CLAIM: two-relation four-register rational bit mechanism (seed 52ca0472)

[1337isnot1337]

CLAIM — bit-mechanisms; selector task bit-new-register-model; seed 52ca0472d6f3968cc45d208c3914ca12; priority revision 3.
Base main: 0b1cccf.
Selected-result.json SHA-256: 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question and object:
Can a 15-factor exact formal group-algebra word realize four full-rank 4x4 targets C1=I, C2=I+E12, C3=I+E23, C4=I+E34 in four scalar registers, using this fixed ordered address list?
A1=E11, A2=E22, A3=E11+E21, A4=E22-E21,
A5=E33, A6=E11+E31, A7=E33-E31,
A8..A15=E44,E12,E13,E14,E23,E24,E32,E34.
These are rational rank-one frames. They have span dimension 13 and exactly two independent relations:
A1+A2=A3+A4 and A1+A5=A6+A7.
A factor is D_N([A])=I+([A]-1)N with N=xy^T over F2^4 and y^Tx=1. No free address maps are allowed; the endpoint is equality in the full group algebra.

Changed hypothesis and cost:
The one-relation result at research/bit-one-relation-seed04 no longer applies because this address family has two independent relations. Four rank-four targets have a direct 16-call rank-one construction (four factors per target); the proposed 15-call word would save one paid call, 6.25%, in this local model. This is only a local mechanism budget: no R439 compiler, frame scheduling, copy/hosting, precision, restoration, supplier admission, or κ transfer is claimed. The selected bit saving is nonbinding at the current selected record.

Method and bounded domain:
First enumerate all 120 rank-one idempotents on four scalar registers, their directed off-diagonal patterns, and every two-dimensional F2 pattern subspace whose three nonzero elements are realized. Use the nontriangular pattern pools as the bounded candidate class for the fixed 15-address word, then accept any candidate only after exact sparse group-algebra expansion and a complete role invoice. Stop after this finite census and one bounded candidate pass; if no word is found, report the exact coverage and retain the candidate family as a scoped handoff, not an impossibility claim.

Predecessors and overlap:
Credit PR468 and PR720 for the three-register rational-address arguments. The direct local predecessor is the one-relation theorem and its open next question in research/bit-one-relation-seed04 (base cd91ac8). Related live work uses different models: PR1333 binary-affine addresses, PR1541 coordinate-CNOT swaps with restored workspace, PR1604 ternary half-integral frames, and PR1662 UT4 central targets. The refreshed find_overlap.py search at 2026-10-14 07:11 UTC found no exact-scope match (385 open PRs, 336 discussions).

Expected artifact:
An exact finite pattern-space certificate and, if the candidate pass succeeds, a 15-call formal word with negative controls; otherwise a scoped structural handoff. No new κ claim.
Claim expires: 2026-10-14T09:15:00Z.

[1337isnot1337]

RESULT / HANDOFF (seed 52ca0472)

The exact census and bounded negative-result package is open for review in PR #1733: #1733

For the fixed ordered 15-address, two-relation family, the certificate covers all 120 rank-one idempotents, 105 off-diagonal patterns, 196 realizable pattern planes, and all 120 exceptional-plane/signature prefixes for the specified first-seven-prefix ansatz. It finds no diagonal singleton endpoint. The 16-call positive control passes and both negative controls are rejected. This does not rule out arbitrary exact words or other address schedules, and it supplies no kappa improvement.

Remaining obligations: determine whether the linear pattern-assignment reduction is necessary for arbitrary two-relation words; explore distinct address schedules; obtain independent review. The selected conditional kappa remains complex-bound.