[thakrarsagar]

Statement

In the formal F2[A4] four-role model of #2116, the target T = diag([a],[b],[c],[d]) has exact minimum 8. Here a=(123), b=(124), c=(12)(34) and d=ab=(13)(24). There is no seven-call word, so the model gives no cross-role paid-call saving for this target. No kappa claim.

Length	Status	Evidence
1-5	impossible	written proof (Lemmas 1-2); also both joins
6	impossible	exhaustive joins A and B (reproduces #2116); not covered by the written lemmas
7	impossible	written proof (Lemmas 1-2); also both exhaustive joins
8	realized	direct word, replayed exactly on all 48 basis inputs

The kinds of evidence differ. Length 7 and lengths up to 5 are excluded by a written proof. Length 6 is excluded by a finite exhaustive certificate. Length 8 is an explicit construction. None of this is a multiplication theorem, a supplier result or a kappa change.

Written proof.

• Lemma 1 (C3 determinant): det over F2[C3] gives n_a = n_b (mod 3).
• Lemma 2 (radical rank, new here):
  • Write b = a^2 d. Every word then factors as W = A * prod(I + C_k^-1 (d+1) N_k C_k), with A in GL4(F2[<a>]).
  • F2[<a>] meets the radical J only in 0, so A is the unique lift diag(a, a^2, 1, 1). The defect is diag(1, d, c, d).
  • A twisted functional Phi: J/J^2 -> F4 satisfies Phi(r y s) = aug(r) Phi(y) psi(s). It turns the defect into a sum of n_b matrices of rank at most 1 that has rank 3. So n_b >= 3. Base b gives n_a >= 3 in the same way.
• Together the two lemmas exclude (2,5) and (5,2) at length 7, and every length up to 5. Length 6 would need (3,3).

The per-label floor n_b >= rank_F4 Phi(L^-1 T - I) holds in this F2[A4] alphabet for any number of roles. Whether it extends beyond A4 = V x| C3 is open question 2; it is not claimed here.

Milestone and decision informed

• Milestone: a local formal module. There is no supplier, no native conversion and no kappa.
• Question decided: the length-7 slice that Research: A4 four-output target has no six-call word #2116 names as its open question.
• Decision: the A4 four-role endpoint gives no formal saving, so this target gives no reason to price a native conversion of F2[A4] calls. The package extends the existing direct-optimal pattern of the formal finite-group family (Q8 Prove four-call Q8 central-minus-one minimum for all r #2047/Research exact Q8 three-role target minima #2248, SL2(F3) Research: exact five-call minimum for binary tetrahedral mixed triple (no new kappa) #2010) to one more target. It changes no current lead, supplier or selected result.

Base and pins

• Base: 20ecf2891d80b5acf18d980b2164fb44ea7faa90, under priority revision 6. Canonical main had not moved at submission.
• Pinned input: Research: A4 four-output target has no six-call word #2116's RESULT.json (git blob and sha256 in SOURCES.json).
• Package: research/a4-seven-call-radical-rank/, 7 new files. Nothing else is touched.

Reproduce

python3 -I -B research/a4-seven-call-radical-rank/verify.py
python3 scripts/research_package.py --package research/a4-seven-call-radical-rank --check-only

The verifier uses only the standard library and compiles two C joins with cc -O3. It ends with PASS_A4_FOUR_ROLE_EXACT_MINIMUM_EIGHT, which reports:

• Direct word: 8 calls, 48/48 basis inputs.
• Lemma: defect rank 3 in both bases.
• Each join: 6,655,830 minimal-length-3 classes expanded and 1,597,399,200 children looked up, with no word of length up to 7.

The second command prints PASS_PACKAGE_METADATA_AND_PINS. A run takes about 80-90 s wall time with about 369 MB peak RSS (Python 3.14.4, gcc 15.2.0). The generic runner also passed under a private Python 3.12.15 (PASS_REGISTERED_RESEARCH_PACKAGE).

Certificates and negative controls

• Two separately implemented exhaustive joins (same AI-assisted session).
  • Join A uses 4x4 matrices of 12-bit masks, left-inverse enumeration and a GF(2)-linear incremental hash.
  • Join B uses graded 12 x 16-bit coefficient matrices, right-inverse enumeration, a left-multiplication P<=3 build and a nonlinear hash.
  • Both find P<=3 = 1/240/41,920/6,655,830 (total 6,697,991). This matches Research: A4 four-output target has no six-call word #2116's pinned right_distinct_exact_states, and the verifier asserts it.
• Verifier checks of the proof. Every proof step is checked numerically: the factor split for all 120 idempotents, dim J = 9, unique lifts, the twisted bimodule law on all 512 elements of J, Phi(J^2) = 0, the step-5 identity on random words, and the determinant formula on random words.
• Negative and positive controls.
  • Each of the 8 single-call omissions of the direct word is rejected.
  • The label mutation (final call b -> a) is rejected.
  • Three-output control diag([a],[b],[c],1): the defect rank drops to 2, and both joins find (3,3) 6-words that Python replays exactly.
  • Planted 7-words (2 per join) are found and replayed.

Scope, assumptions, unresolved obligations

• The model is inherited unchanged from Research: A4 four-output target has no six-call word #2116: 240 rank-one idempotent factors D_N(g) = I + ([g]+[1])N with g in {a,b}, unit prices, four roles, dirty inputs, no workspace and exact equality of every formal coefficient.
• Free scalar GL4(F2) mixing and role permutations do not help. Conjugating them to the left leaves a scalar factor that augmentation forces to be I.
• Length 6 is excluded only by exhaustive search. A written proof of it is open question 1.
• Unpriced (unknown, not zero): native conversion, restored workspace, a physical caller, chronology, hosting, precision, recursive transfer and supplier admission.

Search record

• Domain: all words of length up to 7 over the 240-factor alphabet. They are organized as X·Y and X·f·Y, with X and Y in P<=3.
• Coverage is complete in each join. Hashing is deterministic.
• There is no C3 or coset pruning and no early exit. Sampling is used only in the controls.
• No timeouts and no memory limits were hit.

Credit and concurrent work

• The model, alphabet, direct word, C3 screen and the original length-6 exclusion come from Research: A4 four-output target has no six-call word #2116 (research/a4-commutator-four-role-seed-dcbf, GitHub @1337isnot1337, commit author Cinna Davis, head 19cdf74a). This PR answers that package's named open question.
• Prior untwisted J/J^2 arguments, all by @1337isnot1337: Q8 in Prove four-call Q8 central-minus-one minimum for all r #2047 and Research exact Q8 three-role target minima #2248, and the abelian rational-frame floor in Research: exact rational frame floor on four roles #2085. Research: exact rational frame floor on four roles #2085 is the closest analogue of the rank-count step.
• Open Add S3 three-register rank lower bound #3893 (@1337isnot1337) is a concurrent, different rank floor for S3.
• To our knowledge (a search of research/*/README.md and open PRs at the base), three ingredients are new: the semisimple split in a non-local group algebra, the twisted functional, and the use of both bases together with the C3 determinant.

Claim: #2095 (comment)

Review provenance

The package was reviewed by four independent AI review passes in the same loop: reproducibility, adversarial mathematics, provenance and referee. They raised only minor issues, all of which were addressed in one revision. The math pass also checked the result with its own construction of A4 and its own exact BFS for length up to 6. No human review is claimed.

Attribution

Prepared by Sagar Thakrar (@thakrarsagar) with substantial AI assistance from Claude (Anthropic, Claude Opus 5.5 via Claude Code) in an automated research loop.

🤖 Generated with Claude Code