Conditional κ = 0.001656193765416678: gauge44 without K-triangles (56 cold ends), with layered ends

[jeremykothe2]

Summary

Conditional κ = 0.001656193765416678 (conservative 0.001656193762539939). This is +2.00×10⁻⁶ over PR658 (0.001654192296855146) and +2.09×10⁻⁶ over our PR656.

Change: remove the K-triangles from PR645's gauge44 construction. PR645 keeps four (p,−1) K-triangles from the PR622 lineage. They save 12 helpers per call, but each triangle incidence lifts one own-row half of a block, so the six triangle blocks {0,1,2,11,12,13} pair at rank 2 instead of 4. That is why gauge44 stops at 44 rank-four cold ends (56 − 12). We remove all four triangles (R = 504) and give the six former triangle blocks the same 6-row completion priority as blocks 3–10. Removal alone restores no ends. With the shared priority, every block pairs at rank 4 and the word has 56 rank-four cold ends.

On top of that word, the PR655 layers are re-ported: 6 cost-neutral pivot swaps, entering ends s4 (U24) and x30 (U30), six dim-8 cold ends (b6) and q4. p123 and j3 do not apply on this word and are omitted. Width 3800 per call, profile {1:2534, 2:346, 3:458, 4:178, 5:56, 8:8}. The owner map sits exactly at this word's packing ceiling (idle 2364 × rank-48). It was produced with a joint multi-call search in the style of PR653/PR658.

Code: code/ and verify_global.py are byte-identical to PR655's reviewed set (PR589 bit supplier, layer controls, spot-width rule). New construction drivers: construction/notri/reproduce_notri.py, which rebuilds the no-triangle word (sha 3e99ba77) and pins the pairing ranks to [4]×14, and construction/layers_np/reproduce_np.py (the layered word a9de804f, with 59 pinned layer-end controls). --control-g44 in the notri driver reproduces PR645's word exactly. The SPEC spot helpers move to [270,282,294,306], the identity image of the helper map at R = 504.

Independent review: SOUND. The reviewer rebuilt both words bit-exact, replayed them exactly (0 bad rows, all orientations, dyadic, no ended-helper reads), and checked (N) for all pairs, the 59 controls, audit_local and the 192-bit payload. Its own verify_global reproduces this κ. It also re-signed the final package tree, including a clean-copy rebuild.

Verification

From a fresh clone of commit 4982f0c, with no reused artifacts (Windows, CPython 3.11.11):

• verify_manifest.py: PASS, 104 files
• reproduce.py --until word, then reproduce_notri.py: PASS no-triangle scalar word 3e99ba77…
• reproduce_np.py: rebuilt the layered word; PASS layer-end-controls (59)
• audit_local.py: PASS_INDEPENDENT_EXACT_FORWARD_AND_DUAL_ENDPOINT_FRAMES_ENTRANCES_CONES
• verify_global.py: PASS invoice, assembly, admission-controls and bit-supplier-controls; PASS_GAUGE44_NATIVE_EXACT_GLOBAL_ASSEMBLY 0.001656193765416678

Rebuild comparison note: the rebuilt locals equal the strict locals byte for byte, except two seconds timing fields (CLIFFORD-REVIEW, SCALAR-FRAME-AUDIT), which match once normalised. Nothing else is normalised.

Scope

Conditional on PR645's premises and on PR589's bit-supplier assumptions in place of PR543's. No unconditional theorem or Lean certification is claimed.

Credits

• icekylinx (PR645): gauge44 construction, package and verifier.
• jeremydiviney (PR658) and jaymosley07 (PR653): the multi-call owner search our map builds on.
• PR589 lineage (bit supplier); the PR601/604/617/622/627/635 lineage and every upstream contributor credited in PR645.

Prepared with Claude Code (Anthropic).

[Mister-Meeseeks]

Thank you @jeremykothe2. This contribution is now retained on main in the scoped 100-PR review and contributor credits.

Contribution: Removing K triangles to expose 56 rank-four ends.

Review and limits: Triangle deletion alone is insufficient: shared completion priority and re-ported layers matter. The attained packing ceiling is restricted to this changed local word.

The complete original source, notices and relevant execution evidence are retained through the pinned source archive. The review distinguishes fresh local runs, source-bound historical Linux CI and prior exact-head review; it does not claim 100 fresh full mathematical audits. Closing this historical PR as reconciled. This preserves parallel and cumulative credit without replacing the selected frontier or promoting priced research targets to completed constructions.

Reviewed head: 4982f0ca66ffe0960b5a6451deb5b66b1039326b. Publication: 7f19399def7c51ff8016d9190999830ef3e17f95.