Research: bound additive idle-axis repair costs

[RossRichardson]

Idle-axis concentration can reduce a concave recursive bill only if its changed physical operator can be repaired affordably. This package tests repairs added beyond fixed original residual rank mass, with all original local/hosted calls, stock and normalization retained.

For 0 <= a <= 1/400, at most twenty original idle slots and residual rank cap 51, even fractional rank-51 packing saves less than 153/160 raw child-power units. One additional positive-rank whole child costs at least one, so the repaired moment increases by more than 7/160. The written proof covers the whole parameter interval. Four exact positive relaxed-invoice controls show why changing mass, group sharing or the integral repair tariff falls outside it.

The source-bound PR3148 replica-zero / port-zero example moves axis 16 from child 10 to child 12. At character 16 its full eight-stream discrepancy has rank two over Q(i), with minor determinant -2. One single-stream phase insertion at a retained caller cut cannot repair it. Two signed rank-one children restore the original factors, including global phase; the inverse is one selected child plus a translation, whose movement the favorable screen grants free.

The result does not cover repairs absorbed into existing children, changed interactions/stock, shared repairs across groups, opaque recursive internals or other saving intervals. No new kappa, supplier/native admission or full theorem is claimed. The selected record is unchanged.

Validation: Python 3.12 registered replay and metadata/pins pass; all 2,717,280 decoded T740 rows authenticate; inventories are reconstructed at 112 owner-replica-zero ports. Independent mathematical-specialist implementations confirm the allowance and operator witness. All 27 publication package/input files match commit 4aa4bd8d53a2bb80fa0c7d6f634c078b33965113; deterministic export integrity and Git-free optimized Python 3.14 replay pass. Continuous claims are written proofs; finite exact checks and AI specialist review are separately scoped, with no formal-kernel or independent human review. Hosted CI is separate.

Base 463113d675af7410ac91d53a4fed51ec6494eb9a, priority revision 6. Pre-submission refresh at 18:16 UTC retains that base. Live overlap covers 265 open PRs including drafts and 2,032 Discussion records with complete pagination and no warnings; matching only our claim is not proof of novelty.

Credit Rohan Arun/#3184/#3190, Gonzalo-Mier/#1252/#3196/#3199, Gareth Davidson/#1387, Nick Petalas/#3187, Tekalign/#3204, Jeremy Diviney/#3148, dcharb78, Marius-Paul Dumitrean, Jeremy Kothe and the retained native/word/geometry lineage. Prepared by Ross Richardson with substantial OpenAI Codex assistance. Apache-2.0. Canonical investigation.