Conditional κ = 0.001600067494703216: width-seven providers and whole-owner packing

[DominikScholz]

Change and mathematical scope

This adds a conditional finite construction at κ = 0.001600067494703216, improving PR564's 0.001593134158181640 by 0.435201%.

Three extra exact cold relations and 32 entrance-frame promotions create width-seven residual helpers. Whole-owner repacking uses 63 width-seven calls and nine plain four-cold calls, allowing 9+7 pairs in rank-sixteen owners. The complete complex profile is {1:169056, 2:41184, 3:25968, 4:13824, 5:4032, 6:4032, 7:48, 8:576, 48:1808}. Dimension 73, live stock 7168, physical stock 7682 and rank deficit 2560 are unchanged.

The native complex rate is 0.001602631813774414678548. The conservative model separately passes at κ = 0.001600067491919801. The unchanged cited PR543 bit supplier remains nonbinding.

The package also records exact capacity/stage dual bounds, bounded negative geometry searches, and explicitly hypothetical helper-cost targets for larger gains. These are distinguished from admitted constructions.

Dependency and attribution: PR564 is pinned at c18417b02b80594b2ca1c8c21185d28befafc935; its source package is an external replay input, not republished in this diff. Root main is pinned at db0e57bc7f36b5ac0fc844ad7ce1396165ee6647. Inherited notices and source/adaptation hashes are retained. Substantial OpenAI Codex assistance was used.

PR567 and PR568 appeared during this work and report further cold-kernel improvements. They were fetched for comparison but are not replayed or composed here; the numerical comparison above remains against the freshly reproduced PR564.

All inherited all-size/compiler, common-label/mutable-basis, normalized residual, correlated whole-owner, precision/tape and analytic-transfer assumptions remain. This is not an unconditional multiplier theorem, complete Lean proof, or worst-case O(1) indexing result.

Attribution and new contribution

This contribution is submitted by Dominik Scholz,
with substantial OpenAI Codex (GPT-6 Astra) assistance in mathematical
exploration, implementation, verification and documentation. The new work is
the additional checked suffix relations, coupled width-seven entrance
promotions, their literal whole-owner assignment, exact model exclusions and
portable replay. It builds on the following contributions.

Contributor	Retained or cited contribution
eumemic	PR564 and PR561/562/558: cold-kernel and zero/four/eight providers and admission; PR550: hardened assembly ancestry; PR543: cited unchanged bit supplier, not freshly rebuilt here.
Rohan Arun / rohanarun	PR559: native exact-complex moment argument; retained complex501 follow-up: stage-search base.
Dugongue	PR552: collective waves and weighted response; PR505: asynchronous whole-owner ancestry; PR563: earlier capacity-pattern LP.
Chafik Boukhalfa / chafreaky	PR551: full phase replay driver and earlier geometry/audit/invoice lineage.
icekylinx	PR531: 504-helper direct delivery, frame coalescing and compiler/controller ancestry; PR535: post-center rank-four kernels.
DreamingOfClouds	PR484 and successors: shared-donor kernels, gauges, normalized wrappers and word/frame machinery.
DaysSky and Zhihao Chen / jacklightChen	Inherited PR442/381 hosting and bridged-column checks, and PR425/386 mutable-source interface and geometry work.

All earlier contributors and source-specific AI disclosures remain credited
in package NOTICE, the repository contributor record and the unchanged
external dependency. These contributions are not independently additive
numerical gains.

Validation

• Complete pinned PR564 reproduction: all 36 stages passed.
• New local providers: all 728 columns in both orientations, dirty restoration, exact frames, phase recipes and endpoint adapters passed.
• New global construction: physical chronology and all 16 owner mutations; complete finite invoice; both independent exact moment engines; 47 strict outer conditions per model; all 89 assembly mutations rejected; two timing-only phase changes accepted and two semantic changes rejected.
• Exact relaxed upper bounds are checked with rational duals and independent engines, including corrupt-certificate controls.
• make gen4-bank-verify passed. The broader make verify run reached an inherited C++ target requiring unavailable Boost headers. This is not recorded as a pass.
• Portable replay completed: all 71 fresh provider stages followed by all seven global stages passed, reproducing both exact kappa values. The final receipts and log hashes are retained in global/replay-evidence/.

Full reproduction from the new package, with the exact PR564 checkout available:

python3 -B research/larger-gain/global/reproduce.py \
  --source /path/to/pinned-PR564/research/complex-allcold \
  --work /tmp/width7-reproduction

The output directory must be new. See research/larger-gain/README.md, global/width7/RESULT.json, and global/README.md for the proof scope, retained checks and source-bound reuse options.

[Mister-Meeseeks]

Thank you @DominikScholz. This contribution at 1424da81263a5305b35e4cf74ef686c275f03f3f is incorporated in the reviewed family through #588, published in 9594b61.

Entrance promotions change only frames under exact nesting/response constraints. Six explicit providers and the global mixed assignment are separately admitted. Exact capacity duals are scoped upper bounds on the nominated inventory/packing class; nomination search optimality is not claimed.

Preserved implementation and research:

• research/larger-gain

The family review and individual disposition record the scope, evidence and attribution. Parallel and intermediate contributions remain credited even where a later composition has a stronger bound. The published multiplication bound remains conditional on the documented compiler, realization, hosting and analytic assumptions; these finite checks are not a full Lean multiplication proof.