CLAIM: exact ceiling for within-group repartitions of #1094 idle rank-48 children (#1252 multi-child follow-up), expires 23:30Z

[PatatePatatePatate]

CLAIM
Question: What is the exact best κ reachable by any repartition of the unhosted idle rank-48 children of the selected #1094 replay within a (replica, port) group (cap 51, no owner-slot children, child count fixed), and how much of that relaxation is realizable under #1252's commuting/phase-preserving move rule generalised to arbitrary axis counts and multi-child moves? This answers the two open questions of #1252 (independent max-matching reproduction; broader multi-child moves) on a bounded class.
Base commit / predecessor: main dadfea6; #1252 head 1b9db2f (Gonzalo-Mier, idle 48+48→45+51 pairs); #1240 (fixed-word packing ceiling, a broader relaxation that allows rank-1/3 hosted idle mass to move).
Method and bounded domain: frozen selected-replay.tar.gz idle-remainder frames (4,866 unhosted rank-48 children, 1,792 groups). Schur-concavity ⇒ per-group optimum is the majorization-maximal vector (51^(k−1), 51−3k) for k eligible children; priced exactly with the pinned multicold moment/outer engines. Then an independent exact search per group over commuting axis transfers (any number of axes, any donor/receiver, chained), separately from #1252's builder.
Expected artifact: scoped research package with a relaxation ceiling for this residual subproblem + independent matching audit; possibly a strictly better finite profile if multi-child moves are feasible (would be labelled a finite candidate, not admitted/selected). No change to the selected record.
Claim expires: 2026-10-13T23:30Z
Prepared in a donated session (GitHub: PatatePatatePatate) with substantial Claude (Anthropic) assistance.

[PatatePatatePatate]

DONE: PR #1276 (research/frontier1094-idle-star-ceiling/, no new selected κ)

FINDING (exact native κ, pinned multicold engines, from the frozen #1094 replay; 4,866 unhosted non-owner idle rank-48 children in 1,792 groups):

• Idle pair exchanges improve the #1094 κ candidate #1252 pair class: an independent recount gives 1,825 pairs and κ 0.001672846574134982, matching Idle pair exchanges improve the #1094 κ candidate #1252 at native and conservative precision.
• Ceiling over any within-group repartition (ranks ≤ 51, under any commuting rule): κ ≤ 0.001673151843213755, i.e. +4.9e-7 over Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094. That is about 3.4% of Research note: packing ceiling (+0.87%) and 1/99 budget for the selected complex local word (no new kappa) #1240's fixed-word packing headroom, so this route cannot move the frontier materially.
• Disjoint-star class (a centre sends ≤ 3 axes per leaf, each axis under Idle pair exchanges improve the #1094 κ candidate #1252's per-axis commuting rule): exact optimum of 2,357 moves from 1,561 centres, κ 0.001672987659462603. This is +1.41e-7 over Idle pair exchanges improve the #1094 κ candidate #1252 and two thirds of the ceiling gain. The overlay is structurally verified (7/7 mutation controls rejected) but not admitted.

HANDOFF @Gonzalo-Mier / anyone with #1252's replay set up: the next step is to generalize audit_allcold_idle_overlay.py from pair records to (centre, leaf, moved_axes) records, re-pin the SPEC, and rerun the controller, invoice and strict assembly on STAR-MOVES.json.gz. Also open: whether a multi-leaf move needs any commuting condition beyond donor and receiver.

Claim released.