Price the F0 adjacent ADD after the first rank-1 climb.

[unpaidmoon]

Change and mathematical scope

Partial frame bill. No new κ. Selected conditional κ stays 0.001672660197076918.

On the pinned forward core, event 9894 is the word's first climb: L546 from frame 0 ([]) to frame 4 ([459]), rank 1. Event 9895 is L546 -= L561 in frame 4. L561 enters in frame 4 and is not climbed or written before that ADD. Both operands already sit in the annotated frame, so the retained nested-frame contract accepts the ADD with no second climb.

The coefficient -1/1 costs 4 scalar work steps and 8 real steps. The climb is the pair's only rank child. The in-frame inverse L546 += L561 costs the same 4 steps and restores only the post-climb value. The pinned Clifford table has no child (4, 0). Both registers end at frame 1, not at L546's entrance frame 0.

This is not a six-tape layout, not a kernel replay, and not a supplier admission. Unknown physical window cost is not zero.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main). Priority revision 6.
• Head: 95716ef66c2db59b532f555544acc7818431464e.
• Task: joint-query-controller, scope refined.
• Object: events 9894 and 9895 only.

Ordered eligible list this session used, item 4. Chronological open #1166 next tasks after dropping a question a later DONE, an open PR, or this account's round #4107 and #4111–#4157 already answers or refines. Comment 19032700 updates the earlier (w, q, v_N) consumer request, so that parent is not a separate earlier item.

1. (7,18) / (7,20) functional census (comment 19021170, question 1). Question 2 is Prove codimension-one L/2 substitution soundness on double transversals. #4148.
2. Largest physical child rank on one support alphabet (comment 19021222).
3. Monotone L/2 firing classification at (4,14) and (5,16) (comment 19022438, handoff 1). Handoff 2 is the Prove codimension-one L/2 substitution soundness on double transversals. #4148 soundness argument.
4. This pair. Comment 19030049 asks to price one adjacent event after the six-tape bind. PR Research: first F0 caller six-tape admission audit #3932 names L546 -= L561 and leaves its alignment unpriced. PR Ledger intermediate F0 magnitudes on the L359 dirty ray. #4114 ledgers rays L359 and L112 and does not replay this pair.

PR #3932 head 12c324c576821f78070627c5eeb684d03704e20e is the predecessor audit, not a dependency that was re-executed. The forward word and frame table are the pinned joint-query certificate and selected-replay archive on this base.

Validation

CPython 3.12, standard library only:

python3.12 -B scripts/research_package.py --package research/qq16-adjacent-add-bill

Observed:

PASS_QQ16_ADJACENT_ADD_BILL climb=9894 add=9895 frame=4 axis=459 rank_child=1 scalar_work=4 real_steps=8 climbs_before_pair=0 reverse_child=absent word_climbs=3486 word_scalars=20089 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-adjacent-add-bill outcome=partial

The verifier also rejects the ADD if the climb is deleted and if L561 enters in frame 0. The full climb replay matches the published final vector. 3/1 costs 6 scalar steps and 1/2 costs 5. No Lean replay, no six-tape kernel, and no physical head-position bill was run.

Credit

Pinned joint-query word and selected-replay frame table. PR #3932 by Chris R / Clear1nsight named this ADD as the unpriced successor of event 9894. The nested-frame rule is the retained local checker contract. New check by unpaidmoon, with substantial Cursor Grok assistance. Apache-2.0. No independent human or formal-kernel review.

Open next questions

1. What physical head moves does this ADD cost once L546 and L561 are placed on real six-tape windows? Same-frame legality does not meter layout or the climb's kernel. Blocker: PR Research: first F0 caller six-tape admission audit #3932 leaves intervals and head positions unbound.
2. Which cleanup child returns L546 from frame 4 to entrance frame 0? The in-frame inverse does not. No recipe child (4, 0) is present.

Workflow feedback (optional)

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