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Change and mathematical scope

Audit of one recovery bill. No new κ and no selected-frontier change.

For the idle-deleted controller in upstream PR CrocSwap#4623, charging the main outputs once and the local helper/paid invoices K>=2 times gives an exact rank-mass loss at t=1:

• complex (b=8, v=112): h(1)=-1856, two-evaluation margin -1920p-3616;
• BIT (b=7, v=111): h(1)=-1617, two-evaluation margin -1680p-3138.

Both added-evaluation steps are negative affine forms, so K=2 is the least negative case. At p=2 the correct bills are -7456 and -6498. Crediting the same main outputs twice yields +160 and +162.

Signed local terms lie on the stated sides of separators 6 and 15/2, and the Jensen base is at least 1+b. Under that hypothesis, negativity at t=1 extends to every smaller t. The derivative of a^t is not formalized. A synthetic profile 5*2^t-3*4^t is negative at t=1 and positive at t=0 when a positive term sits below the separator.

Remaining obligations: the copied-input decoder, the precision lemma, tangent enclosures already checked in PR CrocSwap#4625, native admission, and the selected cold replay.

Task and overlap

Base commit ef5f39403610d231a0c27e8a6b4f08c10209bcd6. Chooser nonce 0ce08b5d1986538ef8eca6579708ff16, track independent-verification, card independent-moment, scope key c6f2494b197df3bc3591. Scope status: changed. The checkout's selected moment already has an independent price; this checks the repeated-evaluation barrier that upstream PR CrocSwap#4625 left open.

Predecessor head fd5d3be12b9d6ee92c8fa089d95bd529c3236563 (PR CrocSwap#4623). Nearest overlap is PR CrocSwap#4625, which checks the idle-mass identity 177408-141568=35840 and the tangent bounds, not these margins. A refreshed overlap search on 2026-10-22 covered 404 open PRs and 3574 discussions with complete pagination.

Discussion writes are forbidden for this account. That is recorded in the package and was not retried.

Validation

python3 -B research/repeated-controller-margin/verify.py
python3 -B scripts/research_package.py --package research/repeated-controller-margin

Python 3.12.3, standard library only, PYTHONOPTIMIZE unset. Result: PASS_REPEATED_CONTROLLER_RANK_MARGINS and PASS_REGISTERED_RESEARCH_PACKAGE. Three controls reject an extra rank-9 helper, a positive term below the separator, and an indivisible BIT target. The t=1 arithmetic is exact integer rank mass. The extension to smaller t is the predecessor's monotonicity argument under the checked coefficient hypothesis, not a new enclosure of the tangent bounds.

Not run: make frontier1094-verify, decoder audit, precision lemma, proof-assistant build.

Credit

Dillon, PR CrocSwap#4623, for the repeated-evaluation policy and published margins. Ryanson Jonathan, PR CrocSwap#4625, for the complementary idle-mass and tangent check. Cinna Davis for the joint-helper invoice and the query-thinned BIT controller lineage. Vendored bytes and SHA-256 pins are in research/repeated-controller-margin/PINS.json. Apache-2.0.

Substantial AI assistance was used. No coding-agent product is named. This is not an independent human or formal-kernel review.

Open next questions

1. Does PR Construct border controllers without idles and bound recovery costs (no κ) CrocSwap/integer-mult-bounds#4623's copied-input X decoder fail every concave moment on these invoices? It is the other open recovery route. Rebuild the claimed per-port query rank p-1 first. The decoder schedule is not in the vendored invoices.
2. Can two evaluations share internal paid calls and avoid the single main-output credit? Name those calls and recompute the t=1 mass. No shared-call schedule is vendored.

Workflow feedback (optional)

The useful gap was a sentence in an adjacent audit that named what it did not check. Refreshing the overlap search immediately before submission avoided repeating the compact-switch and tangent-bound work already on the board.