Research: exact ceiling for #1094 idle rank-48 repartitions, #1252 recount and star-move lead (no new kappa)

[PatatePatatePatate]

Scoped finding; no new selected κ and no admitted construction. This adds only research/frontier1094-idle-star-ceiling/.

Task card

• Card: selected-packing (incumbent-optimization track, chooser seed 709c25d61571e40787fc6379741fb2da, base dadfea6e). The scope is refined to one named residual subproblem, using the card's method "exact relaxation or exact optimum".
• Object: the frozen Conditional kappa 0.001672660197076918: rare-pattern T16 exchanges #1094 idle remainder has 4,866 unhosted, non-owner idle rank-48 children in 1,792 (replica, port) groups. These are the children Idle pair exchanges improve the #1094 κ candidate #1252 repartitions.
• Question: this answers both of Idle pair exchanges improve the #1094 κ candidate #1252's open next questions on a bounded class. (a) Does an independent recount reproduce its 1,825-pair maximum? (b) What can coordinated multi-child moves gain, and what can any within-group repartition gain?

Result (exact native κ, pinned multicold engines)

Profile	κ	Gain over #1094
#1094 selected	0.001672660197076918	—
#1252 pair class, independent recount = 1,825	0.001672846574134982 (matches #1252)	1.86e-7
Disjoint-star class optimum (2,357 moves, 1,561 centres)	0.001672987659462603	3.27e-7
Ceiling: any within-group repartition, ranks ≤ 51	0.001673151843213755	4.92e-7

• Ceiling. Children can't be deleted (48k > 51(k−1)), splitting only adds cost, and the moment is Schur-concave at a fixed child count. So (51^(k−1), 51−3k) is optimal per group under any commuting rule. This bounds the multi-child route at about +4.9e-7 κ, which is 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. Four random repartition controls price below it.
• Star class. A centre sends ≤ 3 axes to each of several leaves. Each moved axis satisfies Idle pair exchanges improve the #1094 κ candidate #1252's per-axis rule: it commutes with the leaf's axes, every stage X/Y, the terminal Y and both terminal translations. This class contains the pair class and captures two thirds of the ceiling gain. The explicit overlay is in STAR-MOVES.json.gz.

Verification

python3 -B research/frontier1094-idle-star-ceiling/analyze.py --check   # ~5 s, byte-identical outputs
python3 -B research/frontier1094-idle-star-ceiling/verify.py            # ~2 s, independent, 7/7 mutation controls rejected
python3.12 scripts/research_package.py --package research/frontier1094-idle-star-ceiling   # PASS (outcome=partial)
python3 -B -m unittest discover -s tests -p 'test_research_*.py'        # 20 OK, 1 skipped

Everything is read directly from the committed selected-replay.tar.gz. Every price checks acceptance at the exact endpoint, rejection at +1e-24, all 47 strict outer constraints, and rejection of the next κ grid point. Baseline and #1252 values match at both native and conservative precision.

Not checked / remaining obligations

• For the star overlay, the serialized controller/owner audit, the mixed-provider invoice and the strict global assembly were not run. The star κ is a priced, structurally verified lead, not an admitted candidate.
• Composing Idle pair exchanges improve the #1094 κ candidate #1252 transfers into stars (with donor rank below 48) inherits Idle pair exchanges improve the #1094 κ candidate #1252's pair argument and is not proved here.
• All inherited conditions of the selected record (compiler, realization, hosting, precision/tape, analytic transfer) remain.

Open next questions

These are structured in TASK.json: admission of the star overlay by generalizing #1252's auditor; the controller contract for multi-leaf moves; and stacking the same moves on the X54 / mixed-T37 candidates (#1218/#1261).

Credit

Gonzalo-Mier's #1252 provides the move, the commuting rule and the pricing recipe. #1240 provides the broader fixed-word ceiling. The #1094 replay, owner map and engines come from the selected review's ancestry. Claim and coordination are in Discussion #1265. Prepared by Guéorgui Maissouradze (GitHub: PatatePatatePatate) with substantial Claude (Anthropic) assistance. Apache-2.0.

Workflow feedback (optional): the overlap helper and package generator worked well. The main friction was that the generator refuses an existing directory, so adopting work started before packaging needs a manual file shuffle. An --adopt flag that only writes the missing metadata files would help.

🤖 Generated with Claude Code

[Mister-Meeseeks]

100-PR batch review at 588425ebfc307dea3a94db9ba4bd4a3c74141b55.

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

Reviewed scope: Fixed within-group idle-cap majorization ceiling and pair recount, with a separately priced 2,357-move star proposal. Exact analysis passes; multi-leaf soundness and whole-supplier admission remain open.

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.