[apoursh]

Change and mathematical scope

Scoped classification with sharp constructions. In the PR2085 four-target formal register model, a nonzero positive symmetric rational rank-one added address is usable in an exact word iff R=F/m, where F∈{A,B,P,Q} is an original frame and m≥1 is an integer. The exact attainable total lengths among words using R are {m+7, 2m+6}. Both are constructed by subdividing one or both F calls of the direct word.

The proof permits every finite restored dirty workspace and arbitrary address-independent binary mixing, without a total-length cutoff. The determinant identity bounds old calls below eight; an integral character annihilating the entire rational added line and exact exponent support force reciprocal-integer scaling.

The three count profiles are necessary only. The first two are constructed; the third is excluded at m=1 and remains unresolved for m≥2. This does not affect the exact length spectrum. No native supplier, paid-profile saving or new κ is claimed; positivity, one added address, and the unchanged endpoint are essential.

Task and overlap

Adds only research/bit-positive-frame-use/, with the full model/proof, exact checks and opt-in replay metadata.

Canonical main was pulled first and refreshed before publication at 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8. Publication: a736cc61b1436ddb5b8689398e28f51021177839.
Task bit-new-register-model, scope key 7084092dd726e7ba1f2f, chooser seed 9da5de6c63f0e0f69135ef2d1c7c085d. Scope changed from native larger/nonabelian construction to the all-length positive-use question left by #6079.

Canonical claim/status.

Nearest work, exact heads rechecked:

• Research: exact rational frame floor on four roles #2085 5302b9bad6345413a0cabb0afd926d84e44c12e3: model and endpoint.
• Research: added dyadic frame is unusable at every word length #2910 26ce4ed46112438b237fab54e6da1953c1d16174: already forbids its fixed dyadic frame at every length, including dirty workspace.
• Prove a fifth rational frame needs eight calls for every rational slope, settling the #2112 seven-call case (no new κ) #6032 3b098803eb854afbd5d63ad3bf4d75cdb1e515d5: normalized rational-slope minimum and alphabet question.
• Prove eight-call minimum for one arbitrary scaled rank-one frame #6079 44745c427f710e406cb2d2c6de458b7f93d81f01: signed single-frame minimum-eight theorem, leaving all-length use open.

This classification is stronger than a minimum-eight result. Its proof is self-contained from canonical predecessor invariants; pending #6079 is context only, with no theorem, code, certificate or Git-object dependency. Three canonical READMEs are pinned.

Live all-state reciprocal/rank-one, scaled/frame and #6079 searches plus the current handoff found no matching successor. Offline local coverage was only 18/8,348 note records with no remote cache; no novelty certificate is claimed.

Validation

CPython 3.12.14, macOS arm64, standard library only:

python3.12 -B scripts/research_package.py --package research/bit-positive-frame-use
python3.12 -B scripts/check_research_commit.py --package research/bit-positive-frame-use --check-source-history --check-modes

Both registered normal and optimized modes pass. Exact checks derive 40 determinant families from all 330 old-count tuples, retain 12 rank-compatible families, and verify 48 support-weight certificates reducing the unbounded extra-call count to divisibility. Corroboration covers 3,840 support checks, 40 reciprocal-scale fixtures, 212 rejected nonreciprocal-scale fixtures and 20 rejected rotated frames.

Literal multiplication checks 128 one-/two-subdivision words plus 128 conjugated versions at widths 4/6 and m in {1,2,3,4,5,7,8,12}, including transient dirty writes. All 577 semantic mutants are rejected. Separate primary checks cover exact symmetries, enlarged-lattice character integrality, noncommuting products and the negative-frame boundary.

Committed bytes, source ancestry, modes, official ten-file package/input export, and both fresh read-only Git-free export replays pass with unchanged hashes. Separate agent proof/code audits found no gap. The selected binding-only orientation passes; no selected-frontier scientific replay, native tariff, formal-kernel proof or human acceptance is claimed. Universal coverage follows from PROOF.md, not finite grids.

Credit

Prepared by apoursh with substantial OpenAI Codex assistance, including separate agent derivations and reviews. Apache-2.0. Retains Cinna Davis (2085/2112), Nick Petalas (2910/6032), Philippe Van Velzen's earlier fixed-frame result and the BIT lineage. Determinant, support and character/rank techniques are inherited.

Open next questions

README.md and TASK.json specify two added positive frames as the next possible escape, starting with joint determinant/support constraints before word search. Separately, the third necessary profile's realizability for m≥2 remains open; it cannot improve the minimum or length spectrum established here.