Idle pair exchanges improve the #1094 κ candidate

[Gonzalo-Mier]

Change and mathematical scope

Adds a finite candidate on the reviewed #1094 replay. Repartitioning 1,825 disjoint, completely idle rank-48 child pairs as 48 + 48 → 45 + 51 preserves each pair's exact Z4 phase polynomial and improves its exact candidate price from κ = 0.001672660197076918 to κ = 0.001672846574134982 (conservative price 0.001672846571282754). The paired axis union is unchanged; the maximum child rank is 51 and the finite halving guard remains 73² > 2·51².

This is a finite conditional construction and is not promoted to the selected result. The full paid profile is in CANDIDATE.json. It retains the 5,120 rank-mass deficit and all existing controller/owner constraints.

Task and overlap

Base: dd6403f7 (main after the requested pull). Scope: user-assigned search for a better integer multiplication κ using a new idle-pair repartition.

This builds on selected PR #1094 and the T16 rare-pattern exchange source. It follows the idle-child search explored in #1042, but uses rank-48 pair exchanges and includes its own pinned pricing support rather than importing the earlier research helper. Open PR #1183 reports a stronger κ from mixed T37 entrance providers; this submission makes no claim to be the current frontier. Its added value is an independent phase-preserving finite construction, full strict admission receipts, and a replayable overlay on the reviewed #1094 checkpoint.

Validation

Reproduction commands and exact inputs are in research/frontier1094-idle-exchanges/README.md. On Python 3.10 with the repository's pinned engine sources:

• Fresh Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 hosted emission reproduced the archived template and related baseline artifacts byte-for-byte.
• build_candidate.py passed with 1,825 pairs and κ = 0.001672846574134982.
• verify_overlay.py independently passed for all 1,825 pairs and 3,650 changed children.
• The strict independent controller/owner audit passed with all 15 mutation controls rejected.
• Retained strict invoice and assembly receipts pass, including 47 positive outer constraints, 69 assembly mutation controls, and 12 bit-supplier controls.

These are finite certificate checks. The all-size compiler, common-label and mutable-source realization, precision/tape, recovery, analytic transfer, and Lean theorem obligations remain conditional and were not discharged here.

Credit

The package retains the #1094 replay and T16 source attribution, identifies the earlier #1042 exploration as a predecessor, and discloses OpenAI Codex assistance in its README.

Open next questions

See the package README: independently reproduce the matching maximum; search broader commuting multi-child moves; and address the inherited end-to-end theorem obligations.

[Mister-Meeseeks]

100-PR batch review at 1b9db2faf82b8866b0bb0164501c9a627c7dca9b.

This remains open after scoped triage. Next review step: Idle-axis repartition changes the selected construction and root README; reproduce baseline emission, exact overlay phase/owner audit and repricing before admission. A standalone overlay script is not the full requested sequence.

This is a specific continuation point, not a rejection or a request to redo already recorded checks. Completed local evidence is retained separately from outstanding whole-head obligations.

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