Geometric coordinate flags: conditional κ = 5.1428783626028e-5

[rohanarun]

This gives a finite conditional witness with κ = 12857195906507 / 250000000000000000 = 5.1428783626028×10⁻⁵, approximately 0.02750% above PR71. It also exceeds PR75's reported 5.14154245470094×10⁻⁵. The bit saving is 25715714341 / 500000000000000.

Geometric change: optimize the coordinate flag of PR71's existing physical words. A coordinate permutation commutes with I+J and preserves I−J/9. Relabeling all source triples, outputs, frame masks and scatter indices preserves the circuit, every containment and the complete data-case multiset. It changes which adjacent pivots form recursive blocks in internal transition profiles.

Roles remain 27,455 / 36,015, width remains 135,075,665, and the literal role operations, paid clearing, endpoint corrections and total recursive rank remain unchanged. Both words are regenerated from the pinned PR71 words by the explicit selected permutations; the general relabeling argument is in research/coordinate-flags/README.md.

Focused verification passed: byte-for-byte word reconstruction, full ordinary-input and arbitrary-dirty-basis replay in both orientations, independent transition reconstruction, complete fixed-I+J profiles with the inherited exact CRT bounds, all 47 strict constraints and seven margins, independent rational moment bounds, both next-grid rejections, exclusion of PR71's complete profile, and six negative controls. Run make coordinate-flags-verify.

Full verification completed. All expanded make verify groups are covered by the recorded original and resumed runs, followed by the new focused checks (80 isolated test modules; 6 new controls). The first run stopped at a local Apple Clang OpenMP configuration error; the failed group and every remaining group passed after enabling installed libomp. No checks were bypassed. Receipt: research/coordinate-flags/validation.json. All 39 GitHub checks passed on the validated research/packaging head; this follow-up commit only adds the receipt. The new verifier is included in the sequential full target and the Python 3.11/3.13/3.14 CI matrix. Root-document source pins and derived digests are refreshed for the inherited joint/split packages; their numerical certificates are unchanged.

This is a finite conditional bound under the inherited all-size compiler, routing, prime selection, exact recovery, finite-alphabet/fixed-tape and analytic-transfer hypotheses. The bounded coordinate search establishes neither global optimality nor measured practical speedup.

Credit: Chafik Boukhalfa (PR71), Rohan Garg (PR59), Avi Eisenberg (PR62), Dominik Scholz (PR63/68), Rohan Arun (PR65/67), eumemic (PR57), icekylinx, James Chang, Zhihao Chen, Aurel Prosz/Paureel, Swapnil Jain, Alejandro Zarzuelo Urdiales, RaD/hipotures, Douglas Colkitt, OpenAI and all inherited contributors. 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: Relabel by permutations commuting with I+J.

Review and limits: Exact-head CI evidence; internal profiles are recomputed, while the full Cartesian data invariant is preserved.

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: 7c3793131155fa5795920eff35057672be38833a. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.