Conditional κ = 0.00166202617947539: U8 with two rank-seven helper ends

[AlexS-scale]

This construction gives conditional κ = 0.00166202617947539, above the pinned selected PR #942 value of 0.001661960547713947. The exact gain is 65631761443/1000000000000000000, approximately 0.00394906% of the preceding κ. This is an improvement to the asymptotic exponent saving, not a measured runtime result.

It combines PR #869's U8 word with two smaller helper ends and a complete owner assignment. Helpers 588 and 589 finish in dimension seven instead of eight. Eight B frames and two early-cleanup frames change with them. The owner assignment combines the released space into exactly 2,400 rank-48 residuals. Scalar gates and their order retain their U8 values; the ports, center interface, physical geometry, and 72-call schedule retain their PR #799 values.

Quantity	Before the end reduction	This construction
Helper width per call	3778	3776
Local rank mass	5634	5632
Local profile	{1:2548,2:338,3:446,4:162,5:72,8:8}	{1:2542,2:352,3:438,4:162,5:72,8:8}
Global idle profile	{1:1,3:1,44:1,48:2396}	{48:2400}

The complete global profile is {1:183024,2:25344,3:31536,4:11664,5:5184,8:576,48:2400}. Rank mass remains 520704, with deficit 2560. The complex endpoint is 0.001664793109206301164750; the retained bit endpoint is 0.001733427039589844234749. The complex supplier remains limiting. The conservative calculation gives κ = 0.001662026176592891.

The certified global prefix bound is 257 bits, within the existing 512-bit allowance. The physical row coefficient is 16467 < 20161. PR #942 uses 333 calls and a 1024-bit allowance; its all-replica bound fixes the previous U8 profile and helper widths. This construction changes both.

The submission comparison is the selected #942 record, unchanged at fetched main 2921874911956718fcaaaf9396fe8267b3cfa9f0 on October 13, 2026. Concurrent open PRs #987, #1004, #1005, #1018, and #1022 report larger κ values through other constructions; #1022 reports 0.001672081255752945. Those claims are not independently validated here. This package contributes a separate, fully replayed change to the end frames and owner assignment. PR #988 audits terminal cones with end-of-B frames fixed and retains rank-eight ends for helpers 588 and 589; this construction changes those frames and supplies the smaller ends and complete packing.

Run from the repository root with Python 3.11+ and Git, using a new output directory:

python3 -B research/complex-rank7-end-pair/reproduce.py --work /tmp/rank7-end-pair

Expected final output:

PASS_COLD_RANK7_END_PAIR_COMPLETE_ADMISSION 0.00166202617947539

The command verifies all source manifests and freshly rebuilds the PR #741 ancestry, PR #799 word, and PR #869 U8 word. No optimizer or Python package installation is needed.

The default fresh-fetch command passed in 914.6 seconds on CPython 3.14.8, macOS 15.7.3, arm64. A separate Python 3.11.14 check passed word construction, point audit, and owner reconstruction with their corruption controls. The validation receipt records the complete replay environment and all 13 unchanged code/data input hashes.

Validation includes:

• All 38 U8 construction controls, all 728 input columns in both orientations, all 2,950 phase edges, exact wrappers, 25 native controls, and 89 applicable inherited cold-end controls.
• An independent 512-point frame audit; 22 targeted mutations with 43 expected raw rejections; five point-audit controls; and eight owner-construction controls.
• All 36,288 literal owner roles, followed by strict global admission, 69 admission controls, 12 bit-supplier controls, and seven independent owner controls.
• Both exact moment engines and all 47 outer inequalities in both native and conservative modes, including the complete precision and tape invoice.

research/complex-rank7-end-pair/ contains the minimal recipe, deterministic owner reconstruction, proof, search scope, exact source pins, expected results, and validation receipt. The selected-result snapshot used for the comparison is retained with its exact upstream commit and content hash.

This remains a finite conditional witness. The inherited all-size/compiler, common-label/mutable-source, whole-call hosting, precision/tape, exact-recovery, analytic-transfer, and bit H1–H5 assumptions remain. The cited PR #589 bit circuit is repriced, not reconstructed by this command. Finite replay does not prove the full multiplication theorem. Repository-wide make verify and Linux CI are separate from the local affected checks.

Credit goes to Jeremy Kothe's PR #869 U8 and PR #799 X4 constructions and their credited NewBArch/JointXF work; Chafik Boukhalfa's PR #741 and ownership work; the PR #777 verification tools; the PR #635 wrapper machinery; and the retained PR #589 bit supplier and predecessors. Jonathan Capalbo's PR #942 supplies the selected comparison. NOTICE.md preserves detailed attribution and discloses substantial OpenAI Codex assistance.

This PR is independent of #784. The research and submission branch begin at 66849e1475c930fd7732b1a0067ffee7e1e95f60; the final development and comparison snapshot is 407d52ff15b2217b6e22bce44728b4c3907b6515. The earlier submission base permits a package-only fork push without unrelated upstream workflow changes. Changes are confined to the new research package. upstream/ and the selected certificate are unchanged.