Adjacent coordinate refinement beyond PR93: conditional κ = 5.2790937858556e-5

[maxime-fleury]

Result: a small exact gain beyond #93

Conditional κ = 13197734464639/250000000000000000 = 5.2790937858556e-5, compared with #93's 5.2790906739644e-5.

• Gain over Coordinate refinement of aligned composition: conditional κ = 5.2790906739644e-5 #93: 3.1118912e-11.
• Gain over Aligned frame composition: κ=5.2789616935221e-5 with Lean arithmetic certificate #91: 1.320923335e-9.
• h23/h25 roles remain 26387 / 34772.
• Width remains 130417912, total rank 74988452500, deficit 1846900.

This is deliberately presented as a small finite refinement, not an asymptotic breakthrough or a practical speedup.

Construction and attribution

Start from #91 at 264f202b52edc04d0c72ca9cb3138282ca68b9ad, with its original files and archives untouched. On the already selected coordinate labels apply:

• h23: swap 5/6.
• h25: swap 22/23 and 5/6.

The h23 5/6 and h25 22/23 choices match #93 at c42039369af319a4ccc249b11efcfedecc2cc72f; the additional disjoint h25 5/6 transposition supplies this increment. The literal XOR sequence, slots and chronology are unchanged. Frames, source triple indices, scatter rows and output targets are transported consistently with the credited PR74/PR91 relabel action.

Credits: Chafik Boukhalfa's #91 compiler, Thomas DiFiore's #74 coordinate construction, Rohan Arun's #93 and #78 refinements and #65/#67 arithmetic, and all inherited contributors, licenses and notices. Search and additive verification by Maxime Fleury with Codebuff assistance.

Verification completed locally

The new verifier:

• Checks local source hashes and all inputs pinned by the Aligned frame composition: κ=5.2789616935221e-5 with Lean arithmetic certificate #91 source manifest.
• Deterministically regenerates both coordinate-transformed words from frozen Aligned frame composition: κ=5.2789616935221e-5 with Lean arithmetic certificate #91 words and compares decompressed bytes.
• Replays complete ordinary/arbitrary-dirty physical words in both orientations.
• Reconstructs transitions independently from the actual XOR word.
• Recomputes actual-frame CRT profiles; both have zero CRT disagreements.
• Rebuilds the entire paid child list with inherited endpoint, cleanup, data-growth and exterior charges unchanged.
• Finds the adjacent accepted/rejected 1e-18 savings, checks two independent rational moment enclosures, 47 strict inequalities, seven margins and next-grid κ rejection.
• Independently recovers Coordinate refinement of aligned composition: conditional κ = 5.2790906739644e-5 #93's exact κ from its supplied profiles and excludes both Aligned frame composition: κ=5.2789616935221e-5 with Lean arithmetic certificate #91 and Coordinate refinement of aligned composition: conditional κ = 5.2790906739644e-5 #93 complete profiles at the new saving.
• Checks the improvement also survives the coarser 1e-11 grid.

--record followed by compare-mode replay passed. 11 controls passed, including invalid/incomplete relabels, terminal corruption, rank-mass corruption, exact baseline comparisons, source closure and next-grid rejection. git diff --cached --check passed. The added Linux workflow runs this interface on Python 3.11 and 3.14; remote CI remains pending.

python3 research/aligned-coordinate-edge/verify.py
(cd research/aligned-coordinate-edge && python3 -m unittest test_controls -v)

Dependency and limits

This branch stacks one additive commit / 16 files on the open #91 head; its diff against main includes #91's dependencies. No inherited tracked file is edited. The #93 profiles are arithmetic comparison inputs; its producer is not independently regenerated here.

The parent #91 producer is inherited, not recompiled by this verifier; the supplied physical words are fully replayed. #91's Lean companion proves its own values, not these new numbers. This increment is rational Python verification, not a new Lean proof. Analytic, all-size framed/residual compiler, routing, recovery, prime selection and fixed-tape hypotheses remain inherited. Full repository verification was not rerun locally. No global optimum or O(n sqrt(log n)) multiplication algorithm is established.

[Mister-Meeseeks]

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

Contribution: Adjacent-flag improvement with inherited formal artifacts.

Review and limits: Exact-head CI evidence. The older #91 Lean package is not a formal proof of this newer kappa.

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: 0c440e40c9d4a5a02ca5bd028a2ebb5ebbca55d7. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.