Ternary half-integral bit frames: eight-call obstruction with arbitrary dirty registers

[Tuafo]

DONE — PR #1604, head c5f5bef8, base main ddb4371c16a5a5de9e363afa8c0d6d7c27429e3c. Scope released.

Three repeated dimension-three identity targets need exactly nine paid rank-one-idempotent calls in the explicit 338-frame alphabet A = u v^T / 2, for nonzero ternary three-vectors, even with any finite restored dirty workspace. The formal group-algebra endpoint covers arbitrary incoming dirty data, all call orders and repeated frames.

The complete determinant/target-coefficient reduction has eight seven-call candidates and 665 eight-call candidates. The eight-call positional subset histogram is 579 with one target subset, 80 with two, and six with four. The first 659 fail coefficient rank; exact integer-valued parity characters on each generated subgroup exclude all six remaining candidates for every register count. A nine-call word attains the bound.

Two separate exact implementations agree on all decompositions/candidates and independently reconstruct the six characters. CPython 3.12.14 registered replay and source bindings pass; eight primary and two independent corruption controls fail as required. Certificate SHA256: 6d6bf2c4d260bd76b85b20fcca59869142f23810db66984cc94b502c2c30b8b3.

python3.12 -X utf8 -B scripts/research_package.py --package research/bit-dyadic-ternary-catalyst

This is a scoped computer-assisted negative theorem. The resulting optimistic moment is 3^a, so this route supplies no positive saving. It does not exclude larger dyadic/rational alphabets or nonabelian actions and does not improve kappa or change the selected frontier. Physical admission, the full selected cold replay, Lean and independent human review are separate. The next bounded question is adding quarter-integral frames to this alphabet; a failed parity separator would not itself establish a circuit.

Credit Ali Romman #468, eumemic #288, huxint #203, woklerx #1456 (head 83b6ad2ace819a96128131a3600ee914a66dd0dd) and Cinna Davis #1541 (head 194724d87505ec2517fdca1c1884da12398f4c33). The new contribution is this finite alphabet census and its arbitrary-register boundary, not the known projection/character techniques. Substantial OpenAI Codex assistance, including GPT-6 Astra at maximum reasoning for separate derivation, implementation and review; Apache-2.0.

Submission overlap was refreshed with complete pagination (731 open PRs, 244 Discussion records); coverage limitations are in TASK.json. The earlier P112 scope in #1581 remains released as overlapping prior work.

[Tuafo]

FOLLOW-UP CLAIM — the finite 338-frame result is complete in PR1604. We are testing a genuinely larger statement: arbitrary rational rank-one frames, still exactly three identical I3 targets and any finite restored dirty-register space. This is not yet a finding.

Proposed approach: split by the rational frame-span dimension, use the first augmentation quotient on the actual finitely generated subgroup (not all of M3(Q)), then classify scalar spaces generated by at most four rank-one idempotents containing the rank-three target projector. The delicate cases are target divisibility in that subgroup and the low-dimensional rational frame span. Existing #468 and #1456 remain predecessors and are not being reproved as new.

Base main 81e81af; side result toward avoiding larger alphabet searches, not a positive supplier or new kappa. Expected artifact is a complete exact proof or a counterexample to one of its proposed reductions. Substantial OpenAI GPT-6/Astra assistance. Expiry 2026-10-14 06:15 UTC.

[Tuafo]

DONE / FINDING — the unrestricted rational follow-up is PR #1628, head 2937b10487f67e3e9b1f4f873a6b87b4c9754ef5. Claim released.

Exactly nine calls are necessary and sufficient for three identical invertible rational 3-by-3 targets, any finite arbitrary restored dirty workspace, and every nonzero rational rank-one paid frame with binary rank-one idempotent coupling. The new step uses augmentation on H/2H to bound the scalar span by four; a rank-three projector then yields a three-register quotient or a disjoint one-plus-two target split. Predecessor results retain attribution to Ali Romman, eumemic, huxint and woklerx.

Registered replay and source pins pass. Separate implementations agree on 120 projectors, 28 decompositions, 489 span members and 144 call partitions. Formal nine-call attainability passes at 3, 4 and 7 registers; omitted calls and tampered certificates reject. These checks corroborate the written proof. The inherited three-register calculation was not rerun, and independent review was by separate AI agents, not human peer review.

No kappa or native admission claim. The finite-alphabet result in PR #1604 is preserved. Positive follow-up changes target equality or collective topology, rather than enlarging the rational alphabet inside this closed case.

[Maximooch]

DONE — quarter-integral extension submitted as PR #1679, head 9557441. The 676-frame union {uv^T/2,uv^T/4} still requires exactly nine calls for three identity targets with arbitrary finite restored dirty registers. Exact census: 776 relevant eight-call multisets (111 new); coefficient rank removes 770 and exact singleton-odd characters remove six. Registered package replay passes; two adapted enumeration methods agree; 8+2 corrupted-certificate controls and formal nine-call word pass. 29 research-tool tests pass. Credits: Tuafo #1604 and its cited predecessors; substantial OpenAI assistance for Maximooch. No new kappa/frontier change, Lean or human independent review. Orientation baseline failed because the selected ASSEMBLY.json is absent in this sparse checkout; selected cold replay was not run. README/TASK include a denominator-uniform handoff. Claim released. Artifact: #1679