Cost-aware live controls: conditional κ = 5.103512721785e-5

[DominikScholz]

This PR composes PR #67's cost-aware retired-slot selection with compatible pending live controls at both axes and refines the assembly backoff. The conditional bound is

$$ T(n)=O\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{1020702544357}{20000000000000000}=5.103512721785\times10^{-5}. $$

The exact gain over PR #67 (10206752116843/(2*10^17) at b3745601e947a94316bf25c2c6263d93c06e3364) is 273326727/(2*10^17). Bit saving: 51037731934993/10^18. Both new physical words, their complete paid profiles and the exact assembly have passed focused independent validation. Newer PRs #69, #70 and #71 claim stronger bounds; this PR is not the current global frontier. All inherited analytic and fixed finite-alphabet multitape hypotheses remain conditional.

Change

At h=23 and h=25, extend Rohan Arun's PR #67 clearing basis with pending live carriers satisfying F ⊆ E ⊆ G, where F is the current carrier frame, E the clearing region and G its assigned next-use frame. PR #67's integer profile-cost heuristic chooses a valid dependency; every chosen carrier is used through the literal XOR routine and both physical transitions remain charged.

Quantity	PR #67	This PR
h=23 auxiliary roles	27,918	27,918
h=25 auxiliary roles	36,586	36,586
Physical wires W	137,151,806	137,151,806
h=23 clearing XORs	1,415	1,420
h=25 clearing XORs	1,950	1,942
Bit saving	51036365162009/10^18	51037731934993/10^18
Conditional κ	10206752116843/(2*10^17)	1020702544357/(2*10^16)

All data pairs, copied centers, endpoints, the complex branch and assembly costs remain charged. The complete rank mass remains 78,860,441,550 with deficit 1,846,900. The discovery cost is a heuristic; acceptance depends on the physical word and full paid moment.

A final parameter-only refinement reduces assembly backoff from 10^-12 to 10^-18, gaining exactly 153/10^18 in κ with the same physical words and bit saving. Its common eventual cutoff increases from 14,000,000,000,000 to 14,000,000,000,000,000,000. No parameter or construction optimality is claimed.

Verification

Independent checks passed: complete serialized dirty-state replay of both words, exact frame/scalar/scatter checks, literal paid transition reconstruction, actual C++ fixed-basis profiles (6,049/7,997 CRT matrices, zero disagreements), complete paid rank identity, exact adjacent moment bracket, all 47 strict constraints and seven margins, and next-grid rejection. The parameter-only refinement was separately reproduced from its pinned physical receipt and exact sources; its certificate matches exactly.

The committed physical candidate preserves its initial publication-time pending flags as a historical snapshot. The separate validation receipt records the completed physical checks. The latest assembly parameters are in research/cost-live-parameters/certificate.json.

make cost-live-verify
python3 research/cost-live-parameters/refine.py
make verify

The first target is integrated into make verify and the Python 3.11/3.13/3.14 CI matrix. It reruns complete physical validation followed by the parameter certificate check. Full current-head repository checks and CI review remain pending. Upstream workflows require maintainer approval; the fork runs automatically.

Inherited joint-package Makefile/README/NOTICE hashes and derived certificate links were refreshed after integration. Ten focused replay/failure controls and the independent PR #60/#61 arithmetic audit passed; the inherited mathematical certificate is unchanged. All existing verification and formal groups remain enabled.

Commit: b1b2af3415ecb494a0552b16c18045bf3ee05e95. Physical proof and scope, physical validation receipt, word/compile receipt, parameter proof, final exact certificate, source pins.

No finite check here proves the inherited all-size compiler, residual transfer, analytic/routing/recovery or fixed-tape multiplication theorem, global optimality or practical speedup. Earlier research snapshots remain separately attributed in their own directories.

Attribution

Rohan Arun's PR #67, with OpenAI Codex assistance: cost-aware selection, profile oracle and refined exact arithmetic. Dominik Scholz, with substantial OpenAI GPT-6 Astra / Codex assistance: compatible live controls, the two-axis composition and parameter refinement. Alejandro Zarzuelo Urdiales's PR #61 supplies the smaller-backoff refinement precedent. Chafik Boukhalfa's PR #60 supplies ranked reclamation; eumemic's PR #57 supplies the joint compiler; Avi Eisenberg / ikeboy's PR #62 supplies the graph (Claude assistance). rfu08's #64 capped-mass comparison format remains credited in the earlier package. All predecessor notices, Apache-2.0 terms, community, OpenAI and Harvey–van der Hoeven credits remain.

[Mister-Meeseeks]

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

Contribution: Cost-aware live frames F subset E subset G and a global-anchor negative argument.

Review and limits: Exact-head evidence for the main construction. Supplemental cost-live-both proof is explicitly a draft with replay outstanding; preserved as research, not an accepted extra bound.

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: b1b2af3415ecb494a0552b16c18045bf3ee05e95. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.