Reordered rank-pair regions: conditional κ = 5.1028985071975e-5

[rohanarun]

Conditional bound

This contribution gives κ = 2041159402879/(4×10^16) = 5.1028985071975×10⁻⁵, approximately 0.00277315% above #63, under the inherited analytic, all-size compiler and fixed finite-alphabet multitape hypotheses. The bit saving is 5103158916233/10^17.

The subsequently posted #64 refines the same #63 profiles to κ = 5.10275840622536575×10⁻⁵; this new schedule witness also exceeds that reported bound by approximately 0.00274559%. This is a numerical bound comparison, not a comparison of the strength of #64’s independent verification and formalization. See the pinned current-comparison.json.

Change

Keep #63's scalar graph and rank-first retired-slot selection. Change the order of regions within each rank: increasing cover/core at h=23 and decreasing minimum node index at h=25. Each schedule is checked to be topological and to preserve forward carrier candidates. This changes the physical frame paths and complete fixed-basis child distribution. It does not remove paid clearing, copied centers or endpoint corrections.

The roles remain 27,918 and 36,586; W remains 137,151,806. The construction gain is combined with tighter exact Taylor enclosures and finer rational grids, retaining the original backoff h=10^-12. The proof note separates these contributions; no global optimality or practical speedup is claimed.

Validation

Both dimensions passed full dirty-basis checks in both orientations, independent serialized-word replay, and fixed-basis profile regeneration. Exact moment enclosures, all 47 strict constraints and seven margins, and rejection of the next grid points pass. An independent logarithm/Taylor computation checks the arithmetic.

Full make verify PASSED on research commit 49e84f939d15b618b50714eb039cabf97c74256a in 23m 30s: 467 core unit tests, all 20 historical patch checks, complete physical word/profile regeneration, six physical negative controls and eight arithmetic tests. All 32 GitHub CI jobs on that head also passed. Receipt: research/reordered-rank-pair/validation.json; log SHA256 accde9848e886d6c7802a14d4ac415c1b0c2e5ce049868d5b8c2c150c1e74f76.

Upstream main was integrated afterward to resolve concurrent merge conflicts. All consumed source code, physical witnesses and numeric proofs are unchanged; focused metadata/hash/profile composition and arithmetic checks passed. Newly imported upstream groups are preserved in the expanded CI matrix and were not part of the earlier local full run.

The first expanded CI run exposed stale joint-dual hashes for Makefile, README and NOTICE. Those three reviewed source pins and their two derived metadata hashes have been refreshed. The complete make verify-joint now passes, including both physical compiler rebuilds and ten negative controls; all numerical and physical certificate fields are unchanged. The replacement GitHub joint checks passed on Python 3.11, 3.13 and 3.14 at b0cea06d7fd5ac10f76169b01d3114f208fc7448 (run).

Run make reordered-rank-pair-verify. See research/reordered-rank-pair/README.md for the general argument, source pins, negative controls and limitations.

Credit

Dominik Scholz (#63), Avi Eisenberg (#62), Chafik Boukhalfa (#60), eumemic (#57), Alejandro Zarzuelo Urdiales (#61, parameter-refinement precedent), RaD/hipotures, icekylinx, James Chang, Zhihao Chen, Aurel Prosz/Paureel, Swapnil Jain, Douglas Colkitt, OpenAI and all inherited contributors. Existing source notices are retained.

Prepared by Rohan Arun with OpenAI Codex assistance.

[Mister-Meeseeks]

Thank you @rohanarun. This contribution is now recorded on main in the scoped reconciliation review and contributor credits.

Contribution: Region ordering subject to the dependency DAG.

Review and limits: Exact-head CI evidence; rank ties alone do not authorize an invalid schedule.

The complete original source, proof/research notes, notices and relevant replay/CI evidence are retained through the pinned source archive. The review distinguishes fresh runs, exact-head historical CI and prior integration evidence. Closing this historical PR as reconciled; its obsolete headline does not replace the current selected bound, and draft/retracted claims retain their explicit qualifications.

Reviewed head: b0cea06d7fd5ac10f76169b01d3114f208fc7448. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.