[unpaidmoon]

Change and mathematical scope

Audit of a fixed two-word invoice. The pooled ceiling from PR #4091 still upper-bounds the recorded 1,090-call mixed fourth-entry root. The local words, bin capacities and stock coefficient are the same in both certificates. The call selector and the idle histogram differ. No new κ. The selected record stays 60403/20000000 = 0.00302015.

The recorded roots satisfy

1080 literal upper < 1090 lower < 1090 upper < pooled ceiling lower.

The separations are

1090 lower − 1080 literal upper = 810594906779329 / 10^24
pooled ceiling lower − 1090 upper = 1140318244487201 / 10^24

The headroom 1.951e-9 measured from the 1,080-call literal fill is a different gap: the later fill has already moved part of the way from that literal root toward the ceiling.

Remaining obligations: feasibility of the 1,090-call idle inside the 841-pattern program, and the four-provider ordinary rebind.

Task and overlap

Base commit ff3cc58e370d0f678e529246f8c73e8094e18fcf (upstream main). Task card independent-moment, priority revision 7. Scope status refined: the card's first step asks to validate word, capacity and stock equivalence before applying the #4091 ceiling, which is pinned to PR #4088 head 76088d49e3ed4142a958659aa996b0ca4da5e88e, to the later 1,090-call fill.

Nearest work:

• research/staged-fourth-mixture-ceiling (Ross Richardson, PR Prove the sharp pooled optimum for fourth staging #4091) supplies the pooled ceiling and the 1,080-call literal bracket.
• research/staged544-mixed-fourth-entry (Cinna Davis, PR Improve conditional κ with a mixed fourth staged entry (+8.696e-9) #4088) supplies the 1,090-call fill.
• research/staged544-mixed-idle-rebind records the idle-cost block. This package uses that fill's recorded idle and does not rerun admission.

An open-PR title search for 4091, 1090, mixture ceiling, staged-fourth and pooled ceiling did not find a pull request that already checks this equivalence.

Validation

Python 3.12 standard library only. From the repository root:

python3.12 -B scripts/research_package.py --package research/staged-fourth-ceiling-equivalence

Observed:

PASS_STAGED_FOURTH_CEILING_EQUIVALENCE words=identical capacities=identical stock=23546880 bins=725760 capacity=17418240 paid_mass=23431680 fourth=1080->1090 above_literal=810594906779329/1000000000000000000000000 below_ceiling=1140318244487201/1000000000000000000000000 controls=3 no_new_kappa=true
PASS_REGISTERED_RESEARCH_PACKAGE research/staged-fourth-ceiling-equivalence outcome=completed

What the checker establishes:

• The four gzip word files are byte-identical. Their gzip SHA-256 values match the ceiling source_sha256 pins. The SHA-256 values of the decompressed JSON match the later INPUTS.json pins. Those two manifests hash different byte strings of the same words.
• Both THREE orientations have local profile 2487 H1 + 713 H2 + 224 H3 + 102 H4 + 20 H5 + 8 H8 and widths {5:28, 6:18, 7:56, 8:93, 9:213}. Both FOUR orientations add 2 H1 + 3 H3 − 3 H4 and use widths {5:28, 6:18, 7:57, 8:92, 9:213}. Helper 601 is at entrance [45] in THREE and [127, 45] in FOUR. Local rank mass drops by 1.
• TEMPLATE.json.gz and ORDER.json are byte-identical. Template ranks {8:672, 16:448, 32:448, 48:448}, 18 children on each of 112 ports, times 360 replicas, are the ceiling bins {8:241920, 16:161280, 32:161280, 48:161280}: 725,760 bins and capacity 17,418,240.
• Rebuilt profiles 2160 L3 + 1080 L4 + {3:22788, 5:1620, 48:138492} and 2150 L3 + 1090 L4 + {3:22783, 5:1625, 48:138492} match the recorded profiles. Both have paid mass 65088*360 = 23431680. The ten extra fourth calls move ten rank units into the idle histogram. Both proofs use stock (896*360)*73 = 23546880.
• Three controls reject a rank-1 profile shift, the 1,080 idle on the 1,090 selector, and a truncated THREE-base word.

Checks not run: the 841-pattern linear program, native stage replay, and the ordinary/outer four-provider release. The 1,090 idle is checked against recorded capacity and rank mass. This audit does not show it is a feasible point of the pooled program. The argument is in research/staged-fourth-ceiling-equivalence/PROOF.md.

Credit

Credit Cinna Davis and PR #4088 for the later fill, and Ross Richardson and PR #4091 for the pooled ceiling. Prepared by unpaidmoon with substantial Cursor Grok assistance. New work is Apache-2.0.

Open next questions

1. Does the 1,090-call idle {3:22783, 5:1625, 48:138492} correspond to a feasible pattern vector of the 841-pattern program? Express the idle delta from the certified eight-pattern witness at x=1080 as an integer combination of tight patterns, then check capacity and the width-5 through width-9 helper totals. Pattern enumeration is not replayed here.
2. Can the pinned ordinary rebind take four provider ids, recompute this idle, and still pass the outer release? That is open question 1 of research/staged544-mixed-idle-rebind. This equivalence does not run it.