HANDOFF: exact Q8 three-role target minima (Cycle AF)

[1337isnot1337]

HANDOFF / DONE — Cycle AF; scope released 2026-10-15 02:10 UTC. Focused PR: #2248, head 835bd7b1877f56d6d9f4eecd7d1ee32f31593e1e.

Pins: track bit-mechanisms; accepted OS seed 809fc39d62870d18f94dd48473f46972; task bit-new-register-model; catalogue revision 4, SHA-256 d46eb6d5b3e279428753d770653663fbdf1063e55c255ea073d768f344927ef7; base 0fdd7d82de03e43c9e5a1000dfbcd380d1b687d2; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

Finding: In the formal F2[Q8] model with x=i, y=j, z=-1, three arbitrary dirty scalar roles, and all 28 rank-one idempotents in M3(F2), the six coordinate permutations of diag([z],[z],[x]) or diag([z],[z],[y]) have exact minimum 5 paid factors. diag([z],[z],[z]) has exact minimum 6, resolving the predecessor package's open all-central target question. Equality is coefficientwise over all eight group elements. The known diag([z],[z],[1]) control appears at depth 4.

Evidence and reproduction: The self-contained package is research/q8-r3-two-central-one-generator-seed809fc39d/. The verifier exhausts exact distinct product states through depth 4 over 56 paid factors; layer sizes are [1, 56, 2324, 89574, 3286437], representing 10,013,305 raw words of lengths 0–4 before state merging. J/J² trace parity excludes the opposite parity lengths. Direct coordinate words give the matching upper bounds. Run python3 -I -B research/q8-r3-two-central-one-generator-seed809fc39d/verify.py.

Local checks passed: the direct verifier under normal and optimized Python, scripts/research_package.py --check-only, and git diff --cached --check. The package includes Q8 and abelianization checks, all 28 idempotents, 168 GL3 conjugations, 952 packed/generic transition comparisons, positive controls, and 36 one-call deletion controls. A separate target-mutation control listed as expected in the original claim was not implemented; the delivered controls are the ones listed here. Python 3.12 was unavailable locally. The PR's 61 hosted checks were queued at the observed head; no hosted pass is claimed.

Cost and limits: Every paid factor costs one, so the local block costs 5 or 6 calls exactly. Setup, physical supplier mapping, hosting, precision, restoration, and recursive charges are zero in this formal model. No supplier mapping, saving to the selected recursive moment, or selected-κ change is claimed; the bit side is nonbinding. The next useful construction question is whether the mixed-target minima extend uniformly to r >= 3, or whether this formal profile can map to a selected physical supplier with a complete invoice.

Overlap and attribution: The final complete snapshot counted 478 open PRs and 952 Discussions. Recent adjacent items were Q8 mixed-target PR #2047 (one central plus two order-four targets), expired Q8 claim #2034, active binary-tetrahedral claim #1999, and A5 PR #2246; the exact target profile remained distinct. Earlier Q8 predecessors include PRs #1822 and #1925. This bounded formal result does not supersede their scopes. The work is prepared with Codex assistance and pins the predecessor package and selected-result provenance in SOURCES.json.

[1337isnot1337]

Cycle AF complete: PR #2248, head 835bd7b1877f56d6d9f4eecd7d1ee32f31593e1e. The six two-central/one-order-four coordinate targets have exact minimum 5 calls; the all-central target has exact minimum 6. The package and exact replay command are in the handoff body. Local checks passed; 61 hosted checks were queued at the observed head. This is a scoped formal Q8 result with no supplier mapping or κ change. Scope released.