[petalas]

Summary

Scoped proof with exact finite corroboration; no new kappa. Any bilinear formula for b independent unrestricted S3 convolutions needs at least 10b scalar products if its coefficient ring admits reduction to a characteristic-three field. Joint formulas may mix blocks freely. This includes dyadic, dyadic-Gaussian and eighth-root dyadic coefficients.

Characteristic-three S3 algebra: dimension 6, nilpotent ideal dimension 4,
quotient K x K, two maximal two-sided ideals.
Alder--Strassen: r >= 2(6b) - 2b = 10b.

Every rational formula using fewer than 10b products therefore has a coefficient whose reduced denominator is divisible by three. This rules out retaining Cinna Davis's PR #2224 nine-product count through a purely dyadic bilinear replacement, even across joint batches. Its rational construction and prior ceiling are cited, not rechecked. The new contribution is this source-specific denominator obstruction. The Alder--Strassen theorem and specialization method are classical and credited in the proof.

The result was produced by Nick Petalas. Credit Cinna Davis's S3 mechanism and retained Strassen/representation-theory lineage, and Alder, Strassen and Bläser for the classical algebra-complexity results.

Evidence

• Before: Research: exact rank-nine S3 convolution block screen #2224 supplies nine rational products with an unpaid denominator-six implementation question. After: a written reduction proof excludes any nine-product replacement over the stated coefficient rings.
• The new standard-library verifier checks all 36 group-basis products, radical dimensions (4,2,0), 64 radical triple products, all 729 elements of F3[S3] and their left-multiplication ranks, 324 units, coefficient residue maps and eight rejection controls.
• Warning-strict normal and optimized direct checks and both registered commands pass locally on CPython 3.12.15. Each registered argv was timed locally at under 0.1 seconds. Normal and optimized generated evidence agrees. Receipt-path rejection, AST parsing, source-history/committed-byte preflight and whitespace checks pass.
• The selected binding-only baseline passes. The new verifier imports no predecessor scientific checker. Only research/s3-three-local-obstruction/ is added.

From the repository root, reproduce with the available interpreter:

/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -B research/s3-three-local-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -W error -O -B research/s3-three-local-obstruction/verify.py
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/research_package.py --package research/s3-three-local-obstruction
/Users/nick/git/integer-mult-bounds/.venv/bin/python3.12 -B scripts/check_research_commit.py --package research/s3-three-local-obstruction --check-source-history

Other machines may substitute their absolute CPython 3.12 interpreter path. Expected markers: PASS_S3_THREE_LOCAL_OBSTRUCTION, PASS_REGISTERED_RESEARCH_PACKAGE, and PASS_RESEARCH_COMMIT_BYTES. PROOF.md separates the written all-block theorem from the finite checks and the cited general theorem. An optional absolute --receipt path outside the checkout emits a generated source-hash-bound receipt.

Limitations and unrun obligations

Ten is a lower bound, not an attained exact dyadic rank. The model is a sum of products of separate left/right linear forms. Exact division by three on known-divisible restricted intermediates, nonlinear straight-line algorithms, arbitrary complex coefficients, approximation and border rank remain outside the result. Free linear scratch and dirty auxiliary inputs do not evade the bound.

The Alder--Strassen theorem is a cited mathematical premise; this package does not formally prove it. No independent human or proof-kernel acceptance, actual supplier placement, full signed invoice, native implementation, precision/tape schedule, selected cold replay or complete multiplication theorem is claimed. The selected frontier is unchanged.

Open next questions

1. Determine the exact bilinear rank over Z[1/2]. Start with a ten-product construction or a stronger characteristic-three obstruction; attainability remains open.
2. Price a known-exact division-by-three implementation of Research: exact rank-nine S3 convolution block screen #2224's numerator schedule. Establish its intermediate divisibility and physical source placement, quotient, storage, precision and restoration costs before claiming a supplier gain.

Neither follow-on is reserved.

Related open work

• #3893 studies an S3 register-word endpoint over F2 and its paid-call floor. This result studies bilinear convolution multiplication and characteristic-three specialization; no endpoint count is recomputed or compared.
• #5947 studies restored dirty roles for another F2[S3] register-word endpoint. Its call minimum is a different quantity from this bilinear product floor; no extra computation was run to check it.

#2224 and the older literal affine-data placement obstruction #3336 are merged predecessors. The full open-PR title/body search found no other open S3 convolution product-rank result. Sparse local-note coverage was supplemented with whole-main README and merged-PR searches; this is overlap evidence, not a novelty certificate.

The completion refresh searched the titles and bodies of all 14 other open account PRs created since the advance CLAIM at 2026-10-27T22:02:10Z. None computes a competing S3 convolution bilinear product count. One adjacent use of convolution is distinguished:

• #6024 removes endpoint reversals from a fixed-tape Fourier convolution caller. It concerns record-permutation routing, rather than multiplication in the nonabelian S3 group algebra, and computes no S3 bilinear product floor.

No extra scientific computations were run to check other contributions or compare numbers.

CI notes

Both hosted checks passed at exact head ec9c186c035bbb94e6750820ea968426aa40ceb9 in run 40592371168: plan in 4m44s and replay (research/s3-three-local-obstruction) in 1m19s. The completed check list was read once; the run is completed/success and its head equals the PR head.

Hosted replay certifies both registered bounded finite checks. All local scientific, metadata and committed-byte/source-history checks listed above also passed. The general Alder--Strassen theorem remains a cited premise, and the all-block statement remains a written proof; independent human and proof-kernel acceptance are unclaimed.

Merge danger

Door: two-way.

Blast radius: isolated.

One opt-in proof/verifier package. Predecessor sources, shared code and the selected record retain their bytes.