[apoursh]

Change and mathematical scope

Scoped proof: one arbitrary symmetric rational rank-one address cannot lower the exact eight-call minimum for the PR2085 endpoint diag([A+P],[A+Q],[B+P],[B+Q]). This includes signed rational scaling, any finite number of exactly restored dirty roles, and interspersed address-independent binary mixing.

The new reduction combines the determinant invariant already used in PR2910 with exponent support and first-order rank. Every hypothetical word of at most seven calls reduces to one of four doubled-axis profiles; each has rank capacity two against the target's rank four. An eight-call word attains the bound. Longer words using the new frame remain possible.

Adds one self-contained package, research/bit-single-scaled-frame-floor/. No native supplier, paid-profile improvement, κ change, or all-length extra-frame classification is claimed.

Task and overlap

Base: 721c03822c421b7f7fcbb2598993b947c4d88419, priority revision 8; publication head: 44745c427f710e406cb2d2c6de458b7f93d81f01.
Task bit-new-register-model, scope key 7084092dd726e7ba1f2f.
Scope changed: the user selected the one-scaled-frame question in PR6032, rather than a native larger/nonabelian construction. Full original card and rationale are retained in TASK.json.

Canonical collaboration claim/status.

Nearest predecessors, with live heads rechecked before publication:

• Research: exact rational frame floor on four roles #2085: 5302b9bad6345413a0cabb0afd926d84e44c12e3 — endpoint and augmentation method.
• Research: added dyadic frame is unusable at every word length #2910: 26ce4ed46112438b237fab54e6da1953c1d16174 — earlier determinant obstruction for the fixed dyadic frame, 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 — all normalized rational slopes; explicitly asks about arbitrary scaling.

Their canonical README bytes are pinned in SOURCES.json. In particular, the fixed t=1/2 result already existed in #2910. The addition here is all rational scales and all finite restored dirty widths. Targeted all-state searches found no general successor; local overlap coverage was sparse (18 of 8,348 note records), so absence of matches is not a novelty certificate.

Validation

CPython 3.12.14, macOS arm64, standard library only:

python3.12 -B scripts/research_package.py --package research/bit-single-scaled-frame-floor
python3.12 -B scripts/check_research_commit.py --package research/bit-single-scaled-frame-floor --check-source-history --check-modes

Both registered normal and optimized commands pass. Independent exact computation checks 462 count profiles → 40 determinant candidates / 20 frames → 160 support conditions → four survivors, each with a rank-two/rank-four contradiction. Literal group-algebra replay checks six words at widths 4/5/7, including non-coordinate projectors and temporary dirty-role writes; 49 mutations are rejected. Additional controls cover negative/fractional longer words, dirty restoration, and hand-computed noncommuting products.

Committed-byte/source-history/mode checks and the official nine-file package/input export pass. Fresh read-only, Git-free export replays pass in both modes with unchanged hashes. Separate agent proof/code audits found no gap. Finite fixtures corroborate the written uniform proof; they do not establish universal coverage alone. No native tariff, formal-kernel proof, or selected-frontier cold replay was run.

Credit

Prepared by apoursh, with substantial OpenAI Codex assistance in proof, implementation, and separate agent audits; no independent human acceptance claimed. Apache-2.0.

Retains Cinna Davis's #2085/#2112 model and augmentation lineage, Nick Petalas's #2910 determinant argument and #6032 support/scaled-frame handoff, Philippe Van Velzen's earlier fixed-frame result, and the retained BIT predecessors. The determinant lemma and first-order/support methods are inherited.

Open next questions

See README.md and TASK.json: first screen two added frames using the joint determinant/support constraints before any word search; an exact saving still needs a native implementation and complete paid invoice. A separate all-length classification of positive extra-frame use is also left open.