Final-coordinate carrier pricing: conditional κ = 5.2791461623702e-5

[rohanarun]

Under the inherited analytic and fixed finite-alphabet multitape hypotheses,

$$T(n)=O\bigl(n(\log n)^{1-\kappa}\bigr),\qquad\kappa=\frac{26395730811851}{500000000000000000}=5.2791461623702\times10^{-5}.$$

This is a small 0.0006927669% improvement over #95, with bit saving 6599281088677/125000000000000000.

Current comparison: #97, opened while this was being published, reports the higher conditional κ = 6.3965813×10⁻⁵ using deferred bit endpoints and a signed complex schedule. This PR is an incremental improvement to the PR95 construction, not a record claim.

At h25, price legal carrier choices in the final coordinate order, including #93 and #95's later relabelings. Previously the producer made these choices using an earlier coordinate order. The unchanged compiler, with this existing parameter updated, reproduces the selected new physical word byte-for-byte. h23 remains #95's word.

Roles remain 26,387 / 34,772, width 130,417,912, and total recursive rank 74,988,452,500. The complete internal child distribution changes. Every exterior, center-copy charge, side-growth contribution and early/late restoration gate remains paid; the bridge and complex layer are unchanged.

Full verification passed: make -j1 verify (92 isolated test modules, 20 historical patch checks), both parent control suites, full inherited producer-chain regeneration and six own controls completed in 6,633.390 seconds on f07c7db7d1fd378b0deae68e33b53421456319ae. All 57 research-head GitHub checks passed; no source drift. The committed validation receipt records the commands, timestamps and evidence hashes. Subsequent changes only add that receipt and update package documentation and its fingerprint.

Checks:

• Fresh h25 compilation with the pristine inherited engine, byte-identical selected word, and dense scalar support audit.
• Complete physical replay for both axes, including arbitrary-dirty restoration, and fresh full CRT profiles with zero disagreements.
• Exact and independent moment enclosures, 47 constraints, seven margins, both next-grid exclusions and exact rejection of Coordinate descent beyond PR94: conditional κ = 5.279109590448e-5 #95's complete profile at the new saving.
• Six adversarial/roundtrip controls and 2,764 pinned source files.

python3 research/final-coordinate-pricing/verify.py
python3 research/final-coordinate-pricing/test_controls.py
python3 research/final-coordinate-pricing/verify.py --regenerate
make verify

The package contains the general parameter-transfer argument, full paid accounting, source pins and a Python 3.11/3.13/3.14 workflow. This candidate emerged from the completed 16-experiment exchange batch, but no long cycles were accepted: the gain is from coordinate-aware pricing alone. No search optimality, new Lean proof or practical speedup is claimed. All inherited all-size and analytic hypotheses remain assumptions.

Credit Chafik Boukhalfa (#91), Maxime Fleury with Codebuff assistance (#94), Rohan Arun (#93/#95 and geometric pricing), Alejandro Zarzuelo Urdiales, Thomas DiFiore, Dominik Scholz, eumemic, Avi Eisenberg, Rohan Garg, icekylinx, James Chang, Zhihao Chen, Aurel Prosz/Paureel, Swapnil Jain, RaD/hipotures, Douglas Colkitt, OpenAI and all predecessors. Original code and notices are preserved.

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: Coordinate pricing with the baseline legal engine.

Review and limits: Exact-head CI evidence. The long-cycle search accepted zero candidates; retained as a scoped negative experiment.

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: 6862c47f5fcb69dfb424e1bb7f32f415d8659063. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.