[petalas]

Summary

The complete PR1624 primary certificate command takes 33.77% less median wall time locally, from 0.628150 s to 0.415995 s. Claim type: scoped implementation-equivalence proof, exact finite conformance and local performance evidence. This contribution was produced by Nick Petalas. No new kappa or selected-frontier change is claimed.

authenticate the unchanged primary source and certificate
run the original complete default entry point
replace rational scalar dot products with exact integer numerators
reuse row normalization only while every scalar object is unchanged

Only research/paired26-integer-dot-replay/ is added. The new contract preserves intermediate scalar values and exact return types, detects in-place row mutations, caps reuse at 1,024 rows and retains the original shape errors and complete rejection controls. There is no result cache. The accelerated command cannot invoke the predecessor's certificate-writing option.

Andrey Mas / an664's PR1624 retains the mathematical theorem, checker and certificate. Credit woklerx / PR1020/1077, Hanson Sun / bbMisk / PR1521, themilkman216 / PR968, DreamingOfClouds / PR717 and the retained E8 lineage. Josie Elliston's PR1663 is a separate mathematical audit, cited rather than replayed as this contribution. Common-denominator arithmetic and caching are standard. PR5002, PR6272 and PR6281 already apply denominator clearing to other commands; their implementations are not imported or extended here. The new result is this bounded mutation-aware scalar adapter, its equivalence contract and its complete-command measurement.

Evidence

• Before: median 0.628149917 s, range 0.620465667 to 0.634765208 s.
• After: median 0.415995042 s, range 0.412711708 to 0.568668250 s.

Nine alternating fresh-process pairs use the same CPython 3.12.15 interpreter on macOS arm64. Every pair improves, including the slower second candidate run. All eighteen complete stdout hashes agree, with zero failures. Startup, authentication, source execution, every original check and the complete saved-certificate comparison are included. Shared-host background load remained uncontrolled; the README retains every raw sample. These are local observations, not a portable speed guarantee. Discovery stopped at the first verified implementation.

The separate discovery profile places 2.015 of 2.474 profiled seconds in rational dot arithmetic. The candidate profile leaves 0.817 of 1.357 profiled seconds in the replacement dot, including row conversion/snapshot checking. Profiling is outside reported wall times.

Local conformance compares all 189,904 reached dot values and exact types, both complete computed certificates and the unchanged saved certificate. Fourteen original rejection controls plus two complex controls remain. Additional coverage is 65,536 rational pairs, ten fallback widths, three fallback scalar types, five mutations, 2,049 capacity inputs and eight rejecting controls. Source authentication precedes execution; each input is declared for CI invalidation.

At 8488bdcb4c905229e5a530b41240b2ea7ca3d54c:

• Normal and optimized direct commands pass in 1.870 s and 1.953 s, below their sixty-second limits. Both registered commands and metadata/source pins pass.
• Exact committed bytes, executable modes and canonical source ancestry pass for all nine package/input files.
• Official nine-payload-file export integrity passes. Normal/optimized conformance and the accelerated default CLI pass in a read-only Git-free export with empty PATH. All ten files, including the export manifest, retain their bytes and modes.
• A generated conformance receipt binds package/input hashes. No receipt is handwritten.

From the repository root, standard library only:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/paired26-integer-dot-replay/replay.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/paired26-integer-dot-replay/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/paired26-integer-dot-replay/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/paired26-integer-dot-replay
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/paired26-integer-dot-replay --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/paired26-integer-dot-replay/benchmark.py --pairs 9 --output /absolute/fresh/outside-checkout.json

Any CPython 3.12 executable can replace the displayed path. Expected markers are PASS_FIXED_PAIRED26_MINIMUM_62, PASS_PAIRED26_INTEGER_DOT, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations

The measured result covers one complete default command. The separate predecessor audit.py is unchanged and outside the optimization. This does not establish a new decoder minimum, physical supplier, all-size multiplication theorem, independent human review or formal proof-kernel acceptance. All five inherited framework assumptions remain. No selected cold replay, Lean, Docker or separate predecessor scientific audit was run. No local obligation for the new bounded adapter is deferred to CI. Python allocation and authenticated source execution/static-dispatch deviations are documented in the README.

Open next questions

Can the representation improve another named complete rational replay? First profile its full command and specify its widths, scalar types, denominator growth and mutation contract. New source-bound conformance and complete timing evidence are required; this measured percentage does not transfer automatically. No continuation is reserved.

Related open work

The completion refresh screened the titles and bodies of all fourteen other account open PRs created since the advance CLAIM. No competing computation of this exact paired26 primary command or its runtime was found. This is bounded overlap coverage, not proof of novelty; no extra computations audited another contribution. The merged PR1624 and PR1663 are predecessor and audit context, respectively.

• Reduce complete E8 root-swap replay time by 13.95% (no new κ) #6569 accelerates the separate E8 root-swap command. Its fixture and quantity differ; no numerical performance comparison is asserted.
• Remove 15,732,800 zero products from complete incidence replay #6570 accelerates the separate frame-incidence command. It does not change this paired26 command or compute its certificate.
• Reduce complete core-section replay time by 79.02% (no κ) #6583 accelerates PR3773's core-section consumer using packed Gray-code updates. It enumerates different objects and has a different complete-command benchmark.
• Reduce complete A2+E6 primary replay time by 27.86% (no κ) #6584 accelerates PR2061's A2+E6 primary replay by caching fixed rational contractions. Its normal family, certificate and complete command differ. Neither PR reports this paired26 timing; no cross-fixture numerical comparison is asserted.

CI notes

Both hosted checks, plan and replay (research/paired26-integer-dot-replay), passed at commit 8488bdcb4c905229e5a530b41240b2ea7ca3d54c in run 41618026723. The completed checks were read once; run metadata separately confirms the matching head, status completed and conclusion success. Planning took 3m43s and the registered replay job took 1m30s.

Every registered command also passed locally. Hosted CI certifies execution of the metadata/registered conformance checks, including normal and optimized modes. The timing percentage remains local evidence; CI does not independently accept the written equivalence proof, re-prove the predecessor theorem or discharge the five framework assumptions. No local obligation for this bounded adapter remains unrun.

Advance claim: #1166 (comment)

Merge Danger

Door: two-way. Blast radius: isolated. This opt-in package adds a new authenticated command; reverting it removes the adapter. Frozen predecessor sources and selected-frontier files are unchanged.