Envelope ordering on balanced frames with geometric pricing: conditional κ = 5.203279888519e-5

[rohanarun]

Conditional improvement

$$ T(n)=O(n(\log n)^{1-\kappa}),\qquad \kappa=\frac{5203279888519}{10^{17}}=5.203279888519\times10^{-5}. $$

0.49049670% above PR #80, under the inherited all-size and analytic hypotheses. Bit saving: 13008876609597/250000000000000000.

Full verification passed: make -j1 verify, inherited combined checks, fresh byte-for-byte producer regeneration, complete physical/exact checks and seven new controls completed successfully. The run covered 82 isolated test modules and took 44m50s. All 45 GitHub checks passed on the tested research head; the follow-up commit only records validation and updates README status. This is not an unconditional multiplication theorem or a global optimum.

Change

PR #80 selected already-compiled axes. Here we compose the methods before compilation on both axes:

• Balanced coarse split frames: conditional κ = 5.1745625472866e-5 #79’s balanced coarse sums, future-compatible basis completion and bounded Profile-aware frame compilation: conditional kappa = 5.10403292611078e-5 above frozen #62 #70-style carrier exchanges;
• Envelope-ordered carry exchanges: conditional κ = 5.1738676045141e-5 #76/Stopped pair-star trees: conditional κ = 7.808981744031e-5 #74’s envelope ordering;
• Geometric coordinate flags: conditional κ = 5.1428783626028e-5 #78’s geometric coordinate flags.

At h25 we also price candidate compiler choices using the final permuted coordinates. This changes heuristic choices among legal operations, while preserving all independence and frame-containment checks. Its engine differs from #79’s only in the oracle frame writer and attribution comment; the portable compiler asserts this exact relationship.

Quantity	#80	This witness
h23 roles	27,256	27,075
h25 roles	35,684	35,500
W	134,031,764	133,289,600
Recursive rank	77,066,417,400	76,639,673,100

All copied centers, endpoints, data growth, cleanup and complex costs remain charged. We derive the width bridge from the actual W. Complete #76/#79/#80 recursive profiles are each excluded at the new bit saving using their own widths.

Validation and reproduction

Passed: full input and arbitrary-dirty-basis replay in both orientations; literal frame transitions; exact bounded-minor/CRT profiles with zero disagreements; complete paid profile/rank reconstruction; independent rational moment bounds; all 47 inequalities and seven margins; adjacent bit/κ grid rejection; six physical/coordinate failure controls and three stale bridge controls. Every source pin was additionally checked against committed Git bytes.

The new portable compiler regenerates original words, then the verifier checks coordinate relabeling byte-for-byte. Dedicated CI runs this regeneration and the controls on Python 3.11/3.13/3.14, preserving every inherited workflow.

python3 research/envelope-balanced/verify.py --regenerate
(cd research/envelope-balanced && python3 -m unittest test_controls -v)
make verify

See research/envelope-balanced/README.md for the general compatibility argument and conditional scope, certificate.json for exact arithmetic, and SOURCE.json for immutable inputs. The completed full rerun is recorded in research/envelope-balanced/validation.json.

Credit

Thomas DiFiore (#74), Dominik Scholz (#76), Chafik Boukhalfa (#79/#71), eumemic (#69/#57), Alejandro Zarzuelo Urdiales (#70), Rohan Arun (#78/#80/#65/#67), Rohan Garg, Avi Eisenberg and every inherited contributor retain credit. Tested #75 zero-circuit variants were not selected. Original licenses, author and AI disclosures, OpenAI and Harvey–van der Hoeven notices remain intact. This composition and coordinate-aware pricing 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: Compose before compiling and use coordinate cost oracles.

Review and limits: Exact-head CI evidence; oracle scores remain candidate-selection heuristics.

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: 67b446ccdb9d23cae9683f6059fb607b52285004. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.