Conditional κ = 0.001674097969905390: an attaining T37 construction and fixed R480 owner optimum

[aliromman]

The seven-slot k8 R480 word has an exact owner optimum across all positive replica counts. This package proves that bound and gives a physical 37-replica construction that attains it, with conditional κ = 0.001674097969905390. It changes only the owner assignment and its finite admission. The local word, bit supplier, and selected-result pointers are unchanged.

Review files: Proof and scope · Evidence · Source pins and credit.

The signed bin inequality covers all 1,163 whole-helper patterns. Two exact rational engines certify its moment support over the complete root interval [0.001676905273619680861300, 0.001676905273619680861301]. Equality forces a multiple of 37 replicas. An integer pattern-flow certificate expands into 159,840 literal roles in 333 complete calls. The independent audit reconstructs source-event windows, 73-axis charts, and disjoint pending children on each physical family.

The full finite invoice charges 1024 complex prefix bits: the actual bound is 775. Its physical row coefficient is 17,495, below 20,161. The complete deficit stays 320 per replica; the gain changes the child-rank distribution. The complete native/bit composition passes all 47 outer inequalities. This is the maximum grid value for the fixed word, capacity inventory, and outer recipe. A changed local word or another assembly recipe lies outside that ceiling.

Reproduce from this branch with a new output directory:

python3 -B research/r480-owner-optimum/check.py --work /tmp/r480-owner-check

Expected final line:

PASS_R480_ALL_REPLICA_OPTIMUM_AND_ATTAINING_CONSTRUCTION 0.00167409796990539

The command needs no solver, network, or third-party Python package. It passed in the author worktree and in a separate coordinator replay. Checks include 18 certificate CLI controls, 15 controller mutations, 7 independent owner controls, 69 admission controls, and 12 bit controls. A separate actual-CLI phase-axis collision is rejected at the intended predicate. evidence/ contains the exact receipts. git diff --check passed. All 18 targets in the experimental base's make verify sequence passed across the original and resumed runs, including the historical PR #741 cold replay, 96 isolated test modules, and 20 patch checks. The continuations used installed GCC16 for OpenMP, the installed Boost include path, and the positive producer's documented fresh-output option. No source or check was changed. This shared baseline is separate from the focused package replay; the current PR #1094 cold replay and standalone formal-verify were not rerun.

The experimental base is abdc6a7002abdbcd9e21bfc5e1a795784deacfdf. The local word is pinned to PR #1074 052afc128954f7b6a2b3c409eb4d72248e4020f2. A concurrent sprint worker rebuilt the unchanged local chain, native scalar program, wrappers, and all 2,896 exact phase edges; this package retains those source-bound inputs and command receipts. The owner replay consumes that local admission. The cited PR #589 bit circuit is not rebuilt here; its full profile and exceptional-residue surcharge are repriced.

At the final submission refresh, main 1b59b624d21924f8e3537cdc1ac52d3be6eda2fe selects PR #1094 at κ 0.001672660197076918. This proposal is about 0.08596 percent higher. The complex side remains binding. PR #1121 (61e816c29165a94f6cfc57d9e195063a5b027bc2, reported κ 0.001672965558289236) changes the local word and is outside this theorem. evidence/OVERLAP.json records the compared heads.

Concurrent PR #1131 (0640fa55) independently attains the same κ and fixed-inventory grid ceiling with a different 333-call order and owner flow. We rebuilt its frozen flow, reran its price certificate, and checked its owners with our independent auditor; those focused checks pass. Its full cold-replay driver complements our interval-wide root certificate and explicit disjoint-pending-axis check.

Credit: Chafik Boukhalfa (@chafreaky)'s PR #795 signed prices; the PR #890, PR #942 all-replica proof method; Hao Yan (@FanPUYen)'s owner and source-event machinery; Jeremy Kothe (@jeremykothe2)'s PR #1074, PR #1018 local word and its ancestry, including Rohan Arun (@rohanarun)'s PR #987; the PR #1034 scoped old-catalogue bound; PR #1069's independent pricing audit; and Josie Elliston (@JosieElliston)'s PR #1091 pending-axis check. Preserved notices, exact source pins, and explicit derived-code diffs are included. New work is by Ali Romman (@aliromman) with substantial OpenAI Codex assistance, under Apache-2.0.

The full tight-face chronology LP timed out. Restriction to the nine-pattern optimal basis gave the exact integer flow. The timeout is not an impossibility result. The optimum follows from the proved bound and the attaining construction, not from solver optimality. All-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, exact-recovery, analytic-transfer, and bit H1-H5 obligations remain inherited. This is not an unconditional multiplication theorem, a full formalization, or a measured runtime gain.

[Mister-Meeseeks]

100-PR batch review at a007dc2e1cc7ae7e8f7ec1343e8883fe43259f21.

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

Reviewed scope: All-replica fixed-inventory owner bound and attaining T37 construction, with exact interval moment certificate and complete physical-owner admission; bit ancestry is explicitly retained.

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.