[petalas]

Summary

Scoped written mathematical proof with exact finite corroboration. A rational ADD/QUERY/APPLY/SWAP word of length L on r arbitrary dirty roles has an exact complete-endpoint identity test in O(L 2^(k+1) (r^2+1)^3) rational operations, where k is the maximum number of signs already seen that occur again. Bit complexity is 2^k times a polynomial in ordinary input size. The test preserves repeated-sign correlations and returns an actual counterexample assignment when identity fails.

branch new signs within each live assignment
transform actual homogeneous prefix matrices
forget signs only at last use; retain a span basis
check every final representative against the complete identity

At each cut, at most 2^k(r^2+1) actual prefix matrices remain. The commuting diagonal-sign native interpretation covers every positive tensor width and every dirty input/return. Two words can be compared by concatenating the first with the inverse second word; k is measured on the concatenation.

This contribution was produced by Nick Petalas. It answers Hiroki KUMAZAKI's PR #5716 Open next questions about a tractable restricted identity-test class. Its CNF audit and Ali Romman's PR #5369 are cited, not replayed as this deliverable. Credit Barrington's classical branching-program construction, Schutzenberger/Tzeng reachable-span methods and Stefan Kiefer's weighted-automata exposition, Section 2. zxv's #5235 independent-operator certificates and #5441 fixed-transfer-table results retain their credit. The scoped application and explicit lifetime bound are the new step; no new general equivalence principle or literature-priority claim.

Only research/sign-frontier-identity/ is added. Canonical base de5ec2e7a6074068452f45b1a191b5c72832df4a, priority 8. Proof, scope and reproduction, source pins. Advance CLAIM.

Evidence

Before, #5716 leaves useful tractable subclasses open. After, the proof supplies a precise sign-lifetime restriction, an exact bound and a concrete-witness algorithm.

Local CPython 3.12.15, standard library only:

• All 1,555 words of lengths zero through four over the declared six-gate alphabet, across 4,097 sign assignments.
• 216 structured words on two, three and five roles, across 9,216 assignments. A separate column-vector interpreter checks all complete columns, verdicts and returned counterexamples.
• Two 128-sign bounded-frontier families, each with exactly 512 candidate updates. The width-zero family has an all-positive identity endpoint and an explicitly detected nonidentity branch; the width-one family checks repeated-sign cancellation. These are not exhaustive assignment enumerations.
• An eight-live-sign identity visits all 256 live keys. Five capacity/inverse fixtures cover the 4,096-gate cap, label 4,095, 64-bit coefficients, empty one-role word and a mixed instruction inverse. Eighteen invalid-input/capacity rejections and three semantic controls pass.

Both registered warning-strict commands passed locally in 1.091 seconds each; these are check durations, not a performance comparison. Their 120-second timeouts are below the CI limit. The registered runner passes PASS_REGISTERED_RESEARCH_PACKAGE; each direct command reports PASS_SIGN_FRONTIER_IDENTITY.

Metadata/source pins, AST compilation, function bounds and exact committed-byte/source-history preflight pass. The preflight checks nine package/input files and canonical-base ancestry. The official committed export passes integrity and declared-input coverage. Normal and optimized direct commands pass in its read-only Git-free tree with empty PATH; all ten exported files retain their bytes and modes. Generated stdout receipts bind every package file by SHA256. No receipt is handwritten.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/sign-frontier-identity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/sign-frontier-identity/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/sign-frontier-identity
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/sign-frontier-identity --check-source-history

Elsewhere substitute CPython 3.12. The verifier authenticates the two declared predecessor documents without importing or replaying their executables.

Limitations and unrun obligations

No small k is proved for the selected P104 caller or the general CNF construction. No source parser, arbitrary affine/quadratic native alphabet, noncommuting address action, rounded implementation, physical tariff, supplier or multiplication theorem is supplied. The executable deliberately caps r at five, L at 4,096 and k at eight; the written theorem allows arbitrary finite parameters. The README documents bounded managed allocation and the fixed local loader/callback deviations. No external static analyzer is claimed.

No new kappa or selected-frontier change. Independent human and formal proof-kernel review are unclaimed. No predecessor scientific suite, selected cold replay, Lean or Docker run was performed. All registered local obligations ran.

Open next questions

1. Does a useful emitted caller in this commuting-sign subclass have small k after legal reordering? Begin with a source-bound label/interval extraction that preserves every dirty return.
2. Can a named correlated family avoid explicit live-assignment enumeration? Prove a replacement invariant first; the current test remains exponential in k.

Related open work

No other open PR studying bounded-live-sign complete-endpoint identity testing appeared in the 470-PR complete cache, current live PR/claim checks, targeted searches or main README searches. The directly related #5716, #5369, #5235 and #5441 are merged and credited above. Sparse local notes, arbitrary historical diffs and unpublished work remain coverage limits. No extra predecessor computations were run to validate other contributions.

The completion refresh found no open PR from this account created since the CLAIM at 2026-10-27T13:29:17Z. No additional same-object or same-quantity line is needed, and no predecessor computations were run for comparison.

CI notes

Registered research packages run 40514192213 finished on exact head b7923ed3312a518534ae94be0cd16b6beb518619 with plan failed; replay skipped. The plan job failed only at Onboarding helper tests. All ten failure/error entries point to unchanged tests/test_research_current_assignment.py, ending FAILED (failures=8, errors=2, skipped=22).

The two errors are KeyError: 'follow_up_steps' and KeyError: 'catalog_first_step'. The eight assertion failures concern frozen/current routing, missing helper/reference checks and sparse omission lists. The ten test headers, exception/assertion messages and final summary match the directly inspected PR #5926 plan log. This confirms the shared main-base onboarding failure at de5ec2e7a6074068452f45b1a191b5c72832df4a; no package scientific replay failed. No shared implementation, test or workflow was changed, and no new commit was pushed for this closeout.

The package replay job was skipped. Hosted replay has not certified this package; only the local scientific and publication checks above have run. The written mathematical proof still awaits independent human and formal proof-kernel review.

A maintainer merged the PR at 2026-10-27T13:43:54Z in bulk archive intake, merge commit aacce1a2c411a2f083b5fd5f1cdcf87661267ca3. Under the research archive policy, this preserves the contribution and credit. It is not scientific approval, independent replay, supplier admission or a selected-frontier update.

Merge Danger

Door: two-way. Blast Radius: isolated. One opt-in package; shared code, frozen predecessors and selected certificates are untouched.