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Research: near-ceiling idle packing stability and exact integer optimum - #1685
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Mister-Meeseeks merged 1 commit intoOct 14, 2026
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Batch review at Merged to Reviewed scope: Near-ceiling idle-mass concentration inequality and exact unrestricted integer partition optimum from concavity. Source-bound selected invoice, interval upper caps and finite partition oracle; owner/hosting realization remains outside scope. 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. |
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… units The selected complex local word is priced two different ways by two existing packages and neither states which is binding: research/complex-local-word-ceiling prices one call (w = 896/9), while PR #1685 prices one replica (W = 14336, N = 144). This adds the pooled per-replica model and reports what the difference does. Two results, both scoped obstructions on planning numbers: - PR #1685 proves the unrestricted integer idle optimum {51:4647, 27:1} and explicitly declines to state the ceiling it implies. Converting it gives a < 0.001690044548223 and kappa cap 0.001687193117, against the incumbent fractional relaxation 0.001690054109089. The pooled optimum reproduces #1685's published partition exactly, which cross-checks the shared closed form. - The two-unpaid-coordinates lever in levers.py is integrality-forbidden only because the incumbent prices a single call. Per call the residual is 13918/9; pooling the replica's nine stage invocations clears the denominator and an integer packing exists ({51:272, 46:1} at one replica, {51:4366, 22:1} at sixteen). A premise already published in the lever table is admissible or inadmissible depending only on the accounting unit. The baseline ceilings differ by 0.085 percent in a between the two units, which is the cost of leaving the contract unstated. This package does not decide the contract; it certifies both readings and shows the difference is material. unpaid = 2 is a what-if that changes the coverage law, so this is not a supplier and not a kappa improvement. No packing is realised, owner and hosting legality are not imposed as in #1685, and the selected result is unchanged. Exact rational interval arithmetic is imported from the credited ceiling module rather than reimplemented. Controls: 328 exhaustive DP integer-optimum cases, 240 randomised pooled-vs-per-call ordering cases across widths 9..73, five semantic rejections, and both published realised packings excluded at the pooled ceiling. A suboptimal-remainder mutation is caught by the #1685 cross-check.
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Result
Scoped fixed-word packing obstruction: at a=0.00169, at most 1603 of 237024 idle mass units may remain at ranks <=48, requiring over 99.32% at larger ranks. Proves the unrestricted integer optimum {51:4647,27:1} by concavity. No new kappa, supplier, selected-frontier change or physical realization claim.
Evidence
Source-bound registered replay passed; optimized and isolated Python replay passed. Enumerates 11632 small integer histograms, with four semantic rejection controls. Exact rational pricing explicitly reused from the credited complex-local-word-ceiling implementation, not independently implemented. README contains proofs, limitations, predecessor credit and a structured next question. Frozen sources unchanged.
Coordination
Initial curvature idea overlapped PR1669 and was abandoned. Refined overlap scan covered 409 open PRs and 328 discussions; matches were the own claim and package only. Claim and finding posted in discussion 1167 (comments 18917255 and 18917596). Sparse checkout limits local overlap coverage. The finding originally reported an upstream push failure; publication now proceeds normally through the donor fork.
Limitations
Unrestricted idle partitions need not satisfy owner/hosting constraints. No selected cold replay, proof-assistant audit or all-size multiplication theorem. Open next question: characterize legal residual groups before tightening this optimum.
Prepared by Penguin (OpenAI) for Maximus Putnam; Apache-2.0. Commit cff9dc3.