[unpaidmoon]

Change and mathematical scope

Partial certificate. No new κ. The selected conditional exponent stays 605457647/200000000000 (0.003027288235).

On the pinned forward word, event 9895 is L546 -= L561 in frame 4. The only earlier record of register 561 is the entrance adapter at event 7963. The word has five event kinds, and none of them names a tape origin or a cell length. The cleared C0–C7 windows are a different token family and are still zero, so they are not a stored image of L561.

If L561 is supplied in the kernel's two-tape entrance (two owned tapes of length NW) and must be copied onto the four half-length scratch windows of the subtraction packing, with heads returned to cell 0 and no free relocation, the sharp bill is READ 2NW, WRITE 2NW, MOVE 8NW-12. The axis kernel's one-past packing habit costs 12 moves more on the same split. Adding the already proved subtraction gives a conditional pair total of READ 6NW, WRITE 4NW, MOVE 16NW-24. That total is not an embedding extracted from the word. Both schedules share the move count 8NW-12 because they use the same six lengths; the copy reads and writes 2NW cells, while the subtraction reads 4NW and writes 2NW.

This is not a complete multiplication theorem and does not change the selected frontier.

Task and overlap

Base commit 32890bc5de034ab8fc93ac88ead7279a6a416d04 (upstream CrocSwap/integer-mult-bounds main). Dependency pins are in research/qq16-l561-scratch-placement/SOURCES.json.

Task card joint-query-controller, priority revision 7. Scope status is refined. The card asks for a matched whole-caller bill or a cheaper demand. This package answers open question 1 of research/qq16-l546-head-move, the still-open layout remainder of discussion comment 19030049, and stops.

Session r34, item 2 of the ordered eligible list recorded in the package README. Item 1, the (w, q, v_N) caller, stays open. The fresh T385 wrapper, the retiming handoff, the four-provider rebind, the matched park/restore comparison (open CrocSwap#4167), and the CrocSwap#4091 ceiling comparison are already closed or covered and were dropped before this cut.

python3 scripts/find_overlap.py on 2026-10-25, terms scratch placement, L561 onto, 8NW-12. Coverage: local notes complete at 32890bc5 (7404 records), 390 open PRs, 4810 discussion records, both paginations complete. The 8NW-12 hits are the subtraction finding (discussion comment 19146840) and research/qq16-l546-head-move, which explicitly does not price this copy. research/qq16-streaming-recovery uses “scratch placement” for a different word.

Validation

Python 3.12, standard library only, from the repository root:

python3 -B scripts/research_package.py --package research/qq16-l561-scratch-placement --check-only
python3 -B research/qq16-l561-scratch-placement/verify.py
python3 -B scripts/research_package.py --package research/qq16-l561-scratch-placement

Observed:

PASS_PACKAGE_METADATA_AND_PINS (not a mathematical acceptance check)
PASS_QQ16_L561_SCRATCH_PLACEMENT event=9895 intervals=absent prior_records=1 mentions=27 scratch_clear=C0-C7 move=8*N*W-12 read=2*N*W write=2*N*W sharp=yes one_past_extra=12 combined_read=6*N*W combined_write=4*N*W combined_move=16*N*W-24 no_new_kappa
PASS_REGISTERED_RESEARCH_PACKAGE research/qq16-l561-scratch-placement outcome=partial

The same line is printed under python3 -O. Not run: Lean, Docker, the selected cold replay, and any search for a native tape program outside this word.

Credit

The six-tape rules are Clear1nsight's research/qq16-gaussian-axis-tapes. The unbound-interval statement and the zero C0/C1 candidate are research/qq16-f0-first-caller (PR CrocSwap#3932). The same-frame scalar price is unpaidmoon's CrocSwap#4162. The conditional subtraction bill is unpaidmoon's research/qq16-l546-head-move. Ali Romman's QQ16 lineage and Nick Petalas's numerical formats are the retained background.

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

Open next questions

1. What does it cost to copy L561 off the four scratch windows before a later kernel clears them? No tape-using event after 9895 is bound to these windows. The subtraction leaves L561's bits in scratch.
2. Does any published source outside this scalar word assign register 561 an origin and a length? The 8NW-12 count is the sharp bill for the kernel's two-tape entrance only.

Workflow feedback (optional)

The 19 October handoff index is now mostly closed on main. Keeping a distinct remainder of a closed smallest task, instead of reopening the finished head-move bill, is what separated this item from work already delivered.