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DONE — inverse-aware universal certificates for complete dirty linear circuits Status: completed and claim released, 2026-10-23 14:42 UTC; priority revision 7. PR #5262, exact head The inverse-frontend question is established at the stated scope: two complete dirty linear programs with r independent inputs and q total unary occurrences have an r+q-state feedback realization. Exact invariant equations certify their full endpoints for every invertible substitution and payload dimension; failure yields explicit unipotent matrices in dimension at most r+q. The written central-parameter argument justifies arbitrary invertible substitutions without evaluating an infinite series numerically. This is a scoped author-checked proof and implementation, with no independent human or proof-kernel review claimed. Both registered normal/optimized replays pass: 504 signed-word cases, 160 differential programs, 64 boundary cases, 36 delayed/infinite-tail cases, 324 exact matrix substitutions and 31 rejecting controls. The 2^64-word endpoint uses 260 states and 64 invariant rows. Committed-byte/source-history and export-integrity checks pass; both read-only Git-free replays pass on the final head, with all 21 file hashes unchanged. Hosted run 39971793532 passed both planning and the focused Linux replay on this exact head. Hosted execution is separate from scientific review. Open handoff, unclaimed: bind one actual inverse-bearing native word to this grammar, retaining every independent input and final helper, and compare against its existing exact reduced-word oracle at a pinned source head. The source/type/phase adapter is still required. This standalone package's full replay takes about 16 seconds per mode on the recorded host. Other operator relations, rectangular typing, native realization, rounding, precision, stock, hosting, tapes and the complete supplier invoice remain separate. No new kappa or selected-frontier change. Credit zxv's #5235 at Freshness: canonical main refreshed and unchanged at {"schema":"integer-mult-coordination/v1","kind":"DONE","task":"formal-interface-lemma","scope":"Compact inverse-aware universal certificates for complete dirty linear circuits; r+q states and explicit invertible witnesses","base":"ff3cc58e370d0f678e529246f8c73e8094e18fcf","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5262"}Original scoped claim (retained)CLAIM — inverse-aware universal certificates for complete dirty linear circuits Status: active, 2026-10-23 14:11 UTC; priority revision 7. No result is claimed yet. This takes the explicitly open inverse-relation frontend in PR #5235 / this discussion, rather than rechecking its independent-symbol theorem. Question. Can a finite linear register program with named invertible operators and their actual inverses be certified in a number of states linear in its input roles plus unary occurrences, without expanding reduced group words? Every dirty input and declared return remains independent. Only the inverse relations are imposed; commutation, Clifford and other relations are excluded. Base and predecessor. Fresh canonical main Method and bounded domain. Expand each generator as Expected artifact / stop rule. One self-contained proof and reusable producer/checker with rejecting controls and a compact stress example, or a precise obstruction if the proposed representation is unsound. Stop after this interface is established; no supplier, physical invoice, new kappa or selected-frontier change. The donor budget is at most four hours including verification/publication; this claim expires 2026-10-23T16:11:00Z. Value and overlap. #5235 explicitly leaves inverse contracts open. Live PR bodies, current Discussions, #1166, recent krzysztofwojt claims and main package notes were checked; the complete cached snapshot covers 317 open PRs, 4,503 Discussion records and 7,091 local notes, followed by a live targeted check immediately before claiming. Targeted historical PR searches found #5235 for weighted automata and no inverse-aware/Magnus implementation. Arbitrary historical diffs and unpublished work remain outside coverage. All donor-excluded topics are outside this scope. Credit: zxv's compact independent-operator bridge and its retained dirty-echo/tensor/dirty-phase lineage; classical recognizable rational-series and weighted-automata theory (Schutzenberger/Fliess, Kiefer), and the classical Magnus substitution. No new general automata/PIT theorem or global literature priority is claimed. Prepared by Krzysztof Wojt with substantial OpenAI Codex assistance; no independent human or proof-kernel review claimed. |
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DONE — PR #5268, exact head Every false complete independent-symbol dirty linear identity over Q or Q(i) has a real symmetric positive-definite contraction witness with condition number at most two. The dimension is D+1 for the highest nonzero word degree. Exact-length reachable spaces find that word without expansion; a separate literal replay checks the nonzero endpoint. The pinned wrong-order echo gives discrepancy 1/512 in dimension four. The 2^64-word stress case uses 132 coefficient states, at most two basis rows per degree, and dimension65. Normal/optimized registered and read-only Git-free replays pass: 240 differential circuits, 384 literal replays, 96 spectral enclosures, fifteen controls and four code mutations rejected in both modes. All16 exported files remain unchanged. Source pins, committed bytes, public links and published body/rendered-HTML readback pass. Both hosted checks now pass at this exact head; full-history wrapper, human/proof-kernel review and native implementation are unclaimed. No kappa or frontier change. Credit zxv's #5235 compiler and retained weighted-automata/PIT and echo/tensor lineage. Pending predecessor bytes are pinned and copied unchanged. Krzysztof Wojt's #5262 inverse frontend is distinct and was not audited. The next unreserved question is a relation-preserving realization inside a specified native child family; independent SPD substitutions do not supply one. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. Correction: the first hosted replay stopped before mathematical execution because an unmerged source Git object was absent. The repair retains the same compiler, scalar code, example, proof and attribution in this PR history and binds all source pins to available ancestors. A fresh clone passes both registered modes with its remote disabled. No mathematical assertion was weakened. Original advance claim follows for provenance. CLAIM — positive-definite counterexamples for complete dirty linear circuits. Status: active, 2026-10-23 14:52 UTC; priority revision 7. I will extend the independent-symbol counterexample interface of PR #5235, pinned at Question: can every failed complete dirty linear identity over Q or Q(i) be separated by real symmetric positive-definite generators with spectra in [1/2,1]? The proposed construction uses a highest-degree coefficient word, a symmetric labelled path and a fixed dyadic identity shift. Exact-length reachable spaces will find that coefficient without expanded words. The deliverable is a written dimension/conditioning proof, a reusable witness adapter and a literal full-program replay with all dirty roles retained. This changes the formal-interface card from a proof-assistant obligation to a constructive algebraic interface. It addresses whether conditioning alone can exclude a generic counterexample. It does not impose inverse relations, commute generators, implement a native caller, change kappa or change the selected frontier. Krzysztof Wojt's active inverse-aware frontend is separate and remains untouched. Stop after the complete bounded-condition witness theorem and exact corroboration; no reservation on other extensions. Claim expires 2026-10-23T16:52:00Z. Overlap: fresh open-PR/Discussion search plus targeted canonical-main proof and README searches, and closed-PR searches. No matching construction was found in the inspected evidence; arbitrary historical code and unpublished work are outside coverage. The underlying circuit compiler and scalar arithmetic will be retained byte-exact from the pending predecessor, with explicit pins and source notices. Credit zxv and the weighted-automata/PIT and dirty-echo lineage documented in that source. No new general PIT algorithm or global literature priority is claimed. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. |
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DONE — PR #5296, exact head Every false complete rational dirty linear identity with true named-generator inverses has an SPD witness in dimension at most r+q. A nonzero Magnus word gives a symmetric path. The endpoint numerator degree is at most L=q_forward+d*q_inverse, so one of L+1 rational grid points must work. The generators can be arbitrarily close to identity; both generators and actual inverses have spectral condition number at most two. Inverse norms may reach two. Proof and exact receipt. Both registered modes pass: 160 reduced-word comparisons, 137 independent dense inverse replays, eight dirty-sandwich deletions, 154 inverse basis checks, fourteen rejecting controls, and four genuine code mutations rejected before receipt/hash checks in both modes. Four cancellation examples force the final grid trial. A 2^32-word leading endpoint needs 66 realization states and a dimension-33 witness. Source bytes and every declared pin's ancestry, committed bytes, export integrity and both Git-free read-only replays pass; all 19 exported files remain unchanged. Public links and published body/rendered-HTML readback pass. Hosted run 39986954491 passed both planning and the focused Linux replay at this exact head. Hosted execution is separate from scientific review. The proof and code cover Q and the named-inverse grammar only. Extra native relations, inverses of sums, practical floating detectability, polynomial coefficient bit growth, native physical implementation and the complete paid invoice remain outside scope. The full-history wrapper was not passed in this shallow checkout; direct checks cover every declared pin. No independent human/proof-kernel acceptance, selected cold replay, new kappa or frontier change is claimed. Credit @krzysztofwojt's PR #5262, source Freshness: canonical main remains Open handoff, unclaimed: bind one actual native inverse-bearing word to this grammar, preserving every dirty role and stating each extra operator relation before applying the generic witness theorem. This contribution does not reserve that adapter work. {"schema":"integer-mult-coordination/v1","kind":"DONE","task":"formal-interface-lemma","scope":"Conditioned SPD witnesses for complete rational dirty programs with actual named generator inverses","base":"ff3cc58e370d0f678e529246f8c73e8094e18fcf","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5296"}Original advance claimCLAIM — bounded-condition witnesses for complete circuits with true named inverses. Status: active, 2026-10-23 16:12 UTC; priority revision 7. Base: current main Question: can every false complete rational dirty linear identity with true named-generator inverses be separated by SPD generators arbitrarily close to identity, with both generators and their inverses of condition number at most two? The highest-degree polynomial method does not apply to inverse series. I will use a nonzero Magnus coefficient, a symmetric labelled path, and a finite rational grid justified by a numerator-degree bound. Literal exact replay will use actual inverses and every returned role. Deliverable: a written dimension/grid/conditioning proof, reusable witness adapter, independent dense inverse checks, a cancellation example which defeats the first grid point, and mutation controls. Budget: 90 minutes including publication. Stop at this algebraic interface; no native-family relations, floating detectability, physical invoice, new kappa or selected-frontier change. The claim expires 2026-10-23T18:12:00Z. This is a changed formal-interface-lemma scope, not a proof-assistant claim or a reservation on native adapters. Overlap: the fresh remote snapshot and live comments show the two predecessors complete. Targeted canonical-main README/PROOF/document scans and historical PR searches found no matching conditioned inverse construction in the inspected evidence. The sparse helper did not search all local notes; full main source scans supplemented it. Historical arbitrary diffs and unpublished work remain outside coverage. Credit @krzysztofwojt for the inverse realization and @zxv for the earlier register bridge, with the classical rational-series/weighted-automata and dirty-echo lineage retained in the source notices. No new general PIT algorithm or global literature priority is claimed. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. |
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DONE — PR #5315, exact head Every false complete rational dirty-register identity with true named inverses now has an exact special-orthogonal witness in dimension at most r+q, arbitrarily close to identity. Inverse equals transpose and condition number one are preserved. A shortest Magnus word, a skew-path Cayley transform and at most dq+1 rational trials give the construction. The proof includes a concrete failure when shortestness is dropped. Both registered CPython 3.12.13 modes pass: 160 programs against a separate free-group oracle, 142 dense full-program replays, 154 inverse-basis checks, four nested commutators, forced grid cancellations and dirty/alias controls. A 33-dimensional witness handles 2^32 lowest-degree words without expansion. Four actual code mutations fail in both modes. Committed bytes, every source pin's direct ancestry, portable archive validation, both read-only Git-free export replays and published body/rendered-HTML readback pass. All 19 exported files remain unchanged. Hosted run 39992699684 passed planning and the registered Linux replay for published head Credit @krzysztofwojt for PR #5262, @zxv for PR #5235, and the retained dirty-echo/tensor/phase and classical rational-series lineage. PR #5296 supplies the SPD inverse-witness architecture adapted here; Cayley transforms are classical. The PR links the exact proof, source bytes and receipt. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. Next handoff: bind one actual native caller while retaining all its extra relations. Orthogonality alone does not impose finite order, involution, commutation or a finite Clifford alphabet. Bit complexity and a numerical separation bound also remain open. No native tape, selected cold admission, Lean/Docker, independent human proof review or complete kappa calculation was run. The full-history wrapper remains unpassed in this shallow checkout. No complete kappa improvement or selected-frontier change follows. Base and final refreshed canonical main: {"schema":"integer-mult-coordination/v1","kind":"DONE","task":"formal-interface-lemma","scope":"Rational near-identity special-orthogonal witnesses for complete inverse circuits","base":"ff3cc58e370d0f678e529246f8c73e8094e18fcf","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5315"} |
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DONE — PR #5339, exact head A rational Cayley endpoint is either zero or has magnitude at least 1/[C(3B)^(dq)]. The proof pays every scalar denominator, operator denominator, scalar gain, coefficient quantization and data truncation. The integer dyadic detector chooses sufficient precision, returns every role, and distinguishes the selected zero/nonzero endpoint at half the separation bound. It does not use an exact endpoint as its decision oracle. The detector's precision and arithmetic bit sizes are polynomial after the word is supplied; the upstream word producer is not reanalyzed. Both registered CPython 3.12.13 modes pass 102 cases: 24 zero and 78 nonzero, 101 separate dense exact comparisons, seven domain controls, tiny parameter/discrepancy 2^-80, gains 2^100, aliases and dirty returns. The 65,536-word inverse-series stress uses dimension 17 and 2,262 fractional bits. Four actual mutations fail in both modes. Committed bytes, every source-pin direct ancestry, official export integrity, both read-only Git-free replays and published body/rendered HTML pass; all 23 exported files are unchanged. Hosted run 39996997226 passed planning and the registered Linux replay for published head Credit PR #5315 for the orthogonal realization, @krzysztofwojt's PR #5262 for the inverse compiler and shortest-word theorem, @zxv's PR #5235, and retained lineage. Cayley, determinant and forward-error methods are classical. Prepared by Ali Romman with substantial OpenAI Codex assistance; Apache-2.0. The next task needs a pinned native-compatible caller and a complete sparse/error/cost comparison. A zero endpoint at one tuple is not a universal identity certificate. There is no optimal-guard, IEEE, native/tape, selected cold replay, independent human/proof-kernel, Lean/Docker or complete kappa claim. The full-history wrapper is unpassed in this shallow checkout. The selected frontier remains unchanged. Final main is ff3cc58, priority 7; the final 17:38 UTC overlap search found no competing result within its documented coverage. {"schema":"integer-mult-coordination/v1","kind":"DONE","task":"formal-interface-lemma","scope":"Explicit separation and finite-precision dyadic detection for rational orthogonal inverse witnesses","base":"ff3cc58e370d0f678e529246f8c73e8094e18fcf","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5339"} |
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DONE — PR #5352, exact head The complete inverse-circuit producer and its separate checker now have explicit polynomial rational-size and bit-cost bounds for explicit integer/fraction inputs. Common denominators bound the compiled model and path rows; pivot minors bound normalized elimination and all certificate coordinates. Raw unreduced operands and arithmetic temporaries are included. The source compiler, search, checker and oracles remain unchanged. The new budget interface rejects compressed scalar syntax before expansion. Python uses the standard per-process integer-string-limit option for large explicit values. Both registered isolated, warning-strict CPython 3.12.13 modes pass all 99 programs: 36 equal and 63 different, with 98 independent reduced-word comparisons, 63 exact matrix witnesses and nine rejecting controls. The weighted 2^64-word stress uses 260 states, 64 invariant rows and at most 254-bit certificate coefficients. A 16,001-bit scalar, unreduced fractions, 1,025-bit near-dependence coordinates, coprime denominators, delayed inverse tails, dirty returns and aliases are included. Three separate edge audits pass. Four actual mutations fail in both modes for mathematical reasons. Committed bytes, every source-pin direct ancestry, official export integrity, both read-only Git-free replays and publication body/rendered HTML pass. All 21 exported files remain unchanged. Hosted run 40002911314 passed planning and the registered Linux replay for published head Credit @krzysztofwojt's PR #5262, exact source The next task remains a pinned native caller with every extra operator relation, type, phase, storage and paid-call obligation. This result closes an explicit rational-size gap; it is not a practical speedup, optimal bound or new general equivalence algorithm. No independent human/proof-kernel review, selected cold replay, Lean, Docker, native tape, supplier admission or complete kappa calculation was run. The full-history wrapper is unpassed in this shallow checkout. Main remains ff3cc58, priority 7. The final 18:21 UTC scoped overlap snapshot found no competing result; sparse notes, arbitrary historical diffs and unpublished work remain coverage limits. The selected frontier is unchanged. {"schema":"integer-mult-coordination/v1","kind":"DONE","task":"formal-interface-lemma","scope":"Explicit polynomial rational-size and bit-cost bounds for the unchanged complete inverse-circuit certificate producer","base":"ff3cc58e370d0f678e529246f8c73e8094e18fcf","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5352"} |
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DONE — complete selected-provider application; all 30,328 fixed pair fusions fail at every positive width. Scope released. PR #5376, exact head The new Gram engine checks all 7,000 native recipe classes and both complete local physical programs selected from PR #4610, with PR #5241's full paid catalogue. All 632 persistent inputs and the arbitrary owned input remain independent. Every output is checked, including zero owned return. Peak expanded support is 22,331 forward and 12,930 backward. This measures the fixed provider; it is not a general support bound or a new outer admission. The search began with adjacent rank-one pairs and now covers every pair of paid climbs on one persistent role, at all ranks, including intervening climbs, warps and whole owned-center windows. There are 7,582 pairs per direction. Two endpoint placements per pair give 30,328 rewrites. Every other physical operation stays fixed. Each pair has a saved exact contact block No kappa gain and no paid saving. The selected conditional value remains 0.0030217758. The obstruction covers fixed early/late placement of one same-role pair only. It does not exclude changed contacts, coordinated fusions, interior placement, cross-role pairs, owned-center pairs, new operators or later providers. It does not subsume the separate PR #5393 frozen-relation argument. The initial invoice target comes from the PR #5243 handoff; its required physical pair was not found. CPython 3.12.13, standard library: python3.12 -B scripts/research_package.py --package research/clifford-tensor-gramAll six registered replays pass. The provider prints Exact-head hosted CI passes. No selected outer cold replay, new controller pricing, Lean, Docker, independent human review or proof-kernel acceptance is claimed. The optional full-history ancestry preflight is omitted in this shallow checkout. Exact source objects and bytes were checked. The five selected framework assumptions remain. Cao Yifan (@intqwq), Chris R (@Clear1nsight), Cinna Davis (@1337isnot1337), zxv and the full native/helper/adapter lineage retain credit in the provider notice. The earlier PR5241 free-group lift is prior work. New proof, importer, application, census and receipts: Ali Romman with substantial OpenAI Codex assistance, Apache-2.0. The final overlap refresh covered 350 open PRs and 4,684 Discussion records with complete remote pagination. Sparse local notes limit historical coverage. No other chat's files or PRs were changed. No selected record was changed, and no PR was merged or closed. Next useful work: start from one saved contact certificate and price a changed contact, or name a small joint rewrite and check its complete endpoint. Individual rejection does not prove joint rejection. Independent review should check the two-sided cut formula, adapter boundaries, owned-cut closure and coefficient census. The coupled-helper proposal in PR #5402 changes the helper boundary and is also outside this fixed-endpoint pair class. Its caller work remains with its existing Discussion claim. |
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Status: DONE (author-checked scoped algebra), 2026-10-23 13:11 UTC. Claim released.
Artifact: PR #5235, exact head
48976daa4dce59b4867ef046da72a370e389925d. Contributor guide. This is a user-directed reusable result, not a κ search.The common question is how a finite dirty-register word can acquire a universal substitution certificate without exponentially expanding its noncommutative words or relying on a few matrix sizes. A program with r arbitrary inputs and q unary occurrences has a compact r+q-state coefficient representation. Exact invariant-subspace closure certifies every returned role under every independent-operator substitution and payload dimension. A failed identity yields a shortest nonzero coefficient word and an explicit finite matrix counterexample; a shifted variant makes every generator invertible and unipotent. Scalars remain scalar under termwise tensor lifts.
The artifact contains the written bridge proof, reusable producer, separately transcribed no-elimination checker, Q/Q(i)/prime-field arithmetic, committed example certificates and a literal source-bound adapter for the existing dirty-echo scheduler. Classical weighted-automata equivalence and noncommutative PIT are credited to Kiefer's 2020 notes and Raz–Shpilka 2005; no new general PIT algorithm is claimed.
Both registered normal/optimized Python 3.12.8 checks pass: 210 complete echo identities, 120 explicit-word differential circuits, 180 literal matrix counterexamples, 15 malformed-certificate rejections, field/tensor controls and three committed examples. A 2^64-word endpoint requires 130 states and 65 invariant rows. Normal/optimized Git-free export replays also pass, with all 19 exported regular files unchanged. Source bytes and export integrity pass. Hosted run 39958986337 passed both planning and the focused Linux replay on exact head
48976daa4dce59b4867ef046da72a370e389925d; its committed-input check also passed. Hosted execution is not independent scientific acceptance. The optional full-history preflight refuses the shallow local checkout; no full-history success is claimed. All declared source pins use fetched canonical mainec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c, which is an ancestor of the contribution.A negative result concerns independent symbols: it need not hold under inverse, commutation, Clifford or other relations. Typed rectangular blocks, nonlinear/rounded leaves, physical/native implementation, precision, stock, hosting, tapes and the complete paid/outer invoice are separate obligations. No supplier, selected-frontier change, new κ, human review or proof-assistant verification is claimed.
Smallest useful next task: adapt one additional actual independent-symbol word, preserving every dirty input/return, and compare its complete endpoint against a pinned symbolic oracle. The guide has a three-role example and the package replay takes roughly 10–30 seconds per mode on the recorded host. A relation-aware frontend needs its own normalization theorem first; do not treat names such as G_inverse as inverse contracts.
Credit: the androolloyd dirty-echo package following Chafik Boukhalfa/chafreaky #628 and Ali Romman #621; Cinna Davis #4100/#4088, ToikY's free-word tensor audit, Sagar Thakrar #3687, roger-sato #3693, and the universal dirty-phase audit following Charles Bethin #1278. Their existing results and wider attribution are retained. Prepared for @zxv with substantial OpenAI Codex assistance; Apache-2.0.
Freshness: canonical main refreshed before submission and unchanged; priority revision 7. Publication overlap snapshot covers 348 open PRs and 4457 Discussion records at 13:07–13:08 UTC with complete remote pagination. Sparse local helper coverage was supplemented by source scans and a full origin/main README/PROOF/doc term search. This is not a novelty guarantee or coverage of unpublished work.
{"schema":"integer-mult-coordination/v1","kind":"DONE","task":"formal-interface-lemma","scope":"Compact universal independent-operator certificates for complete dirty linear circuits, with constructive nilpotent/unipotent witnesses and source-bound echo applications","base":"ec2780e8aeb1c88acaf5e9e470ef8c68e7fbbb6c","expiry":null,"artifact":"https://github.com/CrocSwap/integer-mult-bounds/pull/5235"}All reactions