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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260 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
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.
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-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-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-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-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-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.
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.