Coordinate refinement of indexed two-cycles: conditional κ = 5.2272192668365e-5

[rohanarun]

A coordinate refinement of Chafik Boukhalfa's PR #84 gives the conditional bound κ = 10454438533673/200000000000000000 = 5.2272192668365×10⁻⁵, approximately 0.00173% above #84 at 88ca39571907343a49e97f328971ec7bcd26fbfd.

Ready for review: full repository verification passed in 42 minutes 53 seconds; all 45 GitHub checks passed at the validated research head.

Exchange labels 3 and 4 (zero-based) in both complete axis words. The scalar circuit, carrier matching, roles, physical XORs, width and recursive rank are preserved. The coordinate order changes the residual child-size profiles; freshly regenerated profiles and exact arithmetic certify the modest gain. This is a finite numerical refinement, not a new asymptotic technique or a global optimality claim.

• Bit saving: 13068731298371/250000000000000000.
• Physical roles: h23 = 26834; h25 = 35348.
• Width: 132466108; total recursive rank: 76166165200.
• Checked: complete arbitrary-dirty replay in both orientations, reconstructed physical transitions, fresh three-prime profiles with zero disagreements, independent exact rational moment bounds, both next-grid exclusions, 47 strict constraints and seven margins.
• PR84's complete child list is strictly excluded at the new saving.
• Six control tests passed, including inverse-coordinate transport, invalid permutations, omitted children, stale width/exponent and assertion-disabled execution.
• Both parent producer regenerations and all inherited repository checks passed. The dedicated CI also regenerates both parent words on Python 3.11, 3.13 and 3.14.

python3 research/indexed-coordinate-refinement/verify.py --regenerate
python3 research/indexed-coordinate-refinement/test_controls.py
make verify

The package includes a general coordinate-transport argument, exact certificate, pinned sources and literal physical words. All analytic, all-size residual compiler, tape, routing, prime-selection and recovery hypotheses remain inherited. Finite tests do not prove these general hypotheses.

Credit: Chafik Boukhalfa (#84 and predecessors), Thomas DiFiore (#74), Rohan Arun (#78), eumemic, Rohan Garg, Avi Eisenberg, Dominik Scholz, Alejandro Zarzuelo Urdiales, OpenAI and all previous contributors. All original source archives, licenses and assistance disclosures remain unchanged. Prepared by Rohan Arun with OpenAI Codex assistance.

CI follow-up: b3de008 fixes the verifier comparison between typed compiler objects and JSON-loaded words by requiring canonical serialized-byte equality. Both producers and the exact certificate passed regeneration locally after this fix, together with six controls. Numerical words, profiles and bounds are unchanged. All 45 replacement GitHub checks and the full repository run passed.

Validation receipt: research/indexed-coordinate-refinement/validation.json. The full run covered 84 isolated test modules and 20 historical patch checks. It began on fb1f2c5; the later b3de008 verifier-only serialization correction was covered by independent producer regeneration and the completed run. Numerical artifacts and compiler code were unchanged.

[Mister-Meeseeks]

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

Contribution: Coordinate relabeling in dimensions three and four.

Review and limits: Exact-head CI profile evidence; no automatic gain from relabeling without recompilation.

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: dd7690a5a7a3027afcb002aa8ef41d0e29bde58f. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.