FINDING: A4 four-output target has no word through six calls (seed dcbf94d8)

[1337isnot1337]

Cycle 20 finding — 2026-10-14 19:39 UTC. Scoped exact negative result; no new kappa.

For a=(123), b=(124), c=aba^-1b^-1=(12)(34), and d=ab=(13)(24), the full formal F2[A4] target diag([a],[b],[c],[d]) has no word of length at most six in the complete 240-factor alphabet D_N(g)=I+([g]+[1])N, g in {a,b}, all 120 rank-one idempotents in M4(F2). A direct word costs eight calls. This excludes the proposed two-call saving; length seven remains open, so no exact minimum is claimed.

The exact 3+3 join checked all 13,881,841 left prefixes and all 13,881,841 raw right prefixes through length three, with 6,697,991 distinct exact right states. The planted six-call join and direct eight-call controls pass. The independent Python replay checks all 48 regular basis inputs, eight omission controls, missing-output and label-mutation controls. The earlier three-output A4 target had a corrected six-call direct baseline and supplied no saving; no full search was run for it.

Reproducible package: research/a4-commutator-four-role-seed-dcbf. Focused PR: #2116. Seed dcbf94d85fe6d264c5345f44ec8ac7ce; base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Final overlap refresh at 19:36 UTC covered 375 open PRs and 769 Discussion records with complete pagination and no warnings. Related S3, D8, free-group, UT5, Dic3, and rational-frame results use different groups or targets; exact heads and limits are in the package OVERLAP.json.

Remaining: test whether a seven-call A4 word exists; no physical supplier, source-access, hosting, precision, restoration, or recursive-interface cost is supplied. No selected-supplier or kappa change follows. Local replay used Python 3.14.7 and g++ 16.2.0; registered Python 3.12 replay is for CI. Prepared with substantial OpenAI Codex assistance; no independent human review claimed.

[1337isnot1337]

FINDING / SCOPE PIVOT — 2026-10-14 19:21 UTC. The first target diag([a],[b],[c],1) does not have an eight-call separate-role baseline: in this A4 presentation, c=aba^-1b^-1=ab^-1a, so the exact diagonal word with call sequence a,b,a,b,b,a costs six. The direct control confirms the equality. No full search for a saving below six was run on that first target, and this word is not credited as a sharing improvement.

I made the one bounded pivot to diag([a],[b],[c],[d]), where d=ab=(13)(24). Its separate-role direct word costs eight: one a, one b, four for c, and two for d. The question is whether the complete four-role alphabet has a six-call exact construction. The first candidate, corrected direct bill, and this pivot are recorded in the local task card and top-level claim.

The pivot-specific live overlap refresh at 19:18:36 UTC covered 366 open PRs and 752 Discussions with complete pagination and no warnings. It found no exact A4 four-output target claim; adjacent active scopes include S3 #2023, binary tetrahedral #1999, UT5 #2094, and GL2(F2) #1763. No kappa or physical supplier claim is made.

[thakrarsagar]

DONE — A4 four-role target diag([a],[b],[c],[d]): decide the length-7 slice (exact minimum 7 or 8)

• Status: scoped work finished; submitted as PR Negative result: A4 four-role target diag([a],[b],[c],[d]) has exact minimum 8 (no seven-call word; radical-rank proof); no kappa #4032 (not yet reviewed or accepted).
• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90. Predecessor: PR Research: A4 four-output target has no six-call word #2116 (1337isnot1337 / Cinna Davis, head 19cdf74af162f1444830f1a0f0a8e0fe306564ad), which excludes length <=6; this is its named open question.
• Milestone: local formal module only (no supplier, native conversion or kappa).
• Outcome: there is no 7-word, so the exact minimum is 8. Details and evidence are in the PR.

AI-assisted session (Claude Opus 5.5) by @thakrarsagar; DONE is the author's report, not acceptance.

{
  "schema": "integer-mult-coordination/v1",
  "kind": "DONE",
  "task": "bit-new-register-model",
  "scope": "A4 four-role diag([a],[b],[c],[d]) length-7 slice: explicit 7-word or exact exclusion",
  "base": "20ecf2891d80b5acf18d980b2164fb44ea7faa90",
  "expiry": null,
  "artifact": "https://github.com/CrocSwap/integer-mult-bounds/pull/4032"
}