Record the idle cost blocking the mixed staged544 endpoint.

[unpaidmoon]

Change and mathematical scope

Audit, not a new κ. The highest pending conditional endpoint is PR #4088's mixed staged544 profile,

κ = 0.001730979115670792472760 − 10^-46 − 10^-70,

with PR #4100 a sharper reading of the same profile that does not run the historical outer-admission CLI. Selected κ stays 0.001672660197076918.

The inherited ordinary rebind (source/new/rebind.py in the PR #3815 replay archive) has only provider ids gauge44 and extra, stamps one width vector on every call of a provider, and adds the shipped idle remainder unchanged. An independent scan of the #4088 rows shows that rule cannot carry this invoice.

Helper 601 (origin index 367) has width 8 on 2,150 calls and width 7 on 1,090 calls, split inside both orientations. The two extra rank-1 cells per fourth-entry call are not divisible by either orientation's call count, so no single integer local profile exists. The scanned idle differs from the #4085 idle by -59 H3 − 265 H5 + 54 H48 (rank mass +1090), which balances the local saving. Keeping the #4085 idle drops those 1,090 rank units. That lighter profile's root lies near 0.00174722463155, more than 1.62455 × 10^-5 above the scanned root. The scanned profile itself does reproduce the #4088 moment bracket. That is a positive control, not an admission.

Task and overlap

Canonical base bb4bccb. Task independent-moment, scope changed: the catalogue's #2979 group audits already exist, and the live admission blocker is this complete-invoice cost. Overlap search on 2026-10-20 covered 857 open PR bodies and 2,686 discussions; the only open PR match for this mixed fourth-entry invoice is #4088 itself. #4085 is the completed two-provider release. #4100 reuses the profile and leaves the outer CLI unrun.

Package: research/staged544-mixed-idle-rebind/. Statement: PROOF.md. Head branch: unpaidmoon:cursor/mixed-fourth-idle-rebind-f457. Base: CrocSwap/integer-mult-bounds main.

Validation

CPython 3.12, standard library only:

python3 scripts/research_package.py --package research/staged544-mixed-idle-rebind --check-only
python3.12 scripts/research_package.py --package research/staged544-mixed-idle-rebind

Both passed (PASS_MIXED_FOURTH_IDLE_REBIND_OBSTRUCTION). The verifier checks SHA-256 pins, walks all 1,321,920 rows, and rejects five mutations: a 108-of-324 selector, a three-entry width on the first fourth-entry call, the frozen-idle profile at the claimed bracket, one extra rank-1 cell, and the #4085 profile at the claimed bracket.

Not run: native five-stage replay, admit544.py, the outer ledger, and #4100's unconditional argument. Scalar semantics of the words are not replayed; local histograms are the pinned word reports, checked against the frame-rank mass.

Credit

Prepared by unpaidmoon with substantial Cursor Grok assistance. Apache-2.0.

Unmodified exports of Cinna Davis's PR #4088; three-entry gzip hashes are zxv's PR #4085 candidate pins; the rebind file is the PR #3815 archive copy used by Dugongue's PR #3945 and Rohan Arun's PR #4076. NOTICE.md records the export.

Open next questions

1. Extend the rebind to four provider ids, recompute idle from the per-call widths, keep every admission predicate, and rerun the ordinary/outer release. That is the step still required before this endpoint can be considered.
2. A different literal fill might place the width-7 helper 601 calls without 54 rank-48 idle cells. It needs its own row scan; a fractional screen is not enough.