implementation evidence; use the symbolic proofs for arbitrary graphs and code relations.
FPRD-SH-B06 The reachable six-state construction of FPRD-SH-X04 presents each pair $(i,2i)$ as one synchronized macro-event. The direct word expression $R_5=(a_0a_0+a_1a_2+a_2a_4+a_3a_1+a_4a_3)^*$ presents the same code as two original-alphabet events along a path. Its length- $2m$ slice is the classical independent language $C^m$ , where $C=\{00,12,24,31,43\}$ . Reachability of the former does not determine the natural-prefix state count of the latter.	Boundary · exact separation of macro-event and path-language realizationsSelf-contained comparison; the graph language is explicit in the primary automata-capacity literature	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Macro-event versus path compiler	FPRD-SH-X04; FPRD-SH-T29; Graph languages	Meiburg (2025); FPRD comparison with the prefix construction of Broda et al. (2021)Reviewed 2026-09-01The graph-language construction is peer reviewed; the FPRD prefix comparison has no external review.	Fix a resource measure that does not count a hard-coded block expression as an automata-specific capacity improvement.
FPRD-SH-X05 For $R_5=(a_0a_0+a_1a_2+a_2a_4+a_3a_1+a_4a_3)^*$ , the Broda--Maia--Moreira--Reis backward prefix construction has the eleven states $\varepsilon$ , $R_5a_i$ , and $R_5a_i a_{2i}$ . Merging $\varepsilon$ with the five completed-block states gives a minimal six-state partial reversible DFA for the same language.	Exact example · natural-prefix-to-reversible-quotient calibrationExact symbolic derivation and independent automaton audit	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Eleven-to-six direct block compiler	FPRD-SH-B06; The prefix-automaton backward recursion	FPRD calculation from Broda et al.'s prefix constructionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Compare natural-prefix and compact-generator size only after fixing a uniform resource model.
FPRD-SH-C14 An independent verifier checks all 488,281 alphabet words of lengths zero through eight, all ten pentagon-code pairs, the eleven-state natural prefix automaton, and its minimal six-state reversible quotient. Exactly 781 tested words are accepted. All checks pass.	Computational finding · direct-expression auditAll exact checks pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Independent direct-expression audit certificate	FPRD-SH-B06; FPRD-SH-X05; Exact enumeration	FPRD Lab independent direct-block verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as regression evidence for the semantic separation; use symbolic proofs for all lengths.
FPRD-SH-X06 The published 367-word independent set in $C_7^{\boxtimes5}$ can be reconstructed without storing its vectors. The cyclic 382-word precursor and conflict filter leave 327 words. The 71-vertex, 85-edge extension graph has exactly eight maximum independent sets of size 40: they share a 37-word core and differ by one endpoint choice on each of three disjoint conflict edges. Three selector bits recover the published extension.	Exact computational theorem · table-free code compilerIndependent reconstruction and exact maximum-set enumeration pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Table-free 367-code compiler	Polak--Schrijver cyclic construction; Strong-product independence; Exact maximum-clique search	Polak and Schrijver (2019); FPRD reconstruction and extension-graph decompositionReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Treat the generated typed package, rather than a 367-vector input file, as the leaf of Gao's recursive compiler.
FPRD-SH-T30 For coordinate order $(0,2,3,4,1)$ , the minimal partial DFA accepting the published 367-word code has residual-layer counts $(1,7,49,61,9,1)$ , hence 128 states and 424 transitions. Across all eight maximum extensions of the fixed 327-word core and all 120 coordinate orders, the published extension is the unique code attaining 128 states; the other minima are 129 or larger.	Theorem · exact residual quotient and finite optimalityMyhill--Nerode proof for each order; all 960 extension/order pairs checked exactly	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Residual-optimal published extension	FPRD-SH-X06; Myhill--Nerode right residuals; Finite coordinate-order exhaustion	FPRD residual compiler for the Polak--Schrijver codeReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Determine whether Gao's typed product operation admits a residual quotient computed compositionally from child packages.
FPRD-SH-C15 A standard-library exact verifier reconstructs the published 367-code without an input table, enumerates all eight maximum 40-word extensions, checks 960 residual compilers, verifies Gao's eight private pairs, complementary transversals, and $(321,26,20)$ auxiliary partition, and reproduces the six-node split recurrence through $M_{40}$ . All checks pass.	Computational finding · C7 base compiler and recursive-interface auditAll exact checks pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	C7 base-compiler audit	FPRD-SH-X06; FPRD-SH-T30; Gao's private-pair product lemma; Exact integer arithmetic	FPRD C7 compiler audit; Gao (2026), arXiv:2607.27869v1Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Use the 204-node typed syntax DAG as the next compiler input; do not enumerate its 200-dimensional output.
FPRD-SH-X08 Retain the ten fixed-length languages $I,B,R,Q,P^{\mathrm H},P^{\mathrm V},X,O,H,V$ of a Gao gadget. The product code is the disjoint union of seven concatenation rectangles, and all ten output types close under a constant union/concatenation template. Twelve base syntax nodes and 32 nodes for each of six product steps give an exact 204-node typed syntax DAG denoting Gao's full $M_{40}$ -word code without enumerating a product word.	Exact example · resource-preserving typed-language translationStructural induction proved; exact cardinality audit reproduces Gao's M40	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Typed product language compiler	FPRD-SH-X06; FPRD-SH-C15; Gao's product lemma; Fixed-length regular-language concatenation	FPRD translation of Gao (2026)Reviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain this resource-preserving translation as the input for any later semantic minimization or comparison.
FPRD-SH-C16 Exact Brzozowski-derivative closure of the 204-node typed expression DAG yields a deterministic partial path compiler with $192{,}771$ reachable canonical syntax states and $618{,}415$ transitions for Gao's $M_{40}$ -word code in $C_7^{\boxtimes200}$ . The computation closes all 201 depth layers without enumerating the code; the largest layer has 6,761 states.	Computational finding · exact canonical syntax-compiler profileComplete canonical syntactic derivative closure; deterministic rerun and independent two-block semantic audit pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Two-hundred-dimensional symbolic compiler audit	FPRD-SH-X08; Brzozowski derivatives; Hash-consed regular-expression DAGs	FPRD typed-language compiler and independent semantic auditReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as exact implementation evidence; do not spend another run minimizing a presentation of the known Gao code.
FPRD-SH-B08 The table-free base reconstruction, constant-template product DAG, and 192,771-state canonical symbolic lowering give a complete path-language presentation of Gao's known 200-dimensional code. They do not improve $\Theta(C_7)$ , prove DFA minimality, or realize the code as a small last-signature image. Further minimization of this presentation is not presently a literature-recognized open problem.	Boundary result · research-line closureExact scope boundary recorded	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	C7 compiler stopping boundary	FPRD-SH-X08; FPRD-SH-C16; Current C7 capacity bounds	FPRD scope audit against Gao and Buys--Polak--ZuiddamReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Study exact XCC algorithms parameterized by support-family diversity.
FPRD-SH-C10 Exact enumeration checks 16,384 signature subsets for XCC compatibility and 1,355 graphs for the one-shot product correspondence, weighted matching formula, and isolated-edge recovery identity. All 21,152 matching receipts and useful product states agree with the theorems.	Computational finding · compiler and reduction auditSource replay and independent implementation pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Dancing Links and matching audit	FPRD-SH-T19; FPRD-SH-T20; FPRD-SH-T21; Exact enumeration	FPRD Lab XCC and one-shot matching verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as implementation evidence; use the symbolic proofs for arbitrary policies and graphs.
FPRD-SH-C09 Exact enumeration checks 65,536 target-set families, 2,924 one-support policies with 28,856 signature intersections, and 3,905 pure-shuffle state-count vectors comprising 809,710 noninitial product states. All collision identities, matching boundaries, closed formulas, and clique-bound gaps agree.	Computational finding · theorem and boundary auditSource audit and independent complete-domain replay pass; strict-gap scope corrected	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Global-overhead exact audit	FPRD-SH-T17; FPRD-SH-T18; Exact finite enumeration	FPRD Lab global-overhead verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as implementation evidence; use the symbolic proofs for arbitrary sizes.
FPRD-SH-C08 Exact enumeration checks 16,452 two-letter support policies through arity three against every homogeneous unused, $a^$ , and $b^$ target assignment, and checks all 256 subinstances of the $2\times2\times2$ three-dimensional-matching universe under the hardness reduction. Every comparison passes.	Computational finding · theorem and reduction auditSource audit and independent complete-domain replay pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Compatibility and matching audit	FPRD-SH-T15; FPRD-SH-T16; Exact clique and matching enumeration	FPRD Lab signature-compatibility verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as an implementation audit; use the symbolic proofs for arbitrary arity.
FPRD-SH-C07 The source enumeration checks 16,448 two-letter support policies at arities two and three, 11,736 rigid-policy/component combinations, 16,382 non-rigid starred witnesses, and 16,448 signature-expansion projections. An independent review covers all 16,452 policies at arities one through three and checks 172,128 same-letter support-pair occurrences at the unmarked boundary. Full synchronization and the pairwise-intersecting support triangle pass; pure shuffle fails uniqueness.	Computational finding · theorem auditSource audit and independent complete small-policy review pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Prefix-rigidity exact audit	FPRD-SH-T12; FPRD-SH-T14; Exact homogeneous product constructor	FPRD Lab Boolean-product prefix-rigidity verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain as an implementation audit; use the symbolic proof for arbitrary arity.
FPRD-SH-C06 Exact enumeration checks 8,192 combinations of two-state one-letter transition structures, all eight Boolean policies on the supports $\{1\},\{2\},\{1,2\}$ , and discrete or universal component partitions. All 5,408 predecessor-compatible cases satisfy product predecessor compatibility and exact quotient/product commutation. An independent boundary audit also checks 56,448 saturated initial-set pairs and 86,528 final-set pairs.	Computational finding · theorem auditSource transition audit and independent initial/final boundary audit pass	Evidence and limits →	Automata and formal languagesPrefix quotients under shuffle	Boolean-product exact audit	FPRD-SH-T11; Exact Boolean-product constructor	FPRD Lab Boolean-product quotient verifierReviewed 2026-09-01No documented external or specialist review of these FPRD results is recorded.	Retain this as an implementation audit; use the all-arity proof for the theorem.
FPRD-RS-D01 Let $\Sigma=\{a,b\}$ and let $f:\Sigma^\to\mathbb Q$ have an explicit $d$ -dimensional linear representation $f(w)=\lambda M_w\gamma$ . Its relative degree is the unique $r\in\mathbb N$ , when one exists, for which $\operatorname{supp}(f)\subseteq a^r\Sigma^$ and $f(a^rb)\ne0$ .	DefinitionPrecisely stated for explicit rational linear representations	Evidence and limits →	Algebra and discrete mathematicsRational series and linear representations	Definitions and input model	No recorded dependencies	Problem proposed by Lorenzo Clemente; FPRD Lab solutionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Ask the problem author to check that the explicit linear-representation input model matches the intended encoding.
FPRD-RS-L01 Put $A=M_a$ , $B=M_b$ , $V=\operatorname{span}\{M_x\gamma:x\in\Sigma^\}$ , and $W=\{p:p\gamma=0\text{ and }pBV=0\}$ . Then $\operatorname{supp}(f)\subseteq a^r\Sigma^$ if and only if $\lambda A^k\in W$ for every $0\le k<r$ .	LemmaProved with finite-dimensional linear algebra	Evidence and limits →	Algebra and discrete mathematicsRational series and linear representations	Safe-row criterion and proof	FPRD-RS-D01	Problem proposed by Lorenzo Clemente; FPRD Lab solutionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Obtain an independent specialist check of the two directions.
FPRD-RS-T01 If a $d$ -dimensional linear representation has a relative degree, then that degree satisfies $r<d$ .	TheoremProved by a Krylov-space argument	Evidence and limits →	Algebra and discrete mathematicsRational series and linear representations	Dimension bound	FPRD-RS-D01; FPRD-RS-L01	Problem proposed by Lorenzo Clemente; FPRD Lab solutionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Obtain an independent specialist check of the first-dependence argument.
FPRD-RS-T02 For an explicit rational linear representation, existence and value of the relative degree are decidable in deterministic polynomial time: compute $V$ and $W$ , test $0\le r<d$ , and return the unique candidate satisfying the safe-row conditions and $\lambda A^rB\gamma\ne0$ .	Theorem · algorithmProved; the reported bounded experiment is tracked separately because its checker is unavailable	Evidence and limits →	Algebra and discrete mathematicsRational series and linear representations	Algorithm and correctness proof	FPRD-RS-L01; FPRD-RS-T01; FPRD-RS-C01	Problem proposed by Lorenzo Clemente; FPRD Lab solutionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Send the public proof to Lorenzo Clemente for review after confirming a contact address and obtaining send-time approval.
FPRD-RS-C01 Whenever a relative degree exists, it is unique.	CorollaryProved directly from the support and witness conditions	Evidence and limits →	Algebra and discrete mathematicsRational series and linear representations	Uniqueness proof	FPRD-RS-D01	Problem proposed by Lorenzo Clemente; FPRD Lab solutionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Include the uniqueness check in any external review of the main algorithm.
FPRD-RS-C02 The originating run artifact reports 100,000 random exact-integer representations of dimensions one through four, with no mismatch between the Krylov algorithm and direct residual-zeroness tests.	Computational finding · unreproducedCaptured from the source artifact; checker and certificate are absent from the available durable and public record	Evidence and limits →	Algebra and discrete mathematicsRational series and linear representations	Reported finite corroboration and evidence boundary	FPRD-RS-T02	fprd-automata-exchange-run-003-relative-degree-rational-series-2026-08-25.mdReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Keep this historical report separate from the new exhaustive certificate unless its original generator and distribution are recovered.
FPRD-RS-C03 For all 531,522 one- and two-dimensional integer linear representations over the binary alphabet with every entry in {−1,0,1}, the Krylov algorithm agrees with a direct definition-level oracle for every candidate r<d; no mismatch or nonunique candidate set occurs.	Computational finding · exhaustive finite certificateReproduced from public code over the complete stated finite domain	Evidence and limits →	Algebra and discrete mathematicsRational series and linear representations	Exhaustive certificate and checker	FPRD-RS-D01; FPRD-RS-T02	FPRD Lab exhaustive exact-arithmetic auditReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Extend only if a new adversarial domain tests a distinct proof boundary; larger random samples alone would not mature the theorem.
FPRD-RE-D01 For a regular expression $B$ , let $Q_B$ be a finite Antimirov partial-derivative carrier containing $B$ and closed under one-letter partial derivatives.	Definition · finite constructionClassical construction stated for the quotient proof	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Finite derivative carrier	Antimirov partial derivatives	Antimirov (1996) finiteness theorem; FPRD construction for Zhuchko's problemReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Make the carrier construction fully explicit in a mechanized implementation.
FPRD-RE-L01 Interpreting $0,1$ , letters, union, concatenation, and star respectively as the empty relation, identity, partial-derivative edges, union, relational composition, and reflexive-transitive closure on $Q_B$ gives $(P,Q)\in\llbracket A\rrbracket_B$ exactly when some $u\in L(A)$ satisfies $Q\in\partial_u(P)$ .	LemmaProved by structural induction	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Relational semantics lemma	FPRD-RE-D01	Problem proposed by Ekaterina Zhuchko; FPRD Lab constructionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Obtain an independent check of the star case and its finite path bound.
FPRD-RE-T01 For regular expressions $A,B$ , form the finite set of states $Q\in Q_B$ reachable from $B$ through $\llbracket A\rrbracket_B$ , and let $f(A,B)$ be their union as regular expressions. Then $L(f(A,B))=L(A)^{-1}L(B)=\{v:\exists u\in L(A),\ uv\in L(B)\}$ .	Theorem · constructive solutionProved under the finite-derivative interpretation of the requested construction	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Construction and correctness proof	FPRD-RE-D01; FPRD-RE-L01	Problem proposed by Ekaterina Zhuchko; FPRD Lab constructionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine whether the finite carrier and relation saturation can be compiled away under the pinned model FPRD-RE-D02.
FPRD-RE-C01 On 405 selected binary regular-expression pairs and all 31 suffixes of length at most four, three saved automata formulations agree in all $12{,}555$ membership comparisons. The Antimirov-versus-Thompson audit removes the first checker's normalization and derivative code, while a review audit replaces the epsilon-NFA oracle with an epsilon-free Glushkov position automaton. The first audit also detects four injected errors.	Computational finding · reproducible finite auditReproduced by three saved checks; zero ordinary mismatches, identical semantic transcripts, four named negative controls detected, and 65 hostile star cases confirmed	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Finite audit and downloadable evidence	FPRD-RE-D01; FPRD-RE-L01; FPRD-RE-T01	FPRD Lab replacement finite auditReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use the exact unbounded star family as the next adversary rather than continuing to enlarge this bounded corpus.
FPRD-RE-D02 A carrier-free constructor-only quotient compiler recurses on ordinary regular-expression constructors and returns an ordinary regular expression, without enumerating a residual carrier, saturating a finite relation, or adding an explicit least-fixed-point constructor.	Definition attempt · strengthened construction disciplineInformal exclusion-based discipline; a formal compiler class is still missing	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Constructor-only formulation boundary	FPRD-RE-T01	FPRD Lab formulation attempt for a stricter construction disciplineReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Specify an admissible compiler grammar, auxiliary-state policy, and equivalence/cost model before testing general existence or impossibility.
FPRD-RE-T02 For languages $K,L$ , the quotient $Q(K^,L)$ is the least fixed point of $\Phi_{K,L}(X)=L\cup Q(K,X)$ , hence $Q(K^,L)=\bigcup_{n\ge0}Q(K^n,L)$ . On a finite derivative carrier of $L$ , this becomes finite reachability saturation.	Theorem · presentation-dynamics decompositionProved directly from left-quotient composition	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Fixed-point theorem and proof	FPRD-RE-L01; FPRD-RE-T01	Elementary language-theoretic deduction; FPRD presentation-dynamics formulationReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine whether the finite star-orbit closure can be compiled uniformly into the stricter model FPRD-RE-D02.
FPRD-RE-X01 For every $k\ge0$ , $\varepsilon\in(a^*)^{-1}\{a^{k+1}\}$ but $\varepsilon\notin\bigcup_{i=0}^{k}\{a^i\}^{-1}\{a^{k+1}\}$ . Therefore no fixed finite quotient-depth truncation computes all star left quotients.	Negative theorem · counterexample familyProved by an explicit unary family	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Unbounded-unfolding proof	FPRD-RE-T02	FPRD Lab direct counterexample familyReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use this family as the first adversary for any proposed carrier-free star compiler; seek a broader lower bound only after its formal interface is pinned.
FPRD-RE-B01 Uniformly bounded star unfolding is impossible and finite derivative saturation is sufficient. A broader carrier-free existence question has not yet been posed as a formal mathematical problem because FPRD-RE-D02 does not define a quantified compiler class.	Unresolved formalization boundaryNot yet a well-posed existence or impossibility problem	Evidence and limits →	Automata and formal languagesRegular expressions and quotients	Interpretation boundary	FPRD-RE-D02; FPRD-RE-T02; FPRD-RE-X01	FPRD Lab strengthened formulation boundary, motivated by the Automata Exchange questionReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Specify an admissible compiler grammar, auxiliary-state policy, and equivalence/cost model before seeking a carrier-elimination theorem or lower bound.
FPRD-TM-D01 For transformations $s_1,s_2\in T_n$ , the marked morphism $\eta_s:\{a,b\}^*\to T_n$ sends $a$ to $s_1$ and $b$ to $s_2$ ; its kernel records exactly which words induce the same element of the generated monoid.	DefinitionPrecisely stated under the identity-preserving convention	Evidence and limits →	Algebra and discrete mathematicsMinimum faithful transformation degree	Marked monoids and their word kernels	No recorded dependencies	Problem proposed by Ismaël Jecker; FPRD Lab analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Retain the marked generators explicitly in any later hardness reduction or minimization algorithm.
FPRD-TM-L01 The marked monoid ⟨s_1,s_2⟩ embeds in (T_m) if and only if there are $t_1,t_2\in T_m$ whose marked word kernel equals the kernel of (s_1,s_2).	LemmaProved by constructing the induced embedding in both directions	Evidence and limits →	Algebra and discrete mathematicsMinimum faithful transformation degree	Kernel criterion and proof	FPRD-TM-D01	Problem proposed by Ismaël Jecker; FPRD Lab analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Seek an independent check of the non-unital reformulation.
FPRD-TM-T01 Given two transformations on n points, deciding whether their generated monoid embeds in T_m for some m<n belongs to PSPACE.	Theorem · complexity upper boundProved; no matching hardness result is claimed	Evidence and limits →	Algebra and discrete mathematicsMinimum faithful transformation degree	Polynomial-space kernel comparison	FPRD-TM-L01	Problem proposed by Ismaël Jecker; FPRD Lab analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Search for a matching lower bound or a sharper algorithm on structural subclasses without importing unrelated semigroup hardness.
FPRD-TM-T02 For every finite group (G), the minimum (m) admitting an embedding $G\hookrightarrow T_m$ equals the minimum faithful permutation degree μ(G).	TheoremProved; standard group-action fact	Evidence and limits →	Algebra and discrete mathematicsMinimum faithful transformation degree	Equality of the two degrees	FPRD-TM-D01	Elementary finite-group action argument; problem proposed by Ismaël JeckerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use standard minimum-faithful-action terminology in later group-specific work.
FPRD-TM-T03 Given two permutations on n points, deciding whether their generated group has a faithful permutation action of degree smaller than n belongs to NP.	Theorem · complexity upper boundReconstructed from standard permutation-group algorithms; externally known	Evidence and limits →	Algebra and discrete mathematicsMinimum faithful transformation degree	Coset-action certificate	FPRD-TM-T02; Polynomial-time permutation-group algorithms	Kantor–Luks (1990) permutation-group algorithms; Levet–Srivastava–Thakkar (arXiv:2501.16039v4) records the general NP boundary; FPRD reconstructionReviewed 2026-09-01The general NP upper bound is externally documented in the minimum faithful permutation-degree literature; no external review of this FPRD exposition is recorded.	Determine whether the unrestricted two-generated group problem has a sharper deterministic bound; do not transfer the Fitting-free NC/RNC result without its hypothesis.
FPRD-LRS-D01 An order-(d) rational linear recurrence sequence is given by binary-encoded initial values $f_0,\ldots,f_{d-1}$ , recurrence coefficients $c_0,\ldots,c_{d-1}$ , and a binary index (n); the decision question is whether (f_n=0).	Definition · input modelPrecisely stated for explicit rational recurrence input	Evidence and limits →	Algorithms and complexityIndexed zeroness for linear recurrence sequences	Input model and normalization	No recorded dependencies	Problem proposed by Lorenzo Clemente; FPRD Lab analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Maintain this as the exact public input model; analyze other succinct representations separately.
FPRD-LRS-T01 If (Q) clears the recurrence-coefficient denominators and (E) clears the initial-value denominators, then (g_k=EQ^k f_k) is an integer LRS of polynomial description size and (f_n=0) exactly when (g_n=0).	PropositionProved by an index-dependent scaling	Evidence and limits →	Algorithms and complexityIndexed zeroness for linear recurrence sequences	Rational-to-integer normalization	FPRD-LRS-D01	Problem proposed by Lorenzo Clemente; FPRD Lab analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Include the scaling explicitly in every later reduction from rational LRS input.
FPRD-LRS-T02 Indexed LRS zeroness is polynomial-time equivalent to deciding whether a specified entry of an integer matrix raised to a binary exponent is zero.	Theorem · polynomial-time equivalenceProved by companion, block-matrix, and Cayley–Hamilton constructions	Evidence and limits →	Algorithms and complexityIndexed zeroness for linear recurrence sequences	Two polynomial-time reductions	FPRD-LRS-T01; Cayley–Hamilton theorem	Problem proposed by Lorenzo Clemente; FPRD Lab analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare deterministic subclasses on the recurrence and matrix-power sides.
FPRD-LRS-T03 Indexed LRS zeroness belongs to $\mathsf{coRP}$ : repeated squaring builds a polynomial-size division-free integer circuit for the queried matrix entry, and random-prime modular evaluation has one-sided error.	Proposition · complexity upper boundDerived from the standard EquSLP upper bound; externally used	Evidence and limits →	Algorithms and complexityIndexed zeroness for linear recurrence sequences	Reduction to arithmetic-circuit zeroness	FPRD-LRS-T02; EquSLP is in coRP	Problem proposed by Lorenzo Clemente; FPRD Lab reconstructionReviewed 2026-09-01Bacik, Ouaknine, and Worrell use the same iterated-powering and arithmetic-circuit zero-test subroutine; no external review of this FPRD exposition is recorded.	Investigate deterministic hitting sets for the structured repeated-power circuits.
FPRD-LRS-T04 Indexed LRS zeroness belongs to $\mathsf{coNP}$ : if the queried integer circuit output is nonzero, some polynomial-bit prime preserves that nonzeroness and is an NP certificate.	Proposition · certificate upper boundProved by a circuit-growth and prime-counting bound	Evidence and limits →	Algorithms and complexityIndexed zeroness for linear recurrence sequences	Escaping-prime certificate	FPRD-LRS-T02; Deterministic polynomial-time primality testing	Problem proposed by Lorenzo Clemente; FPRD Lab analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Seek an independent complexity-theory review of the escaping-prime proof.
FPRD-LRS-F01 Repeated squaring does not yield a deterministic polynomial-time output-producing algorithm in the bit model: the order-one sequence (u_n=2^n) has (n+1) output bits when (n) is supplied in binary.	Negative result · failed approachProved output-size obstruction	Evidence and limits →	Algorithms and complexityIndexed zeroness for linear recurrence sequences	Output-size obstruction	FPRD-LRS-D01	FPRD Lab analysis of indexed LRS evaluationReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Keep compact circuit construction separate from exact integer expansion in future algorithms.
FPRD-LRS-B01 The present reductions do not establish a deterministic polynomial-time zero test, EquSLP hardness, or equivalence with derandomizing general arithmetic-circuit identity testing.	Open boundaryUnresolved under the explicit variable-order input model	Evidence and limits →	Algorithms and complexityIndexed zeroness for linear recurrence sequences	Limits of the current reductions	FPRD-LRS-T03; FPRD-LRS-T04; FPRD-LRS-F01	FPRD Lab analysis of indexed LRS evaluationReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Test fixed order, diagonalizable characteristic polynomials, nondegenerate recurrences, and restricted coefficient semigroups for deterministic algorithms.
FPRD-LT-D01 Given a minimal DFA whose transition monoid $M\le T_Q$ is $\mathcal L$ -trivial and a target transformation $f:Q\to Q$ , the membership problem asks whether $f\in M$ .	Definition · decision problemPrecisely stated; exact complexity remains open here	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Problem statement and conventions	No recorded dependencies	Problem proposed by Andrew Ryzhikov; FPRD Lab analysisReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Determine whether positive instances always admit polynomial-length word witnesses in the supplied transformation action.
FPRD-LT-L01 For $M=\langle\tau_a:a\in\Sigma\rangle$ , membership of $f$ in $M$ is equivalent to membership of the same underlying element in $M^{\mathrm{op}}$ after reversing a witnessing word; moreover $\mathcal L$ -triviality of $M$ is equivalent to $\mathcal R$ -triviality of $M^{\mathrm{op}}$ .	Lemma · algebraic dualityProved directly from opposite multiplication	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Opposite multiplication and proof	FPRD-LT-D01	Problem proposed by Andrew Ryzhikov; FPRD Lab analysisReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Use the duality only together with an explicit size analysis of any representation of the opposite monoid.
FPRD-LT-T01 If an $n$ -state L-trivial instance admits a polynomial-time computable faithful poDFA action $\rho:M^{\mathrm{op}}\hookrightarrow T_P$ of polynomial degree, explicit generator images, and a computable target code $t_f$ that equals $\rho(f)$ for positive instances and lies outside $\rho(M)$ for negative instances, then membership is in NP and positive instances have word certificates of length at most $\|P\|(\|P\|+1)/2$ .	Theorem · conditional complexity transferProved conditionally; the compact dual representation is not known in general	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Conditional NP theorem	FPRD-LT-L01; Ryzhikov–Wolf 2024, Proposition 9	FPRD Lab conditional transfer using Ryzhikov–Wolf 2024, Proposition 9Reviewed 2026-08-25The poDFA representative bound is a published MFCS 2024 result; no external review of this conditional FPRD transfer is recorded.	Construct a polynomial-degree faithful action of the opposite monoid or prove that no uniform construction exists.
FPRD-LT-T02 Let $M=\langle S\rangle$ and let $f:Q\to Q$ be the target. A polynomial-time computable poDFA action $\rho:M^{\mathrm{op}}\to T_P$ and code $t_f\in T_P$ suffice for the conditional NP transfer when, for every $g\in M$ , $\rho(g)=t_f$ if and only if $g=f$ ; full faithfulness of $\rho$ is unnecessary.	Theorem · conditional quotient transferProved conditionally; a uniform compact target-isolating construction remains open	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Target-fibre theorem and proof	FPRD-LT-L01; Ryzhikov–Wolf 2024, Proposition 9	FPRD Lab conditional theorem using Ryzhikov–Wolf 2024, Proposition 9Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Construct target-isolating poDFA observations directly from the supplied L-trivial action, beginning with fixed alphabets and principal-left-ideal features.
FPRD-LT-X01 For the L-trivial monoid $M=\{1,a,0\}$ with $a^2=0$ , the target $f=1$ has a two-state target-isolating poDFA quotient that maps $a$ and $0$ to the same transformation; hence the quotient is nonfaithful while its target fibre is a singleton.	Example · strict quotient compressionProved by a complete multiplication-table and action check	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Three-element monoid example	FPRD-LT-T02	FPRD Lab exact exampleReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Search for canonical target-isolating congruences that retain poDFA order while collapsing large non-target regions.
FPRD-LT-C01 The minimal two-state DFA for words ending in $a$ has L-trivial transition monoid $\{1,c_0,c_1\}$ , but reversing its transitions yields a nondeterministic automaton with a two-state cycle and a self-looping state having another outgoing transition on the same letter.	Counterexample · failed reductionProved by an explicit two-state example	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Two-state reversal counterexample	FPRD-LT-D01	Problem proposed by Andrew Ryzhikov; FPRD Lab counterexampleReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Exclude graph reversal and matrix transposition from future dual-representation arguments unless functionality and order are proved.
FPRD-LT-E01 Exhaustive enumeration of unordered pairs of distinct transformations on $n=2,3,4$ states found maximum shortest-word diameters $1,2,3$ , respectively, among the binary L-trivial semiautomata that admit a minimal DFA final set.	Computational findingFinite exhaustive evidence through four states; the inferred pattern is refuted at five states	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Enumeration method and results	FPRD-LT-D01	FPRD Lab finite enumerationReviewed 2026-08-25No documented external or specialist review of this FPRD result is recorded.	Retain these counts as the completed four-state baseline; use FPRD-LT-E02 for the five-state classification.
FPRD-LT-T03 For every n ≥ 5, there is a minimal binary n-state DFA with J-trivial transition monoid and a target transformation whose shortest representative has length exactly n. Thus the conjectured n − 1 bound for binary L-trivial inputs is false already inside the J-trivial subclass.	Theorem · exact witness familySelf-contained construction and proof; no external review	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Binary family and exact lower bound	Masopust–Krötzsch 2021, confluent-poDFA characterization; Simon 1975, piecewise testability iff J-trivial syntactic monoid	FPRD Lab construction using the Masopust–Krötzsch and Simon characterizationsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Seek a polynomial upper bound for fixed binary alphabets or a family with superpolynomial shortest representatives; the linear lower bound alone does not settle complexity.
FPRD-LT-E02 Among binary five-state semiautomata with L-trivial transition monoid that admit a minimal-DFA final set, exhaustive enumeration finds maximum shortest-word diameter 5. Up to state relabelling fixing the initial state and interchange of the two letters, there are 1,148 retained generator-pair classes and exactly three extremal classes.	Computational finding · exhaustive classificationExact finite computation with an independent extremal audit	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Five-state exhaustive classification	FPRD-LT-D01	FPRD Lab exact enumerationReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use the explicit family in FPRD-LT-T03 as the corrected lower-bound baseline; do not infer a general upper bound from five states.
FPRD-LT-X02 For the 14-element five-state extremal generated by a=(2,1,2,3,1) and b=(0,1,3,4,4), minimizing the left-Cayley poDFA with sole accepting element f=(4,1,1,1,1) gives a nine-state poDFA action whose transformation fibre over f is exactly {f}.	Example · exact target-isolating quotientProved by exact partition refinement and fibre audit	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Nine-state extremal quotient	FPRD-LT-T02; FPRD-LT-E02	FPRD Lab exact extremal analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Find a quotient construction whose state space and fibre proof are obtained directly from the supplied n-state action rather than from the enumerated monoid.
FPRD-LT-T04 For $f\in M=\eta(\Sigma^*)$ , let $P_f(x)=\{z\in M:zx=f\}$ and $d_f=\|\{P_f(x):x\in M\}\|$ . The minimal DFA of $K_f=\{w^R:\eta(w)=f\}$ has exactly $d_f$ states. If $M$ is L-trivial, this DFA is partially ordered, so every generated target has a representative of length at most $d_f-1$ .	Theorem · canonical target observerSelf-contained Myhill–Nerode proof; no external review	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Canonical observer and path bound	FPRD-LT-L01; Myhill–Nerode theorem; R-trivial language characterization	FPRD Lab target-specific synthesis using classical automata theoryReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine whether target-residual degree is polynomially bounded in the degree of every supplied L-trivial transformation action.
FPRD-LT-T05 For the binary J-trivial family of FPRD-LT-T03, the source transition monoid has exactly $\|M_n\|=\binom n3+n-1$ elements, while the canonical target-residual observer for $f_n=(n-1,1,\ldots,1)$ has exactly $d_{f_n}=\binom n2-1$ states.	Theorem · exact quotient familySelf-contained normal-form and residual proof with exact computational audits	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Exact monoid and observer formulas	FPRD-LT-T03; FPRD-LT-T04	FPRD Lab exact analysis of the binary lower-bound familyReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Generalize the fibre classification or construct an L-trivial family with superpolynomially many target profiles.
FPRD-LT-C02 Among all 32,896 unordered transformation-generator pairs on four states, 4,473 generate L-trivial monoids. Their maximum left-Cayley height is 6, while the maximum shortest right-Cayley distance of any target across the same search is 3; ideal height is therefore a strict overestimate of target certificate length.	Counterexample · complexity proxyExact finite computation; semigroup control rather than a minimal-DFA classification	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	Four-state Green-height control	FPRD-LT-D01	FPRD Lab exhaustive four-state semigroup computationReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use extension-distinguishable target profiles or actual target geodesics rather than principal-ideal height as the obstruction measure.
FPRD-LT-B01 It remains open whether every positive L-trivial transformation-membership instance has a polynomial-length word witness in the degree of the supplied action. The canonical target-residual degree d_f gives a witness of length at most d_f−1; whether d_f is always polynomial is the current sufficient-bound question.	Open problem · representation boundaryUnresolved; the n-minus-one conjecture is refuted, but neither NP membership nor PSPACE hardness is established	Evidence and limits →	Automata and formal languagesL-trivial transformation membership	What remains open	FPRD-LT-T04; FPRD-LT-T05; FPRD-LT-C02; FPRD-LT-T03; FPRD-LT-E02	Problem proposed by Andrew Ryzhikov; FPRD Lab boundary analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Prove a polynomial upper bound on target-residual degree from L-trivial fibre evolution, or construct a structured superpolynomial target-profile or geodesic family.
FPRD-LRN-D01 For $F(q)=\sum_{n\ge0}a_nq^n\in\mathbb Q[[q]]$ , an exact-prefix learner receives $a_0,\ldots,a_{N-1}$ , while a revisable-snapshot learner receives polynomials $P_t$ satisfying only $\forall n\,\exists T_n\,\forall t\ge T_n:[q^n]P_t=a_n$ .	Definition · observation modelsPrecisely stated; the two models have different learnability	Evidence and limits →	Automata and formal languagesLearning rational sequences	Observation models	No recorded dependencies	Problem proposed by Benjamin Kaminski; FPRD Lab model analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine which observation model is supplied by the probabilistic-program application motivating the revisable examples.
FPRD-LRN-T01 A rational series of positive Hankel rank $r$ is determined and reconstructible from exactly $2r$ initial coefficients, and this count is sharp. With a known bound $R\ge r$ , $2R$ coefficients give certified exact reconstruction; without a known rank bound, repeated reconstruction identifies every rational series in the limit. The rank-zero class contains only the zero series.	Theorem · exact learningClassical recurrence reconstruction, restated and proved for this model	Evidence and limits →	Automata and formal languagesLearning rational sequences	Exact-prefix theorem	FPRD-LRN-D01; Massey 1969 shift-register synthesis; Berstel–Reutenauer 2010 Hankel minimization	James L. Massey (1969); Jean Berstel and Christophe Reutenauer (2010); FPRD sharpness proofReviewed 2026-09-01The recurrence-reconstruction theorem is classical; no external review of this FPRD exposition is recorded.	Quantify coefficient bit growth for the intended application and add a certified-prefix extractor if its approximants provide one.
FPRD-LRN-C01 After any prefix through degree $N-1$ , the distinct rational series $F(q)$ and $F(q)+cq^N/(1-\lambda q)$ , with $c\ne0$ , agree on every observed coefficient. Therefore unrestricted exact-prefix learning cannot certify from finite data that its current rational hypothesis is final.	Counterexample · finite identifiabilityProved by an explicit rational tail	Evidence and limits →	Automata and formal languagesLearning rational sequences	Rational-tail indistinguishability	FPRD-LRN-D01	Problem proposed by Benjamin Kaminski; FPRD Lab boundary proofReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Identify a usable rank bound, equivalence query, or finality certificate in the intended application.
FPRD-LRN-T02 No learner identifies every rational target in the limit from arbitrary polynomial snapshots $P_t$ whose coefficients merely stabilize pointwise; the impossibility already holds using the zero series and single monomials $q^k$ .	Theorem · impossibilityProved by a diagonal presentation	Evidence and limits →	Automata and formal languagesLearning rational sequences	Diagonal impossibility theorem	FPRD-LRN-D01	Problem proposed by Benjamin Kaminski; FPRD Lab diagonal argumentReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Test the intended approximation semantics for a computable stabilization modulus, finality markers, monotone bounds, or active validation.
FPRD-LRN-B01 A positive exact-learning theorem for revisable polynomial approximants requires more than pointwise stabilization, such as a computable stabilization modulus, finality markers, certified monotone bounds with an effective gap test, a detectable revision bound, or active coefficient and equivalence queries.	Research boundaryNecessary direction isolated; application semantics remain unknown	Evidence and limits →	Automata and formal languagesLearning rational sequences	What would make learning possible	FPRD-LRN-T02	Problem proposed by Benjamin Kaminski; FPRD Lab boundary analysisReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Recover the semantics of the probabilistic-program approximants and state the strongest available certificate before designing a learner.
FPRD-MN-D01 For a rational matrix $M$ , power nonnegativity asks whether $M^n\ge 0$ for some $n\ge 1$ ; eventual nonnegativity asks whether $M^n\ge 0$ for every sufficiently large $n$ ; everywhere nonnegativity asks whether this holds for every $n\ge 0$ . Inequalities are entrywise.	Definition · quantifier distinctionPrecisely stated and used consistently	Evidence and limits →	Algorithms and complexityNonnegative powers of rational matrices	Three nonnegativity definitions	No recorded dependencies	FPRD Lab terminology calibrated against D'Costa, Ouaknine, and WorrellReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Keep the exponent domain and quantifiers explicit in every later complexity or certificate statement.
FPRD-MN-T01 For every rational square matrix $M$ , the exponent set $S(M)=\{n\in\mathbb N:M^n\ge0\}$ is effectively computable and ultimately periodic.	Theorem · external resultPublished theorem; FPRD exposition reconstructed from the source	Evidence and limits →	Algorithms and complexityNonnegative powers of rational matrices	The theorem and proof architecture	Nondegenerate LRS residue decomposition; Perron–Frobenius/Berstel domination	D'Costa, Ouaknine, and Worrell, STACS 2024, Theorem 7Reviewed 2026-09-01Published in the peer-reviewed STACS 2024 proceedings. This does not constitute external review of the FPRD exposition.	Extract explicit certificate and complexity bounds from the effective spectral construction.
FPRD-MN-C01 Power nonnegativity and eventual nonnegativity of a rational matrix are decidable by testing, respectively, whether the effective ultimately periodic set $S(M)$ contains a positive exponent and whether it is cofinite.	Corollary · decision consequenceProved as an immediate consequence of the published theorem	Evidence and limits →	Algorithms and complexityNonnegative powers of rational matrices	Two decision consequences	FPRD-MN-D01; FPRD-MN-T01	D'Costa, Ouaknine, and Worrell, STACS 2024; FPRD implication auditReviewed 2026-09-01The underlying ultimate-periodicity theorem is published; no external review of this FPRD consequence note is recorded.	Extract explicit certificates and complexity bounds for both decision tests from the effective periodic representation.
FPRD-MN-C02 The scalar matrix $M=[-1]$ is power nonnegative because $M^2=[1]$ , but it is not eventually nonnegative because every odd power is negative.	Counterexample · quantifier separationProved by direct calculation	Evidence and limits →	Algorithms and complexityNonnegative powers of rational matrices	Smallest quantifier counterexample	FPRD-MN-D01	FPRD Lab statement auditReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Use this example as the standing guardrail against conflating an arithmetic progression of good powers with a cofinite tail.
FPRD-MN-R01 For a $d\times d$ rational matrix $M$ , all $d^2$ entry sequences of $M^n$ can be interleaved into one effectively constructed rational LRS of order at most $d^3$ , whose ultimate nonnegativity is equivalent to eventual entrywise nonnegativity of $M^n$ .	Proposition · reductionReconstructed and proved; useful upper-bound reduction but not the decision argument	Evidence and limits →	Algorithms and complexityNonnegative powers of rational matrices	Generating-function interleaving	Cayley–Hamilton theorem	Akshay, Chakraborty, and Pal 2022; FPRD reconstructionReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Keep this reduction separate from the stronger structured matrix theorem and from any unproved reverse reduction.
FPRD-MN-F01 There are signed permutation matrices of dimension $d$ whose least positive entrywise-nonnegative power is $\exp(\Theta(\sqrt{d\log d}))$ ; hence checking only polynomially many powers cannot decide power nonnegativity.	Counterfamily · failed algorithmProved by signed prime-cycle blocks	Evidence and limits →	Algorithms and complexityNonnegative powers of rational matrices	Signed-cycle lower-bound family	Elementary cycle arithmetic; Prime number estimates	FPRD Lab counterfamilyReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Determine sharp certificate-size and least-hit bounds for restricted spectral classes.
FPRD-RDV-D01 A reversible data VASS uses finitely many place types and orbit-finitely many multiset-rewrite rules over a data domain; every rule is available in both directions, and reachability asks whether one finite marking rewrites to another.	Definition · decision problemPrecisely stated with reversibility and effectiveness separated	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	Model and conventions	No recorded dependencies	Problem proposed by Arka Ghosh; FPRD Lab reconstructionReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Obtain an author check of the intended effective presentation and embedding-action conventions.
FPRD-RDV-T01 For an orbit-finitely presented reversible system with transition binomials $v_i-u_i$ , source and target monomials $s,t$ satisfy $s\leftrightarrow^* t$ exactly when $t-s$ belongs to the ideal generated by all embedding-renamed $v_i-u_i$ . On a homogeneous data structure, the same finite renamings can be realized by automorphisms.	Theorem · external algebraic reductionPublished theorem; reduction reconstructed in the FPRD exposition	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	Exact ideal-membership reduction	Reversible commutative rewriting; Embedding action on variables	Ghosh and Lasota, Equivariant ideals of polynomials, Section 8 (Theorem 3 in arXiv v3; cited as Theorem 64 by Ghosh--Lopez)Reviewed 2026-09-01The underlying theorem appeared at LICS 2024. No external review of the FPRD exposition is recorded.	Seek a specialist check of the automorphism-versus-embedding presentation used on the page.
FPRD-RDV-T02 Let $\mathcal A$ be an $\omega$ -well-structured, totally ordered data domain. For every Noetherian commutative coefficient ring $K$ , each embedding-equivariant ideal of $K[\mathcal A]$ has a finite equivariant basis.	Theorem · external finite-generation consequenceSupported by the published equivariant Hilbert-basis theorem	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	What the two structural hypotheses imply	Equivariant Hilbert-basis theorem; FPRD-RDV-D01	Ghosh and Lasota, LICS 2024Reviewed 2026-09-01The finite-basis theorem appeared at LICS 2024. No external review of this FPRD specialization is recorded.	Keep existence of a finite basis separate from effective computation of one.
FPRD-RDV-T03 Under the paper's computability assumptions, reachability in reversible Petri nets with data is decidable for every nicely orderable group action.	Theorem · external strengthened caseProved in a 2025 preprint under stronger effective hypotheses	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	The decidable strengthened case	FPRD-RDV-T01; Computable equivariant Gröbner bases; Effective oligomorphism; Labelled WQO for every WQO label set	Ghosh and Lopez, Computability of Equivariant Gröbner bases, Corollary 39Reviewed 2026-09-01The source is an arXiv preprint; no peer-reviewed publication or external review of the FPRD exposition is asserted.	Track the publication status and test whether the binomial case needs less than the full labelled-WQO hypothesis.
FPRD-RDV-B01 It is not established here that WQO for finite substructures labelled by $\mathbb N$ implies the $\mathbb N\times\mathbb N$ -labelled WQO used by the core equivariant Gröbner computation, still less the all-WQO-label condition of nicely orderable actions.	Open problem · exact implication gapUnresolved under the stated weaker hypothesis set	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	The remaining implication	FPRD-RDV-T02; FPRD-RDV-T03	FPRD Lab comparison of the 2024 and 2025 hypothesis setsReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Prove the lifting for homogeneous ordered finite-signature structures or specialize the algorithm to binomial ideals.
FPRD-RDV-X01 The pure Rado graph has no distinguished invariant linear-order relation, and the ordered random graph fails the required labelled-WQO condition: its age contains the chordless cycles $C_n$ as an induced-subgraph antichain. Thus neither structure satisfies the displayed conjunction of hypotheses.	Counterexample · source-example correctionProved by an induced-cycle antichain; consistent with the later undecidability theorem	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	Rado-graph hostile boundary	Induced-substructure ordering; FPRD-RDV-B01	Ghosh and Lopez, Example 54; elementary induced-cycle antichainReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Keep dense rational order as the positive example and do not list the Rado graph as satisfying both hypotheses.
FPRD-LL-D01 An $s$ -state rational linear loop is $z(n+1)=Mz(n)$ with $M\in\mathbb Q^{s\times s}$ . Its first $d$ coordinates form the visible orbit $u(n)$ , and a polynomial invariant $p$ requires $p(u(n))=0$ for every $n\ge0$ . Nontriviality means that the visible orbit contains infinitely many distinct points.	Definition · modelPrecisely stated; homogeneous updates and literal visible coordinates are explicit	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	Model and quantifiers	No recorded dependencies	Problem proposed by Anton Varonka; FPRD Lab reconstructionReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Keep homogeneous and affine updates, literal coordinates and output maps, and infinite time and infinite orbit distinct.
FPRD-LL-L01 For each fixed polynomial $p$ , the bare existence of a finite bound on witness dimensions is automatic: if any witness exists, the set of its possible state dimensions has a least element. The substantive questions ask for a computable $N(p)$ , a uniform $N(d,D,L)$ , or a dimension-only $N(d)$ .	Lemma · quantifier distinctionProved by well-ordering; the intended effective bound remains open	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	Which bound is being asked for?	FPRD-LL-D01	FPRD Lab logical calibration of the linear-loop synthesis questionReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Determine which effective or uniform parameterization is intended before treating the general question as a decision problem.
FPRD-LL-T01 Let $V_u=\operatorname{span}_{\mathbb Q}\{S^ju_k:j\ge0,1\le k\le d\}$ , let $r(u)=\dim V_u$ , and let $q(u)=\dim\operatorname{span}_{\mathbb Q}\{u_1,\ldots,u_d\}$ . If $r(u)<\infty$ , the least state dimension of a rational linear loop whose first $d$ coordinates are exactly $u_1,\ldots,u_d$ is $d+r(u)-q(u)$ .	Theorem · minimal realizationProved with matching lower bound and construction	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	Coordinate-realization formula	Finite-dimensional shift-invariant spaces	FPRD coordinate-cost formulation and elementary shift-space proofReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Relate the coordinate-cost formula to standard minimal-realization statements with an external output map.
FPRD-LL-T02 For $f(x)=\sum_k f_kx^k\in\mathbb Q[x]$ of degree at least two, the graph $y=f(x)$ has a nontrivial rational linear loop with $r_f=\|\{1\}\cup\{k:f_k\ne0\}\|$ state variables. Hence $r_f\le\deg f+1$ , and $r_f\le\deg f$ when $f(0)=0$ .	Theorem · constructive upper boundProved by a diagonal exponential construction	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	Sparse-degree construction	FPRD-LL-T01; Vandermonde independence	FPRD Lab graph-polynomial constructionReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Extend the construction from polynomial graphs to rational parametrized curves and higher-dimensional varieties.
FPRD-LL-C01 For the scaling orbit $x(n)=2^n$ , $y(n)=f(2^n)$ used in FPRD-LL-T02, the minimum coordinate-state dimension is exactly $r_f=\|\{1\}\cup\operatorname{supp}(f)\|$ .	Corollary · orbit-specific lower boundProved from block-Hankel rank and Vandermonde independence	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	Minimality of the constructed orbit	FPRD-LL-T01; FPRD-LL-T02	FPRD Lab application of minimal realizationReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Use finite Hankel minors as checkable certificates for proposed orbit dimensions.
FPRD-LL-T03 For $\deg f\ge2$ , the graph $y=f(x)$ admits a nontrivial rational two-variable homogeneous linear loop if and only if $f(x)=\alpha x^m+\beta x$ for some $m\ge2$ , $\alpha\ne0$ , and $\beta\in\mathbb Q$ .	Theorem · exact classificationProved by polynomial identity and degree comparison	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	Two-state classification	Infinite graph orbit gives infinitely many x-values; Rational roots of unity are ±1	FPRD Lab graph-polynomial classificationReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Determine the corresponding classification for affine two-variable updates and for larger ground fields.
FPRD-LL-C02 No nontrivial rational two-variable homogeneous linear loop remains on $y=x^3+x^2$ , while the orbit $x(n)=2^n$ , $z(n)=-4^n$ , $y(n)=8^n+4^n$ gives a three-state witness.	Corollary · strict dimension separationBoth the lower bound and witness are proved	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	A strict two-versus-three separation	FPRD-LL-T02; FPRD-LL-T03	Anton Varonka's example; FPRD Lab proof reconstructionReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Seek independent review of the exact classification proof that supplies the two-state lower bound.
FPRD-LL-F01 The rank certificate for a chosen infinite C-finite orbit in $V(p)$ cannot be promoted to a lower bound for $p$ itself without proving that every admissible infinite orbit in $V(p)$ has at least that rank.	Negative result · failed lower-bound methodQuantifier failure isolated	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	Why orbit minimality is not variety minimality	FPRD-LL-T01	FPRD Lab hostile quantifier auditReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Find a family for which every infinite rational C-finite orbit has growing rank, or prove a uniform compression theorem.
FPRD-LL-B01 It remains open here whether one can compute, from an arbitrary rational polynomial $p$ , a finite state bound sufficient to find a nontrivial rational linear loop whenever one exists; no uniform bound $N(d,D,L)$ or $N(d)$ is established either.	Open problem · general boundaryOpen after checking the graph, quadratic, pure-difference-binomial, and bit-bounded subclasses	Evidence and limits →	Algebra and discrete mathematicsLinear loops and polynomial invariants	What remains open	FPRD-LL-L01; FPRD-LL-F01	Kenison–Kovács–Varonka (ISSAC 2023); Hitarth–Kenison–Kovács–Varonka (STACS 2024); Ait El Manssour–Kenison–Shirmohammadi–Varonka (POPL 2025); source problemReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Formalize the target parameterization, then minimize d+r(u)-q(u) over all infinite rational C-finite orbits in V(p).
FPRD-CPA-C01 A formal series is input-commutative when its value depends only on the Parikh vector of the input word; a commutative-control presentation requires every control series, not only the output, to have this property.	Definition · model boundaryDefined and used consistently in the proofs	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Problem and model	No recorded dependencies	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Keep input commutativity distinct from commutativity of a product rule in later comparisons.
FPRD-CPA-C02 A commutative polynomial-automaton series has an equivalent presentation whose control series are all commutative if and only if it belongs to a finitely generated, derivative-stable Hadamard algebra consisting entirely of commutative series.	Proposition · exact reformulationProved by a written two-way construction; internally audited	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Control-algebra equivalence and proof	Clemente 2025: Hadamard/polynomial equivalence up to reversal	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Seek an author or specialist check of the presentation convention and reversal step before publication-ready status.
FPRD-CPA-C03 If every control series is a Hadamard polynomial in finitely many left derivatives of the commutative output series, then every control series is commutative; the displayed recovery identities form an effective sufficient certificate.	Proposition · sufficient certificateProved; effectiveness relies on published decidability of polynomial-automaton equivalence	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Polynomial recovery criterion	FPRD-CPA-C02; Clemente 2025: equality for polynomial/Hadamard automata is decidable	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Develop practical search bounds or structural hypotheses that produce recovery identities.
FPRD-CPA-C04 For a commutative rational series, a finite basis chosen from its derivative span yields an equivalent weighted automaton whose control series are all commutative.	Reconstruction · classical weighted caseReconstructed from the classical finite derivative-space theorem	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Weighted-case reconstruction	Classical finite derivative-space theorem for rational series	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Seek an independent specialist check of the classical derivative-basis reconstruction.
FPRD-CPA-C05 Finite generation of the derivative Hadamard algebra D(f) is not necessary for a positive commutative-control presentation; the failure already occurs over a unary alphabet and also over the displayed two-letter alphabet.	Counterexample · negative resultProved by an explicit factorial automaton; internally audited	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Factorial automaton counterexample	FPRD-CPA-C02	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Retain the written grading proof as the general justification and the bounded checker only as corroboration.
FPRD-CPA-C06 It remains unresolved whether every commutative polynomial-automaton series belongs to a finite derivative-stable Hadamard algebra consisting entirely of commutative series.	Open problem · exact residual obligationOpen; no general yes/no answer claimed	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Unresolved proof obligations	FPRD-CPA-C02; FPRD-CPA-C05	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Search for finite commutative over-algebras or an invariant violated by a commutative output with chronology-sensitive controls.
FPRD-CPA-C07 Finite generation of an ambient commutative algebra does not imply finite generation of each of its subalgebras.	Negative result · algebraic obstructionProved here by the explicit factorial subalgebra; applied to rule out an invalid proof step	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Failure of the subalgebra argument	No recorded dependencies	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Keep finite over-algebras separate from the output's derivative-generated subalgebra in future arguments.
FPRD-CPA-C08 Injectivity of a polynomial observation map does not imply polynomial recovery of the hidden coordinate: the map t ↦ (t²,t³) is injective, but t does not belong to Q[t²,t³].	Counterexample · failed abstractionProved by the cusp coordinate ring	Evidence and limits →	Algebra and discrete mathematicsCommutative polynomial automata and series	Polynomial-recovery boundary	No recorded dependencies	Commutative polynomial automata and series; originating question by Lorenzo Clemente (2025)Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Require explicit polynomial recovery identities rather than finite separation alone.
FPRD-T96 A causal resolver emits one call, return, or local tag in the same round as each observed input letter. Its one-stack product with a deterministic visible-stack machine has exactly the resolver state and tagged-machine configuration after every prefix. A fixed finite family of such products executes on parallel stacks and accepts exactly their finite disjunction. If the family is complete for every accepted tagged lift, its executor and the scaffold produced by the verified compiler decide exactly the forgotten-action projection, including at epsilon.	Mechanically checked resultmechanically checked	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	formalizations/frontier-derivative/FrontierDerivative/ResolverCover.lean	FPRD-T89	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Expose the formal/checker scope, reproducibility instructions, and the gap to a complete mathematical proof.
FPRD-T98 A finalizable delayed resolver emits append-only online tagged batches and an explicit finish batch at end of word. Its semantic visible-stack evaluator has exactly the resolver state and direct visible run on the online output after every prefix, and after finish it has exactly the direct run on the completed annotation. A lawful fixed finite Complete Cover therefore recognizes exactly the forgotten-action projection, including at epsilon. This theorem does not invoke the strict compiler.	Mechanically checked resultmechanically checked	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	formalizations/frontier-derivative/FrontierDerivative/DelayedResolver.lean	FPRD-T96	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Expose the formal/checker scope, reproducibility instructions, and the gap to a complete mathematical proof.
FPRD-T18 For every finite conventional context-free grammar G, nonterminal A, and input word w, the least relational suffix behavior of the canonical translation returns exactly the suffixes v for which A derives a terminal prefix x with w=xv; consequently complete-match observation of the translated start variable is exactly the classical language of G.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D07; FPRD-D09; FPRD-T14; FPRD-T15	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T19 The canonical labeled translation of every finite conventional context-free grammar generates an M12 compact relational recognition dynamic reconstructed from its finite reachable graph shadows, and the complete-match call observation at its distinguished initial state is exactly the grammar's classical language.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D08; FPRD-D09; FPRD-T16; FPRD-T17; FPRD-T18	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T20 Every finite positive relational presentation has a finite conventional CFG word normal form whose translated equation operator agrees in every environment, hence whose finite stages, exact and bounded least suffix solutions, and complete-match language agree exactly; together with the M13 lift, the complete-match languages of finite positive relational presentations are exactly the context-free languages.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D07; FPRD-D09; FPRD-D10; FPRD-T14; FPRD-T15; FPRD-T18	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T21 For every labeled positive relational presentation, distributive CFG normalization induces a finite origin relation on call occurrences, and at every exact and bounded observation level each normalization-surviving source right-success frame behavior is the relational union of the frame behaviors of its normalized call clones; split, merge, and elimination witnesses show that raw presentation dynamics need not be invariant.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D03; FPRD-D08; FPRD-D10; FPRD-T15; FPRD-T16; FPRD-T20	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T22 A language is regular exactly when its residual recognition dynamic is finite; in that case residual actions and empty-word output recognize it exactly, every reachable deterministic recognizer maps uniquely and surjectively onto the residual dynamic by accepted-future language, and the residual dynamic is the unique minimal exact recognizer up to labeled isomorphism, recovered after finite bounded observational refinement.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D12	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T23 For languages A and B with positive minimum word length m in A, the operator F(X)=AX union B is at most 2-to-the-minus-m contractive in the shortest-distinguishing-word ultrametric, has the unique fixed point A-star B, and its finite unfoldings become exact on bounded words; under the suffix-recognition lift the same fixed point solves the corresponding relational composition-and-union equation with exact remaining-suffix and complete-match readout.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D06; FPRD-D11; FPRD-T13	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T24 Every finite Greibach recognition presentation induces one-half contractive operators on the finite products of exact language behaviors and full set-valued suffix behaviors; both have unique fixed points, bottom unfoldings at depth N plus one are exact on words and input rows through length N, the vector suffix lift identifies the two solutions, and their readout equals ordinary finite CFG derivability.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D06; FPRD-D11; FPRD-D13; FPRD-T13; FPRD-T14; FPRD-T15; FPRD-T18; FPRD-T23	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T25 A language is context-free exactly when it is the complete-match observation of the unique suffix fixed point of a finite Greibach recognition presentation; thus every context-free language has a finite terminal-delayed presentation with coherent exact bounded observations and uniform remaining-suffix readout.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D09; FPRD-D13; FPRD-T18; FPRD-T24	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T26 For a finite cycle-guarded positive relational presentation with r variables, the occurrence-delay matrix bounds componentwise semantic first differences by min-plus multiplication, some power p at most r of the exact suffix operator is one-half contractive, the original operator has a unique fixed point, and bottom depth p times N plus one is exact on all input rows through length N with coherent inverse-limit recovery.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D06; FPRD-D14; FPRD-T13	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T27 For every finite cycle-guarded positive relational presentation, the unique contractive-iterate fixed point is its M11 least fixed point and finite-derivation semantics, its bounded suffix and complete-match readouts are exact by the certified block depth, and its labeled M12 continuation actions are evaluated in that same uniquely determined behavior vector.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D07; FPRD-D08; FPRD-D14; FPRD-T14; FPRD-T15; FPRD-T16; FPRD-T26	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T28 Every finite generalized Dyck-control presentation has a finite positive state-pair suffix equation system whose components recognize exactly the well-nested traces carrying the selected DFA between their endpoint states; every recursive dependency has positive delay, so the system has one exact suffix solution recovered at depth N plus one through trace length N, and letter-to-letter decoding gives exact remaining-suffix behavior on the output alphabet with the same bounded observation index.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D15; FPRD-T18; FPRD-T22; FPRD-T26; FPRD-T27	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T29 For every nonempty finite output alphabet Sigma there are one finite generalized Dyck alphabet and one fixed letter-to-letter readout, depending only on Sigma, such that the exact suffix behaviors of context-free languages over Sigma are precisely the decoded observations obtained by varying a finite regular controller; every such observation is coherently reconstructed from the controller-product's finite shadows.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D15; FPRD-T25; FPRD-T28	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T30 Every capacity-witnessed fixed-envelope recoding induces a constant length injective homomorphism tau that sends each matching source pair to reversed matching envelope blocks, preserves and reflects Dyck membership on encoded words, and commutes with the fixed projection readout; for every source trace language K, direct block readout of K and fixed projection readout of tau of K have identical exact suffix behaviors, and whenever source languages K and H agree through floor of N over m, their decoded output suffix behaviors agree through length N.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D16	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T31 For every finite complete source controller, the fixed-envelope recoder constructs a finite complete deterministic trie controller recognizing exactly its tau image, sending malformed blocks and unused envelope symbols to a rejecting sink; intersection with envelope Dyck geometry then equals the tau image of the source Dyck-control intersection, so the abstract M18 theorem supplies one exact decoded suffix dynamic with same-index envelope reconstruction and the block-scaled source observation modulus, while every instance admitted by an explicit finite one-character codec is conjugately realized by the executable M18 implementation.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D16; FPRD-T28; FPRD-T30	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T32 For every injective width-m block code and finite complete source controller, the compiled boundary macro-actions are conjugate to the source actions, the right language at each boundary is exactly the code image of the corresponding source right language, and source residual equality is equivalent to boundary residual equality; bounded future-language and full suffix-lift partitions correspond exactly by N maps to floor of N over m, shortest distinguishing lengths multiply by m, and the source residual dynamic embeds into the image residual dynamic under the encoded macro-actions.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D17; FPRD-T22; FPRD-T31	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T33 The fixed-block compiler allocates exactly one sink plus one boundary and one state for every distinct nonempty proper code prefix at every source state, has the corresponding exact reachable-state and transition counts, transports every surjective source-controller quotient functorially, and preserves source residual count as a lower bound on image residual count; for the fixed envelope, q at most j to the m is exactly the positional-code capacity condition, yielding explicit minimal width or base, envelope-size, trie-overhead, residual, and observation-lag coordinates.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	OPEN_PROBLEMS.md	FPRD-D17; FPRD-T22; FPRD-T31; FPRD-T32	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T95 If the source and cover carriers of an FPRD-T94 stationary principal derivative cover are finite and its frontier language is X, then X reversed belongs to PEL. Equivalently, to obtain a PEG for L by this route, construct the finite stationary cover for L reversed.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/notes/stationary-principal-derivative-cover-proof.md	FPRD-T94	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T97 Let a deterministic visible-stack language have a complete family of a fixed positive finite number of causal resolvers. If the input, visible control, stack alphabet, resolver control, and family are finite, and X is the resulting forgotten-action projection, then X reversed belongs to PEL.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/notes/finite-causal-resolver-cover-proof.md	FPRD-T96	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T99 Let a deterministic visible-stack language have a complete family of a fixed positive finite number of lawful finalizable delayed resolvers, with finite input, visible-control, stack, resolver-control, and family carriers. Each branch has a deterministic PDA realization: a right endmarker may drain finish explicitly, and a finite augmented top-window construction eliminates that marker by predicting the same final readout. The projected language is therefore a finite union of DCFLs and belongs to PEL by Rubtsov--Chudinov Theorem 10, with complete-match semantics and no reversal.	Written-proof claiminformal proof	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/notes/finalizable-delayed-resolver-cover-proof.md	FPRD-T98	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T01 For a finite alphabet Sigma, the semantic quotient of finite nonrecursive PEG expressions embeds as the subalgebra of suffix-decreasing partial endofunctions of Sigma-star generated by identity, failure, terminal left-cancellations, parser-order composition, preferential union, and antidomain.	Written-proof claiminformal proof	Evidence and limits →	Models of computationFPRD foundations	docs/specs/m1-semantic-algebra.md	FPRD-D01	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T02 Equality of suffix-decreasing partial-function behaviors on all inputs of length at most N is a finite-index congruence for the deterministic PEG operations, and the ambient behavior algebra is the inverse limit of its bounded observational quotients.	Written-proof claiminformal proof	Evidence and limits →	Models of computationFPRD foundations	docs/specs/m1-semantic-algebra.md	FPRD-D02	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T03 Every legal finite suffix-behavior table is denoted at its bound by a synthesized finite nonrecursive PEG expression; consequently the term- generated PEG algebra is dense in the ambient behavior algebra and its observational profinite completion for the bounded-input congruence chain is canonically isomorphic to that ambient algebra.	Written-proof claiminformal proof	Evidence and limits →	Models of computationFPRD foundations	OPEN_PROBLEMS.md	FPRD-T01; FPRD-T02	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T04 Every finite strongly input-guarded deterministic prioritized recognition equation system induces a contractive operator on a specified suffix-behavior ultrametric and therefore has a unique denotational fixed point.	Written-proof claiminformal proof	Evidence and limits →	Models of computationFPRD foundations	OPEN_PROBLEMS.md	FPRD-T01; FPRD-T02	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T05 For the strongly input-guarded fragment, the unique denotational fixed point agrees with ordinary operational evaluation, and sufficiently deep finite unfoldings are exact on every bounded input domain.	Written-proof claiminformal proof	Evidence and limits →	Models of computationFPRD foundations	OPEN_PROBLEMS.md	FPRD-T04	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T06 A finite uniquely labeled positive strongly guarded presentation admits a well-defined extraction of finitely many right-success call-site context actions and a graph-generated observational dynamic, functorial under variable and call-label renaming and invariant under sequence reassociation.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D03; FPRD-T05	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T07 The M3 continuation orbit generated by K_n=b^n has pairwise distinct exact states, compact closure obtained by adjoining failure, N+2 states in its depth-N observational shadow, linear observational growth, an inverse- limit reconstruction from those shadows, and a one-counter realization with uniform prefix-removal readout; its transition also has a bounded rational state coordinate that is not by itself a complete recognition realization.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D04; FPRD-T02; FPRD-T06	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T08 For every finite context rank r at least one, a canonical strongly guarded r-call presentation generates the exact free-monoid continuation orbit, whose one-point compact closure has 1 plus the geometric sum through N states in its depth-N shadow, coherent inverse-limit reconstruction, rank-dependent observational growth, and a readout-aware free-word realization.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D04; FPRD-T02; FPRD-T06; FPRD-T07	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T09 For every finite context rank r at least one, the polynomial actions F_i(p) = End / (terminal-c_i ; p), generated from End, have the exact finite prefix-tree orbit and a compact closure indexed by finite and infinite context words, with longest-common-prefix metric, Cantor boundary at rank at least two, explicit finite node/cylinder shadows, inverse-limit reconstruction, and a uniform prefix readout.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D04; FPRD-D05; FPRD-T01; FPRD-T02; FPRD-T04; FPRD-T08	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T10 Every well-formed finite unary-polynomial recognition-action presentation has a compact invariant orbit closure canonically homeomorphic and action-conjugate to the inverse limit of its finite bounded orbits, with each bounded orbit equal to the projection of the exact closure and with exact remaining-suffix behavior recovered by a uniform thread readout.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D04; FPRD-D05; FPRD-T01; FPRD-T02; FPRD-T09	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T11 Every finite uniquely labeled positive strongly input-guarded recognition presentation has a graph-directed exact continuation dynamic whose orbit closure is compact and canonically homeomorphic and edge-conjugate to the inverse limit of its finite bounded M3 orbits, with exact location, continuation, and call behavior recovered by uniform thread readouts.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D03; FPRD-D04; FPRD-T01; FPRD-T02; FPRD-T04; FPRD-T05; FPRD-T06; FPRD-T10	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T12 For any finite-observation recognition doctrine, a finite family of continuous restriction-natural actions from one distinguished state has a compact invariant orbit closure canonically homeomorphic and action-conjugate to the inverse limit of its finite bounded orbits, with the doctrine's exact word readout recovered uniformly from threads.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D04; FPRD-D06; FPRD-T10	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T13 Set-valued suffix relations form a finite-observation recognition doctrine under relational composition, commutative union, preferential nonempty choice, and antidomain; deterministic PEG behavior embeds under singleton outcomes and preferential choice, while a nonfunctional optional-prefix family shares M7's actions, compact coordinate space, finite shadows, growth, and complete-match acceptance but has different full remaining-suffix readout.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D06; FPRD-T01; FPRD-T02; FPRD-T09; FPRD-T12	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T14 Every finite positive relational recognition presentation induces a monotone operator preserving joins of increasing omega-chains on the complete lattice of relational suffix-behavior vectors; its least fixed point is the union of finite unfoldings and agrees exactly with finite derivation suffix semantics.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D06; FPRD-D07; FPRD-T13	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T15 For every finite positive relational presentation and input bound N, restriction of the exact least solution is the least solution of the finite bounded equation system, finite iteration stabilizes within the number of variable/input/suffix membership facts, and the coherent bounded solutions reconstruct the exact solution with uniform readout.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D06; FPRD-D07; FPRD-T13; FPRD-T14	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T16 Every finite uniquely labeled positive relational presentation admits a finite complete extraction of right-success call sites whose frame semantics in the M11 least solution induces continuous relational continuation actions commuting with every bounded restriction; variable and call-label renaming and sequence reassociation preserve the resulting dynamics in the specified conservative senses.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D03; FPRD-D06; FPRD-D07; FPRD-D08; FPRD-T06; FPRD-T13; FPRD-T14; FPRD-T15	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-T17 The graph-generated relational continuation orbit of every M12 presentation has a compact closure canonically homeomorphic and edge-conjugate to the inverse limit of its finite bounded reachable graph orbits, with exact location, set-valued continuation, and set-valued call behavior recovered by uniform thread readouts.	Written-proof claiminformal proof	Evidence and limits →	Models of computationRepresentation dynamics	OPEN_PROBLEMS.md	FPRD-D02; FPRD-D03; FPRD-D06; FPRD-D08; FPRD-T11; FPRD-T12; FPRD-T13; FPRD-T15; FPRD-T16	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Audit the proof against its dependencies and publish a self-contained evidence summary before increasing maturity.
FPRD-D09 A finite conventional context-free grammar has finite disjoint terminal and nonterminal sets, finitely many productions, and a distinguished start nonterminal; its canonical relational translation maps terminals to relational terminal actions, nonterminal occurrences to uniquely labeled calls, concatenation to sequence, alternatives to commutative union, epsilon to identity, and an absent production set to failure.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m13-cfg-relational-lift.md	FPRD-D03; FPRD-D07; FPRD-D08	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D10 The finite distributive word normal form of a positive relational term maps failure to no words, identity to the empty word, terminals and calls to singleton words, union to finite set union, and sequence to finite word concatenation; for labeled presentations, an annotated enrichment records the finite origin relation from canonical normalized call occurrences to source call labels.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m14-cfg-normal-form.md	FPRD-D03; FPRD-D07; FPRD-D08; FPRD-D09	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D11 The suffix-recognition lift Lambda maps a language L to the set-valued suffix behavior that returns every v for which the input factors as xv with x in L; complete-match readout recovers L, and the lift is intended to preserve empty language, epsilon, union, and language concatenation as relational failure, identity, union, and composition.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m15-classical-regular-dynamics.md	FPRD-D01; FPRD-D02; FPRD-D06	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D12 The residual recognition dynamic of a language L has the left residuals u-inverse L as states, one-letter left differentiation as its named input actions, L as its distinguished state, empty-word membership as output, and agreement on continuations of length at most N as its bounded observational equivalence.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m15-classical-regular-dynamics.md	FPRD-D01; FPRD-D02; FPRD-D11	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D13 A finite Greibach recognition presentation is the canonical labeled relational translation of a finite context-free grammar whose productions have the form A to terminal-a followed by a finite word of nonterminals, or A to epsilon; it retains the finite language and suffix equation operators, labeled call-context actions, bounded restrictions, and remaining-suffix and complete-match readouts.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m16-greibach-recognition-dynamics.md	FPRD-D06; FPRD-D07; FPRD-D08; FPRD-D09; FPRD-D11	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D14 The delay profile of a finite positive relational recognition presentation assigns each variable occurrence its guaranteed consumed prefix length, collects the minima in a min-plus dependency matrix, and calls the presentation cycle-guarded when its zero-delay dependency graph is acyclic, equivalently when every dependency cycle has positive total delay.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m17-delay-graph-guardedness.md	FPRD-D02; FPRD-D06; FPRD-D07	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D15 A generalized Dyck-control recognition presentation consists of a finite alphabet of matching brackets and neutral symbols, a finite complete DFA on that alphabet, the finite state-pair equation system recognizing well-nested DFA runs, a distinguished union over final states, and a total letter-to-letter map whose decoded observation returns every suffix remaining after an encoded trace prefix.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m18-universal-dyck-control.md	FPRD-D06; FPRD-D09; FPRD-D11; FPRD-D12; FPRD-D14	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D16 A capacity-witnessed fixed-envelope recoding consists of a nonempty finite output alphabet Sigma, a positive block width m, a base j at least two, a finite matched source-bracket alphabet with at most j to the m pair types, one m-letter output block for each source bracket, and one distinct m-digit code for each matched pair; its envelope alphabet has symbols (polarity, current output, mate output, digit), matches by reversing polarity and the two output fields, reads the current output field, and replaces each source bracket by one length-m envelope block.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m19-fixed-envelope-recoding.md	FPRD-D15	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D17 The fixed-block boundary residual profile of an injective uniform code tau and a finite complete source controller records each source-state right language and its compiled boundary right language, their exact and bounded suffix-lift observations, shortest separation depths, the distinct proper code-prefix counts, raw and reachable compiled-control sizes, and, for a fixed-envelope witness, the digit capacity, envelope size, and block-width observation lag.	Definitiondefinition	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/m20-fixed-block-residual-profile.md	FPRD-D11; FPRD-D12; FPRD-D16	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D01 The deterministic FPRD foundation distinguishes four layers: semantic algebra, observational completion, presentation-generated dynamics, and readout-aware coordinate realization.	Definitiondefinition	Evidence and limits →	Models of computationFPRD foundations	docs/design/foundation.md	No recorded dependencies	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D02 Exact behavior maps preserve all remaining-suffix observations, whereas an observational abstraction preserves only a named interface such as inputs of length at most N, a test family, reachable inputs, or complete acceptance.	Definitiondefinition	Evidence and limits →	Models of computationFPRD foundations	docs/design/foundation.md	FPRD-D01	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D03 Extensional equivalence, presentation equivalence, and dynamic equivalence are distinct relations unless a bridge theorem proves an implication for a specified class of presentations and transformations.	Definitiondefinition	Evidence and limits →	Models of computationFPRD foundations	docs/design/foundation.md	FPRD-D01	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D04 A coordinate realization includes a coordinate domain, an injective state encoding, representations of each named action, and a uniform readout that recovers remaining-suffix behavior from a coordinate and an input word.	Definitiondefinition	Evidence and limits →	Models of computationFPRD foundations	docs/design/foundation.md	FPRD-D01; FPRD-D02	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D31 The FPRD v2 kernel consists of a separating bounded-observation tower; a set of presentations with a semantic map not assumed surjective; a target- and horizon-independent resource map; the set-valued finite-horizon feasible field as the primary object; constrained sections as core demands and unconstrained projections as derived core quantities; an eligibility gate requiring field form and a non-triviality witness; and an auxiliary register for retained rank- and limit-type descriptors. Compact sewing and escape are optional modules, not automatic consequences of admission to the kernel.	Definitiondefinition	Evidence and limits →	Models of computationFPRD foundations	docs/specs/fprd-v2-kernel.md	FPRD-D28	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D05 A finite polynomial recognition-action presentation consists of a finite PEG behavior as distinguished state and finitely many named unary polynomial actions built from the deterministic recognition-algebra constructors; its dynamic is the closure of their generated orbit with exact and bounded word observations.	Definitiondefinition	Evidence and limits →	Models of computationRepresentation dynamics	docs/design/foundation.md	FPRD-D01; FPRD-D03; FPRD-D04	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D06 A finite-observation recognition doctrine supplies an exact Hausdorff behavior algebra, finite bounded behavior algebras with compatible surjective restrictions and inverse-limit reconstruction, restriction-compatible recognition constructors, and uniform word readout from sufficiently deep coordinates.	Definitiondefinition	Evidence and limits →	Models of computationRepresentation dynamics	docs/design/foundation.md	FPRD-D01; FPRD-D02; FPRD-D04	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D07 A finite positive relational recognition presentation has a finite alphabet, finitely many variables with one equation each, a distinguished start variable, and terms generated by relational failure, identity, terminals, variables, commutative union, and relational sequence, with recursion interpreted by a selected least solution.	Definitiondefinition	Evidence and limits →	Models of computationRepresentation dynamics	docs/design/foundation.md	FPRD-D01; FPRD-D03; FPRD-D06	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D08 A finite uniquely labeled positive relational presentation is an M11 presentation in which every recursive call occurrence has a globally unique nonempty label, so its source, target, diagnostic path, and ordered right-success frames can be retained as presentation metadata.	Definitiondefinition	Evidence and limits →	Models of computationRepresentation dynamics	docs/design/foundation.md	FPRD-D03; FPRD-D07	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D24 For a finite controlled-Dyck presentation, balanced controller distance is the shortest length of a balanced word realizing a controller-state pair; raw arrival is the shortest prefix reaching a saved stack/controller coordinate; bounded semantic-class arrival is the minimum raw arrival inside one M24 future-signature class; and B_M is the maximum finite balanced distance whose source is start-reachable in the underlying controller DFA of the chosen presentation.	Definitiondefinition	Evidence and limits →	Algorithms and complexityResource bounds and finite observation	docs/specs/m29-controlled-access-calculus.md	FPRD-D21; FPRD-D23	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D25 For a controlled-Dyck presentation M, P_M is the finite saturation of controller states reachable by legal Dyck prefixes, E_M consists of the start state and the states immediately after an opener from P_M, B_M^ent is the maximum finite balanced distance sourced in E_M, regular languages are Dyck-trace equivalent when their intersections with the fixed generalized-Dyck language agree, and beta_D(L) is the minimum entry certificate over minimal regular controllers in that trace class. The balanced residual access gamma_D(L) is the maximum, over semantic right-residual classes restricted to balanced histories and tests, of the shortest balanced representative length. Entry-context access eta_D(L) takes the supremum of the analogous factor-residual access over the start and all legal histories ending immediately after an opener; it is exactly computable by finite completion-summary saturation in any controller presentation.	Definitiondefinition	Evidence and limits →	Algorithms and complexityResource bounds and finite observation	docs/specs/m30-dyck-trace-minimax.md	FPRD-D24	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D26 For a controlled-Dyck trace L and a nonnegative access budget b, the state-budget function nu_D(L;b) is the minimum number of states in a finite regular controller whose intersection with the fixed Dyck envelope is L and whose legal-entry balanced diameter is at most b, with value infinity when no such finite controller exists.	Definitiondefinition	Evidence and limits →	Algorithms and complexityResource bounds and finite observation	docs/specs/m31-entry-atlas-sewing-tradeoffs.md	FPRD-D25	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-D27 For every modulus m at least two, K_m is the one-bracket Dyck trace containing epsilon and the nonempty words whose terminal consecutive closer run has length m-1 modulo m; its state/access atlas records the nondominated controller-state and legal-entry-diameter pairs.	Definitiondefinition	Evidence and limits →	Algorithms and complexityResource bounds and finite observation	docs/specs/m32-terminal-descent-moduli.md	FPRD-D26	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add a self-contained definition page with examples, nonexamples, and dependency boundaries.
FPRD-C02 There exists a deterministic visibly pushdown language whose call, return, and local annotations forget to a language outside PEL. Forgotten-action deterministic visible projections are exactly CFL by FPRD-T91, while Kim and Park exhibit a linear context-free language outside PEG. The universal question is therefore resolved negatively.	Source verificationsource verified	Evidence and limits →	Automata and formal languagesFormal languages and scaffolding automata	docs/specs/post-m33-dyck-annotation-forgetting.md	FPRD-T91; FPRD-T92; FPRD-T93; FPRD-T94; FPRD-T95; FPRD-T96; FPRD-T97; FPRD-T98; FPRD-T99; FPRD-T88; FPRD-T100; FPRD-T101; FPRD-T102; FPRD-T103; FPRD-T104; FPRD-T105; FPRD-T106; FPRD-T107; FPRD-T108; FPRD-T109; FPRD-T112; FPRD-T164; FPRD-T165; Kim-Park 2026, Theorem 4.19	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Add the exact primary-source locator and keep source verification separate from proof of the FPRD consequence.
FPRD-C01 Every Ford-well-formed finite PEG presentation is eventually contractive under some equivalent weighted bounded-observation metric.	Conjectureconjecture	Evidence and limits →	Models of computationFPRD foundations	OPEN_PROBLEMS.md	FPRD-T04; FPRD-T05	FPRD governed claims ledgerReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Preserve open status; add known partial results, obstructions, and exact source locators.
PUB-fprd-program-textbook--siteauto-chapters-01-recognition-problem-tex-2 Proposition 1.2 . If $m\ne n$ , then $K_m$ and $K_n$ have different complete-match behavior.	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 1.2	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--lem-state-future Lemma 3.2 (State equality preserves every future). If $\delta^(s_0,u)=\delta^(s_0,u')$ , then for all $v\in\Sigma^*$ , $uv\in L(\mathcal A)\quad\Longleftrightarrow\quad u'v\in L(\mathcal A).$	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Lemma 3.2	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-03-finite-automata-tex-5 Proposition 3.5 (Regular languages are closed under Boolean operations). If $L_1$ and $L_2$ are recognized by finite DFAs, then so are $L_1\cap L_2$ , $L_1\cup L_2$ , and $\Sigma^*\setminus L_1$ .	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 3.5	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-04-residuals-nerode-tex-1 Theorem 4.1 (Residual recognition). The residual dynamic recognizes $L$ exactly.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 4.1	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-04-residuals-nerode-tex-2 Theorem 4.2 (Myhill–Nerode theorem, residual form). For a language $L\subseteq\Sigma^*$ , the following are equivalent: $L$ is recognized by a finite DFA; $L$ has finitely many left residuals; the relation $\equiv_L$ has finite index. When these conditions hold, the residual automaton is the unique minimal reachable DFA up to isomorphism.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 4.2	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-06-topological-spaces-tex-11 Proposition 6.11 . The composition of continuous maps is continuous.	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 6.11	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-07-products-compactness-tex-5 Theorem 7.5 (Finite intersection characterization). A space $X$ is compact if and only if every family of closed sets $\{C_i:i\in I\}$ with the finite intersection property has nonempty total intersection. The finite intersection property means $C_{i_1}\cap\cdots\cap C_{i_k}\ne\varnothing$ for every finite subfamily.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 7.5	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-07-products-compactness-tex-6 Theorem 7.6 . Let $X_n$ be a nonempty finite discrete space for every $n\in\N$ . Then $\prod_{n\in\N}X_n$ is compact.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 7.6	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-08-inverse-limits-tex-2 Proposition 8.2 (Universal property of the inverse limit). Let $Y$ be a set and suppose maps $f_n:Y\to X_n$ form a compatible cone: $\rho_{n+1,n}\circ f_{n+1}=f_n \qquad\text{for every }n.$ There is a unique map $f:Y\longrightarrow\varprojlim_nX_n$ such that $\pi_n\circ f=f_n$ for every coordinate projection $\pi_n$ . If the spaces are topological and every $f_n$ is continuous, then $f$ is continuous.	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 8.2	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-08-inverse-limits-tex-5 Proposition 8.5 (Closedness of compatibility). For continuous maps between Hausdorff spaces, the set of compatible threads is closed in $\prod_n X_n$ .	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 8.5	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-08-inverse-limits-tex-6 Theorem 8.6 (Finite discrete inverse-limit existence). Let every $X_n$ be a nonempty finite discrete space, with arbitrary bonding maps $\rho_{n+1,n}:X_{n+1}\to X_n$ . Then $\varprojlim_n X_n$ is nonempty.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 8.6	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-10-recursion-guarding-tex-2 Theorem 10.2 (Contraction fixed point). If $\mathcal B$ is complete and $F$ is a contraction, then $F$ has a unique fixed point. Starting from any $x_0$ , the iterates $x_{n+1}=F(x_n)$ converge to it.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 10.2	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-10-recursion-guarding-tex-3 Proposition 10.3 . If $F^r$ is a contraction on a complete metric space, then $F$ has a unique fixed point.	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 10.3	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-13-dyck-residuals-tex-1 Theorem 13.1 (Dyck residual classification). The residuals of $D_k$ are precisely $\{R_s:s\in\{1,\ldots,k\}^*\}\cup\{\varnothing\}.$ Distinct stacks give distinct residuals.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 13.1	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-16-information-complexity-tex-1 Theorem 16.1 (Fixed-cut state minimum). The minimum number of reachable cut states in a deterministic strict layered recognizer is the number of distinct rows of $C_{m,n}$ .	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 16.1	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-17-local-global-sewing-tex-2 Theorem 17.2 (Unique language from compatible cuts). Every compatible Boolean cut field determines a unique language $L_C=\{w:C_{\|w\|,0}(w,\eps)=1\}$ such that $C_{m,n}(u,v)=\mathbf 1[uv\in L_C]$ for every cut. Conversely, every language produces a compatible field.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 17.2	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--thm-global-plateau Theorem 18.1 (Global adjacent plateau). If $q_L(N)=q_L(N+1)$ for some $N$ , then $L$ is regular and $Q_N(L)$ is the state set of its minimal residual DFA.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 18.1	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-1 Proposition 19.1 (Basic access laws). For every language over a finite alphabet: $a_L(N)$ exists; $a_L(N)$ is nondecreasing in $N$ ; and if $L$ has a reachable $k$ -state DFA, then $a_L(N)\le k-1$ .	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 19.1	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-2 Theorem 19.2 (Observation-growth gap). If $L$ is nonregular, then $q_L(N)$ is strictly increasing and $q_L(N)\ge N+2$ for every $N\ge0$ . Equivalently, if $q_L$ ever fails to increase at an adjacent pair of horizons, then it remains bounded.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.2	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-3 Theorem 19.3 (Bounded access characterizes regularity). For a language $L$ , the following are equivalent: $L\text{ is regular} \quad\Longleftrightarrow\quad \sup_N a_L(N)<\infty \quad\Longleftrightarrow\quad a_L(N)\text{ is eventually constant}.$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.3	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-5 Theorem 19.5 (Exact access calculus for controlled Dyck presentations). For every finite controlled-Dyck presentation and every horizon $N$ , $a_{L_M}(N)=\max_{K\in S_N}\alpha_{M,N}(K).$ The quantity is effectively computable by finite balanced-distance relaxation, the stack recurrence above, and the finite controlled-Dyck shadow quotient. Moreover, $\boxed{ a_{L_M}(N)\le\max\{1,(N+1)B_M+N\}. }$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.5	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-6 Theorem 19.6 (The linear access certificate is pointwise sharp). For every horizon $N\ge0$ , some finite controlled-Dyck presentation attains the displayed upper bound. For every $N\ge1$ , one can arrange $B_M=2$ and $a_{L_M}(N)=3N+2=(N+1)B_M+N.$ The presentation is allowed to depend on $N$ .	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.6	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-8 Theorem 19.8 (Entry diameter is monotone under DFA quotient). For every such morphism, $B_{M'}^{\mathrm{ent}}\le B_M^{\mathrm{ent}}.$ Consequently the reachable minimal DFA of one fixed regular language has no larger entry diameter than any DFA recognizing that regular language.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.8	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-9 Theorem 19.9 (Dyck-trace minimax certificate). For every controlled-Dyck language inside the fixed envelope $D$ , $a_L(N)\le\max\{1,(N+1)\beta_D(L)+N\}.$ The number $\beta_D(L)$ depends only on $L$ and the selected Dyck envelope, not on one chosen regular controller.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.9	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-10 Theorem 19.10 (Balanced residual access lower bound). Every regular controller presenting $L$ satisfies $B_M^{\mathrm{ent}}\ge\gamma_D(L),$ and therefore $\beta_D(L)\ge\gamma_D(L).$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.10	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-11 Theorem 19.11 (Entry-context residual lower bound). Every regular controller presenting $L$ satisfies $B_M^{\mathrm{ent}}\ge\eta_D(L).$ Consequently $\boxed{ \beta_D(L)\ge\eta_D(L)\ge\gamma_D(L). }$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.11	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-12 Proposition 19.12 (Finite completion-summary saturation). For every controlled-Dyck language, $\eta_D(L)$ is exactly computable from any finite controller presentation by a terminating finite saturation.	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 19.12	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-13 Proposition 19.13 (Minimax sandwich stopping rule). If $b_M=\eta_D(L)$ , then the supplied presentation attains the global Dyck-trace minimum and $\beta_D(L)=\eta_D(L)=b_M.$ Both endpoints are computed by terminating finite procedures.	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Proposition 19.13	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-14 Theorem 19.14 (First-flip lower bound). Every regular controller presenting $L$ satisfies $B_M^{\mathrm{ent}}\ge\tau_D(L),$ and therefore $\beta_D(L)\ge\tau_D(L).$ Moreover, $\beta_D(L)=0$ exactly for $L=\varnothing$ and $L=D$ .	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.14	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-15 Theorem 19.15 (Punctured-threshold separation). For every $r\ge2$ , $\boxed{ \tau_D(L_r^\circ)=2 <\gamma_D(L_r^\circ) =\beta_D(L_r^\circ)=2r. }$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.15	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-16 Theorem 19.16 (First-component contextual separation). For every $r\ge3$ , $\boxed{ \tau_D(H_r)=2, \qquad \gamma_D(H_r)=4, \qquad \eta_D(H_r)=\beta_D(H_r)=2(r-1). }$ In particular, $\beta_D(H_r)>\gamma_D(H_r)$ for $r\ge4$ .	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.16	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-17 Theorem 19.17 (Terminal-descent sewing collapse). For the trace $K_3$ , $\boxed{\eta_D(K_3)=\beta_D(K_3)=4.}$ The three-state presentation is state-minimal but not access-optimal.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.17	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-18 Theorem 19.18 (Exact state/access frontier of $K_3$ ). The minimum size of a controller presenting $K_3$ is three, and $\nu_D(K_3;4)=6.$ The nondominated state/access pairs are exactly $\boxed{(3,6)\quad\text{and}\quad(6,4)}.$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.18	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-19 Theorem 19.19 (Terminal-descent family law). For every $m\ge2$ , $\eta_D(K_2)=\beta_D(K_2)=4, \qquad \eta_D(K_m)=\beta_D(K_m)=2(m-1)\quad(m\ge3).$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.19	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-20 Theorem 19.20 (Even positional construction). For every even $m$ , an $(m+2)$ -state controller presents $K_m$ at the optimal entry cost $\beta_D(K_m)$ . At $m=4$ it is isomorphic to the six-state controller displayed above.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.20	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-23 Theorem 19.23 (Exact access radius for square lengths). For every $m\ge0$ , $a_{L_{\square}}(2m)=(m+1)^2+1, \qquad a_{L_{\square}}(2m+1)=(m+1)^2+2.$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.23	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-fprd-program-textbook--siteauto-chapters-19-growth-access-tex-24 Theorem 19.24 (Exact depth-threshold profiles). Let $D_{k}^{\ge r}$ be the balanced words over $k$ bracket types whose stack height reaches at least $r\ge1$ . Then $q_{k,r}^{\ge}(N) =1+\sum_{h=0}^{N}k^h +\sum_{h=\max(0,2r-N)}^{r-1}k^h,$ $a_{k,r}^{\ge}(N)=\max(2r,N).$ For the complementary balanced language $D_k^{<r}$ that stays below $r$ , $q_{k,r}^{<}(N) =1+\sum_{h=0}^{\min(N,r-1)}k^h, \qquad a_{k,r}^{<}(N)=\max(1,\min(N,r-1)).$ The formulas remain unchanged when neutral letters are added.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Models of computationFPRD foundations	FPRD program textbook, Theorem 19.24	Theorem-level dependency graph not yet atomized	FPRD program textbookReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--thm-transfer-intro Theorem 1.1 (Real-time transfer). Let $L$ be recognized by a finite deterministic strict real-time multitape Turing machine with a fixed positive number of ordinary single-head work tapes. The machine receives one input symbol and makes one global transition per step, has no input-length advice, and performs no post-input computation. Then $L^R\in\mathsf{PEG}.$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Theorem 1.1	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--cor-epal-intro Corollary 1.2 . $\mathsf{EPAL}_2\in\mathsf{PEG}$ . Hence Conjecture 7 of Loff–Moreira–Reis is false.	corollaryreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Corollary 1.2	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--thm-lmr Theorem 2.1 (Loff–Moreira–Reis). For every language $K$ , $K\in\mathsf{PEG} \quad\Longleftrightarrow\quad K^R\text{ is decided by a scaffolding automaton}.$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Theorem 2.1	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--lem-fresh-initial Lemma 2.2 (Fresh initial state). Every finite scaffolding automaton $A$ has an equivalent finite scaffolding automaton $A^\star$ whose initial state occurs only before the first input symbol. The construction preserves acceptance of the empty word and every nonempty word without inserting an input step.	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Lemma 2.2	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--lem-stack-update Lemma 3.1 (Persistent-stack update). Suppose the old scaffold is strictly backward and satisfies $\mathsf{CleanEmpty}$ . After appending the node specified by the table, the decoded stack at $v^+$ is exactly the result of the selected stack operation. The new scaffold is strictly backward and again satisfies $\mathsf{CleanEmpty}$ .	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Lemma 3.1	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--thm-parallel-stacks Theorem 3.2 (Parallel persistent stacks). Every deterministic letter-synchronous finite-control machine with a fixed positive number $s$ of stacks, performing at most one push, keep, or pop per stack and per input symbol, is simulated exactly by a degree- $s$ , distance-two scaffolding automaton.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Theorem 3.2	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--lem-zipper Lemma 4.1 (Tape zipper). Every strict real-time $m$ -tape machine is simulated letter-for-letter by a finite-control machine with $2m$ stacks, performing at most one push, keep, or pop on each stack per input symbol.	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Lemma 4.1	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--thm-machine-scaffold Theorem 4.2 (Strict machine-to-scaffold simulation). Let $M$ be a strict real-time $m$ -tape machine, $m\geq 1$ . There is a scaffolding automaton $A_M$ of degree $2m$ and distance two such that $L(A_M)=L(M).$ After every input prefix, the two persistent stacks associated with each tape, together with its scanned symbol in finite control, reconstruct the source tape exactly. In particular, the reconstructed and source tapes agree at every integer offset from the head, including their implicit blank tails, and the source control states agree.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Theorem 4.2	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--thm-transfer Theorem 5.1 . If $L$ is recognized by a strict real-time multitape machine, then $L^R\in\mathsf{PEG}.$ Equivalently, $\mathsf{rev}(\mathsf{RT\text{-}MTTM})\subseteq\mathsf{PEG}.$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Theorem 5.1	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--lem-strict-online Lemma 5.2 (Strict real time is online constant time). Every language recognized by a strict real-time multitape machine belongs to $\mathsf{Online}(O(1))$ in the sense of Loff–Moreira–Reis Definition 22.	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Lemma 5.2	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--cor-proper Corollary 5.3 (Proper class containment). $\mathsf{rev}(\mathsf{RT\text{-}MTTM})\subsetneq\mathsf{PEG}.$	corollaryreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Corollary 5.3	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--thm-classical-pal-interface Theorem 6.1 (Classical palindrome interface (classical, imported)). There are finite $m,Q,T$ and a total strict real-time $m$ -tape machine $P$ such that, for every $x\in\{0,1\}^*$ , the state reached after exactly $\|x\|$ transitions is accepting if and only if $x=x^R$ . The statement includes $x=\varepsilon$ : the initial state is accepting.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Theorem 6.1	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--lem-epal-identity Lemma 6.2 (Even-palindrome identity). For every binary word $x$ , $x\in\{ww^R:w\in\{0,1\}^*\} \quad\Longleftrightarrow\quad x=x^R\ \text{ and }\ \|x\|\text{ is even}.$ The equivalence includes $x=\varepsilon$ .	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Lemma 6.2	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--lem-epal-reversal Lemma 6.3 (Reversal invariance). $(\mathsf{EPAL}_2)^R=\mathsf{EPAL}_2.$	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Lemma 6.3	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--lem-parity-product Lemma 6.4 (Parity product). Suppose a strict real-time machine $P$ reports after every prefix $x$ , including the empty prefix, whether $x=x^R$ . Then a strict real-time machine $P_{\mathrm{even}}$ recognizes exactly $\mathsf{EPAL}_2$ .	lemmareleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Lemma 6.4	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--thm-epal Theorem 6.5 . The even-length binary palindrome language has a parsing expression grammar: $\boxed{\mathsf{EPAL}_2=\{ww^R:w\in\{0,1\}^*\}\in\mathsf{PEG}.}$	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Theorem 6.5	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-stacks-scaffolding--siteauto-main-tex-19 Corollary 6.6 . Conjecture 7 of Loff, Moreira, and Reis [4] is false.	corollaryreleased publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Real-time multitape languages transfer to parsing expression grammars, Corollary 6.6	Theorem-level dependency graph not yet atomized	Real-time multitape languages transfer to parsing expression grammarsReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-gapcvp-parameter-retuning--thm-main Theorem 1 (Retuned direct reduction). For every fixed real number $c$ with $0\le c<1/28$ , there is a deterministic polynomial-time mapping that assigns to each $3$ SAT formula $\varphi$ a nonsingular matrix $B\in\mathbb Z^{n\times n}$ , a target $t\in\mathbb Z^n$ , and a radius $r\in\mathbb Q_{>0}$ such that $\begin{aligned} \varphi\text{ satisfiable} &\implies \operatorname{dist}_2(t,L(B))\le r,\\ \varphi\text{ unsatisfiable} &\implies \operatorname{dist}_2(t,L(B))>n^c r. \end{aligned}$ Consequently, $\mathsf{GapCVP}^{(2)}_{n^c}$ is NP-hard under deterministic polynomial-time many-one reductions. The same construction gives approximation factor $n^{2c}$ for binary nearest codeword and binary syndrome decoding.	theoremreleased publication statement; not independently promoted by this index	Evidence and limits →	Algorithms and complexityGapCVP parameter retuning	Parameter retuning in the direct reduction for the Euclidean closest vector problem: 1/30-hardness and a 1/28 retained-parameter frontier, Theorem 1	Theorem-level dependency graph not yet atomized	Parameter retuning in the direct reduction for the Euclidean closest vector problem: 1/30-hardness and a 1/28 retained-parameter frontierReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-gapcvp-parameter-retuning--prop-binary-soundness Proposition 2 (Retuned binary soundness). Under the exact conditions above, if equation (10) has a solution $x$ satisfying $\operatorname{wt}(x)\le4M^{2c}R,$ then $\varphi$ is satisfiable.	propositionreleased publication statement; not independently promoted by this index	Evidence and limits →	Algorithms and complexityGapCVP parameter retuning	Parameter retuning in the direct reduction for the Euclidean closest vector problem: 1/30-hardness and a 1/28 retained-parameter frontier, Proposition 2	Theorem-level dependency graph not yet atomized	Parameter retuning in the direct reduction for the Euclidean closest vector problem: 1/30-hardness and a 1/28 retained-parameter frontierReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-gapcvp-parameter-retuning--cor-binary Corollary 3 . Binary nearest codeword and binary syndrome decoding are NP-hard to approximate within $n^{1/15}$ under deterministic polynomial-time many-one reductions.	corollaryreleased publication statement; not independently promoted by this index	Evidence and limits →	Algorithms and complexityGapCVP parameter retuning	Parameter retuning in the direct reduction for the Euclidean closest vector problem: 1/30-hardness and a 1/28 retained-parameter frontier, Corollary 3	Theorem-level dependency graph not yet atomized	Parameter retuning in the direct reduction for the Euclidean closest vector problem: 1/30-hardness and a 1/28 retained-parameter frontierReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-lmr Theorem 2.7 (Loff–Moreira–Reis [7, Theorem 16, p. 21] ). A language $L\subseteq\Sigma^*$ is in $\mathsf{PEG}$ if and only if its reversal $L^R$ is decided by some scaffolding automaton. In particular, if a finite scaffolding automaton decides $L$ , then $L^R$ is recognized by a total complete-match PEG.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 2.7	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-context Theorem 5.4 (Occurrence-context factorization). After reading $a$ , the new longest even suffix $P'$ and reversed context $\ell'$ are exactly $(P',\ell')= \begin{cases} (aPa,\ell_0), &\ell=a\ell_0,\\[1mm] (aD_a(P)a,\omega_a(P)\ell), &\ell\notin a\Sigma^,\ D_a(P)\text{ defined},\\[1mm] (\epsilon,aP\ell), &\ell\notin a\Sigma^,\ D_a(P)\text{ undefined}. \end{cases}$ Moreover, $S\in\mathsf{EPAL}_2\quad\Longleftrightarrow\quad \ell=\epsilon.$	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 5.4	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--lem-chunk Lemma 5.6 (chunk recursion). Let $P$ be a nonempty even palindrome with suffix link $L=\lambda(P)$ , bridge $b$ , and bridge block $B_P$ . Then $D_b(P)=L,\qquad \omega_b(P)=B_P,$ and for every letter $a\ne b$ , $D_a(P)=D_a(L),\qquad \omega_a(P)=\omega_a(L)\,b\,B_P,$ with the same definedness on both sides (for $L=\epsilon$ the right-hand sides are undefined).	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Lemma 5.6	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-mirror Theorem 6.1 (Typed mirror receipt). Suppose an update selects a longest suffix occurrence $Q$ of the old input that is immediately preceded by $a$ , and creates $Y=aQa$ . Then $\lambda(Y)= \begin{cases} aD_a(Q)a,&D_a(Q)\text{ defined},\\ \epsilon,&D_a(Q)\text{ undefined}. \end{cases}$ The palindrome $\lambda(Y)$ has a complete occurrence in the old input and a pre-existing birth record. One old typed pointer can therefore install the suffix-link field of $Y$ .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 6.1	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--prop-endpoint Proposition 6.2 (Endpoint-only escape). For $m\ge2$ , let $S_m=0^{2m}1100$ . Its longest even-palindromic suffix is $001100$ and the suffix link is $00$ . The mirror occurrence of $00$ ends at the historical prefix $0^{2m}$ , whose longest suffix-link chain is $0^{2m},0^{2m-2},\ldots,00,\epsilon.$ Recovering $00$ from that untyped endpoint takes exactly $m-1$ suffix-link steps.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Proposition 6.2	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--prop-type-birth Proposition 7.1 (Future transition after type birth). For $r\geq1$ , put $A_r=0^{2r}$ , $P_r=A_r11A_r, \qquad T_r=1A_r1.$ Then $D_1(P_r)=A_r$ , so the transition selected by $1$ has target $T_r=1D_1(P_r)1$ . The type $P_r$ is born by the end of the prefix $P_r$ , but $T_r$ is not a factor of that prefix. It becomes available only in a later history such as $P_r1P_r$ . Thus the immutable first-birth record of $P_r$ cannot already point to every future transition out of $P_r$ .	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Proposition 7.1	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--prop-anchor-dependence Proposition 7.2 (Anchor dependence). For the fixed one-letter block $w=0$ and anchors $A_r=0^{2r}$ , $\lambda(0A_r0)=A_r.$ Consequently the same raw rope cell $0$ requires infinitely many distinct typed receipt targets as its anchor varies.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Proposition 7.2	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--prop-bridge-chase Proposition 7.3 (Bridge-chasing escape). For $r\geq1$ and $d\geq0$ , let $P_{r,d}=A_r(11A_r)^d, \qquad A_r=0^{2r}.$ These are even palindromes. For $d>0$ , $\lambda(P_{r,d})=P_{r,d-1} \quad\text{and}\quad b_{P_{r,d}}=1,$ whereas the bridge of $A_r$ is $0$ . The search for the $0$ -transition from $P_{r,d}$ therefore makes exactly $d$ failed bridge tests before reaching the variable target $A_r$ .	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Proposition 7.3	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--lem-shadow Lemma 8.3 (Bridge-shadow identity). Suppose $H=RbP$ , $\lambda(H)=P$ , and $P$ is the longest even-palindromic suffix of $w^RP$ . If $w=\epsilon$ , or if the first letter of $w$ differs from $b$ , then $\Lambda(H,w)=\Lambda(P,w).$	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Lemma 8.3	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--lem-mirror-prefix Lemma 8.5 (Mirror-prefix lemma). Let $Y$ be a nonempty even palindrome and $c$ a letter. If $D_c(Y)$ is defined, the final palindrome of $\Pi(cD_c(Y)c,\omega_c(Y))$ is $F=Yc\,\omega_c(Y)$ , and $\lambda(F)=Y$ . If $D_c(Y)$ is undefined, the final palindrome of $\Pi(\epsilon,cY)$ is $F=YccY$ , and $\lambda(F)=Y$ . In both cases the last receipt of the rope is the anchor’s source $Y$ itself.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Lemma 8.5	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--cor-anchor-links Corollary 8.6 . With $Y=LbB$ as above and $c\ne b$ : if $D_c(Y)$ is defined, write $L=ZcD_c(L)$ ; then $\lambda(ZcL)=L$ . If $D_c(Y)$ is undefined, then $\lambda(LccL)=L$ .	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Corollary 8.6	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-ropes Theorem 8.7 (Receipt-rope recurrence). Let $\rho_b$ be the header receipt of $\Omega_b(Y)$ . For $c\ne b$ , $\Omega_c(Y)= \begin{cases} \text{undefined},&\Omega_c(L)\text{ undefined},\\ \Omega_c(L)\cdot(b,\rho_b)\cdot\operatorname{body}\Omega_b(Y),&\text{otherwise}, \end{cases}$ and, whenever $Z_c(Y)$ is defined, $Z_c(Y)=Z_c(L)\cdot(b,\rho_b)\cdot\operatorname{body}\Omega_b(Y).$	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 8.7	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-closure Theorem 8.9 (Full abstract receipt closure). Over a fixed alphabet, a fresh occurrence record, its complete direct/reset rope table, and the updated live context are obtained from old receipts by a fixed number of head, tail, singleton, inject, and catenation operations.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 8.9	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-compiler Theorem 10.2 (Simultaneous batch compiler). Every transducer of Definition 10.1 whose initial record graph is fixed, finite, immutable, and pointer-closed compiles exactly and letter-synchronously into a finite scaffolding automaton of degree $d=\max\{1,s+Aq\}$ and observation/update distance at most $r+1$ . The source transition uses only bounded labelled rooted unfoldings and does not branch on pointer identity; acceptance is in finite control.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 10.2	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-kt Theorem 11.1 (Kaplan–Tarjan [6, Theorem 5.1, p. 592] ). There is a purely functional regular catenable steque whose push, pop, inject, and catenation operations take worst-case constant time and preserve the representation.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 11.1	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-staging Theorem 11.2 (Opaque-atom staging). Let a strict purely functional sequence operation, parameterized by a base element type $A$ , use finite constructor tests and a worst-case bounded number of $\mathsf{car}$ , $\mathsf{cons}$ , and $\mathsf{cdr}$ operations. It may inspect the representation’s own pairs, buffers, and recursive child structures, but it may not inspect, compare, hash, traverse, or identity-test values of the base type $A$ . With new elements replaced by symbolic atoms, its branch and fresh structural allocation graph are determined by old structure alone. The graph is finite, acyclic, and uniformly bounded. A symbolic element may denote a designated fresh payload record, and a fresh owner may store a fixed tuple of resulting roots. Adding those payload and owner slots to the structural skeleton forms one bounded simultaneous batch. Substitution is sound even when a payload field points back to the owner and creates a cycle.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 11.2	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-whole-client-normalization Theorem 11.3 (Whole-client bounded symbolic normalization). Fix a finite immutable record signature. Let one deterministic strict client transition have a uniform bound on calls, constructor tests, loops, recursion, and retained fresh records. Suppose it observes only finite tags, finite nonpointer data, old records reached by bounded rooted addresses, and constructors of ordinary fresh structural records already known to its symbolic evaluation. It neither observes pointer identity or allocation state nor inspects an unresolved opaque atom, designated fresh payload, or designated fresh owner. Suppose also that every old pointer later read or retained has a bounded rooted provenance, temporary control packages do not escape, and unreachable failure leaves are completed by a fixed finite plan. Then the whole client normalizes to one bounded simultaneous record-transducer plan determined only by finite control, the input letter, and a bounded labelled unfolding of the old roots.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 11.3	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--thm-direct Theorem 12.2 (Direct palindrome construction). For a persistent sequence implementation satisfying the contract of Section 11.1, the receipt construction recognizes the binary even-palindrome language with a finite scaffolding automaton. If the complete client has source-record bounds $(A_,q,R_)$ , it has degree $\max\{1,2+A_q\}$ and distance at most $R_+1$ .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Theorem 12.2	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-direct-palindrome-scaffolding--cor-peg Corollary 12.3 (Direct PEG route). Under the sequence contract in Theorem 12.2, the construction yields a PEG for $\mathsf{EPAL}_2$ .	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Automata and formal languagesScaffolding automata and PEGs	Persistent receipts and simultaneous records: a direct scaffold for even palindromes, Corollary 12.3	Theorem-level dependency graph not yet atomized	Persistent receipts and simultaneous records: a direct scaffold for even palindromesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-lmr Theorem 2.3 (Loff–Moreira–Reis [12, Theorem 16, p. 21] ). A language $L$ is in $\mathsf{PEG}$ if and only if $L^R$ is decided by some scaffolding automaton.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 2.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-full-abstraction Theorem 3.6 (Future-context full abstraction). For rooted degree- $d$ scaffolds over one working alphabet: equal unfoldings, equality of every finite rooted neighborhood, and deterministic labelled bisimilarity are equivalent; unfolding equivalence is preserved by every common scaffold row and hence by every common continuation; and unequal unfoldings are distinguished after finitely many steps by a distance-two unary scaffold continuation.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 3.6	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--cor-right-congruence Corollary 3.7 . Both $\equiv_{\mathrm s}$ and $\equiv_{\mathrm f}$ are stable under a common right suffix of rows.	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Corollary 3.7	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-minimal-figure Theorem 3.8 (Minimal residual figure). Every finite rooted scaffold has a canonical finite behavioral figure $M(\mathcal U)$ whose states are $\mathcal R(\mathcal U)$ , whose root is $\mathcal U_{\epsilon}$ , and whose port $i$ sends $\mathcal U_p$ to $\mathcal U_{pi}$ when defined. The physical reachable graph maps onto $M(\mathcal U)$ by a surjective functional bisimulation. Two rooted scaffolds are behaviorally equivalent if and only if their minimal figures are isomorphic.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 3.8	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-chain Theorem 4.1 (Generated-chain theorem). Let $S$ be a legally generated chronological scaffold with current top $t$ . Its physical graph and its minimal figure are weakly acyclic. The nodes reachable from $t$ , and the states of the minimal figure, are totally ordered by reachability. Conversely, every finite behaviorally reduced labelled rooted reachability-chain port graph, with an optional least unlabelled edgeless state, has an exact legal scaffold presentation at some finite distance.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 4.1	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--cor-rtrivial Corollary 4.2 (Regressive navigation). After adjoining a least missing sink to a generated minimal figure and numbering its chain, every port acts regressively. Its port-word transition monoid is a submonoid of the full regressive transformation monoid and is $\mathcal R$ -trivial.	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Corollary 4.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-guarded-fold Theorem 5.2 (Guarded folds generate bisimilarity). On every finite weakly acyclic deterministic labelled partial port graph, the equivalence generated by elementary guarded folds is exactly the greatest bisimulation equivalence.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 5.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--cor-static-nf Corollary 5.3 (Static normal form). Orient each guarded fold toward its quotient. The system terminates and is confluent up to rooted labelled port-graph isomorphism. Its normal form is the minimal residual figure.	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Corollary 5.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--prop-semantic-sewing Proposition 6.1 (Semantic causal sewing). Two strongly equivalent length- $n$ histories are connected by at most $n$ semantic step replacements.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Proposition 6.1	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-extension Lemma 6.2 (One-state extension dichotomy). Suppose two such fresh roots have the same behavior. If the behavior is not already a state of $Q$ , their missing and $\mathsf{SELF}$ positions agree, and their old target agrees at every other port. If the behavior is an old state $q\in Q$ , each row is a guarded unroll of $q$ : it copies every missing and strict-successor port, while a self-loop of $q$ may target either $q$ or the fresh variable $X$ .	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 6.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-canonical-exists Lemma 6.6 (Canonical history exists). Every realizable finite strong trace has a unique lexicographically least history. It is obtained from left to right by appending the least row that realizes the next figure over the already chosen prefix.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 6.6	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-reduced-prefix Lemma 6.7 (Reduced live prefixes). Every root-reachable component at every prefix of a SELF-eager history is behaviorally reduced.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 6.7	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-causal-nf Theorem 6.8 (Convergent strong-trace presentation). For fixed finite $d,k,\Gamma$ , oriented endpoint detours and literal guarded unrolls form a terminating confluent graph-local presentation of strong equivalence. Two histories have the same strong trace if and only if they reduce to the same SELF-eager history. A length- $n$ history normalizes in at most $dn$ row moves when $U$ counts as one whole-row move.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 6.8	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-support Theorem 7.3 (Replay invariance). Suppose an equal-length history $H'$ agrees with $H$ on the labels of $R_H$ and on every descriptor cell in $C_H$ . Then the final rooted concrete scaffolds of $H$ and $H'$ are identical. Moreover $R_{H'}=R_H$ and $C_{H'}=C_H$ .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 7.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-garbage-cube Theorem 7.5 (Causal-garbage cube). Every garbage fibre is the Cartesian product of the finite value sets of its unsupported label and descriptor coordinates. Simultaneous $G$ factors into single-coordinate changes. Fill a triangle for three values of one coordinate and a square for changes of two distinct coordinates. The resulting path two-complex is simply connected.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 7.5	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-two-row Theorem 8.1 (Unexposed two-row sewing). Fix $d$ , $k\geq1$ , and a behaviorally reduced old prefix. Consider two unexposed acyclic two-row extensions with the same consumer label, the same valid nonmissing path-demanding consumer coordinates, and literal agreement as missing or $\mathsf{SELF}$ on every other consumer coordinate. If corresponding path demands have the same final old residual behavior, the blocks are connected by endpoint detours, causal-garbage moves, continuation-aware routing crossings, and dedicated relation exchanges. Every common suffix is preserved.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 8.1	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-relay Lemma 8.3 (Unique relay). If consecutive states $u>v$ of that final reduced chain are carried at nonadjacent construction times, the supported descriptor segment between them is, up to endpoint detours at a recursive bottom, $\epsilon\,0\,0\cdots0.$	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 8.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-unary-final Theorem 8.4 (Unary final presentation). Fix a length $n$ , a nonempty finite working alphabet, degree one, and distance one. Two histories have the same final behavioral figure if and only if they are connected by endpoint detours, absorption, causal-garbage changes, and temporal carrier slides. Every final fibre contains one literal right-packed normal form determined by $n$ and the final figure.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 8.4	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--prop-carrier-lattice Proposition 8.5 (Carrier-surgery CAT(0) geometry). For a fixed unary distance-one final figure and history length, the temporal slide graph on strong/garbage chambers is the undirected Hasse graph of the order-ideal lattice of the rectangle above. Filling every Boolean family of independent slides gives the CAT(0) cube complex $X(P)$ . If the rectangle has $r$ rows and $c$ columns, then the complex has $rc$ hyperplanes, dimension $\min(r,c)$ , edge distance $d(t,s)=\sum_{i=1}^{r}\|t_i-s_i\|,$ and coordinatewise-median carrier tuples. In particular, its diamond-filled two-skeleton is simply connected.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Proposition 8.5	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-batch Theorem 9.2 (Simultaneous-batch compiler; fixed virtual initialization). Every bounded simultaneous record transducer with parameters $(s,A,q,r)$ , one fixed finite immutable pointer-closed initial graph, and plans determined by finite control, the input letter and bounded labelled rooted unfoldings has an exact letter-synchronous native scaffolding-automaton implementation of degree $d=\max\{1,s+Aq\}$ and observation/update distance at most $r+1$ . Physical pointer identity is not an observable test; fixed initial records are virtual finite decoder data.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 9.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-staging Theorem 9.4 (Opaque-element staging). Let a strict purely functional sequence operation: use only finite constructor tests and a worst-case bounded number of fixed-arity constructor, projection, and pointer steps; treat elements of its base type as opaque atoms, with no equality, hashing, pattern match, identity test, or traversal through a base element, while permitting inspection of the representation’s own pairs, buffers, and recursive child structures; and return a fixed finite tuple of roots and removed elements. For fixed old roots and symbolic new elements, its control branch and fresh structural allocation graph are determined independently of the symbolic values. If one symbol denotes a fresh owner that stores a resulting sequence root, the owner and structural graph form one bounded simultaneous batch and compile by Theorem 9.2 .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 9.4	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--prop-feedback Proposition 9.5 (Feedback programs compile). Let a persistent program, at each input letter, start from a fixed number $s$ of roots and perform a fixed finite composition of strict purely functional sequence operations satisfying the hypotheses of Theorem 9.4 on opaque pointer elements. In addition it may create a bounded number of fresh fixed-arity records whose layouts are determined by the same old observation and whose fields target old records, resulting sequence roots, or records in that fresh finite batch. At most one designated fresh owner’s pointer is stored as a base-type element of those sequences. Suppose the complete per-letter computation, including those sequence operations, follows old logical fields only within some fixed dereference radius $r$ . Then the program is a bounded simultaneous record transducer, and it has an exact letter-synchronous implementation by a finite scaffolding automaton of degree $s+Aq$ and distance at most $r+1$ for suitable finite $A,q$ .	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Proposition 9.5	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--prop-rectangles Proposition 3.1 (Corrected carrier rectangles). For an ordinary missing or base terminal and $m\ge1$ , carrier placements are the ideals of $[m-1]\times[n-m]$ . For a recursive terminal, $m=0$ is a singleton; for $m\ge1$ , placements are the ideals of $[m-1]\times[n-m-1]$ .	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Proposition 3.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-carrier-cube Theorem 3.2 (Carrier cube). Filling every Boolean family of independent temporal slides gives the rooted CAT(0) order-ideal cube of Proposition 3.1 . For a rectangle with $r$ rows and $c$ columns it has $rc$ hyperplanes, dimension $\min(r,c)$ , edge distance $d(t,s)=\sum_{i=1}^{r}\|t_i-s_i\|,$ and coordinatewise-median carrier tuples.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 3.2	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--lem-normal Lemma 4.1 (Unique quotient normal class). Every final fibre has one irreducible garbage class, the right-packed class $\mathsf{RP}_n(F)$ . Hence oriented effective horizontal rewriting terminates and is object-confluent modulo garbage.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Lemma 4.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--lem-collapse Lemma 4.2 (Garbage-trivial collapse). If a legal $D$ or $A$ edge is trivial modulo garbage, it equals one descriptor garbage edge. If a legal $T$ edge is trivial modulo garbage, it equals a two-descriptor garbage path; the two orders agree by the garbage square.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Lemma 4.2	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--lem-descriptor Lemma 5.1 (Descriptor-support monotonicity). For every effective oriented horizontal move $R$ , $\operatorname{DescSupp}(RH)\subseteq\operatorname{DescSupp}(H).$ Every coinitial horizontal/garbage-descriptor branching is therefore a strict naturality square modulo garbage.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Lemma 5.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--prop-captures Proposition 5.2 (Support captures). The only mixed branches without a strict horizontal residual are $A/GL(0),\qquad T/GL(0),\qquad T/GL(-1).$ They require no new generator: the garbage-first path returns along the inverse garbage edge and then takes the original reduction, so $G;G^{-1};R=R$ by vertical cancellation.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Proposition 5.2	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-hexagon Theorem 6.1 (A/T hexagon). The following paths have exactly the same raw physical endpoint: $A_y;A_x = T_{yz};A_x;A_y;D_z.$ No garbage quotient is used in this equation.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 6.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-local Theorem 7.1 (Local cell classification). Every coinitial effective horizontal pair is aspherical, Peiffer/cubical, D/A triangular, A/T hexagonal, or D/T derived. Every horizontal/garbage pair is a strict naturality square or one of the three cancellation captures. Every garbage-trivial horizontal edge has a collapse cell from Lemma 4.2 .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 7.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-main Theorem 8.1 (Uniform guarded path presentation). Fix a finite alphabet and history length in degree one and distance one. In each final behavioral fibre, all parallel paths generated by endpoint detours, literal absorptions, causal-garbage changes, and temporal slides are presented by the following length-independent guarded cell families: garbage same-coordinate triangles and different-coordinate squares; collapse cells for garbage-trivial D/A/T edges; strong Peiffer squares and the same-row D/A triangle; carrier-cube and inessential temporal Peiffer squares; strict mixed naturality squares; and the bounded A/T hexagon. The three support captures use vertical cancellation, and D/T is derived.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 8.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.