Research: Q8 mixed targets retain four-call minimum through one extra role

[1337isnot1337]

Result

Adds research/q8-mixed-three-register-seedcbd5ebc4/, an exact scoped result in the formal group algebra F2[Q8]. For the 36 ordered diagonal triples with positive {i,j} word-length profile (2,1,1), the exact local minimum is four paid rank-one calls at r=3 and r=4. A three-call construction would save one call (25%) relative to the direct word; no such construction exists in the enumerated domains.

There is a uniform argument for the 24 triples whose length-two entry is k or -k: quotienting by the commutator subgroup gives first-order coefficient ranks 1+n_i and 1+n_j, where n_i+n_j=2. Each paid i or j factor contributes a rank-one matrix, so the total call count is at least four for every finite role count r >= 3. The other 12 triples have central entry -1; the same first-order invariant is insufficient, so this package claims the four-call minimum for those only at r=3 and r=4, by exact search.

Exact search and cost

The complete paid alphabet consists of both labels i,j and every binary rank-one idempotent: 56 factors at r=3 and 240 at r=4. Formal group-algebra equality is represented by eight-bit coefficient masks. The search covers all exact depth-one and depth-two states and streams every depth-three extension of each distinct depth-two state, checking all 36 target matrices. Counts are in EXPECTED.json and recomputed by verify.py.

Each target has a direct four-call upper bound. The verifier includes a two-call positive control, full-convolution cross-checks against the optimized rank-one update, direct replay of all 36 targets, and dropped-call/wrong-target controls. Free scalar GL_r(F2) mixing is already covered because conjugation permutes the full rank-one-idempotent alphabet; augmentation forces any terminal scalar map to be the identity.

This is a local formal call count. An additional dirty role has no paid calls in this model, but its physical storage/hosting cost is not priced. The selected bit supplier remains nonbinding, and no supplier placement, finite-size runtime, or kappa improvement is claimed.

Scope, pins, and overlap

Cycle 13; track bit-mechanisms; accepted seed cbd5ebc4789dc7dac5f7f824ab9561fc; task bit-new-register-model; priority revision 3; base 0b1cccf86e2f5132ed286a26ef0a681a495740ea; catalogue SHA-256 6686ad58f1fbdbc87134a78901b60fd53a20d3984355e91defbbebc9e9fa8075; selected-record SHA-256 10b9e340fd8e354a37a0c4813c90e5de1dd42044b5f6aca4a3fc5a42b685fa32.

The nearest predecessor is PR #1822, by navinor, head ee736b4b11bbd63b04d94c8edafb3790161ebdea. It studies a two-role repeated central-commutator target; this package changes to three mixed generator/product targets and checks one additional arbitrary dirty role. It imports no predecessor code. The final complete overlap refresh at 2026-10-14T13:57:45Z covered 573 open PRs and 542 Discussions with no warnings; it found this package's own Discussion #1167 claim, but no remote same-scope claim. Other nearby bit results use distinct targets or groups: Discussion #1727 and PR #1716/#1763/Discussion #1774 use AGL2, Discussion #1759 uses UT4, and Discussion #1818 uses D16.

The maintainer-reviewed selected result remains conditional finite and complex-binding at 836330098538459/500000000000000000. Its review, selected-source manifest, assembly certificate (SHA-256 1122cb3a575f8c788345fc1bf4c108ac7c3250fb81a5988671f5eb04b3ab3045), and make frontier1094-verify reproduction are pinned in SOURCES.json; the selected assembly was not rerun because none of its inputs changed.

Coordination claim and result: Discussion #1167 comment.

Verification

• PYTHONDONTWRITEBYTECODE=1 python3 -I -B research/q8-mixed-three-register-seedcbd5ebc4/verify.py passes both role counts and the algebraic split.
• python3 -B scripts/research_package.py --package research/q8-mixed-three-register-seedcbd5ebc4 --check-only passes source pins and package metadata.
• Local environment: CPython 3.14.7; Python 3.12 is unavailable. The registered package runner specifies Python 3.12, so its replay was not run locally.

Attribution

Credit navinor's PR #1822 for the Q8 central-commutator predecessor and woklerx for the pinned bit-dihedral-commutator package and group-algebra model lineage. No predecessor source code is copied. Prepared by Cinna Davis with substantial OpenAI Codex assistance; no new kappa or selected-frontier change is claimed. Apache-2.0.

[Mister-Meeseeks]

Batch review at 009ed860f34ed5d7b0d626098ce93c4c2a7eb4df.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact formal F2[Q8] enumeration excludes shortcuts through3calls for36 mixed targets with3/4roles; all attain4. Full-convolution crosschecks and direct/omission controls pass. First-order coefficient argument extends24noncentral cases to arbitrary dirty roles;12central cases remain finite-scope only.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.