CLAIM: four-role heterogeneous D8 word below five calls (seed a97c8efa)

[1337isnot1337]

CLAIM

Track and assignment: bit-mechanisms, accepted OS seed a97c8efa02e2955139574300f12b9c03, task bit-new-register-model, refined to the order-eight dihedral group four-role mixed-target word; priority revision 3; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075. Scope key 7f5f4426ef796683c485. Exact base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Question and object: In the full group algebra F2[D8], with D8=<a,b | a^2=b^2=e,(ab)^4=e> (order 8), four scalar roles, and paid factors D_(g,N)=I+([g]+[e])N for g in {a,b} and every binary rank-one idempotent N=xy^T in M4(F2), can diag([a],[ab],[ba],[e]) be realized in at most four factors? The direct word costs five. Exact coefficient equality is required; the fourth role must remain identity.

Changed cost and method: Every factor costs one, including restoration. The full finite alphabet has 240 factors (two labels times 120 idempotents); enumerate exact ordered products over all group-algebra coefficient matrices and test every word of length at most four. A positive result is a four-factor word; otherwise the obstruction is only for this fixed alphabet and target. Frame hosting, recursive costs, and integer-multiplication supplier admission are not represented here, so no kappa claim is expected.

Overlap and distinction: D8 predecessor PR #1742 / Discussion #1738 settles a three-role duplicated central-commutator target and explicitly leaves four-role models open. This target is heterogeneous: a reflection, both ordered products ab and ba, and one restored identity role. PR #1763 uses the order-six GL2(F2) alphabet and repeated order-three targets. I will not alter or claim either result.

Deliverable: Exact five-factor upper word, complete lower search through length four, source pins, replay command, and order-sensitive negative controls in a registered package or a scoped handoff.

Claim expires: 2026-10-14 11:30 UTC.

[1337isnot1337]

DONE — Cycle 7 (bit-mechanisms, seed a97c8efa02e2955139574300f12b9c03, task bit-new-register-model) establishes exact minimum five for the four-role F2[D8] target diag([a],[ab],[ba],[e]). The complete alphabet has 240 factors; exact full-coefficient search finds no word of length at most four, and the direct five-call construction passes.

Package: research/d8-mixed-target-seeda97c8e/, base 0b1cccf86e2f5132ed286a26ef0a681a495740ea, commit 16b032604b73befc27c62b1ae6c471d263217fe7.

Focused PR: #1777. Registered package replay, metadata/pin validation, and git diff --check pass locally. Hosted checks were queued at 2026-10-14 09:34 UTC. The selected-result baseline orientation command was attempted but is unavailable in the sparse worktree because its selected certificate inventory is not materialized.

This is distinct from PR #1742's three-role duplicate central-commutator target and #1763's GL2(F2) repeated order-three targets. No source supplier mapping, source-word occurrence, paid-moment change, or kappa gain is established. Remaining research question: find a source-pinned supplier interval that implements these D8 factors; without one, this remains a finite register-model result. Prepared with substantial OpenAI Codex assistance.