research: trace V45 recovery and price the remaining margin

[RossRichardson]

Change and mathematical scope

The bounded V45 comparison finds no new fully costed improvement to #2979. It explains where the apparent extra recovery went: PR2894 and PR2917 already have the same four algebraic directions on the fixed twelve-register readout U12. The later paid frames make them accessible; after the four PR2942 extractions, every coefficient from a fresh eligible input in U12 is zero. PR2979 retains that exact map.

This package adds the complete before/after coefficient comparison for both providers at five pinned stages, a written nonlinear consequence over every dyadic module, and an exact necessary cost screen. It distinguishes algebraic readouts from shipped literal V45 supports. In the old readout, a dyadic-unit completion identifies the quotient by nuisance inputs with the four existing quantities. In the later readout, the zero eligible block excludes any nonconstant fresh eligible-input function under arbitrary nonlinear decoding or eligible-only recoding. The stated suffix extension fixes the complete dependency graph, output identities, monotone frames, entrance cut and source-access contract.

At PR2979's rational complex parameter, all 2,551 fixed-bin dual patterns certify the marginal fractional idle charge D=(H48-H3-5H5)/20. A hypothetical fresh 9-to-7 extraction has only about 0.0246870954232 tariff units per call left for its extra local charge, plus the separately stated old fractional allocation slack. This is a necessary screen, not a physical invoice or a new outer κ. No candidate survives the information comparison, so this bounded route stops before construction expansion.

The result does not establish optimality of the existing four-direction implementation or exclude changed middles, other cuts, incomparable endpoint combinations, or changed source access. It does not replay PR2979's exploratory lowering search, whose local dependencies are absent.

Task and overlap

User-assigned follow-up to #3172 under complex-new-helper-inventory, with the scope change recorded in TASK.json: exact transfer and a decisive scoped negative result, rather than the card's positive supplier acceptance gate. Canonical main and guidance were refreshed unchanged at 2026-10-17 14:52 UTC, base 997e6a9ca7b4f95d50e366f61412fd640d72a88a, priority revision 6.

Canonical coordination: #2612 (comment)

Nearest work: #2918/#3172 fixed-word obstruction; #2917/#2942/#2979 existing V45 constructions; #3014 frozen-owner negative; #3112/#3119/#3126 native/invoice audits. Their results and credit are retained. The separate active #3133/#3148 whole-word investigation is outside this comparison. Presubmission overlap scan covered all 295 open PR bodies and 1,956 discussion records with complete pagination, plus relevant local notes/ledger; it was not a review of every diff or external literature.

Validation

Commit 9c0399238a0a93defb7123799978bfb33689396b adds only research/v45-recovery-transfer/ (12 files). Fourteen live inputs have SHA256/Git-blob pins; all eight provider copies were checked against their original scientific heads.

• CPython 3.12.12 registered replay: python3.12 scripts/research_package.py --package research/v45-recovery-transfer — PASS_V45_RECOVERY_TRANSFER and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=completed.
• Separately implemented reverse-covector/inverse-column and finite-span checker, importing no predecessor Python, agrees on every U12 coefficient, complete bases, pairing entries, frames and minors.
• Exact cost basis inversion and all 2,551 dual patterns; eleven named rejection controls, four noncommuting fixtures, and six detected development sensitivity mutations.
• Source-identical Git-free isolated optimized CPython 3.14.6 replay passes with the same stored result.
• python3 scripts/check_research_commit.py --package research/v45-recovery-transfer — PASS_RESEARCH_COMMIT_BYTES for all 26 package/input files at the committed head; git diff --cached --check passed before commit.
• Separate source/compliance, cost, mathematical and independent-verifier reviews found no unresolved defect within the declared scope. These were AI-assisted reviews, not independent human or formal-kernel review.

No full native chain, new chronological owner construction, physical invoice, selected cold replay or new κ is claimed. Hosted CI is separate from local evidence; no manual check rerun or no-op retrigger was requested.

Credit

Jeremy Kothe's words/frames/cleanup, Mason Hale's extension, Jeremy Diviney's paired extraction, and Marius-Paul Dumitrean's coordinated V45 construction and rational enclosure kernel remain credited. Nick Petalas and jwr404's existing audits delimit overlap. The retained owner/native lineage includes Yiyang Lu, Tekalign, icekylinx, Dugongue, themilkman216 and the frozen packages' contributors. The authenticated PR2918 forward engine is reused unchanged; the new reverse checker and exact cost basis/dual code are separately implemented. New comparison and proof by Ross Richardson with substantial OpenAI Codex assistance. The underlying quotient and linear-programming principles are standard.

Open next questions

Can a changed middle, changed observation cut, or cheaper realization of the four already-used directions beat the complete cost? The next useful step is an exact changed program with paired residual, compensation/restoration proof, and complete local histogram screened against an applicable bound. Any private-buffer, chronological-owner, native, stock, precision and outer-admission costs remain explicit obligations. Repeating nonlinear decoding on unchanged U12 cannot supply a fifth fresh helper.