Scoped S4 result: four roles do not compress noncommuting 3-cycles below four calls

[1337isnot1337]

Result

In the formal four-role F2[S4] model below, each of the eight ordered noncommuting target pairs

diag([a], [b], [e], [e]), with a,b distinct noncommuting elements of {s1*s2, s2*s1, s2*s3, s3*s2},

has exact minimum paid length four. Every target has a direct four-call word; a complete exact search excludes words of length at most three. This is a scoped negative result for the stated alphabet, not a multiplication-bound improvement.

Model and exact evidence

• Four arbitrary dirty scalar roles; the last two are restored exactly by the target.
• Address group S4, paid labels s1=(12), s2=(23), s3=(34).
• Each paid factor is I+([s]+[e])N for one paid label and any of the 120 rank-one idempotents in M4(F2), giving 360 factors. All are involutions; the complete inventory includes the normalized free scalar GL4(F2) mixing in this model.
• Exact enumeration checks all 129,600 ordered two-factor products, yielding 87,016 distinct matrices including the identity. For each of the eight targets, all 360 final-factor meets miss, excluding every three-call word because T=ABC and C^2=I imply TC=AB.
• The sorted complete radius-two state stream has SHA-256 cc18ab4f9e58731f43f42ed0fd791531696836385f9435c6328cbbe36651f2f7.
• All eight direct four-call words, a one-call positive control, 32 single-call deletion controls, a changed-target control, and the noncommutativity checks pass under exact F2[S4] multiplication.

Cost and limits

The direct cost is four paid calls; the proposed improvement would have been three calls, a one-call (25%) reduction. In the formal model, scalar mixing and the added role's hosting/interface overhead have no paid-call charge; endpoint equality enforces exact restoration. No physical supplier mapping, recursive-moment change, or new kappa is claimed. The current selected binding-only baseline remains complex-primitive-bound.

The package pins base 0b1cccf86e2f5132ed286a26ef0a681a495740ea, the selected-record hash, selected review, source manifest, assembly certificate, and the three-register abelian predecessor. The selected frontier and frozen sources are unchanged.

Reproduce

python3.12 scripts/research_package.py --package research/s4-noncommuting-threecycle-r4-seeda27b

Expected marker:

PASS_REGISTERED_RESEARCH_PACKAGE research/s4-noncommuting-threecycle-r4-seeda27b outcome=negative-result

The local registered replay and python3 -B scripts/research_baseline.py pass. Baseline marker: PASS_RESEARCH_BASELINE_BINDINGS_ONLY, with selected kappa=836330098538459/500000000000000000 and binding_primitive=complex.

Overlap and attribution

This scope is distinct from PR #1720 and PR #1734, which use the same S4 labels at three roles but identical 4-cycle or double-transposition targets. It is also distinct from PR #1742, which uses UT3(F2), labels {x,y}, three roles, and a repeated central-commutator target with eight-call floor. PR #1705 uses another address group. The scoped claim and overlap notes are in Discussion #1740.

Prepared with substantial OpenAI Codex assistance. No independent human review is claimed.

[Mister-Meeseeks]

Batch review at 90405b9498e4abfdd7d0eececcf11c7e83c6d13b.

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

Reviewed scope: All120 rank1 binary idempotents and three adjacent-transposition labels give360 full F2[S4] factors. Complete129600 pair products and2880 target meets exclude three calls for eight noncommuting target pairs at four dirty roles; explicit four-call words attain. No broader alphabet or supplier claim.

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.