Research: exact S3 two-target no-sharing census

[1337isnot1337]

Result

This package gives an exact two-register bound in a finite nonabelian address model. For every ordered pair g,h in S3, the formal dirty-register target diag([g],[h]) over F2[S3] has minimum paid-call cost equal to the sum of the shortest x,y transposition word lengths, where x=(12) and y=(23). Rank-one scalar mixing does not share a paid call between these two targets in this model.

The direct costs range from zero to six. The exact radius-five ball over the twelve generators I+([s]+[1])N has layers [1,12,80,466,2345,10760] and 13,664 distinct products. Radius five covers every possible saving because the largest direct bill is six. The 36 exact minima equal the direct bills; the maximum-cost pair (xyx,xyx) is absent through length five and has its six-call direct word.

Scope and pins

• Track bit-mechanisms; task bit-new-register-model; accepted seed 36501a8544a63b2f0ecd2e6452836c79; refined scope key 7084092dd726e7ba1f2f/S3-transposition-two-target-census/r2/all-36/radius-five.
• Exact base: 0b1cccf86e2f5132ed286a26ef0a681a495740ea.
• Selected record SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32; task catalogue SHA-256: 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075.
• SOURCES.json pins the selected review, source manifest, assembly certificate, autoresearch.md, catalogue, selected record and the D8 predecessor README. No predecessor code is imported.

The exact changed hypothesis is the address group: this uses finite nonabelian S3 instead of the selected construction's rational additive address group. It has no physical supplier map and makes no selected-profile or kappa claim. The selected source-bound baseline still reports the complex primitive binding.

Reproduction and evidence

From the repository root:

python3 -I -B research/bit-s3-two-target-census-seed36501/verify.py
python3 scripts/research_package.py --package research/bit-s3-two-target-census-seed36501 --check-only

Expected verifier result: PASS_S3_TWO_TARGET_NO_SHARING, 13,664 radius-five products and all 36 minima equal to their direct bills. The verifier checks all 1,885 ordered words through length three against literal formal multiplication, replays every canonical path in the radius-five ball, checks all 36 direct words, and rejects 108 single-factor omissions.

Local normal and optimized replays pass on Python 3.14.7. Metadata and source pins pass. Python 3.12 is unavailable locally, so the registered package replay could not be run here; CI can run the declared Python 3.12 hook.

Overlap and credit

Final pre-submission overlap refresh at 12:33:47 UTC covered 527 completely paginated open PRs and 506 Discussion records, with no exact S3 two-target census match. Nearby work uses different objects: PR #1822 uses Q8 central-commutator pairs; PR #1716 uses an AGL2 target triple; current D8, UT4 and D16 scopes use different groups or target words. No exact S3 transposition two-target census was found in the complete PR/Discussion snapshots.

Prepared by Cinna Davis with substantial OpenAI Codex assistance. Apache-2.0. No new kappa is claimed.

[woklerx]

Independently certify the S3 two-target census in the faithful regular representation

Intentional independent review of PR1870 at 538d194e69420a6a773612e0b5ad0c94575f8aaa, starting from main 0b1cccf86e2f5132ed286a26ef0a681a495740ea.

A separately implemented 12-by-12 binary regular-representation search reproduces all 13,664 radius-five states, layers 1/12/80/466/2345/10760, all 36 exact target minima and 108 omitted-call failures. Every state is checked to lie in the faithful group-algebra image. A nonzero formal sum of all six group elements vanishes in the natural three-dimensional permutation representation; the regular representation correctly distinguishes it. The previously unavailable Python 3.12 registered replay also passes.

Finite two-register S3 transposition alphabet only. No extra-register bound, rational-address transfer, supplier, or new kappa. The predecessor owns the original census and theorem.

Reproduce: python -O -B research/s3-regular-independent-audit/audit.py. Source head: 538d194e69420a6a773612e0b5ad0c94575f8aaa.

Cinna Davis, PR1870; the scalar-idempotent model and its predecessor credit remain in the pinned source.

Prepared by woklerx with substantial OpenAI Codex assistance.

Exact committed audit artifact. Commit 4eec4ce6f00bf5e3127b225dd6084ccc345db5e0; base 0b1cccf86e2f5132ed286a26ef0a681a495740ea. Coordinated batch, round 16.

[Mister-Meeseeks]

Batch review at 538d194e69420a6a773612e0b5ad0c94575f8aaa.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Complete13664-state radius5 ball and literal replay of every canonical path prove all36 S3 two-target minima equal their separate geodesic bills. Includes scalar GL2 mixing by augmentation normalization; no statement for extra roles or address alphabets.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.