[petalas]

Summary

The complete PR4345 literal header-census command spends most of its time constructing conjugated requests with dense products. A new source-bound kernel uses the requests' rank-one/rank-two factors instead. Local median command time falls 47.30%, from 9.341 s to 4.923 s, with identical complete output.

all q^(m*m) matrices -> unchanged inverse -> direct low-rank conjugate
                                        -> unchanged guards and counters

Claim type: scoped implementation-preservation proof with finite conformance and local performance evidence. The contribution was produced by Nick Petalas. Credit Ryanson Jonathan / pinterbanget for PR4345's complete engine and the retained header/BIT lineage. The conjugation identities are classical. Existing counts are conformance outputs, not new mathematical results or audit deliverables. No new kappa or selected-frontier change.

Only research/low-rank-header-conjugation/ is added. The authenticated predecessor source is unchanged. The scope update records this implementation target.

Evidence

• Before: two dense matrix products per request. After: O(m^2) outer-product formulas for all eight canonical requests. The written proof preserves each conjugated entry, guard, complete matrix enumeration and output counter.
• Five alternating fresh-process pairs on the complete q=3,m=4 census, with 24,261,120 invertible headers each. Baseline range 9.005–10.028 s; candidate range 4.759–5.112 s. All ten outputs are byte-identical, with zero errors. Generated BENCHMARK.json binds the exact source, adapter, kernel, flags and raw samples. Compilation is excluded; the full census process and final output are timed. Measurements describe this macOS arm64 / Apple clang 21 host, with other research processes active.
• A separate three-second profile puts 1,474 of 2,572 samples at dense conjugation. The unchanged matrix scan, inversion and guard elimination limit the gain. Profiling is excluded from timing samples.
• Direct normal/optimized and the registered warning-strict command pass all eleven complete historical workloads over F2/F3/F4/F5. An entrywise harness checks 14,746,536 entries in 1,515,444 request matrices on 213,714 headers. Four semantic kernel mutations and altered retained source bytes reject.
• Each registered scientific command finishes locally in about 26 seconds, below its 240-second limit. PASS_LOW_RANK_HEADER_CONJUGATION, PASS_REGISTERED_RESEARCH_PACKAGE and metadata/pin checks pass. Committed-byte/source-history validation passes at e808800e784756a0eabf438e269bbe5005a0c75b for ten package files plus the declared C input. C builds use -O2 -std=c11 -Wall -Wextra -Werror -pedantic.

From the repository root, with CPython 3.12 and a C11 cc compiler:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/low-rank-header-conjugation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -O -W error -B research/low-rank-header-conjugation/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/low-rank-header-conjugation
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/low-rank-header-conjugation --check-source-history
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -I -W error -B research/low-rank-header-conjugation/benchmark.py

Another CPython 3.12 path can replace the displayed interpreter. Temporary C sources and binaries use an external temporary directory and are removed. Set TMPDIR to choose it. Timing is optional and has no CI performance threshold.

Limitations

This is a verifier implementation improvement, not independent validation of the predecessor's counting theorems. The proof covers the eight stated requests; executable conformance covers the eleven retained domains. The legacy engine's larger-size counter and CLI limitations are outside scope. No selected cold replay, native/compiler, Lean or Docker build ran. The five framework assumptions remain. Independent human and formal proof-kernel review are unclaimed.

Open next questions

Can a real consumer supply low-rank request factors directly and use this kernel without first constructing dense requests? Start with one complete caller while retaining its field representation, independent inputs and every guard. That caller and its benchmark are missing. Optimization stopped after this first measured improvement.

Related open work

No open PR studying PR4345's engine or this low-rank header-conjugation kernel was found in the refreshed source/title/body searches.

• #6396 studies a different G20 census, with no shared numerical quantity.
• #6414 accelerates PR6396's complete member-bank census by caching exact subspace containment. Its source, kernel and numerical quantities differ from this full-guard conjugation optimization.
• #6409 proves an adaptive corank-one pivot-swap bound. This package preserves the fixed full-guard census and makes no adaptive pivot claim.

The completion refresh inspected all ten account PRs still open and created since the CLAIM at 2026-11-02T21:22:08Z, including this PR. The other nine are #6406 through #6414. None computes a conflicting value for this same kernel or benchmark. No extra computation audited another contribution. Main README searches, recent account work and local reservations were checked; unpublished work and arbitrary historical diffs remain coverage limits.

CI notes

Both hosted checks, plan and replay (research/low-rank-header-conjugation), passed at exact head e808800e784756a0eabf438e269bbe5005a0c75b in run 41529685209. The completed checks were read once: plan took 4m25s and replay took 4m30s. Run metadata independently confirms the matching head and successful conclusion. All local checks above passed. Hosted CI certifies the registered finite conformance and source bindings; it does not certify the local timing percentage, independent scientific acceptance or a proof-kernel check.

Merge Danger

Door: two-way. One isolated package can be reverted.

Blast Radius: package. Shared tooling, predecessor sources and selected-frontier files are unchanged.