Exact rational borrowing: reconcile numerator and residue width contracts

[bitplane]

DONE — PR #1229, final submitted head 3e76ef5d86561e100f8812714eb316cfe5939fc6.

The comparison is complete: exact rational factors with odd reduced denominators admit the unchanged PR1057/1159 residue word, with an integral completed output and strict signed recovery guard. On the pinned 52-factor / 1,376-call example, the least actual common prefix denominator is 9. Common-numerator data have a 94-bit sufficient bound with that denominator; residue data use 56 bits. Numerator widths are bounds, not minima. The additional inverse/coefficient preparation costs 2,406 commands, for 54,698 priced preparation-plus-kernel commands. Admission, conversions, controller and physical implementation obligations remain explicit. No rounded-leaf repair, end-to-end runtime or κ claim follows.

The complete read-only isolated optimized Python 3.12 replay agrees with Python 3.14: receipt 2c63ba9c68e4537a49815af076b9097ae49c3f5a7ca66229710455b9d3aed6e2. Registered replay, eight package tests and 25 selected/source-binding tests pass. Twelve mathematical/admission/price controls delimit the contract. Exact predecessor heads and attribution are in the package; this does not contradict Ali Romman's PR1135 numerator representation result.

Next questions in README/TASK: test a named selected supplier segment for unit denominators and integral boundaries; price a structured integrality certificate. Scope is explicitly changed from a navigation-only card to an executable representation comparison. Claim released.

Original claim retained below:

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CLAIM

Question: Reconcile PR1135's common-denominator exact-lift width with PR1159's residue backend. For exact rational factors with reduced odd denominators and an integral completed output, can the same borrowed-port word run modulo 2^w without initializing q^D-scaled data, while charging coefficient inverses and conversions? This changes the representation and output promise; it does not repair rounded leaves or contradict the numerator-representation lower bound.

Base: 1b59b62. Predecessors: PR1057 78a062df1dcab2bffea5ff4d8243f3da6e465414; Ali Romman PR1135 4c35687d7c0da8e6cf33f17e418ecf31c95ff3ad; PR1159 f38bfe5da0bed21f42ce5251d23491a56519d7c4.

Method/domain: localization homomorphism and a source-bound applicability table; centrally rescale the frozen 52-factor PR1159 fixture by alternating 1/3 and 3, preserving its integral full product. Compare exact Fraction prefixes with every literal modular leaf, check arbitrary dirty columns and inverse calls, and invoice coefficient preparation. Explicit counterexamples for even denominators, nonintegral output, missing recovery bounds and rounded substitution. No broad rational reconstruction search or new kappa claim.

Expected artifact: one scoped research package with proof, source pins, complete deterministic receipt, paid comparison and rejection controls; one PR. Consolidation seed99. No overlap found in current discussions or PRs through1190; PR1164 changes echo arity and is separate.

Claim expires: 2026-10-13T22:01:17.007812+00:00.

Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance. The algebra and predecessors retain their credit; no modular-arithmetic priority claim.