[josecarlosgonzalezroque-design]

Change and mathematical scope

The exact rational-rank certificate producer still used Fraction arithmetic for every forward-elimination entry after #6272 accelerated verification products. Use denominator-cleared primitive integer rows for pivot discovery, preserving the first-pivot policy and complete serialized certificate. This is verifier engineering; no new kappa or selected-frontier change.

On the unchanged #6272 48-by-64 rank-eight fixture, seven alternating fresh-process generate/save/check pairs reduce median local wall time from 727.3981 ms to 673.6825 ms (7.38%). All seven paired after times are lower; all 32 CLI commands, including warmups, pass with identical certificate bytes. This is one Windows/CPython workload, not an all-input or supplier speed claim.

Task and overlap

Started from and refreshed against canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, card replay-runtime, scope key b1dd41940c60d240b47a (refined scope). Checked live handoffs/Discussions and targeted all-state PR-body overlap. Local sparse-note/cache coverage is incomplete and explicitly recorded. #5789 supplies the interface; #6272 changes the product checker, while this change accelerates pivot discovery.

The review note records the proof, exact sources, overlap limits and open questions.

Validation

• Eight rank tests pass normally and with -I -O -W error, including 154 new exact pivot/full-certificate comparisons against the Fraction policy. Existing independent minor-enumeration and damaged-certificate/input-byte controls remain.
• Reduce complete rational-rank certificate time by 54.94% (no κ) #6272's unchanged verifier passes normally and optimized: 591 complete certificate equalities, 802 product/mutation cases, 24 rejections.
• The selected binding-only baseline and git diff --check pass.
• The added opt-in benchmark and committed JSON receipt bind both source versions, fixture, parser and output hashes. Published tree ca8b340c747b1b5dd752cd76a8e562b1e7f73687 exactly matches the staged local tree.

python -B -W error -m unittest discover -s tests -p test_research_rational_rank.py -v
python -I -O -W error -B tests/test_research_rational_rank.py -v
python -W error -B research/integer-rank-product-checks/verify.py
python -O -W error -B research/integer-rank-product-checks/verify.py

Environment: Windows 11 build 26300, CPython 3.12.10, 16 logical CPUs. No full supplier replay, complete multiplication proof, independent human review, or hosted CI success is claimed.

Credit

Credit Ali Romman's #5789 interface, Nick Petalas's #6272 product optimization and benchmark fixture, and classical exact elimination. Prepared for josecarlosgonzalezroque-design with substantial OpenAI Codex assistance, one agent; Apache-2.0. Frozen packages and the selected frontier are unchanged.

Open next questions

Measure a named consumer's source-bound matrices before making a production-throughput claim. Separately compare high-rank matrices with unrelated denominators to assess peak memory and runtime; the present measurement establishes neither a worst-case bound nor a fixed-memory backend contract.