Seeded saturated-frame tie order: conditional κ = 1.09285663466e-4

[rohanarun]

Verification completed: Full verification passed on research commit 06f174c48b06856a3d6c96ee232dc3fc44145c39: make -j1 verify and package verify.py --full completed in 3663.98 seconds with no source drift. All 45 GitHub checks on that research commit passed. See research/saturated-priority-refinement/validation.json for commands, log/archive hashes and scope. The inherited general interfaces remain conditional.

Under the inherited analytic, stopped-product and fixed-tape interfaces,

$$T(n)=O(n(\log n)^{1-\kappa}),\qquad \kappa=\frac{54642831733}{500000000000000}=1.09285663466\times10^{-4}.$$

This is a 0.00418096% increase over updated #114's 27320273617/250000000000000, and exceeds #120's 1363953833/12500000000000. This is a small refinement of the existing construction, not a new asymptotic architecture. PR122 appeared concurrently with submission and claims the larger κ = 1.0941620808215e-4; this PR therefore does not claim the current record.

The producer is eumemic's unchanged #117 DAG. We retain #114/#120 saturated deferred-frame placement, its exact rational priority 2^dimension / reached_targets^2, and its dimension tie-break. A bounded search over 16 seeded permutations changes only the final equal-priority tie order. Seed 1 gives 4,708 deferred roles instead of #114's 4,706; all 28,705 physical roles remain paid. The complex saving is 27327417/250000000000 = 1.09309668e-4. The complete child histogram, rather than deferral count alone, determines the gain.

The bit supplier, carrier matching and scalar network are unchanged. PR118's paid unit-shear repair is retained: coefficients such as 55/42 are split into unit-magnitude shears at the same frame, with every additional scalar operation charged.

Validation:

• Fresh producer regeneration checks root identities, legal matching, arbitrary dirty scratch, exact binary frames, every target chain and complete rank accounting.
• Independent exact integer-coefficient and literal forward/reflected frame audit, including paid unit shears and three reflection mutation controls.
• Exact moments, all 47 strict constraints and seven margins, with rejection of the next complex and κ grid points.
• Six frame/predecessor-profile/optimized-execution controls and eight inherited unit-shear controls. Both Shared completed cores and physical frames: conditional κ = 1.23988197806e-4 #114 and Deferred readouts on the replayed h24 complex network: conditional κ = 1.0911630664e-4 #120 profiles fail at the new complex saving.
• Frozen source pins and a focused CI matrix for Python 3.11, 3.13 and 3.14. Full make -j1 verify and the package full replay passed; evidence is in the validation receipt.

Reproduce with python3 research/saturated-priority-refinement/verify.py --full, followed by make -j1 verify (CXX="c++ -include algorithm" on Linux). General interfaces remain conditional; finite checks and a bounded search establish neither the full all-size theorem nor global optimality.

Credit: eumemic (#117 producer and #114/#120 saturation), Avi Eisenberg (#110 compiler), Swapnil Jain (bit word and deferred readouts), icekylinx (#104 stopped/rational-center interfaces), Zhihao Chen, RaD/hipotures, Aurel Prosz/Paureel, Rohan Gupta and all retained predecessors. This tie-order search and integration 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: Seed-1 tie-order search with independent frame validation.

Review and limits: Exact-head CI evidence; retains the unit-split coefficient repair and bounded heuristic scope.

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: 4634e11162691478556c1ef88fa6a369c2622abf. Publication: b84ba67156a01f9c2cb34fe1e3cb1dcdeab00d9c.