Gauge44 factory and whole-family assignment for conditional κ = 0.001649483702558979

[icekylinx]

This extends the pinned #635 construction with a changed arbitrary-helper-input factory and 44 rank-four cold endpoints. It is one new commit based on main at 17ce0c79b7b51c2634b439ea5a2b754f09909693.

Under the retained all-size compiler, common-label/mutable-source realization, normalized wrappers, whole-call hosting, precision/tape, exact-recovery, analytic-transfer and bit H1–H5 interfaces,

$$ T(n)=O!\left(n(\log n)^{1-\kappa}\right),\qquad \kappa=\frac{1649483702558979}{10^{18}}=0.001649483702558979. $$

The saving is 0.80931% above #635. This compares exponent savings, not practical runtime. The conservative complex model separately certifies κ 0.001649483699681650.

Construction

Within each eight-source factory, exchange four source registers with equal clean copies at their common rank-one frame. This changes how arbitrary initial helper values propagate while preserving the clean operands. Complete the four coordinate totals to an invertible retired-row basis using frozen low-frame forms, and place available totals on source carriers.

The resulting protected-column and inverse-support calculation exposes 44 rank-four cold directions. Exact dyadic input premixes, low-frame tails and a recomputed FULL inverse realize them. Both row and column isolation are checked before removing an ended helper from cleanup. All scalar gates and dirty-input compensations are included in the finite invoice.

Quantity per invocation	Complete #635 JP8	Gauge44
Source / target ports	112 / 112	112 / 112
Helper registers	492	492
Rank-four cold endpoints	22	44
Total helper residual width	3,894	3,794
Local paid rank mass	5,750	5,650

All 72 invocations use the new supplier. An explicit assignment covers 35,424 helper roles, retaining 7,168 live and 7,670 physical families. The whole-family matcher concentrates idle remainders using a capacity exchange argument; 288 integer majorization receipts justify its updates. All 72 final callwise remainder histograms are optimal with the other calls fixed. No joint global optimality is claimed.

Every idle remainder is charged. The final global rank mass remains 520,704, with deficit 2,560 and largest child rank 48. The improved rank distribution certifies complex saving 0.001652208994368411824947.

The unchanged #543 bit supplier gives 0.001653826988146798944268; the complex side binds. No bit improvement is claimed.

Contribution sources and prior PRs

Source	Contributor	Retained contribution
#635, a6895a69f8ad98ab45777b03e9e6f48888c39f6f	jeremykothe2, DreamingOfClouds and credited upstream contributors	K-triangle/Port480 construction, R492 allocation, transported stages, HPE machinery and global admission baseline.
#604 and the #591/#592/#593/#599/#601 lineage	DreamingOfClouds, jeremykothe2 and credited contributors	Cold endpoints, exact premixes, low-frame tails and native endpoint interfaces.
Joint-prefix and frame/identity lineage	Dominik Scholz, chafreaky and credited contributors	Joint-prefix construction, frame coalescing, artifact identities and retained controls.
#543	eumemic and credited predecessors	Unchanged bit supplier and its conditional interfaces.
Retained framework and exact engines	Credited upstream contributors	Whole-call hosting, precision/tape bounds, rational moments and analytic assembly.

My #535 contributed post-center rank-four response kernels; #531 supplied the SAME504 factory and global package; #523 supplied the 112-port restriction; and #433 supplied split centers and source-basis reuse. Earlier #10, #18, #24, #32, #36, #104, #115, #130 and #144 concern batched boundaries, partial swaps, gauges, certified assembly and frame/cover constructions. These are provenance, not additive numerical gains. Original source-specific notices, including asynchronous-scheduling credits, remain in the ancestry.

The new source-branch gauge, retired-row completion, source-carried totals, exact 44-end construction and whole-family matching are contributed by icekylinx, with GPT-6 Astra assistance. Local review and packaging used OpenAI Codex. Original licenses, notices and assistance disclosures are retained.

Proof and validation

The incremental package is research/gauge44-global/.

• CONSTRUCTION.md: exact local construction, protected directions and full cleanup.
• hosting/PROOF.md: capacity, exchange and integer majorization proof.
• independent/CENTRE-GUARD-REVIEW.md: changed source carriers with the same ordered ports and center frames.
• evidence/ASSEMBLY.json: two exact rational moment engines, both complex models and all 47 strict outer inequalities per model.
• construction/INPUTS.json, data/UPSTREAM-PINS.json, MANIFEST.json: construction inputs, inherited dependencies and submitted-file hashes.
• LOCAL-VALIDATION.json: local incremental verification record.

After preparing the pinned #635 supplier as described in the README:

python -B research/gauge44-global/verify_manifest.py
python -B research/gauge44-global/construction/reproduce.py \
  --main . --pr635 /tmp/gauge44-pr635 \
  --out /tmp/gauge44-local --until phases
python -B research/gauge44-global/verify_global.py \
  --main . --local /tmp/gauge44-local/local --work /tmp/gauge44-global

Local incremental verification passed on Python 3.14.8: exact construction and recipe reconstruction; both native orientations on all 716 input columns; 3,019 directed phase edges and 224 endpoint adapters; independent scalar and retained-guard audits; exact owner reproduction and all 72 callwise fixed points; fresh controller, finite invoice and global assembly. All 15 controller mutations and 69 assembly admission mutations were rejected. The component rates and complete global profile match the frozen evidence; separate rational pricing reproduces the same κ.

Fresh phase timing changes the runtime plan hash and its template reference. Every other serialized template field matches the frozen template exactly. LOCAL-VALIDATION.json records both hashes; each runtime reference is checked against its actual file.

The dedicated CI workflow runs the changed construction, independent checks, owner replay and strict global admission on Python 3.11 and 3.14. Raw logs and optional research outputs are excluded from the submission. No predecessor full replay or PDF generation was performed.

Package manifest SHA256: ad36e2c9802a8da4a4c70180eeb710d9f314b1e8cc521b5749d3fbebc299e932.

Global template content SHA256: 0c2aab7dc120f5882a0171496cbbe819cac81373dcb425802a15f45e36898a46.

This remains a finite conditional witness under the cited realization and analytic contracts. It does not claim an unconditional multiplication theorem or complete Lean formalization.

[Mister-Meeseeks]

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

Contribution: Gauge44 dirty-input factory and 44 rank-four cold endpoints.

Review and limits: Source/copy exchanges alter arbitrary dirty propagation. Complete row/column isolation, exact cleanup, actual owners and native phases remain part of the finite contract.

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: 638a5171dd1aca89ff50d07dcbe288b567769804. Publication: 64039f1bd3b6a523b96083fd858ad040a75e3030.