Price one moved-ADD crossed-frame pair at histogram delta zero.

[unpaidmoon]

Change and mathematical scope

Scoped identity and local invoice. No new κ. Selected conditional κ stays 0.001672660197076918.

On three scalar registers, let A be y += x and B be y += z. Swapping them and retargeting both frames from span(e1) to span(e2) is a legal final word under fixed entrances {0} and exits span(e1,e2). Each one-move intermediate is frame-illegal, because those lines are not nested. Both orders implement (x, y+x+z, z) for an arbitrary initial triple, and the cleanup y -= z, y -= x restores that triple. The charged histogram is six rank-1 children either way, so the delta is zero and the moment change is zero for every real a.

A control that widens both moved frames to the exit has delta -6 H1 + 3 H2, negative for every a in (0,1), but one one-move intermediate is legal. That saving is sequential. It is not a provider occurrence.

This is not a search of the PR #3734 words, not a supplier admission, and not a multiplication theorem. Native, centre, stock, precision, tape, all-size and supplier costs are absent, not zero.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main). Priority revision 6.
• Task: scalar-to-moment, scope refined. The card asks whether one scalar simplification changes a paid histogram. The object is the moved-ADD pair in discussion Handoffs and requests for independent checks #1166 comment 19034220, not an R439 helper-star word.
• Handoff: Handoffs and requests for independent checks #1166 (reply in thread)
• Predecessor: open PR Close stationary simultaneous-frame loophole on staged544 #3964, head 8a8bfb8d36cf5c9f7681b9e7f5d7876a4ba0f5a0, which proves the stationary case has no such pair on the pinned providers and leaves the moved-ADD question open. Its synthetic crossed-line control is not a gate move. This package does not import that code.
• Also nearby and not redone: PR research: certify local-first two-ADD obstruction on staged544 #3942 (legal-sequential two-ADD obstruction), PR Certify neutral fixed-frame two-edit obstruction at displacement two #3970 and PR Certify losing-first two-edit obstruction at displacement two on staged544 #3957 (fixed-frame displacement), PR Quantify paid overhead budgets for concave rank-profile replacements #3863 (the H1-2H2+H3 exchange this witness does not realize).

Ordered eligible handoffs used to choose this item, chronological on #1166 after dropping tasks already delivered or claimed by an open PR, and after dropping this account's round (#4107, #4110/#4111, #4112–#4120):

1. Typed actual-width physical H (comment 19016699).
2. Single-hub two-bearing factorization of Classify all thirty two-port support transfers #3573 (comment 19021149).
3. QQ16 six-tape adjacent event (comment 19030049).
4. Integer-shear supplier/recurrence inequality left by Prove minimum binary carrier and native XOR bill for the integer-shear target #3947.
5. Independent review of the research: certify local-first two-ADD obstruction on staged544 #3942 reduction (comment 19032637).
6. P104 adjacent completed-call pair (comment 19034066).
7. This moved-ADD pair.

The #3605 portable replay was dropped because open #3681 says it answers comment 19016211.

Validation

CPython 3.12.3, standard library only:

python3 -B research/moved-add-crossed-invoice/verify.py
python3.12 -B scripts/research_package.py --package research/moved-add-crossed-invoice

Both passed here:

PASS_MOVED_ADD_CROSSED_INVOICE registers=3 adds=2 cleanup=2 histogram_H1=6 delta=0 rank_mass_delta=0 both_intermediates=illegal scalar=commuting_identity favorable_sequential=-6H1+3H2 one_intermediate=legal no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/moved-add-crossed-invoice outcome=completed

The checker also rejects a noncommuting swap, a coefficient-2 mutation, and a dependent pair that fails the reorder test. The same-line swap stays legal. These are exact rational and subspace checks, not a provider census. Not run: PR #3734 replay, Lean, Docker, and the selected cold replay.

Credit

Chris R / Clear1nsight, PR #3964, for the stationary theorem, the synthetic crossed-line control, the reorder test, and the open question. PR #3942 for the legal-sequential predecessor. Dugongue, PR #3863, for the exchange used only as motivation. dataisfire, PR #3734, for the providers that were not replayed.

Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent human review is claimed. Apache-2.0.

Open next questions

Does any moved-ADD pair on the two pinned PR #3734 providers have a legal final frame assignment, two illegal one-move intermediates, and a strictly negative moment for every real a in (0,1)? The first step is one scalar segment and the crossed-line assignment on pairs the reorder test already allows to cross. The provider trace is in open PR #3964, and a full pair census is a separate budget.

Workflow feedback (optional)

Handoff-first selection against open PR titles was enough to separate this question from the stationary census. No separate session report.