CLAIM: projected helper three-cycle with arbitrary dirty extras

[an664]

CLAIM: minimum unit scalar-XOR loop realizing a projected helper three-cycle.

Question: exactly one arbitrary dirty target y and k>=3 independent dirty helpers; raw endpoint I; deleting y-destination gates chronologically leaves z0->z1->z2->z0 and fixes every other register. Determine the minimum number of gates for all finite k, with an attaining word and universal lower bound, or explicitly bounded partial evidence. All ordered distinct-register XORs cost one; no clean allocation, copying, erasure, free mixing, free permutation or address-group coefficients.

This refines Gareth Davidson/@bitplane's PR1420 handoff, head d19a944897714f2ecae72720b7dafc3e256f2f8d. Existing constructions, finite censuses and elementary minima remain credited prior results; @woklerx PR1105 is earlier lineage. Our completed SWAP finding concerns a different endpoint and will only be used as an attributed reduction idea. A short corollary will be a Discussion finding rather than a padded package.

Base main ef5f39403610d231a0c27e8a6b4f08c10209bcd6, priority revision2. This changes the session's complex-masked-decoder selection to an explicit bit-helper-contract prerequisite; it is not a paid-frame construction or selected-supplier improvement. No hosting admission, new kappa or multiplication theorem.

Overlap: complete01:41 UTC snapshot, latest30 updated Discussion bodies and direct source checks through01:49, plus focused current PR search found no matching scalar projected-three-cycle claim. PR1216/1303 use different raw targets/rational primitive alphabets; PR1414/1478 use formal addresses and free scalar mixing. No claim to their results.

Prepared by Andrey Mas with substantial GPT-6 Pro research through ChatGPT and Codex orchestration/local verification. Retain predecessor credit and Apache-2.0.

Claim expires: 2026-10-14T03:54:18+00:00.

[an664]

DONE - PR1546, head 0980a7c266532b42ccda3d77bf61b5cf18be9054.

The projected helper three-cycle has exact minimum 13 unit scalar XORs for every finite k >= 3, with one arbitrary dirty target, arbitrary independent dirty helpers, raw endpoint I, and projected endpoint z0 <- z1 <- z2 <- z0 fixing all other registers. Projection deletes target-destination gates and replays chronologically. All restoration is counted.

The package supplies an attaining word and a computer-assisted lower bound: the written rank/write-count/coefficient-restriction argument reduces every hypothetical word of length at most twelve to five registers. Two representations independently construct the complete radius-six ball: 1,421,983 states and 4,012,300 expanded directed edges each. Exact transposed-state comparison passes; both right-translated intersections are empty. This is a complete bounded-radius calculation, not a full paired-group census.

Local Python 3.12.14 normal and optimized replay produced the same certificate. Dirty-input, restoration, projection and legal same-length mutation controls passed. A separate GPT-6 Pro mathematical review corroborated the universal reduction conditional on the finite nonintersection; that reviewer did not receive the search code or replay its census. Its explicit noncommuting fixture now replaces the overstatement about identity-only translation coverage. The final registered optimized full replay passed with the same certificate and state hash. Local execution and separate AI review are distinct evidence, not independent human or proof-assistant verification.

Credit remains with Gareth Davidson / @bitplane's PR1420, pinned d19a944897714f2ecae72720b7dafc3e256f2f8d, for the joint range, elementary loops, write-count observation, small censuses and SWAP witness; @woklerx's PR1105 is earlier lineage. The separate SWAP finding concerns a different endpoint.

The PR contains only the nine-file research/projected-three-cycle-minimum/ package. No paid-frame/hosting admission, selected supplier, new kappa or multiplication theorem is claimed. A specific next question is formalizing the target-write rank bound for arbitrary finite register sets; details and reproduction commands are in the package. This scoped claim is complete; maintainer review and hosted CI remain separate.

Prepared by Andrey Mas with substantial GPT-6 Pro research and review through ChatGPT, plus Codex coordination, execution audit and local replay. Apache-2.0.