Prove binary cold-factor and separate-reader cost floors for R429

[RossRichardson]

Summary

For the fixed R429 correction matrix on 111 E8 root lines, prove two separate binary reader bounds: every ten- or eleven-term scalar factorization has rational pairing tariff at least 56, and any separately materialized source-reader family has total rational span cost at least 53. Factors outside the retained label code are allowed. The latter bound rejects the specified three-seed caller at primitive exponent a ≥ 1/500 under its complete fixed invoice premise.

One isolated package contains the written proofs, frozen inputs, independent AI checks and source-integrity controls. The nine-term value 63 is an attributed source-binding recheck of an existing result.

Scope and limits

The two tariffs are different and cannot be interchanged. Neither 53 nor 56 is claimed attainable. The cold-factor claim covers ten and eleven terms; twelve or more remain open. The caller exclusion assumes its stated bulk histogram, common-axis credit 222 and nonnegative omitted pressure. Shared native queries, nonlinear/operator readers, dirty restoration, physical owners, tape costs and complete supplier admission remain open. No selected-frontier change or new κ is claimed.

The proofs combine all-factor algebraic arguments with complete finite certificates and an explicit E8 parabolic reduction. This is neither a formal-kernel proof nor independent human certification. Older 49/55 receipts are preserved as dependencies and controls, with their relation to the stronger bounds explained.

Test plan

• Full registered Python 3.12.14 replay: seven mathematical stages, two independent checkers and integrity controls; all nine scientific outputs compared completely.
• All 2,503,900 coset-rank cases, 56,721 unit/codeword toggles, 638 dual bases, 61,320 near-cap cases and 340,260 bounded-cost rank partitions.
• Independent exact source propagation, parabolic capacities, rational pair ranks, fixture correspondence and rational pressure; separate AI mathematical and source-integrity reviews.
• Source/receipt mutation, omitted coverage, scalar-type substitution, normalized-path collision and valid poisoned-cache rejection. Actual-launcher controls pass normally and optimized; committed source-export controls pass from a relocated path.
• Metadata and full source-history preflight at 44f3519ec9758eb0fa3952582fa8835d21a4c5c9; scientific code and registered replay are unchanged from the complete local replay at 6579ca4071428680a0ea83db50792ed9d161ae1e.
• Automatic GitHub CI passed at 44f3519ec9758eb0fa3952582fa8835d21a4c5c9, including the complete registered replay and unchanged-input gate. No manual rerun requested.

Sources and credit

Scientific pin: 20ecf2891d80b5acf18d980b2164fb44ea7faa90. SOURCES.json, ORIGINS.json and FROZEN-SHA256.json bind canonical references and all 33 retained inputs. The package is self-contained and requires only Python's standard library.

R429 exactly matches eumemic's PR555 fixture covered by Russell Seligmann's PR757; the fixed-label value 63 is prior work. Credit also to Joel Pulikkan / GamingPuzzled PR620, and Cody Harris PR761, whose different PR545 fixture supplies a method antecedent. Sagar Thakrar supplied the transported R438 word, building on Ali Romman and the E8 lineage of eumemic and DreamingOfClouds. Chi Zhao, Philipp, Ross Richardson and 3xcalibur-tech supply the flat-capacity/support-union lineage; Hanson Sun supplies related outside-code analysis over a different field. Detailed references and overlap distinctions are in the README.

Prepared by Ross Richardson with substantial OpenAI Codex assistance and separate AI reviews. Original notices retained; Apache-2.0.