Research: exact selected idle-axis relaxation and stream counterexample (no new kappa)

[bitplane]

The selected PR1094 idle-axis domain has a provable structural optimum, but its attaining reassignment changes the physical operator despite passing the inherited controller, invoice and assembly checks. This package proves the exact relaxation optimum and supplies a source-bound counterexample; no improved kappa is claimed.

The proof covers all 4,866 eligible rank-48 children in 1,792 groups under PR1324’s final-list rule and rank cap 51. Nine compatibility patterns give attaining vectors that minimize the formal moment for every real 0<a<1. An independently implemented integral-flow census checks all 1,049 labelled rank vectors. The relaxation price is 0.001673074238324615, a tighter ceiling than the prior rank-only relaxation.

An exact Gaussian-integer replay preserves the eight data streams, actual event order and all 30 relevant hosted helper occurrences. At replica 0, port 0, character 32, the proposed optimum differs from the baseline i*I_8 in three entries. A single legal axis move also fails. The new guard rejects both; --full deliberately reproduces legacy acceptance of the invalid optimum, including all 96 inherited controls.

PR1324 was closed during this investigation for the same cross-family issue, crediting Rohan Arun. That prior finding is explicitly credited: the contribution here is the complete selected structural optimum, explicit matrices and reproducible regression. It extends PR1276’s star-domain analysis; it does not claim that every nontrivial reassignment fails. Frozen sources and the selected record are unchanged.

Validation:

• Registered full replay and all eight test groups pass on Python 3.12.3.
• A read-only isolated Python 3.14.4 full replay produces the identical canonical receipt; inputs remain byte-identical.
• Both rational moment engines, endpoint rejection checks, and all 47 outer constraints pass as relaxation pricing only.
• Original local/native and bit artifacts are authenticated and reused; no fresh complete ancestry rebuild or Lean proof is claimed.

Package: research/selected1094-idle-reassignment/ (README, proof, exact sources, saved certificates and replay manifest). Reproduce with python3 -I -B scripts/research_package.py --package research/selected1094-idle-reassignment.

Coordination: #1339. One recorded random selection; scope changed in response to the semantic counterexample, with no reroll. Credit to Gonzalo-Mier (#1252), Guéorgui Maissouradze / PatatePatatePatate (#1276), Jeremy Kothe (#1324), Rohan Arun and the retained source lineage. Gareth Davidson (@bitplane), with substantial OpenAI Codex assistance; Apache-2.0.

[Mister-Meeseeks]

100-PR batch review at 1d427e04346c4cce6430220c2e8ec1146ad4c396.

Merged to main, preserving the original contributor commits and package attribution.

Reviewed scope: Exact selected-word structural relaxation optimum, independent complete flow census and all-exponent majorization proof. Full legacy acceptance is reproduced, but an exact eight-stream Fourier counterexample rejects the attaining witness. No improved κ is admitted; cross-family flaw credit retained.

The pinned scoped command passed; the batch ledger contains its command, input hashes, environment and log. This incorporates the research/tooling contribution and does not change the selected κ or discharge the inherited multiplication-theorem assumptions.

Batch review, attribution and evidence. Thank you for the contribution.