Congruent splits with geometric prices and null-circuit clearing: conditional κ = 5.2445080818074e-5

[rohanarun]

Conditional improvement over #87

κ = 26222540409037/500000000000000000 = 5.2445080818074×10⁻⁵, approximately 0.0250263% above PR #87 (52431959054651/10^18), under the inherited analytic and fixed finite-alphabet multitape hypotheses. Bit saving: 52447831448991/10^18.

Combine @gupt1156's #87 congruent [4,4,1] splits with #86 final-coordinate pricing, @alejandrozu's #75 null-circuit clearing idea, and bounded two/three-carrier exchanges. Null relations let the compiler substitute a cheaper legal clearing expression; every actual XOR and frame raise remains paid. The selected runs accept four and two three-carrier exchanges, respectively.

Quantity	#87	This
h23 auxiliary roles	26705	26695
h25 auxiliary roles	35137	35132
Physical width	131795727	131763872
Total recursive rank	75780696125	75762379500

The eight cloud experiments all passed. Exact full-profile comparison included all their axis combinations plus 48 independently tested coordinate-only alternatives. No global-optimality, exact weighted-matching, unconditional-theorem or practical-speed claim is made.

Live comparison

At publication, PR #88 reports κ = 5.2514124198891e-5 and the updated PR #74 (428b5c167686b42a83f1abf7003c6d17aec561ea) reports κ = 5.278214423139e-5. Both exceed this witness. The comparison above is specifically against pinned #87; this is not a current-record claim.

Validation

Full repository verification passed; ready for review. Both dimensions pass scalar checks, complete arbitrary-dirty physical replay in both orientations, fresh CRT profiles with zero disagreements, independent exact moment bounds, next-grid rejection, 47 strict constraints and seven margins. The complete #87 child list is excluded at the new saving.

Nine focused tests include four adversarial actual-source null-circuit cases, four carrier-exchange controls, derived-source byte hashes, coordinate inversion, omitted charges, stale width/exponent and optimized-Python rejection. Independent local regeneration also passed in 240.771 seconds: both words matched the cloud outputs byte-for-byte, fresh physical profiles and exact arithmetic passed, and all nine controls passed again. Full make -j1 verify, parent and new compiler regenerations, seven inherited controls and nine new controls passed in 3367.923 seconds (56m 8s), including 84 isolated test modules and 20 historical patch checks, on research commit b2df9de3147006cb6a652e5409ce2f44bbd71926. There was no source drift. All 48 GitHub checks passed on the tested research commit; the subsequent commit adds only the validation receipt and documentation. The committed research/congruent-geometric-pricing/validation.json records exact commands, timing and evidence hashes. The 608-file source closure pins the graph, configurations, compiler derivation, physical words, profiles, proofs and checks.

python3 research/congruent-geometric-pricing/verify.py --regenerate
python3 research/congruent-geometric-pricing/test_controls.py
make verify

PROOF.md explains general grouping contraction, kernel-expression invariance, frame eligibility and paid physical execution. Inherited all-size compiler, scalar/routing/recovery, finite-alphabet multitape and analytic-transfer obligations remain explicit. No topological cancellation theorem is claimed.

Credit

@gupt1156 (#87), Chafik Boukhalfa (#84), Alejandro Zarzuelo Urdiales / @alejandrozu (#75), Rohan Arun (#86), Thomas DiFiore (#74), eumemic, Dominik Scholz, Rohan Gupta, icekylinx, James Chang, Zhihao Chen, Aurel Prosz/Paureel, Swapnil Jain, Avi Eisenberg, RaD/hipotures, Douglas Colkitt, OpenAI and all predecessors. Original sources, licenses and AI-assistance disclosures remain attached.

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: Null-circuit Gaussian relations preserve XOR action.

Review and limits: Exact-head CI evidence includes physical validation; scalar cancellation alone is insufficient.

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: 6e3d667ee119cc630eb5ffc69ea4c222f967ce2d. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.