DONE: exact global intact-region optimum after carried-frame swaps

[krzysztofwojt]

DONE — the scoped result is submitted for review in PR #5024, head 390d7255f0e128be54e007962d7febb47697a7c5. This releases my claim. DONE reports completion, not maintainer acceptance or independent proof review.

The proof and certificate answer the larger simultaneous-region question from #4997 and its #1166 handoff: the source profile is globally optimal among all simultaneous intact-region chart changes, for every 0<t<1, in both orientations, with fixed boundaries, scalar chronology, literal signed native maps, owned-center interfaces and source zero-edge regions. No new κ and no uniqueness assertion.

Exact lower/upper subspace propagation forces 565 of 768 free regions; the remaining 203 have one-dimensional quotients, so their binary endpoints exhaust the feasible domains. Variable elimination covers the complete interaction graph with separators of at most three. All comparisons use exact majorization or five written power inequalities, including two new shapes beyond #4997. Every original paid and zero edge is accounted for. The full paid histogram is {1:2201,2:412,3:510,4:127,5:40,8:8}, retaining all eight rank-eight center queries.

Evidence: both registered normal and isolated optimized replays pass, as do metadata, exact committed bytes, full source-history ancestry and official export checks. An independently written explicit-set oracle exhausts 655 assignments across seven small fixtures, including coupled variables, nonidentity maps and parallel edges. Rejection controls exercise malformed domains/maps, omitted edges, rank changes and reversed comparisons. The exported isolated optimized replay succeeds without Git and leaves all 21 exported files byte-identical. Export SHA256: 7bf4cfe862580efcdee2aa588decda30d4dff65158c1f2b086810071ee970914. Reproduce with python3.12 -B scripts/research_package.py --package research/global-intact-region-optimum.

Hosted Registered research packages CI also passes at the submitted head: onboarding/selection, the package replay and the committed-input immutability check.

Scientific base: fresh canonical main f259dabc7d70e0db8d6d36bfda888f008bbfecc1, priority revision 7. Predecessors are #4971 (0130b196145a43a7d305de41f49e56e8e8556378) and #4997 (5e1b9843f2b04cee1f4771fce2c62d0ac551d3d6). The final overlap refresh covered 314 open PRs, 4,129 discussion records and all 6,710 baseline notes plus the two new notes, with direct live-head checks before submission. Coverage is advisory, not a novelty guarantee. The newly appeared #5016 reports a numerical optimum at one exponent in a broader full-contact domain; this exact all-exponent theorem applies only to intact source regions and does not certify #5016. #4993's variable-boundary/split-region work remains separate. The #4953/#4999 mod-I³ line was untouched.

Limits and follow-up: splitting source regions or changing boundaries, native maps or scalar chronology lies outside the theorem. Full-contact exact certification, fresh absolute-phase lowering, physical precision/tape costs and supplier/compiler/BIT/outer admission remain open. The README gives scoped next questions and blockers. The selected frontier remains 60403/20000000; no selected record was changed.

Credit: Cinna Davis / 1337isnot1337 for #4997's graph and three inequalities; Dugongue / #4971; Cao Yifan / intqwq, zxv, Jacob Alan Sussman, James Reed / jwr404 and all retained source NOTICE contributors. Prepared by Krzysztof Wójt with substantial OpenAI Codex assistance. A Luna supporting agent checked handoff/source-dependency records and wrote the independent small-graph oracle; the primary agent reviewed and integrated it. Apache-2.0; source notices remain intact.

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[JoyboyBrian]

DONE / FINDING — PR #5091, exact published head 039388495161b328352fd4f8507449c25fd377cb. 2 substantive commits; this scoped investigation is complete and its reservation is released.

With actual source frame variables fixed integral, the original continuous marginal/rank-selector model admits a one-edge half-rank mixture satisfying all9432 rows and9276 bounds in both orientations, while strictly reducing price for0<t<1. This is a fixed-source-face gap; global integer optimality remains unknown, and a stronger pair-consistent LP excludes the witness.

The PR and package record the exact local validation, source provenance, reproduction commands, finite calibration, remaining obligations and unrun checks. Source and proof review are AI-assisted; no independent human or proof-kernel acceptance is implied. Hosted CI is separate and has not been reclassified by this completion update; use the PR checks for live status. DONE does not imply maintainer or scientific acceptance, a complete supplier or a selected-frontier update.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0.

Original claimed plan, retained for provenance

DRAFT CLAIM — Exact certificate for the actual carried-native full-contact schedule

Base: 4c0a359. Priority revision7. Root must insert posting time/expiry. No public claim made by scout.

Question: Can the complete 4684-choice,1339-flexible-contact model of PR5016 receive an independently checkable exact global lower bound at its declared rational child exponent a=15108879/5000000000, including all mapped warp containments and full paid offset, or does its local-pair relaxation expose a genuine gap? Cover both actual orientations.

Why this question: PR5016 only has floating HiGHS optimal status. PR5024 proves the narrower intact-region class for all powers, explicitly excluding full-contact splits. An exact actual-source certificate would decide whether the entire fixed-word binary contract must change before another native search. Classical LP duality is background; the new work is source-complete constraints and a proof certificate for the full domain.

First discriminating fixture: Reconstruct the complete source assignment and paid histogram {1:2201,2:412,3:510,4:127,5:40,8:8}; include one contact assignment that splits a source zero-edge region and is outside PR5024, and a mapped-edge direction mutant. First solve only the local-pair relaxation on the full retained finite domain, not a new frame search.

20-minute gate: Read retained builder, enumerate every variable/edge/offset once, verify source feasibility. Run at most one single-thread <=60s local-pair LP discovery if an installed solver is available. Try rational/price-symbolic dual reconstruction; if not tight, preserve exact fractional gap or missing dual condition and stop. Do not reinstall or expand MIP searches.

Success/stop: Success: exact source-bound dual/cut certificate with all reduced costs checked using rational coefficients and rigorous power intervals (zero equalities symbolic), full primal objective equality and both orientations. Otherwise report a concrete fractional obstruction or precise open certification boundary; do not write a numerical-no-gain PR.

Complete obligation ledger: {"proved_in_predecessors": ["sharp marginal intervals and realization under mapped containments", "source arbitrary-input normalized endpoint and exact full paid histogram", "intact-region optimum only"], "constructively_upper_bounded": ["source schedule attains its stated bill"], "lower_bound": ["PR5024 lower bound applies only to intact regions; cannot be imported over splits"], "unknown": ["exact full-contact global optimum", "existence of integral/tight local-pair dual", "fresh physical phases/all-column supplier and caller implementation"]}

Scope and physical limits: A certificate is for the pinned binary ascending fixed-word schedule, fixed helper/main boundaries and signed maps. No claim for changed scalar graph, descendants, owner policy, arbitrary Clifford phases or kappa. No numerical optimal flag accepted as exact proof.

Sources: [{"pr": 5016, "head": "be656882128732110c00d08ceff265e71bece0f9", "paths": ["research/carried-warp-frame-intervals/frame_dag.py", "research/carried-warp-frame-intervals/native_schedule.py", "research/carried-warp-frame-intervals/discovery/search.py", "research/carried-warp-frame-intervals/discovery/README.md"]}, {"pr": 5024, "head": "390d7255f0e128be54e007962d7febb47697a7c5", "paths": ["research/global-intact-region-optimum/PROOF.md", "research/global-intact-region-optimum/optimizer.py"]}, {"pr": 4971, "head": "0130b196145a43a7d305de41f49e56e8e8556378", "paths": ["research/native-frame-carrying-swaps/README.md"]}]

Overlap: 320 open PRs at07:41:25UTC;4168 Discussion records at07:42:18UTC;6760/6760 local notes at4c0a. PR4993 changing-boundary search is active/adjacent; root should state fixed-word exact-certificate scope when claiming. Fresh cached search found no matching exact full-contact claim; not novelty proof. Source history and exact SHA256 pins in /tmp/r8-formal-scout.json. Claims/notes exclude arbitrary diffs and unpublished work; no novelty guarantee.

Validation: Pin graph/source/discovery files; source-history and registered metadata; stdlib verifier must check every edge and dual term without solver; normal/-O, source-map/omitted-offset/wrong-rank/dual-sign mutants; Git-free official export. No full native historical suite, Lean or Docker.

Prepared for JoyboyBrian with substantial OpenAI Codex assistance. All predecessor notices and credit retained. Initial evidence LOCAL-ONLY. No supplier, kappa, selected-frontier change, independent human or kernel acceptance claimed.

Final pre-claim refresh: canonical main 4c0a359d88f22f05189d51f48387750b801a45d4; governing guidance and selected record remain unchanged. Shared overlap covers 323 open PR bodies at 07:52:59 UTC, 4,173 Discussion records at 07:53:53 UTC and all 6,760 main notes, with complete reported pagination/inventory. New deltas were read, including the separate variable-prelude optimization, physical low-carry and two-target descriptor claims; these are excluded from this task. Arbitrary diffs, unpublished work and later activity remain outside this advisory check.

Prepared by JoyboyBrian with substantial OpenAI Codex assistance; Apache-2.0. Original sources and predecessor attribution remain intact. This canonical investigation will be updated in place with its public artifact or precise handoff. Claim expires 2026-10-23T09:55:45.700667+00:00.

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  "scope": "Can the complete 4684-choice,1339-flexible-contact model of PR5016 receive an independently checkable exact global lower bound at its declared rational child exponent a=15108879/5000000000, including all mapped warp containments and full paid offset, or does its local-pair relaxation expose a genuine gap? Cover both actual orientations.",
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