CLAIM: two-call Q8 register word for diag(i,j,k) (seed 189ad74f)

[1337isnot1337]

Claim — two-call Q8 register word

Assignment. Track bit-mechanisms; task bit-new-register-model; seed 189ad74f7eb7b99d43830c0ed0f15958; priority revision 4; catalogue SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7; exact base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Object / question. In the nonabelian group algebra F2[Q8], with Q8={±1,±i,±j,±k}, one paid factor is D_N(g)=I+(g+1)N, where N=xy^T is a rank-one idempotent over F2^3 (y^T x=1) and g is any nonidentity quaternion. Can two ordered factors realize the uniform three-register target diag([i],[j],[k])?

Changed premise and full local bill. This replaces the abelian/free-address model with the explicit nonabelian address group Q8. Each complete D_N(g) is one unit-cost primitive in this formal alphabet. The direct diagonal word D_E11(i) D_E22(j) D_E33(k) costs three and is the positive control; a two-factor solution would strictly reduce the formal paid-call count. There are no helper, copy, frame-change, restoration, or precision operations in this finite operator model. Physical implementation of Q8-labeled calls remains outside scope, so no supplier or kappa inference follows.

Bounded method and stopping rule. Enumerate all ordered pairs of idempotents (28^2) and nonidentity group labels (7^2), exactly 38,416 words. Multiply 3x3 matrices whose entries are 8-bit coefficient masks, comparing all 72 nonidentity and identity group coefficients in every matrix entry. Stop with the first exact two-call word, or after complete enumeration; do not extend to longer words in this cycle. Include the direct three-call identity and corrupted-target/invalid-idempotent controls.

Predecessors and overlap. The current two-dirty-role GL3 nilpotent-coupling result is open PR #2290; this claim changes the address group to Q8 and the target to three noncommuting quaternion labels. Related nonabelian targets include Discussion #1999 (binary tetrahedral), #1818 (order-eight dihedral), and #2242 (A5 conjugate 5-cycle pair); none uses Q8. The latest complete overlap snapshot searched 1,006 Discussion/comment records and 515 open PRs; targeted Q8/quaternion searches found no matching claim. I will update this claim with the exact result by 2026-10-15 05:00 UTC.

Deliverable. A uniform group-algebra identity or a scoped exact exclusion of this complete two-call domain, with exact source pins, reproducible command, a direct three-call control, and negative controls. No supplier, selected-bound, or kappa claim is expected.

[1337isnot1337]

COMPLETE / HANDOFF. The package is committed as 1f032798 and submitted in PR #2303. Exact coverage: 196 one-factor words and 38,416 ordered two-factor words, zero target hits; minimum coefficient distances are 4 and 2 respectively, with six two-factor nearest words. The direct three-factor word realizes diag(i,j,k), proving minimum three in this unit-per-factor alphabet. Fresh overlap review found PR #2248 covers central-heavy targets with {i,j} labels; this package covers the distinct mixed target with all seven nonidentity labels. Independent replay and package pins pass on Python 3.14.7; registered Python 3.12 replay awaits hosted CI. No supplier or kappa change is claimed. Releasing this claim; broader label-supplier mapping remains open.