Research: nine-call minimum for dyadic ternary frames with arbitrary dirty registers

[Tuafo]

Three repeated dimension-three identity interchanges cannot be completed in eight paid calls using the 338 frames A = u v^T / 2 with nonzero ternary three-vectors, even with any finite restored dirty workspace. This package proves the scoped nine-call minimum and gives an alphabet-valid attaining word. It adds only research/bit-dyadic-ternary-catalyst/.

The determinant and target-coefficient conditions reduce every word of length at most eight, in any order and with repetitions, to eight seven-call multisets and 665 eight-call multisets. Coefficient rank excludes all but six; exact additive characters on each generated subgroup exclude the six for every register count. The characters are not asserted to extend to the divisible group M3(Q).

Evidence and replay:

• Two separately implemented exact enumerations agree: decomposition counts 8 / 473 / 1 / 24 and eight-call subset histogram 579 x 1, 80 x 2, 6 x 4.
• Primary formal matrix replay checks the nine-call construction, including restored dirty registers; eight corrupted-certificate controls fail. The independent checker reconstructs the characters, checks positional multiplicities a second way, and rejects two additional corruptions.
• Registered replay and source bindings pass on Windows / CPython 3.12.14, standard library only. Isolated optimized primary replay also passes from outside the repository.

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

The independent checker deliberately refuses -O. Certificate SHA256: 6d6bf2c4d260bd76b85b20fcca59869142f23810db66984cc94b502c2c30b8b3.

Scope: a computer-assisted finite-alphabet theorem, not a lower bound for arbitrary rational frames or integer multiplication. The optimistic nine-call moment is 3^a, so this particular catalytic route cannot yield a positive saving. No new kappa, selected-frontier change, physical supplier admission, full selected cold replay, Lean formalization, or independent human review is claimed. Larger alphabets remain open.

Started from current main ddb4371, priority revision 2, refined bit-new-register-model card. Submission overlap refresh at 2026-10-14 03:50 UTC covered 731 open PRs and 244 Discussion body/comment/reply records with complete pagination; exact terminology and neighboring arithmetic scopes were inspected. It excludes arbitrary code diffs and unindexed closed work and is not a novelty certificate. Scoped claim and handoff.

Credit Ali Romman / #468 and its eumemic #288 / huxint #203 lineage, woklerx #1456 (83b6ad2), and Cinna Davis #1541 (194724d). Their known broader methods and differently scoped obstructions are not claimed as new. Exact input pins and scope distinctions are in SOURCES.json, TASK.json and README.md. Prepared for Tuafo with substantial OpenAI Codex assistance, including GPT-6 Astra at maximum reasoning for derivation, separate implementation and proof review. Apache-2.0.

[Mister-Meeseeks]

Batch review at c5f5bef84f6648d5573f801c283e01a333ea5f89.

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

Reviewed scope: Determinant and positional coefficient-rank conditions reduce the explicit 338-frame dyadic alphabet to six survivors, each excluded by an exact subgroup character. A nine-call formal witness attains the arbitrary-register floor; general rational frames remain outside scope.

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.