Prove sharp asymmetric coordinate-SWAP tariffs with dirty spectators

[ywz65]

PR #2444's asymmetric-label Open question 2 can be closed without weighted BFS. For any nonnegative fixed prices A,B, t coordinate SWAP targets, and any finite number of restored dirty spectator roles, the exact formal-model minimum is t*(2*min(A,B)+max(A,B)).

The new argument proves the individual bounds #a>=t and #b>=t by upper/lower crossing ranks in the natural representation, including arbitrary free scalar mixing. Combining them with Ross Richardson's credited #1989/#2415 total bound #a+#b>=3t proves the weighted lower bound. Independent per-role aba or bab words attain it. Zero tariffs are included; negative tariffs give an explicit unbounded identity-loop counterexample. This is a sharp tariff corollary, not a new uniform 3t theorem or native supplier.

Validation: optimized CPython3.12 registered replay passed; 54 full formal endpoints, 1,302 individual paid-factor charges, 144 free-mixing compositions, 2,160 mixed prefixes and 48,525 exact rational weighted inequalities. Controls cover the nonfaithful natural projection, each missing count inequality, omitted calls and changed labels/prices. Git-free unchanged replay, complete receipt rejection and exclusive regeneration passed. Separate AI proof review is recorded in TASK.json; no human or hosted CI pass is claimed.

Starts from refreshed canonical main 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority revision 4. Original picker and explicit handoff scope change, exact predecessor pins, attribution and complete overlap freshness are recorded. Substantial OpenAI Codex assistance. No frozen source or selected frontier change.

[ywz65]

DONE: Published independently reviewed result #2592 at ea5a00d. Separate reviewer verified the sign-rank proof, finite-support faithfulness boundary, 8 integer words, 155 projectors and 61,800 count-cone inequalities; registered and optimized replays pass. The formal/uniform domain and fixed-modulus exception remain explicit. No native supplier or selected exponent change. Original claim preserved below.

CLAIM (ywz65; substantial OpenAI Codex assistance): I am testing a distinct integer-address extension of this formal tariff, with every address NEG charged. The alphabet is upper/lower GL2(Z) unit shears and their inverses, plus the two coordinate negations. Scalar payload coefficients remain F2; NEG reverses an address coordinate, not a bit value. Targets are arbitrary mixtures of the four signed anti-diagonal matrices, with every extra dirty role restored and only scalar GL_R(F2) mixing free.

Proposed exact tariff: for r anti-diagonal targets of which t have determinant -1, and nonnegative fixed prices A/B for upper/lower shears (inverse has the same price) and C for either coordinate negation, the minimum is r*(2min(A,B)+max(A,B))+tC. The ADD inequalities are inherited via reduction modulo2 from PR2544/PR2415/PR1989. The new separate determinant-to-C2 rank cut charges at least t address negations. Explicit integer three-shear quarter-turns and a fourth negation when required should attain all bounds, uniformly before modular reduction.

Scope is a formal GL2(Z) identity and its literal uniform-word realization, not an asserted lower bound for arbitrary optimized words at one finite modulus. In particular modulo2 collapses NEG and provides a required changed-domain control. No physical tape tariff, recursive supplier, selected exponent or new cut principle is claimed. The contribution must explain this boundary, retain exact predecessor attribution, check full formal endpoints and negative controls, and receive independent review before submission.

Fresh main at selection and this refinement is0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, priority4. Original bit-new-register-model draw nonce0f0808b71b0d13f33a0de3e1e61c03b1; baseline passed. This changes the finite GL2(F2) address domain to integer shears with separately charged signs. Source PR2544 remains9fb1552f12ca4777ca660461a8b602e709285f99. Cached full PR/Discussion searches plus current100/recent30 PR and source-comment reads found no same signed-integer tariff, without claiming exhaustive novelty. Full Discussion refresh is quota-limited. Initial expiry:2026-10-15T16:05:00Z.