[1337isnot1337]

Result

Adds an exact rational rank-9 bilinear construction for regular convolution in the nonabelian group algebra Q[S3], then screens its conditional local profile against the pinned selected complex ceiling. The algebra decomposes as Q × Q × M2(Q): two scalar products plus Strassen's seven-product 2×2 matrix formula compute the six-coordinate convolution, which has 36 separate pair products. No minimum-rank theorem, selected-provider placement, supplier adoption, or new κ is claimed.

Exact evidence

RESULT.json records all nine left and right linear forms and six-coordinate output vectors. The verifier checks the standard-representation homomorphism and exact tensor identity on all 36 group-basis pairs, 256 deterministic integer and dirty-output fixtures, output restoration, and an omitted-product negative control. The input/output transform matrices each have rational rank 6; the output coefficients have denominator at most 6. The explicit linear invoice is recorded separately from recursive child charges.

From repository root:

python3 -I -B research/complex-s3-convolution-seed8b17/verify.py
python3 scripts/research_package.py --package research/complex-s3-convolution-seed8b17 --check-only

Expected direct marker:

PASS_S3_GROUP_ALGEBRA_RANK9 basis_pairs=36 integer_dirty_fixtures=256 controls=1 rank1_capacity=2412 max_blocks=67 zero_overhead_threshold=8 local_only

Local direct verification passed under CPython 3.14.7, including -O. The repository's registered research replay requires Python 3.12; that interpreter is unavailable in this environment, so only package metadata/pin check and direct replay were completed locally.

Conditional ceiling screen and limits

One S3 block would replace 36 rank-one products by a rank-nine child, releasing 27 rank-mass units to the favorable rank-51 idle remainder. Under the pinned m=73, 896-family, nine-stage, centre-64 relaxation, seven ideal blocks fail the 5% ceiling target and eight pass: the candidate ceiling bracket is [0.001775158769926, 0.001775158769927], against target 0.00177455681454345. After eight blocks only one added rank-one-equivalent child total still passes; two fail.

This is a zero-implementation-charge screen. The selected trace has 2,412 rank-one children per orientation, which gives a count-only capacity of 67, but this package establishes no actual S3 support occurrence. The selected source/target map, scratch and frame schedule, output fanout, exact division/precision handling, hosting, restoration, and interface charges are open. The conditional profile must not be read as a realized improvement.

The package pins the exact assignment base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4, seed 8b17cb8b1a12a123db4fca0f9356197b, and selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Predecessors and overlap

This uses a nonabelian matrix component, distinct from the rational C3 cyclic-convolution claim in Discussion #2057 and the Karatsuba block claim in Discussion #1987. The fixed selected source lineage is PR #1094. Final overlap refresh at 2026-10-15 00:27 UTC covered 455 open PRs and 911 discussions with complete pagination; exact S3/nonabelian-convolution terms found no other matching PR or discussion.

The active scope claim and handoff are in Discussion #1167, comment 18936773. Substantial OpenAI Codex assistance is disclosed in the package. New package content is Apache-2.0; frozen sources are unchanged.