[1337isnot1337]

Change and mathematical scope

A fourth staged entry need not be used in every call. Staging helper 601 in exactly 1,090 of the 3,240 T360 calls gives a complete integer owner fill whose paid moment improves #4085. The remaining 2,150 calls use the exact #4085 three-entry words. Both orientations include its receiver-sharing rewrite.

The certified complete moment root is

[0.001730979115670792472760,
 0.001730979115670792472761].

Under the retained transfer hypotheses and padding convention, the conditional endpoint is κ = 0.001730979115670792472760 − 10^-46 − 10^-70. It exceeds #4085's certified upper endpoint by more than 8.696471628770020 × 10^-9.

This revision improves the first PR head 76088d49e3ed4142a958659aa996b0ca4da5e88e by more than 8.10594906779328 × 10^-10. Ten additional fourth-entry calls and a new complete fill remove the earlier rank-16 and rank-32 residual terms. The four local words and native source closure are unchanged. The previous full budget and bracket are retained in inputs/PR4088-FIRST-RESULT.json and independently repriced by the verifier.

Status: author-checked finite construction, exact phase/operator argument, and conditional moment improvement. The uniform native realization, common-cover/hosting, analytic, precision, tape, recovery and BIT transfer hypotheses remain inherited. The full ordinary/outer release engine has not been rerun with the four-provider mixed dispatch. No selected-frontier change, independent human acceptance or formal-kernel verification is claimed.

The new local variant has profile {1:2489,2:713,3:227,4:99,5:20,8:8}; its change from #4085 is 2 H1 + 3 H3 − 3 H4. Its hosted width for helper 601 drops from 8 to 7. A literal fill accounts for all 1,321,920 roles and 725,760 bins, including every residual term. The complete paid profile is

{1:8060060, 2:2310120, 3:751813, 4:327210, 5:66425,
 8:25920, 48:138492}.

Its total paid rank mass remains 65088*360. The finite selector is (109*t) mod 324 < 109; the claim does not depend on a fractional packing objective or an optimality claim.

Construction and replay · Proof and scope.

Task and overlap

• Canonical base: bb4bccbf42844d2801487450727c2f63e10d1d0f; task complex-new-helper-inventory.
• Immediate source Conditional κ = 0.001730970419199163702738 (+1.614e-7 over #4076): receiver sharing on the staged T360 supplier #4085: c5854c77620723aae6914c32122870519a6f820d.
• Three-entry predecessor Conditional κ = 0.001730809033576552081734 (+1.458e-7 over #3945): three staged helper entries and an LP-exact T360 fill #4076: 9744f9df268590295a9872b1952c695a678192b9; terminal-width-five predecessor Raise conditional staged544 κ by filling idle owner slots #3945: 91f81d131252db704f945891da1ea54e29d46b8b.
• Scope is refined to a nonuniform four-provider schedule. Conditional κ = 0.001730809033576552081734 (+1.458e-7 over #3945): three staged helper entries and an LP-exact T360 fill #4076's uniform fourth-entry rejection leaves this mixed supply balance open. Conditional κ = 0.001730970419199163702738 (+1.614e-7 over #4076): receiver sharing on the staged T360 supplier #4085 supplies the receiver-sharing improvement; the additional contribution here is the fourth-entry composition, selector, literal owner fill and complete price certificate.
• The later Prove the sharp pooled optimum for fourth staging #4091 studies the optimal fraction in the pooled relaxation of this invoice. This revision improves the actual chronological integer fill, stays below that reported pooled ceiling, and makes no claim that its selector is globally optimal.
• Independent audit: rank-two defect and dirty cleanup of three staged entries #4087 independently audits the earlier three-entry defect and cleanup. It does not supply this fourth-entry frame/owner construction or the composition with Conditional κ = 0.001730970419199163702738 (+1.614e-7 over #4076): receiver sharing on the staged T360 supplier #4085. Latest overlap refresh also read Prove a sharp native cost floor for six-bank cleanup #4086; its six-bank cost floor does not change this supplier.

EXPORTED-SOURCES.json pins 56 byte-identical predecessor exports by original PR, commit, path and SHA256; each was compared directly with its Git source. The package carries its runtime source closure. The source-history check covers the canonical pins declared in SOURCES.json; it does not claim pending predecessor heads are ancestors.

Validation

Tested head: 72d7efc181edb21806a878f174847c96dd410ab0, CPython 3.12.14.

python3.12 -B scripts/research_package.py --package research/staged544-mixed-fourth-entry
python3.12 -B scripts/check_research_commit.py --package research/staged544-mixed-fourth-entry --check-source-history

Both pass. The official package export and integrity check also pass for all 86 committed files.

• All four exact words pass 723-input forward, dual, reverse-pullback and cleanup-boundary checks, including all 91 arbitrary dirty buffers, plus an independent binary-frame ledger.
• The unchanged four-entry orientations retain the five-stage native replays performed for the first PR revision: transport, ledgers, local replay, wrappers and exact phase recipes. This revision authenticates identical word/source bytes and phase receipts; those native stages were not redundantly rerun. Three-entry native receipts are source-pinned imports from Conditional κ = 0.001730970419199163702738 (+1.614e-7 over #4076): receiver sharing on the staged T360 supplier #4085. Commands to rerun the fresh native checks are in the README; those checks additionally require NumPy.
• Every owner role, bin, selected helper width, family, offset and capacity is checked. The separate exact phase audit checks strict scalar-observation windows, productive-data nonaliasing, and disjoint slices of the original phase axes with their exact residual complements. No axes move between child slots.
• The inherited Z4 normalizer checks 10,416 symbolic identities across all 112 ports and the complete signed controller, with the all-character and positive-tensor-multiplicity argument retained explicitly.
• Rational enclosures certify both the new root and Conditional κ = 0.001730970419199163702738 (+1.614e-7 over #4076): receiver sharing on the staged T360 supplier #4085's bracket. Controls reject omitted/flipped releases, old entrances, wrong sources, owner violations, omitted fourth providers, changed global phases/controllers, unpaid cost and unsupported endpoints. The word checks also pass under -O.

Hosted CI is separate and was not claimed successful at submission. The full ordinary/outer release integration remains open.

Credit

Cinna Davis, with substantial OpenAI Codex assistance; Apache-2.0. Retained credit includes Rohan Arun/#4076; Dugongue/David Ganz/#4085 and #3945; Nick Petalas' shared-covector construction; Ross Richardson/#3605; dataisfire/#3734; ChaloupeMusic/Erwan Secousses/#3949 and #3815; Marius-Paul Dumitrean's frame solver; Sagar Thakrar and roger-sato's audit engines; zxv and the complete earlier geometry/native/compiler/owner lineage. NOTICE.md and the source-specific notices preserve attribution and licenses.

Open next questions

1. Extend the pinned ordinary/outer provider registry to the four providers and mixed dispatch, preserve every admission predicate and mutation control, and rerun the complete release. This is the next integration obligation.
2. Improve the selector or integer packing. The periodic 109-of-324 schedule is sufficient, not proved optimal; every further gain must again produce a complete literal fill and pass the word, native, phase/operator and price checks.

[tekuuu]

CLAIM — source-word chronology experiment, canonical base bb4bccb. I am testing joint commutation retiming of the existing staged-entry releases on exact PR4088 head76088d49e3ed4142a958659aa996b0ca4da5e88e (THREE/FOUR providers), keeping entrance/terminal inventories and every native interface fixed. Bounded domain: the six608/609/600 releases, legal scalar/native commuting windows and coupled relocations, with finite-domain frame optimization only nominating candidates. A winner must pass exact dirty semantics, complete same-owner invoice, native/phase checks and rational pricing. This is distinct from the active ternary covector/transpose source-aggregation claim on4085; no source sums or new carriers are introduced. Three candidates considered: fixed-invoice repacking (4091 leaves under2e-9 headroom), new source aggregation (claimed), and changed release chronology (selected). Expected artifact: a verified improving construction or a scoped search record, without assuming a larger kappa. Finish target18:25UTC. Tekalign (tekuuu), with substantial OpenAI Codex assistance; staged-entry/receiver/frame/native predecessors retained.