Record nonnegative raw-moment invoices for two-neighbour ADD moves.

[unpaidmoon]

Change and mathematical scope

Scoped negative result. No new κ. Selected κ stays 0.001672660197076918.

On both pinned PR #3734 staged544 providers, every legal first ADD cut that crosses at least two scalar neighbours has nonnegative raw rank moment sum n_r r^(1-a) for every real 0 < a < 1, at every feasible common-frame dimension. Each provider has 5,501 ADD gates, 4,979 with a long cut, and 1,761,760 dimension cases (239,103 nonzero, 1,522,657 neutral). The first-move price depends only on dimension.

Witness on the base word: A[595] is add 459, 510 with coefficient -1 and empty frame. Cut 1214 crosses 618 gates, including an ADD and a SCALE. The only legal rank-1 frame is span{85}. The frame bill changes by +3 rank 1 and -1 rank 3, with caps [2, 1, 0, ...], so the move is strictly losing. After the scale-adjusted coefficient and rederived cleanup, all 1,767,681 single-ADD retimings of the edited word keep the combined histogram certified nonnegative.

This is not a review of the PR #3942 interaction reduction, not an extension of the radius-one theorem to every long first move's continuations, and not a second long move. Native, stock, precision, tape, all-size and supplier costs are not repriced. Unknown costs are not zero.

Task and overlap

Base: bb4bccbf42844d2801487450727c2f63e10d1d0f. Priority revision 6. Task scalar-to-moment, scope refined: one frame-path change on the pinned providers, not an R439 scalar-count reduction.

Dependency head: PR #3942 scientific head 1416f8cd6bfb2f2eb9aa5cc22f0388a85fdbac91 for vendor/retiming.py, vendor/audit_core.py, and both BORROW-WORD-43 words. Those bytes are vendored and SHA256-checked. PR #3734 / dataisfire is the word source named by that package.

Nearest overlap: PR #3942 (radius one), PR #3970 (neutral fixed-frame displacement two), and PR #4122 (one synthetic moved-ADD pair with illegal intermediates). This census starts at two crossed neighbours and prices feasible frame dimensions. It does not repeat those proofs. Sibling note 19057045 (cursor/two-add-gap-review-337a) is the separate review of the #3942 fixed-frame new-gap reduction; this census does not repeat it.

The assigned list had 9 eligible handoffs, so item 22 is the refinement of item 4 (comment 19032637). The review requested there is not this package.

Validation

CPython 3.12.3, standard library only.

python3 -B research/staged544-two-neighbour-invoice/verify.py
python3 -B scripts/research_package.py --package research/staged544-two-neighbour-invoice

Observed: PASS_STAGED544_TWO_NEIGHBOUR_INVOICE base_cases=1761760 extra_cases=1761760 witness_second=1767681 improving=0 no_new_kappa
and PASS_REGISTERED_RESEARCH_PACKAGE research/staged544-two-neighbour-invoice outcome=negative-result.

The curvature test rejects a hand-built negative histogram. Vector 1 is checked to lie outside the witness line. Checks not run: the 632-input free-operator interpreter, a continuation census for every long first move, a second long move, and native/physical admission.

Credit

dataisfire (PR #3734); Chris R / Clear1nsight (PR #3886 and PR #3942); Sagar Thakrar (frame auditor). Vendored files keep their headers. New census and witness by unpaidmoon, with substantial Cursor Grok assistance. Apache-2.0.

Open next questions

1. Does every long first move, not only A[595], keep every single-ADD continuation nonnegative? Repeat the continuation census on a move whose frame interval has quotient rank greater than 1. One witness cannot extend the radius-one theorem.
2. Can a second ADD that itself crosses two neighbours improve the combined raw moment? Freeze one edited word and one such second cut first.