Research: prove four-site idle-rank rigidity under the fixed schedule

[tekuuu]

Scoped negative result. Every operator-preserving replacement of idle children 8, 10, 12 and 13 in the fixed data schedule preserves each child's rank separately, under the inherited completed-call contract. This covers arbitrary simultaneous independent-axis replacements, including matched swaps and cycles. It answers the four-position question left open by #3196/#3199 without enumerating assignments. No new κ or supplier admission is claimed.

The proof has two steps. The intervening schedule preserves the receiver's two-dimensional coordinate subspace, so E_B M E_A = M forces both phase corrections to be identity. Polarization of a child's Z4 Fourier phase then recovers a binary matrix whose rank equals the number of independent axes. A complete phase signature uses 2,701 coefficients at dimension 73; this shortcut applies to individual child phases, not arbitrary circuit equality.

The checker authenticates the source schedule and all 448 full recipe bases across 112 ports. It does not reconstruct the T740 fill: rank 48 describes full recipe bases, while a hosted instance retains its own original residual ranks. Additional changed positions, changed intervening interactions and new local contracts are outside the theorem. Rank-preserving frame savings remain possible. Native execution and supplier/outer obligations are inherited, not reverified.

Reproduce:

python3.12 -B scripts/research_package.py --package research/idle-four-site-rigidity

Expected: PASS_IDLE_FOUR_SITE_RIGIDITY, PASS_IDLE_FOUR_SITE_EXPORT, and PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result.

Registered Python 3.12.14 checks pass. Git-free optimized replay also passes on Python 3.14.7. Controls cover 1,447 small independent axis sets, 22,584 character evaluations, 8,192 phase tuples, five source-schedule mutations and three export/rejection cases. Positive controls demonstrate nontrivial equal-phase bases and a transfer after changing the invariant cut. All eleven committed package/input byte comparisons pass. The unrestricted statement rests on the written proof, not the finite replay; no proof-kernel or independent human review is claimed.

Base and refreshed main: 463113d675af7410ac91d53a4fed51ec6494eb9a. Exact input/provenance pins are in SOURCES.json; scope, overlap, seed 40 and next questions are in TASK.json. Final overlap refresh covered 249 open PRs, 2,630 closed PRs and 2,004 Discussion records. #3200 involves additional positions and is outside this result.

Credit Gareth Davidson/#1387 and Nick Petalas/#3187 for the physical-stream obstruction and interval argument; Gonzalo-Mier/#3196/#3199 for the census and simultaneous-assignment question; Jeremy Diviney/#3148 and the retained word/geometry lineage. Ross Richardson's concurrent handoff reports a broader twenty-slot obstruction with local-only evidence. This submission supplies a separate, source-bound four-position proof and rank/phase representation; it claims neither exclusive priority nor review of that unpublished result.

Tekalign (@tekuuu), with substantial OpenAI Codex assistance. Apache-2.0. Only the new research package changes; selected-frontier and frozen predecessor files are untouched. Canonical claim.