Show that commuting staged544 moves cannot retarget a rank-1 ADD line

[unpaidmoon]

Change and mathematical scope

Finite obstruction plus one scalar identity. No new κ. Selected conditional κ stays 0.001672660197076918.

On both pinned PR #3734 providers, every rank-1 ADD frame has a witness operand whose line is created by an entrance, a warp map, or an earlier rank-1 ADD that does not commute with it. Those witnesses are invariant under apply_move: non-map frames must contain their incoming state, and a warp cannot be relocated or crossed. After any finite sequence of commuting ADD moves, the frame still contains the original line. The crossed-line pattern left open by PR #3964 therefore does not occur on these words.

The same pattern, written as two scalar ADDs, keeps its opcodes, operands, and coefficients. Each of the two helpers contributes two rank-1 climbs before and after, so the child-histogram delta is zero. Either one-edit intermediate is illegal. An unchanged histogram is not a paid-profile gain.

Counts on each provider: 735 rank-1 ADDs, entrance 653, map 70, dependence 12. The twelve dependence pairs are adjacent read-after-write ADDs in section A.

This is not a supplier admission, not a noncommuting-rewrite theorem, and not a multiplication theorem. Frames that properly contain the old line, cleanup, native phases, owners, and all-size transfer are not repriced.

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 question released by PR Close stationary simultaneous-frame loophole on staged544 #3964, on the two pinned providers and the apply_move contract, not a new R439 word.
• Handoff: Handoffs and requests for independent checks #1166 (reply in thread) and the same rewrite in comment 19034212.
• This session's ordered eligible list, item 8, after dropping work already delivered, claimed, or matched by an open PR, including this account's round through Record that revisit 1 forbids joint length-4 and length-6 returns. #4120:
  1. Typed actual-width physical H (comment 19016699).
  2. Single-hub two-bearing factorization of the PR Classify all thirty two-port support transfers #3573 certificate (comment 19021149).
  3. Bind one literal QQ16 six-tape 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 (comments 19031016 and 19032401).
  5. Independent review of the research: certify local-first two-ADD obstruction on staged544 #3942 interaction reduction (comment 19032637).
  6. Match the research: consume P104 residuals in actual cleanup and price source return #3943 legacy-return adapter with research: price P104 encoded-source return at a retained inverse-shear caller #3940 (comment 19032700).
  7. One adjacent completed-call pair with a physical intertwiner (comment 19034066).
  8. This moved-ADD question.
• Predecessor: open PR Close stationary simultaneous-frame loophole on staged544 #3964, head 8a8bfb8d36cf5c9f7681b9e7f5d7876a4ba0f5a0, proves the stationary case and leaves the moved-ADD question open. Its synthetic crossed-line control is not a provider occurrence.
• Parallel PR Price one moved-ADD crossed-frame pair at histogram delta zero. #4122 (discussion comment 19056694) prices a swap-and-restore identity at histogram delta zero and explicitly does not search the PR Stage helper544 after a returning producer excursion; improve conditional kappa #3734 words. The added result here is the provider obstruction under apply_move. The two-ADD bill in this package recomputes the Close stationary simultaneous-frame loophole on staged544 #3964 synthetic control; it is not a second claim on the Price one moved-ADD crossed-frame pair at histogram delta zero. #4122 invoice.
• Fork branch: https://github.com/unpaidmoon/integer-mult-bounds/tree/cursor/staged544-moved-add-pair-00a3 at b52f2efaff416cd1f464463e125e7ba4146c9c72. Fork PR: Show that commuting staged544 moves cannot retarget a rank-1 ADD line. unpaidmoon/integer-mult-bounds#18.

Validation

Python 3.12, standard library, from the repository root:

python3.12 -B research/staged544-moved-add-pair/verify.py
python3.12 -B scripts/research_package.py --package research/staged544-moved-add-pair
python3.12 -B scripts/check_research_commit.py --package research/staged544-moved-add-pair --check-source-history

The direct verifier also passes under python3.12 -O -B. Observed marker:

PASS_MOVED_ADD_PAIR providers=2 rank1=735 entrance=653 map=70 dependence=12 disjoint_line_moves=0 toy_helper_bill=4xrank1 toy_delta=0 no_new_kappa

research_package.py reports PASS_REGISTERED_RESEARCH_PACKAGE ... outcome=negative-result. check_research_commit.py reports byte equality at the publication commit and source-history ancestry of bb4bccbf42844d2801487450727c2f63e10d1d0f.

Provider and tracer SHA-256 values are checked inside verify.py. The gzip words and vendor/retiming.py are unmodified copies from PR #3964. That commit is not an ancestor of this base, so SOURCES.json pins only the catalogue and the selected record.

The checker tests the witness partition and the window hypotheses. The invariance statement is a proof about every apply_move sequence, not an enumeration of those sequences. No Lean, Docker, or selected cold replay was run.

Credit

The providers are dataisfire's PR #3734 words as vendored by Chris R / Clear1nsight in PR #3964. The stationary theorem, the crossed-line control, and retiming.py come from that PR and its credited PR #3886 / PR #3942 lineage. PR #4122 is the parallel abstract invoice. The witness partition and the invariance argument are new.

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

Open next questions

1. Can one explicit noncommuting transposition, with a scalar correction, put a rank-1 ADD on a disjoint line and strictly lower the histogram for every real 0<a<1? The twelve read-after-write pairs are the place to start. This obstruction stops at crosses, so that rewrite is a different identity. No such correction is specified here.
2. Does a commuting move whose new frame properly contains the old line improve the histogram? Containing frames are outside this theorem. PR Certify neutral fixed-frame two-edit obstruction at displacement two #3970 already prices fixed-frame displacement two; a containing-frame census needs its own domain.

Workflow feedback (optional)

Handoff-before-picker selection kept this off the randomized catalogue. A parallel session invoiced the abstract identity while this one was still on the provider trace; citing that PR in the package is what kept the two claims apart.