[petalas]

Summary

Scoped implementation-equivalence proof and local performance result. An opt-in symmetric associativity kernel reduces the complete PR5502 primary recognizer caller's median wall time by 65.44%, from 1.415539 s to 0.489242 s, in five alternating fresh-process pairs. Every complete report is byte-identical, with zero errors. Source authentication, startup, all recognition/witness checks and JSON output are timed. The companion exhaustive caller is compared separately and is not timed.

commutativity check remains first
ordered basis triples -> unordered index multisets
all distinct: compare three bracketings
two equal: compare two bracketings
all equal: commutativity already implies the equation

Only research/frobenius-symmetric-associator/ is added. The written proof gives exact internal-kernel behavior at every finite dimension and preserves complete recognition return values. At dimension 16, the instrumented passing-table contraction count is 8,192 -> 2,160. This retains dense O(n^5) complexity. No algebra classification, physical supplier, new kappa or selected-frontier change is claimed.

This contribution was produced by Nick Petalas. Credit apoursh / PR5502 for the original recognizer, complete callers, witnesses and finite oracles; Ali Romman / PR5459 and PR5316 for the algebra family and recognition handoff; Nick Petalas / PR5252 for the separate etale comparison; David Ganz and the retained algebra/tensor lineage. Symmetry of commutative associators is classical. Frozen source files and credit lines retain their bytes.

Evidence

• Original complete-caller profile: the algebra-law kernel takes 3.126 of 3.452 profiled seconds. Profiling is excluded from timing. The timed baseline range is 1.369582 to 1.432597 s; the adapted range is 0.477110 to 0.504267 s. Both arms use identical fixtures and flags. The host is CPython 3.12.15 / macOS arm64, 18 logical CPUs, with other work present. No universal speed guarantee is claimed.
• All 16,418 commutative two-dimensional F2/F3/F5 tables agree with the original kernel and retain their input bytes; all 64 binary complete API return values agree. Another 576 tables, 480 noncommutative precedence controls, both repeated-index cases, a distinct-middle-bracketing trap, invalid selectors and all passing contraction counts through dimension 16 pass.
• Both original complete callers are retained. The primary retains its 46 positive presentations, 3,217 independent product equations and every nonmember, malformed-input, normalization and witness-mutation check. The companion retains its 525 tiny tables and 392 positive four-dimensional presentations. These are retained behavioral coverage, not new audits of predecessor mathematical results.
• Four warning-strict development-mode normal/optimized commands pass locally in 2.337, 2.322, 0.728 and 0.777 seconds. The registered launcher passes. Syntax/function-bound checks and whitespace checks pass.
• Committed bytes, modes and declared source ancestry pass at 2a507b2dfaf97903c5c5d7e7e03a22c05f594ad9, covering eleven package/input files. Official export/input integrity and all four read-only Git-free commands with empty PATH pass. Paired normal/optimized reports agree, and all twelve export files retain their hashes and modes. Actual mutations of each of the four source inputs reject in both modes; removing the middle-bracketing equation also rejects in both modes.

Exact reproduction from the worktree root, using the existing interpreter:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/frobenius-symmetric-associator/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/frobenius-symmetric-associator
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/frobenius-symmetric-associator --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/frobenius-symmetric-associator/benchmark.py

Any clean CPython 3.12 installation can replace the shown interpreter. Expected statuses are PASS_FROBENIUS_SYMMETRIC_ASSOCIATOR and PASS_REGISTERED_RESEARCH_PACKAGE. The benchmark prints a generated hash-bound timing receipt and has no performance acceptance threshold.

Limitations

The reduction requires the internal normalized-table contract and a successful commutativity check. Public API adoption and other callers remain separate work. The universal equivalence proof is written; executable corroboration is finite. Independent human and formal proof-kernel review are unclaimed. The companion caller is outside the timing measurement. No Lean, Docker, selected cold supplier replay, native implementation, physical resource invoice or complete multiplication theorem was run or supplied. All five selected framework assumptions retain their status.

Open next questions

• Bind a public opt-in backend to one actual external consumer, including its resource limits and diagnostics, before adoption.
• Profile another actual tensor recognizer before attempting reuse. A noncommutative input contract can prevent the reduction.

Related open work

• Reduce complete extension-field tensor replay time by 16.56% (no κ) #6303 improves finite-field row elimination in the separate PR5637 extension-field tensor-recognition caller. This changes associativity equations in PR5502 prime-field algebra tables. The callers and timing workloads differ; no same-instance number is compared or newly computed.

Completion refresh inspected all 13 open account PRs created since the advance CLAIM, including this PR. The related extension-field caller #6303 is listed above; the other contributions study different objects or workloads. No same-instance numerical disagreement appeared in this coverage, and no extra computations checked other results.

CI notes

Both hosted checks, plan and replay (research/frobenius-symmetric-associator), passed at head 2a507b2dfaf97903c5c5d7e7e03a22c05f594ad9 in run 41107554596. The completed checks were read once; separate run metadata confirms success and the matching head. Hosted replay certifies the registered finite conformance and source bindings. The written universal equivalence proof retains its stated review limits. Timing remains local evidence and is excluded from the CI gate.

Merge Danger

Door: two-way.

Blast Radius: package.

The adapter is isolated and opt-in. Shared code, source archives and the selected frontier are unchanged.

Generated local timing receipt

{
  "code_sha256": {
    "SOURCES.json": "11886bdec1888b12613326ce66bf99c0ca29be636911ce4268783568d555433c",
    "adapter.py": "3ea2ff3a011aa3a77bbd09853e619719596b545b3790db2b1f86c6bb64f0272b",
    "benchmark.py": "6f62fc965e7d25c15ddfd0ba2e238aa544f595ba7d339481b1fe837b4850b701",
    "verify.py": "96892697757d9f97360b9e173491b1b80d0d4481f1b372f4a1581773dcb1e1f7"
  },
  "complete_report_sha256": "4a7cfa64d12291492a20bde284bcb9520ed7b916a97b0f5cccb1d85ce7a36155",
  "cpu_count": 18,
  "errors": 0,
  "executable": "/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12",
  "load_average_at_finish": [
    2.65283203125,
    3.6494140625,
    2.80615234375
  ],
  "median_seconds": {
    "baseline": 1.415539083071053,
    "symmetric": 0.48924245801754296
  },
  "order": [
    "baseline",
    "symmetric",
    "symmetric",
    "baseline",
    "baseline",
    "symmetric",
    "symmetric",
    "baseline",
    "baseline",
    "symmetric"
  ],
  "platform": "macOS-27.0.1-arm64-arm-64bit",
  "python": "3.12.15 (main, Oct 14 2026, 00:55:37) [Clang 23.1.3 ]",
  "range_seconds": {
    "baseline": [
      1.3695817498955876,
      1.4325970420613885
    ],
    "symmetric": [
      0.4771100420039147,
      0.5042671249248087
    ]
  },
  "samples_per_engine": 5,
  "seconds": {
    "baseline": [
      1.415539083071053,
      1.4205422920640558,
      1.4325970420613885,
      1.3695817498955876,
      1.3735971658024937
    ],
    "symmetric": [
      0.4961120840162039,
      0.4855923750437796,
      0.48924245801754296,
      0.5042671249248087,
      0.4771100420039147
    ]
  },
  "status": "PASS_COMPLETE_FROBENIUS_TIMING",
  "timed_scope": "Fresh process, source authentication, full primary run, JSON stdout; oracle companion not timed.",
  "wall_time_reduction_fraction": 0.6543772871631934
}