Conditional κ ≈ 0.00171937887049554: P53 width-7 vacancy relay

[mdumitrean]

Composes Philippe Van Velzen’s #2853 dirty-A words with two width-7 vacancy paths in its existing T74 allocation. Four helper-owner moves replace two rank-7 idle children with one rank-14 child, preserving all native words, 666 calls and physical stock 66794.

The fully replayed conditional target is:

κ = 0.001719378870495538841812 − 10^-46 − 10^-70

This exceeds #2853’s full exact rational by 1.21179795218276466e-9, or 0.0000704788704%. The complete paid delta is H14 − 2H7 < 0 for every 0 < a < 1. All 271,728 helper placements are included in the literal checks.

Validation:

• Two independent owner/event implementations reconstruct source windows, actual word profiles, active-family exclusion, masks and the directed rational moment bracket.
• Complete candidate admission passed using native preparation freshly generated this round: 119 admission, 12 BIT, 25 controller, 8 owner and 6 geometry controls; all 10 new dirty-A source mutations reject; 63 modern strict inequalities and 17 rejecting controls pass.
• A separate default fresh cold replay passed from a Git-free export at a path containing spaces, using Python 3.12.15. Both modes agree on the exact headline.
• The registered Python 3.12 package checks regenerate the relay, verify exact source adapters and both packing caps, and authenticate 167 complete-admission evidence files.

The exact fixed-bin packing cap is 0.001719637819314309441628..629; this relay realizes about 0.466% of the remaining headroom above #2853. The looser paid-local-word ceiling is 0.001758353980844..845 (97.7834% realized). Three larger bounded flow searches returned UNKNOWN; their diagnostics are retained without an infeasibility claim.

Scientific dependency: #2853 d01045dfb67428679618027dff552f1d032fa999. Admission transport: #2848 c50d12e4720cd1fef2b5a65e337e37de9731879a, retaining the #2843 transport repair and #2839 block-norm contract. These pending heads are merged into the branch, together with canonical main 2a0d3aeab3016f5b5d85a5c0f956b9db5376c77d. New work is in research/aphase-residue-composition/.

The final census covered 555 open PRs and 1500 Discussion records, then inspected new #2854/#2855. #2855’s T372 headline remains below #2853 and its direct P53 compaction loses. Discussion #2612’s newer delayed-mixing lead is explicitly local-only, with no inspectable new word; it is recorded as a possible later composition.

Reproduce the bounded package checks:

python3.12 -B scripts/research_package.py --package research/aphase-residue-composition

The package README gives the tested complete fresh admission command, exact source pins, proof, evidence and open next questions. Inherited all-size/compiler, common-label/mutable-source, hosting, precision/tape, analytic and bit H1-H5 obligations remain. This does not change the selected record or claim a full Lean multiplication theorem.

Credit: Philippe Van Velzen (#2853); Jeremy Kothe (#2833); Yiyang/Lyy-iiis (#2848); Tekalign and Dugongue’s independent-checker lineage; and all retained source contributors. Marius-Paul Dumitrean, with substantial OpenAI Codex assistance. Apache-2.0.