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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50 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-RW-1a In the occurrence-labelled rewriting system $R=\{aa\to a\}$ , the complete reductions $a^n\Rightarrow^*a$ are in bijection with the $(n-1)!$ orders in which the original gaps are deleted. After identifying consecutive contractions with disjoint residual supports, the equivalence classes are the $C_{n-1}$ planar full binary trees on $n$ leaves.	Theorem · exact enumerationProved in the stated occurrence-labelled model	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Enumeration and binary-tree proof	No recorded dependencies	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-1aReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Compare the classification with standard trace-monoid and associahedral formulations.
CRD-RW-1b After adjoining the contextual overlap move $((AB)C)\leftrightarrow(A(BC))$ to the disjoint-square moves, every pair of complete occurrence-labelled reductions $a^n\Rightarrow^*a$ is connected.	Theorem · connectednessProved for complete reductions in the stated model	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Rotation connectedness proof	CRD-RW-1a	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-1bReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Determine which higher cells are required for coherence beyond connectedness.
CRD-RW-2a For complete occurrence-labelled reductions of $a^4$ , the graph generated by disjoint and overlap moves is the six-cycle $C_6$ . Contracting its unique disjoint-scheduling edge gives the five-cycle $C_5$ , the rotation graph of the five planar binary trees on four leaves.	Proposition · finite classificationProved by complete enumeration and edge classification	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Six-history enumeration	CRD-RW-1a; CRD-RW-1b	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-2aReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Retain the raw and schedule-quotient graphs as distinct presentations in later coherence arguments.
CRD-RW-2b The unique loop in the arity-four transformation graph survives backtrack cancellation and residual-interchange squares. In the fixed presentation, attaching one 3-cell along the raw hexagon—or equivalently along the quotient pentagon—is necessary and sufficient to fill it.	Proposition · coherence cellProved for the fixed arity-four component	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Loop and cell-attachment proof	CRD-RW-2a	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-2bReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Test whether the arity-four filler participates in a uniform all-arity presentation.
CRD-RW-3a The five-leaf binary-tree rotation complex $K_5$ has 14 vertices, 21 edges, six contextual pentagons, and three residual-interchange squares. Its nine faces are indexed by the proper contiguous clusters of the five leaves.	Theorem · cell-complex classificationProved for the schedule-quotient five-leaf complex	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Face enumeration and completeness proof	CRD-RW-2b	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-3aReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Generalize the cluster-indexing argument without assuming the five-leaf enumeration.
CRD-RW-3b Every edge of $K_5$ lies in exactly two native faces. Removing one pentagon leaves a collapsible complex; consequently $K_5\simeq S^2$ , with $H_1(K_5)=0$ and $H_2(K_5)\cong\mathbb Z$ .	Theorem · homotopy typeProved by an explicit elementary-collapse certificate	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Edge-incidence and collapse proof	CRD-RW-3a; CRD-RW-F5	Comparative Recognition Dynamics checkpoint and claim record · CRD-RW-3bReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Investigate whether comparable collapse certificates exist in higher associahedral dimensions.
CRD-RW-F1 In the terminating confluent system $aa\to a$ , swapping consecutive disjoint redex contractions does not connect all complete reductions with the same source and normal-form endpoint: the two reductions of $a^3$ begin at overlapping redexes and lie in different square classes.	Counterexample · failed completeness claimRefuted in the stated occurrence-labelled model; repaired by adding contextual overlap moves	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Self-contained counterexample and repair	CRD-RW-1a; CRD-RW-1b	Comparative Recognition Dynamics claim ledger · CRD-RW-F1; failure ledger · CRD-F-5Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Retain the counterexample as the gate against square-only history calculi; audit the all-arity overlap calculus before any coherence claim.
CRD-RW-F2 For $aa\to a$ , the sequence of underlying words is too coarse to recover redex independence or overlap: every complete occurrence-labelled reduction of $a^n$ projects to the same chain $a^n\to a^{n-1}\to\cdots\to a$ .	Representation obstructionRepresentation doctrine refuted for history classification	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Projection obstruction and replacement representation	CRD-RW-1a	Comparative Recognition Dynamics claim ledger · CRD-RW-F2; failure ledger · CRD-F-6Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Keep occurrence, redex-position, and residual-provenance data whenever the question concerns independence of histories.
CRD-RW-F3 For complete occurrence-labelled reductions of $a^4$ under $aa\to a$ , the graph generated by disjoint-square and contextual-overlap moves is connected but has $\pi_1\cong H_1\cong\mathbb Z$ ; connectedness therefore does not establish coherence.	Negative result · coherence obstructionConnectedness-to-coherence inference refuted; one higher cell repairs the fixed component	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Hexagon/pentagon loop and higher-cell repair	CRD-RW-2a; CRD-RW-2b	Comparative Recognition Dynamics claim ledger · CRD-RW-F3; failure ledger · CRD-F-7Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Require an explicit higher-dimensional coherence criterion and audit it beyond the fixed arity-four component.
CRD-RW-F4 Disjoint leaf intervals are not a complete test for independent contextual rotations: at five leaves, an inner rotation on $((AB)C)$ commutes with an outer rotation $((XD)E)\to X(DE)$ even though the inner leaf support is strictly contained in the outer support.	Counterexample · definition obstructionLeaf-support criterion refuted; active-constructor support survives	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Nested interchange witness and finite graph audit	CRD-RW-2b; CRD-RW-3a	Comparative Recognition Dynamics claim ledger · CRD-RW-F4; failure ledger · CRD-F-9Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use residual commutation or disjoint active-constructor support when enumerating interchange squares at five leaves and beyond.
CRD-RW-F5 For the five-leaf square–pentagon complex, the calculation $H_1=0$ cannot by itself establish $\pi_1=0$ , because first homology records only the abelianization of the fundamental group.	Failed proof methodHomology-only proof withdrawn; conclusion recovered by an explicit collapse certificate	Evidence and limits →	Logic, semantics, and rewritingComparative Recognition Dynamics · rewriting histories	Proof-method boundary and collapse repair	CRD-RW-3a; CRD-RW-3b	Comparative Recognition Dynamics claim ledger · CRD-RW-F5; failure ledger · CRD-F-10Reviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Use the explicit elementary-collapse certificate as the proof; retain the homology calculation only as corroboration.
FPRD-D40 A finite adaptive work history consists of independent source cells, well-founded derived cells computed by terminating deterministic oracle procedures over earlier cells, and a terminating deterministic observer. Each execution records an ordered finite query transcript, including repeated queries; its queried-cell set is the support of that transcript. Replay support is the least set containing the observer's queried cells and closed under the queried-cell sets of every supported derived cell.	Definition · execution-specific dependency modelDefined self-containedly and exercised by an exhaustive finite model	Evidence and limits →	Logic, semantics, and rewritingAdaptive replay slices	Adaptive work-history model	No recorded dependencies	Adaptive replay-slice theoremReviewed 2026-10-05No documented external or specialist review of this FPRD result is recorded.	Compare the model more closely with dynamic slicing, provenance traces, and self-adjusting computation before making any priority claim.
FPRD-T150 In every finite adaptive work history, changing only source cells outside one execution's replay support preserves every supported derived value and ordered query transcript, the observer result and ordered query transcript, and the replay support itself. The support is the least trace-closed certificate. At a downward-closed history cut, later demands pull back through the earlier dependency graph of the same execution. With independent source domains, fixing supported source coordinates gives a product cylinder contained in a replay-equivalence class; equality with the whole class is not asserted.	Theorem · replay stability, composition, and support cylindersSelf-contained proof and exhaustive small-model falsification audit	Evidence and limits →	Logic, semantics, and rewritingAdaptive replay slices	Adaptive replay-slice theorem	FPRD-D40	Adaptive replay-slice theoremReviewed 2026-10-05No documented external or specialist review of this FPRD result is recorded.	Formalize the well-founded replay induction and determine which additional conditions turn per-execution slices into a stationary bounded-memory representation.
FPRD-RDV-D01 A reversible data VASS uses finitely many place types and orbit-finitely many multiset-rewrite rules over a data domain; every rule is available in both directions, and reachability asks whether one finite marking rewrites to another.	Definition · decision problemPrecisely stated with reversibility and effectiveness separated	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	Model and conventions	No recorded dependencies	Problem proposed by Arka Ghosh; FPRD Lab reconstructionReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Obtain an author check of the intended effective presentation and embedding-action conventions.
FPRD-RDV-T01 For an orbit-finitely presented reversible system with transition binomials $v_i-u_i$ , source and target monomials $s,t$ satisfy $s\leftrightarrow^* t$ exactly when $t-s$ belongs to the ideal generated by all embedding-renamed $v_i-u_i$ . On a homogeneous data structure, the same finite renamings can be realized by automorphisms.	Theorem · external algebraic reductionPublished theorem; reduction reconstructed in the FPRD exposition	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	Exact ideal-membership reduction	Reversible commutative rewriting; Embedding action on variables	Ghosh and Lasota, Equivariant ideals of polynomials, Section 8 (Theorem 3 in arXiv v3; cited as Theorem 64 by Ghosh--Lopez)Reviewed 2026-09-01The underlying theorem appeared at LICS 2024. No external review of the FPRD exposition is recorded.	Seek a specialist check of the automorphism-versus-embedding presentation used on the page.
FPRD-RDV-T02 Let $\mathcal A$ be an $\omega$ -well-structured, totally ordered data domain. For every Noetherian commutative coefficient ring $K$ , each embedding-equivariant ideal of $K[\mathcal A]$ has a finite equivariant basis.	Theorem · external finite-generation consequenceSupported by the published equivariant Hilbert-basis theorem	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	What the two structural hypotheses imply	Equivariant Hilbert-basis theorem; FPRD-RDV-D01	Ghosh and Lasota, LICS 2024Reviewed 2026-09-01The finite-basis theorem appeared at LICS 2024. No external review of this FPRD specialization is recorded.	Keep existence of a finite basis separate from effective computation of one.
FPRD-RDV-T03 Under the paper's computability assumptions, reachability in reversible Petri nets with data is decidable for every nicely orderable group action.	Theorem · external strengthened caseProved in a 2025 preprint under stronger effective hypotheses	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	The decidable strengthened case	FPRD-RDV-T01; Computable equivariant Gröbner bases; Effective oligomorphism; Labelled WQO for every WQO label set	Ghosh and Lopez, Computability of Equivariant Gröbner bases, Corollary 39Reviewed 2026-09-01The source is an arXiv preprint; no peer-reviewed publication or external review of the FPRD exposition is asserted.	Track the publication status and test whether the binomial case needs less than the full labelled-WQO hypothesis.
FPRD-RDV-B01 It is not established here that WQO for finite substructures labelled by $\mathbb N$ implies the $\mathbb N\times\mathbb N$ -labelled WQO used by the core equivariant Gröbner computation, still less the all-WQO-label condition of nicely orderable actions.	Open problem · exact implication gapUnresolved under the stated weaker hypothesis set	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	The remaining implication	FPRD-RDV-T02; FPRD-RDV-T03	FPRD Lab comparison of the 2024 and 2025 hypothesis setsReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Prove the lifting for homogeneous ordered finite-signature structures or specialize the algorithm to binomial ideals.
FPRD-RDV-X01 The pure Rado graph has no distinguished invariant linear-order relation, and the ordered random graph fails the required labelled-WQO condition: its age contains the chordless cycles $C_n$ as an induced-subgraph antichain. Thus neither structure satisfies the displayed conjunction of hypotheses.	Counterexample · source-example correctionProved by an induced-cycle antichain; consistent with the later undecidability theorem	Evidence and limits →	Logic, semantics, and rewritingReachability in reversible data VASS	Rado-graph hostile boundary	Induced-substructure ordering; FPRD-RDV-B01	Ghosh and Lopez, Example 54; elementary induced-cycle antichainReviewed 2026-09-01No documented external or specialist review of this FPRD exposition is recorded.	Keep dense rational order as the positive example and do not list the Rado graph as satisfying both hypotheses.
PUB-persistent-scaffold-causal-algebra--thm-lmr Theorem 2.3 (Loff–Moreira–Reis [12, Theorem 16, p. 21] ). A language $L$ is in $\mathsf{PEG}$ if and only if $L^R$ is decided by some scaffolding automaton.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 2.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-full-abstraction Theorem 3.6 (Future-context full abstraction). For rooted degree- $d$ scaffolds over one working alphabet: equal unfoldings, equality of every finite rooted neighborhood, and deterministic labelled bisimilarity are equivalent; unfolding equivalence is preserved by every common scaffold row and hence by every common continuation; and unequal unfoldings are distinguished after finitely many steps by a distance-two unary scaffold continuation.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 3.6	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--cor-right-congruence Corollary 3.7 . Both $\equiv_{\mathrm s}$ and $\equiv_{\mathrm f}$ are stable under a common right suffix of rows.	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Corollary 3.7	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-minimal-figure Theorem 3.8 (Minimal residual figure). Every finite rooted scaffold has a canonical finite behavioral figure $M(\mathcal U)$ whose states are $\mathcal R(\mathcal U)$ , whose root is $\mathcal U_{\epsilon}$ , and whose port $i$ sends $\mathcal U_p$ to $\mathcal U_{pi}$ when defined. The physical reachable graph maps onto $M(\mathcal U)$ by a surjective functional bisimulation. Two rooted scaffolds are behaviorally equivalent if and only if their minimal figures are isomorphic.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 3.8	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-chain Theorem 4.1 (Generated-chain theorem). Let $S$ be a legally generated chronological scaffold with current top $t$ . Its physical graph and its minimal figure are weakly acyclic. The nodes reachable from $t$ , and the states of the minimal figure, are totally ordered by reachability. Conversely, every finite behaviorally reduced labelled rooted reachability-chain port graph, with an optional least unlabelled edgeless state, has an exact legal scaffold presentation at some finite distance.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 4.1	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--cor-rtrivial Corollary 4.2 (Regressive navigation). After adjoining a least missing sink to a generated minimal figure and numbering its chain, every port acts regressively. Its port-word transition monoid is a submonoid of the full regressive transformation monoid and is $\mathcal R$ -trivial.	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Corollary 4.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-guarded-fold Theorem 5.2 (Guarded folds generate bisimilarity). On every finite weakly acyclic deterministic labelled partial port graph, the equivalence generated by elementary guarded folds is exactly the greatest bisimulation equivalence.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 5.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--cor-static-nf Corollary 5.3 (Static normal form). Orient each guarded fold toward its quotient. The system terminates and is confluent up to rooted labelled port-graph isomorphism. Its normal form is the minimal residual figure.	corollaryreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Corollary 5.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--prop-semantic-sewing Proposition 6.1 (Semantic causal sewing). Two strongly equivalent length- $n$ histories are connected by at most $n$ semantic step replacements.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Proposition 6.1	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-extension Lemma 6.2 (One-state extension dichotomy). Suppose two such fresh roots have the same behavior. If the behavior is not already a state of $Q$ , their missing and $\mathsf{SELF}$ positions agree, and their old target agrees at every other port. If the behavior is an old state $q\in Q$ , each row is a guarded unroll of $q$ : it copies every missing and strict-successor port, while a self-loop of $q$ may target either $q$ or the fresh variable $X$ .	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 6.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-canonical-exists Lemma 6.6 (Canonical history exists). Every realizable finite strong trace has a unique lexicographically least history. It is obtained from left to right by appending the least row that realizes the next figure over the already chosen prefix.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 6.6	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-reduced-prefix Lemma 6.7 (Reduced live prefixes). Every root-reachable component at every prefix of a SELF-eager history is behaviorally reduced.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 6.7	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-causal-nf Theorem 6.8 (Convergent strong-trace presentation). For fixed finite $d,k,\Gamma$ , oriented endpoint detours and literal guarded unrolls form a terminating confluent graph-local presentation of strong equivalence. Two histories have the same strong trace if and only if they reduce to the same SELF-eager history. A length- $n$ history normalizes in at most $dn$ row moves when $U$ counts as one whole-row move.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 6.8	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-support Theorem 7.3 (Replay invariance). Suppose an equal-length history $H'$ agrees with $H$ on the labels of $R_H$ and on every descriptor cell in $C_H$ . Then the final rooted concrete scaffolds of $H$ and $H'$ are identical. Moreover $R_{H'}=R_H$ and $C_{H'}=C_H$ .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 7.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-garbage-cube Theorem 7.5 (Causal-garbage cube). Every garbage fibre is the Cartesian product of the finite value sets of its unsupported label and descriptor coordinates. Simultaneous $G$ factors into single-coordinate changes. Fill a triangle for three values of one coordinate and a square for changes of two distinct coordinates. The resulting path two-complex is simply connected.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 7.5	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-two-row Theorem 8.1 (Unexposed two-row sewing). Fix $d$ , $k\geq1$ , and a behaviorally reduced old prefix. Consider two unexposed acyclic two-row extensions with the same consumer label, the same valid nonmissing path-demanding consumer coordinates, and literal agreement as missing or $\mathsf{SELF}$ on every other consumer coordinate. If corresponding path demands have the same final old residual behavior, the blocks are connected by endpoint detours, causal-garbage moves, continuation-aware routing crossings, and dedicated relation exchanges. Every common suffix is preserved.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 8.1	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--lem-relay Lemma 8.3 (Unique relay). If consecutive states $u>v$ of that final reduced chain are carried at nonadjacent construction times, the supported descriptor segment between them is, up to endpoint detours at a recursive bottom, $\epsilon\,0\,0\cdots0.$	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Lemma 8.3	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-unary-final Theorem 8.4 (Unary final presentation). Fix a length $n$ , a nonempty finite working alphabet, degree one, and distance one. Two histories have the same final behavioral figure if and only if they are connected by endpoint detours, absorption, causal-garbage changes, and temporal carrier slides. Every final fibre contains one literal right-packed normal form determined by $n$ and the final figure.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 8.4	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--prop-carrier-lattice Proposition 8.5 (Carrier-surgery CAT(0) geometry). For a fixed unary distance-one final figure and history length, the temporal slide graph on strong/garbage chambers is the undirected Hasse graph of the order-ideal lattice of the rectangle above. Filling every Boolean family of independent slides gives the CAT(0) cube complex $X(P)$ . If the rectangle has $r$ rows and $c$ columns, then the complex has $rc$ hyperplanes, dimension $\min(r,c)$ , edge distance $d(t,s)=\sum_{i=1}^{r}\|t_i-s_i\|,$ and coordinatewise-median carrier tuples. In particular, its diamond-filled two-skeleton is simply connected.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Proposition 8.5	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-batch Theorem 9.2 (Simultaneous-batch compiler; fixed virtual initialization). Every bounded simultaneous record transducer with parameters $(s,A,q,r)$ , one fixed finite immutable pointer-closed initial graph, and plans determined by finite control, the input letter and bounded labelled rooted unfoldings has an exact letter-synchronous native scaffolding-automaton implementation of degree $d=\max\{1,s+Aq\}$ and observation/update distance at most $r+1$ . Physical pointer identity is not an observable test; fixed initial records are virtual finite decoder data.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 9.2	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--thm-staging Theorem 9.4 (Opaque-element staging). Let a strict purely functional sequence operation: use only finite constructor tests and a worst-case bounded number of fixed-arity constructor, projection, and pointer steps; treat elements of its base type as opaque atoms, with no equality, hashing, pattern match, identity test, or traversal through a base element, while permitting inspection of the representation’s own pairs, buffers, and recursive child structures; and return a fixed finite tuple of roots and removed elements. For fixed old roots and symbolic new elements, its control branch and fresh structural allocation graph are determined independently of the symbolic values. If one symbol denotes a fresh owner that stores a resulting sequence root, the owner and structural graph form one bounded simultaneous batch and compile by Theorem 9.2 .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Theorem 9.4	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-persistent-scaffold-causal-algebra--prop-feedback Proposition 9.5 (Feedback programs compile). Let a persistent program, at each input letter, start from a fixed number $s$ of roots and perform a fixed finite composition of strict purely functional sequence operations satisfying the hypotheses of Theorem 9.4 on opaque pointer elements. In addition it may create a bounded number of fresh fixed-arity records whose layouts are determined by the same old observation and whose fields target old records, resulting sequence roots, or records in that fresh finite batch. At most one designated fresh owner’s pointer is stored as a base-type element of those sequences. Suppose the complete per-letter computation, including those sequence operations, follows old logical fields only within some fixed dereference radius $r$ . Then the program is a bounded simultaneous record transducer, and it has an exact letter-synchronous implementation by a finite scaffolding automaton of degree $s+Aq$ and distance at most $r+1$ for suitable finite $A,q$ .	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	Persistent scaffold computation: causal presentations, sewing, and feedback, Proposition 9.5	Theorem-level dependency graph not yet atomized	Persistent scaffold computation: causal presentations, sewing, and feedbackReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--prop-rectangles Proposition 3.1 (Corrected carrier rectangles). For an ordinary missing or base terminal and $m\ge1$ , carrier placements are the ideals of $[m-1]\times[n-m]$ . For a recursive terminal, $m=0$ is a singleton; for $m\ge1$ , placements are the ideals of $[m-1]\times[n-m-1]$ .	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Proposition 3.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-carrier-cube Theorem 3.2 (Carrier cube). Filling every Boolean family of independent temporal slides gives the rooted CAT(0) order-ideal cube of Proposition 3.1 . For a rectangle with $r$ rows and $c$ columns it has $rc$ hyperplanes, dimension $\min(r,c)$ , edge distance $d(t,s)=\sum_{i=1}^{r}\|t_i-s_i\|,$ and coordinatewise-median carrier tuples.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 3.2	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--lem-normal Lemma 4.1 (Unique quotient normal class). Every final fibre has one irreducible garbage class, the right-packed class $\mathsf{RP}_n(F)$ . Hence oriented effective horizontal rewriting terminates and is object-confluent modulo garbage.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Lemma 4.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--lem-collapse Lemma 4.2 (Garbage-trivial collapse). If a legal $D$ or $A$ edge is trivial modulo garbage, it equals one descriptor garbage edge. If a legal $T$ edge is trivial modulo garbage, it equals a two-descriptor garbage path; the two orders agree by the garbage square.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Lemma 4.2	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--lem-descriptor Lemma 5.1 (Descriptor-support monotonicity). For every effective oriented horizontal move $R$ , $\operatorname{DescSupp}(RH)\subseteq\operatorname{DescSupp}(H).$ Every coinitial horizontal/garbage-descriptor branching is therefore a strict naturality square modulo garbage.	lemmareview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Lemma 5.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--prop-captures Proposition 5.2 (Support captures). The only mixed branches without a strict horizontal residual are $A/GL(0),\qquad T/GL(0),\qquad T/GL(-1).$ They require no new generator: the garbage-first path returns along the inverse garbage edge and then takes the original reduction, so $G;G^{-1};R=R$ by vertical cancellation.	propositionreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Proposition 5.2	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-hexagon Theorem 6.1 (A/T hexagon). The following paths have exactly the same raw physical endpoint: $A_y;A_x = T_{yz};A_x;A_y;D_z.$ No garbage quotient is used in this equation.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 6.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-local Theorem 7.1 (Local cell classification). Every coinitial effective horizontal pair is aspherical, Peiffer/cubical, D/A triangular, A/T hexagonal, or D/T derived. Every horizontal/garbage pair is a strict naturality square or one of the three cancellation captures. Every garbage-trivial horizontal edge has a collapse cell from Lemma 4.2 .	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 7.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.
PUB-unary-scaffold-coherent-cubical-algebra--thm-main Theorem 8.1 (Uniform guarded path presentation). Fix a finite alphabet and history length in degree one and distance one. In each final behavioral fibre, all parallel paths generated by endpoint detours, literal absorptions, causal-garbage changes, and temporal slides are presented by the following length-independent guarded cell families: garbage same-coordinate triangles and different-coordinate squares; collapse cells for garbage-trivial D/A/T edges; strong Peiffer squares and the same-row D/A triangle; carrier-cube and inessential temporal Peiffer squares; strict mixed naturality squares; and the bounded A/T hexagon. The three support captures use vertical cancellation, and D/T is derived.	theoremreview publication statement; not independently promoted by this index	Evidence and limits →	Logic, semantics, and rewritingScaffold coherence and causal sewing	A coherent cubical algebra for unary scaffold histories, Theorem 8.1	Theorem-level dependency graph not yet atomized	A coherent cubical algebra for unary scaffold historiesReviewed 2026-09-01No documented external or specialist review of this FPRD result is recorded.	Map this publication statement to any governed claim, verify its hypotheses and proof dependencies, and remove duplicate inventory rows only after an exact mapping exists.