Conditional κ = 0.001600848528154892: four-provider joint-prefix packing

[chafreaky]

Conditional κ = 0.001600848528154892, 0.01037836% above #571. The complex supplier binds at b = 0.001603415353263184744465. The conservative complex model separately admits κ = 0.001600848525369710; the unchanged cited #543 bit supplier remains nonbinding with its surcharge retained.

This selects among four verified local suppliers within one newly reconstructed literal owner allocation: 58 joint-prefix width-seven eight-cold calls, eight width-seven eight-cold calls, four plain eight-cold calls and two plain four-cold calls. The scalar words come from #571, #572/#568 and #569. The new contribution is their joint selection, exact owner expansion and complete admission; no predecessor gains are added.

H = {1:169296, 2:41043, 3:25943, 4:13824,
     5:4032, 6:4032, 7:3, 8:576, 48:1817}

Dimension 73, live stock 7168, physical stock 7682, rank mass 520704 and deficit 2560. The assignment checks all 36,288 helper roles, 6,222 actual charts and 3,038 positive idle remainders. A pattern/time flow model produces literal bins through exact bipartite matching. The solver stopped with a feasible nomination at its time limit; no solver or packing optimality claim is used.

Package: research/mixed-prefix-fourway. Python standard library only:

python3 -B research/mixed-prefix-fourway/verify.py --work /absolute/new/fourway

The cold verifier rebuilds the complete pinned #572/#568 chain, then reconstructs the #569 plain four-cold supplier and #571 joint-prefix supplier. All four local providers are freshly verified before the new owners and invoice. Checks include both 728-column orientations, arbitrary dirty restoration, exact frames, normalized wrappers, complete phases/endpoints, kernel omissions, 16 owner mutations, a complete finite invoice, two rational moment engines, both complex models, seven bootstrap levels and 47 strict outer conditions per model. All 109 assembly mutations, four timing accept/reject pairs, four actual recompressions and 72 proof/reference mutations are checked. Every proof-relevant provider field is frozen; generated gzip equality uses exact decompressed content.

Final frozen-source cold replay passed on macOS/Python 3.14.8 in 541.075 s and Linux/Python 3.12.3 in 607.330 s, each with 30 top-level stages and the complete nested dependency replay. Results are byte-identical; all 28 proof-content identities and every numerical/semantic invoice field agree. Receipts and full provenance are in research/mixed-prefix-fourway-validation. The unchanged repository baseline reuses PR572’s published 18-group passing evidence, including gen4-bank-verify; it is not claimed as a new aggregate execution. The initial packaging-path failure and final fresh reruns are documented.

Concurrent #574 (f3e0af901f88f2fd3a213224ad3b59a16b5f4688) admits κ = 0.00160073442456167 (independently replayed all 24 stages in 237.3 s) by upgrading #571's nine plain calls and repacking. This four-provider construction is higher. #574's owner map is not used here. Concurrent #575 (eac6f7e378ab622dac50a2fd4b7ef6eab014de44) reports κ = 0.001600770286423653 using joint prefixes in all 72 calls. Its full 39-stage independent replay passed in 300.912 s; its rate is also below this construction. No #575 source is used here.

Concurrent #576 (269490b0553e59bf224ab96009809ae8908d8f74), discovered at publication, reports the higher κ = 0.001600951400768614 using 12 further cold kernels and a new three-provider allocation. Its independent full 31-stage Mac/Python 3.14 replay passed in 250.217 s. A subsequent #576 head f4e9f7e85e9211cf6ba6c5789da2aa0902d9121e reports κ = 0.00160109005167629 after changing only its owner map, plan, witness specification and manifest; that updated allocation is not yet independently replayed here. This PR does not incorporate those new kernels and does not claim the strongest reported frontier; its contribution is the four-provider packing and fully reproduced admission described above.

Attribution: Dominik Scholz (DominikScholz, #571/#569) supplies the joint prefix, width-seven promotions and four-cold provider; Jeremy Kothe (jeremykothe2, #568) supplies the eight-cold family; Jonathan Capalbo (jon314159, #567) supplies portable identities and controls; our #572 supplies pattern/time flow expansion and the cold-replay package. The retained ancestry includes eumemic, rohanarun, Dugongue, icekylinx, DreamingOfClouds, DaysSky, jacklightChen and earlier contributors credited in the original notices. All 273 vendored files match their pinned Git commits. Substantial OpenAI Codex assistance was used; source-specific AI disclosures and licenses are preserved.

The all-size compiler, common-label/mutable-source realization, normalized recursive residuals, whole-call hosting, precision/tape and analytic-transfer assumptions remain, including the bit H1–H5. This is conditional finite evidence, not an unconditional multiplication theorem, complete Lean proof, practical speedup or κ = 1 result.

[Mister-Meeseeks]

Thank you @chafreaky. This contribution at 086bd33ff5c1ce46f3ff23298b8dc756a54414db is incorporated in the reviewed family through #588, published in 9594b61.

Four-provider packing changes dispatch and whole owners, while each chosen local word is independently rebuilt. The four predecessor improvements are alternatives, not quantities to add.

Preserved implementation and research:

• research/mixed-prefix-fourway
• research/mixed-prefix-fourway-validation

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.