[petalas]

Summary

The complete exact rational-rank certificate produce/save/check path takes 54.94% less local wall time on a dense 48-by-64 rank-eight rational interface fixture. This contribution was produced by Nick Petalas. Claim type: scoped exact-arithmetic implementation proof with finite conformance and measured implementation improvement. No new kappa or selected-frontier change.

 verify all four certificate products
-  multiply and add Fraction values at every dot-product step
+  clear row/column denominators once
+  compare integer dot products by exact cross-multiplication
+  use the original predicate when a common denominator exceeds 16384 bits

The output format, pivot choices, factorization, complete left/right kernel certificates and rejection predicates remain. The first diagnostic can differ when several fields are corrupt. Ali Romman / #5789 retains the exact interface, complete certificate and product checker; classical denominator clearing and elimination retain their credit. The original script is retained byte-for-byte for comparison. The change is implementation work, not a predecessor scientific audit.

Evidence

Before: median 440.23 ms, range 434.41 to 461.09 ms. After: median 198.39 ms, range 195.43 to 210.60 ms. Five alternating fresh-process pairs, CPython 3.12.15 on macOS arm64, shared 18-logical-CPU host. Each timed bundle produces the complete certificate, saves it and checks it in a second process. All twenty commands succeed and all ten certificate byte hashes agree. The untimed profile attributes most original certificate cost to Fraction product checks. Fixture generation and profiling are outside timing.

Raw samples, milliseconds:

original: 434.406917, 450.502708, 461.090000, 436.349834, 440.234625
integer:  198.391583, 201.852167, 210.603875, 197.723625, 195.430375

The generated benchmark report binds measured sources to hashes. Certificate SHA256: 48b99c3926885f307aef55c25bfed6336fe8c734834b3ce9805b24f63be81f26. The package README contains the arithmetic equivalence proof, complete environment and measurement details.

Local checks pass:

• 591 complete certificate equalities, 802 product/mutant cases, 24 corrupted or malformed certificate rejections; warning-strict normal and optimized registered checks take 0.95 and 0.96 seconds respectively.
• Seven relevant unit tests in normal and optimized Python.
• Committed bytes, executable modes and declared source ancestry at 46c937e504abc9f463dd66c34aabfc33a5abaa05.
• Official ten-file export integrity; two matching read-only Git-free direct replays; two source-byte rejection controls and one disabled-product-check rejection. Exported bytes and modes remain unchanged after replay.

From the repository root:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/integer-rank-product-checks/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B research/integer-rank-product-checks/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/integer-rank-product-checks
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/integer-rank-product-checks --check-source-history --check-modes
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B -m unittest discover -s tests -p test_research_rational_rank.py -v
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -O -W error -B -m unittest discover -s tests -p test_research_rational_rank.py -v
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/integer-rank-product-checks/benchmark.py --scratch /absolute/existing/external/scratch

Another CPython 3.12 executable can replace the absolute interpreter path. The direct verifier prints PASS_INTEGER_RANK_PRODUCTS with a generated source-bound receipt. The registered replay prints PASS_REGISTERED_RESEARCH_PACKAGE. Timing remains opt-in and writes only to the supplied external scratch directory.

Limitations

The universal claim is exact product-predicate equivalence. The speed result covers one synthetic rational interface fixture and this local environment; no production supplier or all-input speed guarantee is claimed. Large coprime denominator products use the previous exact arithmetic. Python allocations remain a documented managed-runtime deviation. No Lean, Docker, selected-supplier cold replay, complete multiplication proof, independent human review or formal proof-kernel acceptance is claimed. All five selected framework assumptions remain.

Open next questions

• Does this full-path improvement persist on a named consumer's source-bound interface matrices? Export one actual matrix and apply the retained paired harness.
• Can a fixed-capacity backend derive its scale threshold from a proved integer-memory budget? Such a backend contract is still needed.

Related open work

The completion refresh inspected twelve other open account PRs created since the advance CLAIM. No other refreshed description studies this exact PR5789 certificate checker or reports a competing value for the retained 48-by-64 benchmark. These are bounded title/body checks, not proof of global novelty; no extra computation checked another contribution.

• #6270 uses denominator clearing and fraction-free elimination in a bounded rational-box optimizer. It changes a different source script and full caller; this PR changes rational-rank certificate product checking. The benchmarks and certificate counts are different and are not compared.
• #6274 replaces literal binary moved-subspace intersections with basis-rank computations in the PR4841 mapped-contact caller. It neither changes this rational-rank checker nor computes this benchmark.
• #6263 speeds a separate order-17 matrix replay, with a different caller and benchmark. Its scientific result is neither replayed nor compared here.

CI notes

Both hosted checks, plan and replay (research/integer-rank-product-checks), passed at commit 46c937e504abc9f463dd66c34aabfc33a5abaa05 in run 41067167314. The completed checks were read once; run metadata separately confirms status completed, conclusion success and the matching head. The plan job took 4m8s and the replay job took 1m26s.

Hosted CI certifies the repository onboarding checks and registered finite replay at this commit. The performance measurement remains local, and passing CI does not constitute independent acceptance of the written universal equivalence proof. All registered commands also passed locally; no proof obligation was delegated to an unrun hosted job.

Merge Danger

Door: two-way. Blast radius: rational-rank certificate checking. The shared tool uses exact arithmetic and retains its serialized certificate contract; reverting restores the previous product implementation. The new package is independently opt-in.