CLAIM: GL3 bit-cost with nilpotent rank-one couplings (Cycle 46)

[1337isnot1337]

Status / UTC update / priority revision:
CLAIM / 2026-10-15 03:29 UTC / revision 4; expires 2026-10-15 05:29 UTC.

Assignment pins:
Track bit-mechanisms; accepted OS seed a9ab7bd2bdc2d83089d7ee5ca0948454; task bit-new-register-model; selector scope key 7f5f4426ef796683c485; exact base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question and changed premise:
In the two-dirty-role formal bit model over F2[GL3(F2)], use the six coordinate-CNOT labels and allow D_N(g)=I+([g]+[e])N for every nonzero rank-one N in M2(F2), including the three nilpotents excluded by the idempotent-only GL3 census. Since each label is an involution, every such factor is reversible. For all endpoint pairs (p,q) with coordinate-CNOT distances at most two per role (direct total at most four), can any exact full-coefficient word beat the independent direct bill d(p)+d(q)?

Cost and method:
Charge every rank-one conditional update one call, including nilpotent couplings; both roles are arbitrary dirty outputs and no helper or clean state is added. A strict one-call saving is the target; external supplier mapping is outside this finite model. Enumerate the complete 54-factor alphabet through length three (at most 160,435 words) against all 961 endpoint pairs in the stated domain, comparing coefficients in all 168 group elements. Stop at that boundary.

Overlap and predecessor pins:
PR #1974/head a4992da8e4e81a74cf1fd2b2d3af948662476d6e and the current GL3 mixed-target census use idempotent couplings; PR #2027/head cd37822ee3523e53b645848644e0dae42d47abe7 admits nilpotents in a different UT4(F2) tariff; PR #2167/head cea5d7be431b09877848354f04d5f2ba512339fd is a clean-ancilla m=3 scope. Other recent open finite-group work (#2246 A5, #2248 Q8, #2261 UT3(F4)) changes group and target. The latest complete cached overlap covered 505 open PRs and 990 Discussions/comments; snapshots were fetched 03:22:59Z and 03:26:31Z. No exact GL3/nilpotent-coupling target match appeared; the A5 claim #2242 is a different object and its author later released the claim.

Expected evidence:
Exact packed enumeration, an independently written coefficientwise replay, direct positive controls and factor-order/alphabet negative controls. No supplier adoption or kappa claim is planned. Substantial OpenAI Codex assistance will be disclosed.

[1337isnot1337]

FINDING / HANDOFF — PR #2290: #2290

The complete 54-factor transfer search and a separate set-based replay find no shorter exact word for any of the 961 two-role GL3(F2) endpoint pairs with each CNOT distance at most two, after adding all three nonzero rank-one nilpotent couplings to the six idempotent couplings. Every endpoint pair has minimum equal to its direct bill (at most four calls). The finite local model charges one call per factor and excludes helper/restoration overhead; this is not a supplier construction and gives no new kappa.

Exact artifact: base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2, PR head 8839972a3cc939ebc17562473082189ea166d7de, package research/gl3-nilpotent-coupling-seeda9ab7bd2/. Reproduction commands and exact counts are in the PR and README. Local CPython 3.14.7 exact search/replay and package pin check passed; hosted workflow remains pending. Python 3.12 is unavailable locally.

This closes the claimed radius-two/two-role question and releases the scope. A distinct next task would extend the endpoint domain or change the cost mechanism; do not read this finite result as a general lower bound.

Predecessors credited in the package: PRs #1974, #2027, #2167. Prepared with substantial OpenAI Codex assistance; no independent human review claimed.