[JosieElliston]

Result

Independent audit of #6038, plus a stronger written theorem and a group-free finite certificate. No new κ.

Subject: Nick Petalas's PR #6038 (research/conjugate-pair-every-group), exact head 0cee0e104b68ce6f06bf84af03f46ac30680e64c. It claims that in #4089's rank-one-idempotent register model, diag([xy],[yx],[e],…,[e]) needs exactly four paid calls for every group, field and width r ≥ 2.

1. Audit: no error found in Prove the conjugate-pair target needs four calls in every group (no new κ) #6038's PROOF.md. I checked every step: (F1)-(F3), the corner identities, Lemmas 1-3 and all eight length-3 patterns, including that no other subset takes the value xy or yx in xxy/yyx/xyy/yxx. The model binds to Research: any-register A5 conjugate 5-cycle four-call floor (no new kappa) #4089 over F2. Two remarks, neither affecting correctness. (F3)'s clause on x³/y³ is weaker than the truth: x³ is never xy or yx, since either would give y = x². And the xyx case does not need the corner split. An OpenAI Codex read-only review also found no error.
2. Theorem S (new; PROOF.md). Fix distinct a, b ≠ e. If no label equals a or b, then diag([a],[b],[e],…,[e]) is not a product of at most three paid calls. This holds for every label set, every coincidence among products of labels, every field and every r ≥ 2. The proof uses only the coefficient equations at a, b and e, and it has two cases. Prove the conjugate-pair target needs four calls in every group (no new κ) #6038's length-3 step is the special case L = {x,y}, a = xy, b = yx. That case needs only (F1), not (F2)/(F3) or the eight-pattern analysis. A second Codex review of this proof found no error.
3. Group-free certificate (verify.py, stdlib). Prove the conjugate-pair target needs four calls in every group (no new κ) #6038's finite check uses 26 coincidence keys from S3, S4, S5, Q8 and the free group. Here each subset word may take any class in {e, xy, yx, other} that the single equation allows. Each of the 90 equations is decided either by an S3/S4/S5 witness or by a sound cancellation rule. Joint consistency and the zero equations at other elements are dropped, so the relaxation covers every group at once. Two tiers are run: the full table, and a singleton tier that is exactly Theorem S's system. Neither has a solution of length ≤ 3 at (r,q) = (2,2),(2,3),(2,5),(3,2) (registered) or at (2,7) and (3,3) (local runs; the (3,3) run covered 1,601,613 tuples). A local (4,2) run timed out and is not claimed.

Controls: the direct four-call word satisfies the relaxation, and the search finds a length-4 solution on its own. Unrestricted linear algebra fits at length 2. Wrongly allowing y = xy reopens length 3.

Reproduce

python3.12 -B -W error research/conjugate-pair-group-free-audit/verify.py      # ~30 s
python3.12 -O -B -W error research/conjugate-pair-group-free-audit/verify.py
python3.12 -B scripts/research_package.py --package research/conjugate-pair-group-free-audit
python3.12 -B scripts/check_research_commit.py --package research/conjugate-pair-group-free-audit --check-source-history

Local results on macOS arm64, CPython 3.12: PASS_CONJUGATE_PAIR_GROUP_FREE_AUDIT in both modes, PASS_REGISTERED_RESEARCH_PACKAGE, PASS_PACKAGE_METADATA_AND_PINS and PASS_RESEARCH_COMMIT_BYTES. Base: canonical main 721c03822c421b7f7fcbb2598993b947c4d88419. The PR changes only its own package.

Limits

This is a formal group-algebra statement. It covers no native address operation, conversion, restoration, caller or BIT supplier, and the selected κ (#6030) is unchanged. The finite tiers cover only small prime fields and widths; prime-power fields and r ≥ 4 rest on the written proofs. No human or proof-kernel review is claimed.

Open next questions

1. Prove the conjugate-pair target needs four calls in every group (no new κ) #6038 OQ1: is there a uniform floor of 6 for diag([g],[h]) when neither entry is a product of at most two labels (e.g. diag([xyx],[yxy]))? The support-only analogue of Theorem S fails. In a local NumPy probe (not registered) at r = 2 over F2, letting only subsets of size ≥ 3 carry g or h gives relaxed solutions at length 4 (20/1,296 tuples) and length 5 (938/7,776). So any floor of 6 must use relations among the labels. A second local probe solves the exact equations in the free group: it finds no word of length ≤ 5 for diag([xyx],[yxy]) at r = 2 over F2, and it does find the four-call words for (xy, yx). Next step: a full-tier table for the target (xyx, yxy).
2. Formalize Theorem S (a short statement about rank-one matrices over a field).

Credit

Nick Petalas (#6038) supplied the theorem audited here, the corner-sandwich method, the expansion and the direct word. Wentao Li (#4089, #4093) supplied the model and the A5/PSL(2,7) results, and Cinna Davis (#2246) the original A5 target. Theorem S, its proof, the relaxation and the verifier are new here. Prepared for Josie Elliston with substantial Claude (Anthropic) assistance; Codex was used only for review. Apache-2.0.

Card: chooser seed c1601a53cbbe874bf9cae8c1d992c763 drew independent-verification; scope changed (reason in TASK.json). Claim: #1166 (reply in thread)

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