Combined envelope/balanced axes and geometric flags: conditional κ = 5.1778825456819e-5

[rohanarun]

Conditional result

Combining Dominik Scholz’s PR #76 h23 axis, Chafik Boukhalfa’s PR #79 h25 axis, and our PR #78 geometric coordinate flags gives

$$ T(n)=O(n(\log n)^{1-\kappa}),\qquad \kappa=\frac{51778825456819}{10^{18}}=5.1778825456819\times10^{-5}. $$

This is 0.06415998% above #79 and 0.077600% above #76, under all inherited all-size and analytic hypotheses. Bit saving is 51781506642569/10^18. Both complete parent child lists are strictly excluded at the new saving using their own widths.

Full verification completed. All expanded make verify groups are covered by the recorded original and resumed runs, followed by the new focused checks (82 isolated test modules; 7 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/combined-flags/validation.json. All 42 GitHub checks passed on the validated research/packaging head; this follow-up commit only adds the receipt. No unconditional theorem, global optimality or practical speedup is claimed.

Combination

Each axis implements the same full triple-indexed map and dirty-role interface. We select the complete #76 h23 implementation and #79 h25 implementation, preserving every endpoint, copied center, data term and complex cost. We then permute coordinates consistently across frames, source triples, outputs and scatter rows. Permutations commute with I+J and preserve the bilinear form, so interfaces and full data-case histograms remain valid. Internal corner profiles are rebuilt from the actual changed words.

Quantity	Combined witness
h23 roles	27,256
h25 roles	35,684
W	134,031,764
Total recursive rank	77,066,417,400
Rank deficit	1,846,900

Axis selection without geometric relabeling gives κ = 51767625345147/10^18; the coordinate search supplies the remaining improvement. Thus reported percentage gains are not simply added together. The width bridge is derived afresh: 27 wire bits, row coefficient 843, degree gap 7007/25.

Checked

• Both relabeled words regenerate byte-for-byte from pinned parent words.
• Independent full ordinary-input and arbitrary-dirty-basis replay in both orientations.
• Complete physical transition reconstruction; exact bounded-minor/CRT profiles with zero disagreements.
• Complete paid cost/rank identity, independent rational moment bounds, 47 strict inequalities, seven margins, and adjacent bit/κ grid rejection.
• Seven focused tests cover six physical/coordinate mutations and three stale bridge fields.
• Dedicated Python 3.11/3.13/3.14 CI; all inherited workflows retained.

Run python3 research/combined-flags/verify.py, then (cd research/combined-flags && python3 -m unittest test_controls -v), plus make verify for the expanded inherited suite. See research/combined-flags/README.md for the general combination/relabeling argument and SOURCE.json for provenance.

The branch inherits #79’s portable source package. #76’s original scheduling package and supplied witnesses are archived unmodified with licenses. Its original compiler regeneration still requires its documented immutable #71 archive; our verifier directly replays the pinned word and does not require that archive. Regenerating our relabeling is not a claim that #76’s original search/compiler was portably regenerated.

Credit

Dominik Scholz (#76), Chafik Boukhalfa (#79/#71), Thomas DiFiore (#74), Alejandro Zarzuelo Urdiales (#70), eumemic (#69/#57), Rohan Garg (#59), Avi Eisenberg (#62), Rohan Arun (#78/#65/#67), and every inherited contributor retain credit. Original author, AI-assistance, OpenAI, Harvey–van der Hoeven and license notices are preserved. This combination and coordinate search were 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: Select compatible axes with the correct stock and bridge.

Review and limits: Earlier full physical replay and stale-width correction retained; exact-head CI evidence included.

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: 4fb50b65e3542592bb1674a0211b0b5faa3ed920. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.