[petalas]

Summary

Eliminate 15,400 exact zero scalar products from the complete fixed q-Toeplitz replay supplied by James Reed's PR3865. The new opt-in multiplier uses 6,650 products instead of 22,050 and preserves every one of 4,550 matrix entries across all 300 multiplications.

 for each matrix entry, in increasing summation-index order
-  multiply and add every operand pair
+  multiply and add only when both exact rational operands are nonzero

Claim type: finite certificate of an implementation improvement, supported by a direct exact-arithmetic equivalence argument. No new factorization theorem, supplier, physical speedup or kappa is claimed. All thirty matrices, five rational parameters, factor/Gaussian/inverse identities, width/norm tests and final receipt checks remain. The predecessor and its default command are unchanged.

The package is produced by Nick Petalas. Credit James Reed / jwr404 for the complete finite replay; Chafik Boukhalfa / PR489, Alexey Kuznetsov and Jose Manuel / PR3851 for the retained factorization and Gaussian context. Sparse multiplication is classical. The new contribution is the fixed replay's exact arithmetic reduction, implementation and evidence.

Evidence

At head dbdf3f7db12de975e76aa4677aa5053ae9153ab4, warning-strict CPython 3.12.15, macOS arm64:

• All 300 complete products agree entry by entry. A second operand enumeration agrees with the closed operation count. Every original scientific check runs.
• All 6,561 ordered rational two-by-two matrix pairs, dense boundary fixtures at orders one through six and an exact tiny nonzero product pass.
• Four invalid shapes, all three actual retained-source corruptions and two faulty replacement kernels reject. Normal and optimized checks use explicit exceptions.
• The four registered commands pass. Direct timed durations are 0.09, 0.10, 0.02 and 0.02 seconds; the complete registered package command took 0.68 seconds. These are timeout checks, not benchmark samples.
• Committed bytes/modes and full source-history preflight pass for all twelve package/input files. The official export passes integrity and three-input coverage checks. Normal and optimized conformance replay pass from its read-only Git-free extraction with identical receipts.
• Fifteen alternating complete fresh-process timing pairs produce zero errors and identical stdout. Medians are 64.291 ms and 58.847 ms, an 8.47% descriptive median reduction. Ranges are 62.343 to 99.169 ms and 57.299 to 69.196 ms; fourteen pairs improve. The exact operation saving is independent of host timing. Startup and concurrent work limit timing precision. Raw generated samples, timing boundaries and code hashes are in BENCHMARK.json.

From the worktree root, exact reproduction commands:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/q-toeplitz-zero-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/q-toeplitz-zero-products/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/q-toeplitz-zero-products/replay.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/q-toeplitz-zero-products/replay.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/q-toeplitz-zero-products
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/q-toeplitz-zero-products --check-source-history --check-modes

Any installed CPython 3.12 may replace the interpreter path. Set TMPDIR to a scratch directory to direct temporary control copies there. The standalone verifier emits a generated hash-bound receipt and leaves no outputs in the checkout. Optional fresh-process timings, using a fresh output path outside the checkout:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/q-toeplitz-zero-products/benchmark.py --rounds 15 --output /tmp/q-toeplitz-benchmark.json

Expected markers are PASS_Q_TOEPLITZ_ZERO_PRODUCTS, PASS_Q_TOEPLITZ_FACTORIZATION, PASS_REGISTERED_RESEARCH_PACKAGE and PASS_RESEARCH_COMMIT_BYTES.

Limitations and unrun obligations

This covers the fixed PR3865 exact rational suite at orders one through six. Its mathematical factorization is cited prior work, not a new audit deliverable. No larger matrix domain, production Gaussian core, native multiplier, physical speedup, selected cold replay, Lean, Docker, independent human review or proof-kernel review is claimed. The five selected framework assumptions remain and the selected frontier is unchanged. No package correctness or registered check remains unrun locally.

Scientific base: canonical main 7c49fe32c55de02c2c873655c804b16e4c3fef0a, priority revision 8, selected PR6030. The replay-runtime card is refined to this fixed complete command. Live handoffs/open PRs, recent account work, local reservations, main and merged q-Toeplitz/formula searches were checked at selection and submission. The complete overlap snapshot covered 555 open PRs, 6,596 Discussion records and 8,917 main note records; arbitrary code diffs and unpublished work remain limits. Advance CLAIM.

Related open work

PR6570 removes exact zero products from binary incidence/kernel dot products in the separate frame-kernel-sparse-products complete replay. This PR applies the same classical sparsity principle to rational matrix multiplication in PR3865's q-Toeplitz replay. The workloads differ, so their counts and timings do not estimate the same quantity.

The completion refresh screened this account's open PRs created since the advance CLAIM, including eleven other contributions. None computes the same fixed q-Toeplitz replay quantity. No cross-workload numerical comparison or additional predecessor computation was performed. PR3865 is the merged predecessor, credited above.

Open next questions

Does exact zero-product omission improve a larger currently used rational replay with comparably sparse factors? First profile one different complete command and enumerate its zero operands. A larger workload and its complete equivalence contract remain to be established. No follow-up is reserved.

CI notes

Both hosted checks, plan and replay (research/q-toeplitz-zero-products), passed at head dbdf3f7db12de975e76aa4677aa5053ae9153ab4 in run 41633790574. The completed checks were read once with gh pr checks 6606 --repo CrocSwap/integer-mult-bounds.

All four registered commands and the standalone verifier passed locally. Hosted replay certifies execution of the registered normal/optimized implementation-conformance checks. The local timing percentage, predecessor factorization theorem and complete multiplication theorem are not independently certified by these CI jobs. No check failed. Timing remains descriptive and is not a CI acceptance threshold.

Merge Danger

Two-way door. Blast radius: isolated. Only a new opt-in research package is added; existing source files, certificates and default commands are unchanged.