Prove the sharp pooled optimum for fourth staging

[RossRichardson]

Summary

• Prove that staging the fourth helper in exactly one third of calls is optimal in the pooled cost model of PR4088, even allowing every real staging fraction and every whole-helper bin pattern.
• Certify the sharp complete-moment root in [0.001730980255989036959962, 0.001730980255989036959963]. The pinned literal fill has less than 1.951e-9 of possible root improvement left within this fixed two-provider invoice.
• Add an exact piecewise primal/dual proof, all 841 pattern inequalities, an integer pooled witness, source-derived local costs, and a separately implemented rational check.

Scope and limits

The model fixes the THREE/FOUR words, 3,240 calls, 360 replicas, every original bin and residual, width 73 and stock 896*360. It relaxes chronology, helper identity, family exclusions, offsets and pattern multiplicities. The pooled witness does not establish a legal owner fill. Changed words, capacities, operator fusions or stock require a new model.

This is a written duality/monotonicity proof backed by finite exhaustive pattern coverage and exact rational arithmetic. The larger-kappa target is not achieved; no new supplier, kappa or selected-frontier improvement is claimed. Native/dirty-state correctness, mixed-provider full release, precision, tape, BIT and outer transfer remain separate. AI specialist review is not human or formal-kernel acceptance.

Test plan

Tested commit: a29bdd8f56cd18b805917775cdec09ff935eddc4, CPython 3.12.12.

• python3.12 -B scripts/research_package.py --package research/staged-fourth-mixture-ceiling — all four registered scientific commands and controls pass.
• python3.12 -B scripts/check_research_commit.py --package research/staged-fourth-mixture-ceiling --check-source-history — all 27 committed files and the retained source ancestry pass.
• Official export_research_package.py export and check; all four standalone commands also pass normally and under -O in a read-only Git-free export with empty PATH. All 28 exported files, including its receipt, retain bytes, modes and inodes. The portability driver/receipts are local-only orchestration; every mathematical checker is included here.
• Independent AI mathematical review checks both affine regions, all three duals, complete accounting, the source's historical/current replica distinction and the global root-ceiling implication.
• Natural CI39293406364 completed successfully on the exact tested commit (plan and registered package replay). No workflow rerun or dispatch requested.
• Independent human review — not observed.

Sources and credit

Proof and reproducible package · canonical investigation.

Credit Cinna Davis/4088; Rohan Arun/4076; Dugongue and David Ganz/4085/3945; Nick Petalas; Ross Richardson/3605; dataisfire/3734; ChaloupeMusic and Erwan Secousses/3949/3815; and the retained solver, native, owner and transfer lineage. SOURCE-EXPORT.json, SOURCES.json and the notices bind exact source/base pins and preserve attribution. Fourteen source files are retained byte-for-byte, without requiring pending-predecessor Git objects in a cold checkout. Canonical main was refreshed at bb4bccbf42844d2801487450727c2f63e10d1d0f, priority 6.

Prepared for Ross Richardson with substantial OpenAI GPT-6.1 Sol assistance. Apache-2.0.