Classify H6 dyadic Smith factors and sharp physical-row repair

[RossRichardson]

Summary

For the inherited H6 complement matrix C=I-K over Z[1/2], this derives the complete localized Smith factors 1^30400, 11^336, 209^208, 4807^16. Its determinant has odd factors 11^560*19^224*23^16, so rational invertibility does not make C a reversible dyadic source map.

Every unimodular dyadic repair has rank at least 560. An explicit set of 560 actual physical row IDs, with small exact modular certificates, proves those rows suffice for algebraic replacement while preserving the other 30,400 rows. The proof supplies existence of a primitive row completion, not its coefficients or a paid circuit. A stable unit extension needs exactly 560 auxiliary directions algebraically; its simple unit blocks mix entering helper dirt into the output and change the helper outputs.

Scope and limits

Only research/h6-dyadic-source-repair/ is added, with six live source files pinned to original Git objects at 20ecf2891d80b5acf18d980b2164fb44ea7faa90. The full spectral and sector theorem is inherited explicitly from its moment proof; the new replay does not construct the dense 30,960-square matrix or rerun a dense Smith reduction.

C is the mathematical complement I-K, not the current native K response. No native complement producer, legal source/output gauges, completion coefficients, basis-change tariff, dirt compensation/restoration, physical banks/copies, precision/tape or joint outer admission is supplied. Exact (x,h)->(Cx,h) with all arbitrary helpers restored remains impossible as a dyadic unimodular map. No new kappa, paid source circuit or selected-record change is claimed.

Test plan

• Fresh source-bound determinant, reduced-annihilator and complete Smith valuation checks; named 560-row support, mixed/harmonic/radial modular minors and actual unit blocks.
• Separate exact checker freshly derives all 240 harmonic, 560 mixed and 32 radial polynomial actions. Two producer matrix mutations, two arithmetic counterchecks and three independent mutations execute; inherited full spectral premises remain explicit.
• Registered isolated optimized Python 3.12 replay with warnings as errors, read-only authenticated inputs and fresh external output.
• Exact commit 7bf23e128e58a33c019268f57129629f603547f7: committed-byte/source-history preflight passed for all 18 package/input files, with only ancestral 20ec source history. Actual read-only Git-free optimized export replay passed in 1.96 seconds with every byte unchanged. VALIDATION.json retains earlier precommit measurements.
• Automatic exact-head hosted CI 39130620314 passed for 7bf23e128e58a33c019268f57129629f603547f7; no manual rerun.

Sources and credit

Credit Dominik Scholz's PR606 geometry/factorization, DreamingOfClouds' H6/Walsh proposal, and PR1830's spectral/compensation/encoder audit with its retained predecessor lineage. This uses their exact moment/sector theorem and adds the localized Smith consequences and actual row-support certificate. No priority is claimed for general Smith theory, primitive-module completion or rank-perturbation bounds.

Ross Richardson, with substantial OpenAI Codex assistance in derivation, separate audit and packaging; Apache-2.0. Separate AI implementations are not human review or formal-kernel verification. The package retains source pins, adaptation records, a complete scoped proof and open implementation questions.