Review the staged544 fixed-frame new-gap reduction.

[unpaidmoon]

Change and mathematical scope

Audit of the fixed-frame new-gap reduction in PR #3942. No new κ and no selected-frontier change. Selected κ stays 0.001672660197076918.

On both pinned PR #3734 providers, every scalar-legal third ADD at the swapped interior cut of a fixed-frame adjacent pair either matches a separately coded bridge formula or equals the original-word price at the cut named in that package's PROOF.md §3. The only nonzero bridge rows are the four published restore rows, and each passes the nonnegative curvature certificate. Every radius-one first-move interval has quotient rank at most 3, and an independent one-vector walk matches the Gaussian subspace counts.

This is not a replay of the 11.9-million-row variable-frame census and not a two-ADD impossibility theorem.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main, refreshed 2026-10-20). Priority revision 6.
• Task: scalar-to-moment, scope refined. The card asks for one scalar simplification's paid histogram. The live handoff instead asks for an independent challenge of the reduction that already says this class does not improve the raw moment.
• Handoff: Handoffs and requests for independent checks #1166 (reply in thread) (task 1 only).
• Predecessor: PR research: certify local-first two-ADD obstruction on staged544 #3942, head 1416f8cd6bfb2f2eb9aa5cc22f0388a85fdbac91. This package vendors that head's two provider words and vendor/retiming.py, checked by SHA256, and does not import search.py.
• Ordered eligible handoffs, chronological on Handoffs and requests for independent checks #1166, after dropping items whose next task is already delivered or matched by an open PR title, and after dropping this round's delivered work (Price flag 9's head-step co-host at exactly +2 rank mass #4107 through Record that revisit 1 forbids joint length-4 and length-6 returns. #4120):
  1. Typed actual-width physical H / rounded return (comment 19016699).
  2. Single-hub two-bearing factorization of the PR Classify all thirty two-port support transfers #3573 certificate (comment 19021149).
  3. Integer-shear supplier/recurrence inequality left by Prove minimum binary carrier and native XOR bill for the integer-shear target #3947 (comment 19032401).
  4. Independent challenge of the research: certify local-first two-ADD obstruction on staged544 #3942 new-gap reduction (comment 19032637, task 1). This session.
• Dropped before that cut: the PR Reduce helpers 644/660 from width 8 to 7; conditional κ > 0.00173 #3605 full-wrapper replay (open Replay the complete PR3605 wrapper with a fresh retained bootstrap #3681); the L22 census and one-point pair form (Bounded length-22 census probe: exclude splits (11,7), (10,8), (11,8); measure the rest (answers #3708 OQ1's blocker; no new kappa) #3731, Record the coset-wide one-point pair ceiling #4117, Record that revisit 1 forbids joint length-4 and length-6 returns. #4120); the P104 residual consumer (open research: consume P104 residuals in actual cleanup and price source return #3943); the T385 full replay (open Replay T385 through release and audit R437 complete-stream COPY #3936); the six-tape F0 caller (open Research: first F0 caller six-tape admission audit #3932); neutral-first retiming (Certify two neutral-first retiming domains on the pinned staged544 providers #3941, Certify losing-first two-edit obstruction at displacement two on staged544 #3957) and the coordinated-edit half of this same handoff (Close stationary simultaneous-frame loophole on staged544 #3964, Certify neutral fixed-frame two-edit obstruction at displacement two #3970, Remove two staged544 scalar ADDs with an A2 graph identity #3998, Audit the separated A2 composition and cleanup split #4112); the tensor-pairing review (Review the common-map tensor-pairing bound independently. #4113).
• Not redone: displacement-two searches, the A2 composition audit, or the variable-frame second-move census.

Validation

CPython 3.12.3. Registered replay:

python3.12 -B scripts/research_package.py --package research/staged544-two-add-reduction-review

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/staged544-two-add-reduction-review outcome=negative-result.

verify.py also passed under python3 -I -B. Both providers report 4709 fixed pairs, 14966 bridge gates, 14693 bridge dimensions, quotient maximum 3, and the same four nonzero rows. A fast corridor test was checked against the explicit crossing walk on every restore pair and on the first 40 fixed pairs (166,691 comparisons). One published nonzero row matches apply_move plus a fresh trace.

Not run: the predecessor variable-frame census, Lean, Docker, and the selected cold replay.

Credit

Chris R / Clear1nsight for PR #3942 and the released review request. dataisfire for the PR #3734 providers. The PR #3886 single-move rules and the retiming line credited in that package are reused as the authenticated trace oracle, not re-authored. Prepared by unpaidmoon with substantial Cursor Grok assistance. No independent human or formal-kernel review is claimed. Apache-2.0.

Open next questions

1. Does every disjoint-register second ADD have raw-moment delta equal to its original single-move delta? The crossing algebra is checked here; the summed histograms are not. Start from one changed-frame first word. Blocked by the size of the 11.9-million-row census.
2. Does choosing the same ADD twice undo every radius-one first move? A failed undo would be a word the reduction never prices. Needs the variable first-move list.

Workflow feedback (optional)

The explicit numbered index in #1166 is smaller than the set of still-open prose handoffs. Keeping those handoffs in chronological order, and dropping a request only when its next task already has an open PR or a later DONE that answers that task, made the fourth remaining item the #3942 review rather than another retiming search. A machine-readable open-handoff list with that filter would let parallel sessions share one ordering.