Price one typed actual-width physical H and its toward-zero return

[unpaidmoon]

Change and mathematical scope

One typed physical normalized butterfly H and toward-zero rounding Q_p on the PR #3623 matched layout. The tape program equals that package's explicit tensor on every round of its ten small layouts, in both slot orders. The read, write and gate counts are a closed form in the record count, coefficient count, selected-bit count and precision. Head travel is a data-independent itinerary. This is not a scalar saving, not a paid-profile improvement, and not a new κ. The diagonal, all-round stock and production-width root remain open.

A parallel package on the same handoff, upstream PR #4123 (research/typed-actual-width-h, head 499427578245cbd3d2b07680945a94f6e927d020, discussion comment 19056745), prices normalized H0 by add/sub count, truncations and bit touches, and proves a Euclidean error below sqrt(2) 2^{-p}. This package does not repeat that invoice. It prices the unnormalized butterfly as a one-shot tape program: sequential split, interleave, adjacent butterfly, deinterleave and merge, then toward-zero division by 2^g.

Task and overlap

Base bb4bccbf42844d2801487450727c2f63e10d1d0f, priority revision 6. Task uniform-transfer-audit, scope changed. Predecessor PR #3623 head c5ab698bba8b35323e40ae3b132e78f26179d393. Vendored check_join.py blob 421ae05f081ec37e4c5fa0b1b7e200a1578ab16d, identical at handoff head 328c40966b650f60b1d62cfb0b1d67b2b755eb62. Handoff #1166 (comment). Inter-round stock left open on purpose.

Ordered eligible list used for this session, item 3:

1. Integer-shear supplier/recurrence inequality left by Prove minimum binary carrier and native XOR bill for the integer-shear target #3947.
2. Single-hub two-bearing factorization of the PR Classify all thirty two-port support transfers #3573 certificate (discussion comment 19021149).
3. Typed actual-width physical H (discussion comment 19016699). Taken. The distinct remaining piece, after PR Add a typed one-round actual-width H and rounded return. #4123, is the one-shot tape itinerary and closed read/write/gate bill.

Validation

At head 83db51a76b57f3f70f566d96af4f5eb786d7b880:

• python3 -B research/typed-actual-width-physical-h/verify.py prints PASS_TYPED_ACTUAL_WIDTH_PHYSICAL_H
• python3 -B -O research/typed-actual-width-physical-h/verify.py prints the same
• python3 -B scripts/research_package.py --package research/typed-actual-width-physical-h --check-only prints PASS_PACKAGE_METADATA_AND_PINS
• python3 -B scripts/research_package.py --package research/typed-actual-width-physical-h prints PASS_REGISTERED_RESEARCH_PACKAGE with outcome partial (run on the parent scientific commit; the follow-up commit changes only the README overlap note)
• python3 -B scripts/check_research_commit.py --check-source-history --package research/typed-actual-width-physical-h prints PASS_RESEARCH_COMMIT_BYTES

Fifty calls (10 layouts × 2 orders × every round) match h_direct plus toward-zero truncation. Totals: 18,871,808 reads, 27,630,304 writes, 94,213,460 moves, 9,636,792 gates. Digest 9de82d583fb882cdaa6f238d764ef8e6b72ff757aa856a4983ad9a889229d176. Two-record witness (ell=2, one short axis, K=1, p=4, g=1): 768 reads, 1200 writes, 620 gates, 3890 moves.

Not run: selected cold replay, Lean, Docker, production ell, the PR #3617 diagonal, all-round tape reuse.

Credit

Clear1nsight PR #3623 and sources #3493, #3593, #3599, #3611, #3617 (Sagar Thakrar's diagonal contract, not replayed). Parallel arithmetic invoice: unpaidmoon PR #4123. New tape schedule, bill and checks by unpaidmoon with substantial Grok 4.7 assistance in Cursor. Apache-2.0. No independent human or formal-kernel acceptance.

Open next questions

Smallest follow-up: a two-round schedule on the ell=2 witness with an explicit live-buffer list. Blocked because auxiliary tapes are per selected bit and no simultaneous-stock bound is stated. See README.

Workflow feedback

The released next task was specific enough to start from the frozen checker without replaying the production band. Separating the one-round bill from all-round stock kept the session inside the handoff. The parallel arithmetic invoice landed while this tape bill was already committed; the README now points at it so the two reviews stay separate.