[apoursh]

Change and mathematical scope

This closes PR6111's short-word signed-frame question: two arbitrary nonzero symmetric rational rank-one additions, of either sign, cannot reduce the four-target BIT endpoint below eight paid calls, with arbitrary finite restored dirty workspace and free binary mixing. No new κ is claimed.

The new result is an elementary signed-to-positive reduction. Determinant conservation leaves a residual M. Indefinite M fails a scalar projection of target support; rank-one M fails support or a denominator-safe character rank bound; positive-definite M forces both rank-one summands positive. Josie Elliston's PR6111 then supplies the positive-case exclusion. PR6079 supplies the one-added-frame case. These are explicit theorem dependencies, archived as partially reviewed, not fresh predecessor audits.

The proof avoids a case census as its foundation. A separate exact checker corroborates it without filtering negative frames:

• All 462 count profiles: 422 definite, 20 indefinite, 16 singular-span and four singular-character cases.
• Complete support classification: 435 empty profiles, seven free families, 32 pinned splits across twenty profiles; every pin is positive.
• 148 opposite-sign split fixtures, 576 two-summand geometry fixtures and 24 positive/rejection/boundary controls.
• A literal nine-call word using 2A and -A shows why this is a short-word result, not an all-length positivity theorem. The identity -A+2A+B=I marks the three-addition boundary.

Adds only research/bit-two-signed-frame-floor/: proof, independent checker, bounded replay and exact source/task metadata. All predecessor sources and the selected record remain unchanged.

Validation

Validation at 375aced, CPython 3.12.14, macOS arm64, standard library:

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

Both registered warning-strict normal/optimized commands pass. Exact committed bytes/modes and source ancestry pass for nine package/input files. Official export integrity and both read-only, Git-free direct replay modes with empty PATH pass; all ten exported files retain their hashes. Separate agent proof/code review found no remaining blockers; this is not human or formal-kernel acceptance. Predecessor scientific suites, native/caller/compiler/precision/recovery and selected cold replay were not run.

Task and overlap

Canonical main was pulled before selection and refreshed before publication at 3f0152a32eeebac0df11946dbd50cabe5a7f7839, priority revision 8. The selected binding-only orientation is reused after checking its unchanged inputs, not presented as a fresh scientific replay. Task bit-new-register-model explicitly changes scope to this formal alphabet question.

Canonical claim/status. Current handoffs and bounded all-state searches found no competing signed reduction. Local sparse-note coverage is incomplete. PR6148 resolves a specific eight-call half-frame candidate; PR6186 accelerates the positive checker. The source theorem heads are PR6111 961b749fa40a1117d0746064528d2dfa290c57e0 and PR6079 44745c427f710e406cb2d2c6de458b7f93d81f01. Scope key: 7084092dd726e7ba1f2f.

Open next questions

The next genuinely changed premise is three or more additions. Start with one explicit three-frame candidate and determinant/support checks; the two-summand sign lemma no longer applies, and an exact word plus full native invoice remain missing. No follow-up search is reserved. See the package README and TASK.json.

Credit

Prepared by apoursh with substantial OpenAI Codex assistance. Credit Josie Elliston/Claude (#6111), JoyboyBrian (#6104), apoursh (#6089/6079), Nick Petalas (#2910), Cinna Davis (#2085/2112), Philippe Van Velzen and the retained BIT lineage. Apache-2.0. Classical determinant, kernel and rank algebra retain their background status.