Add a typed one-round actual-width H and rounded return.

[unpaidmoon]

Change and mathematical scope

Partial construction, no new κ and no selected-frontier change. Selected κ stays 0.001672660197076918.

One typed round of H0 = [[1, 1], [1, -1]] / 2 on the matched actual-width bits, then one toward-zero return Q_p and an MSB-first component bridge. Signed temporary width p+g+2 holds the sharp numerator 2^{p+g}; width p+g+1 does not. The return lands on the p-grid unit disk in width p+2, with Euclidean error strictly below sqrt(2) 2^{-p}. Toward-zero commutes with a signed permutation; floor division does not.

The degree-2 witness (K_d = 2, K_g = 1, p = 3) returns (1, 0) at address 2 on bits {0, 1, 2} and (0, 0) on {0, 2, 4}. The 32-address invoice is 192 real add/sub operations, 64 truncations, and 960 bit reads plus 960 bit writes, counting the input load, the bridge, and the read of the returned tape.

This does not prove multi-round stock, join PR #3617, or execute a production cube. The extent ratio 2^{k^{500}-1} is checked as an integer logarithm only.

Task and overlap

• Base: bb4bccbf42844d2801487450727c2f63e10d1d0f (canonical CrocSwap/integer-mult-bounds main; priority revision 6).
• Head: 499427578245cbd3d2b07680945a94f6e927d020.
• Task: uniform-transfer-audit, scope changed. The card asks for one all-size or precision implication. This session takes handoff comment 19016699's first construction question only.
• Handoff: Handoffs and requests for independent checks #1166 (comment)
• Predecessor: PR research: match actual-width twiddle layout and validate diagonal caller #3623, heads 328c40966b650f60b1d62cfb0b1d67b2b755eb62 and c5ab698bba8b35323e40ae3b132e78f26179d393 (same frozen blobs). The (1, 40) guard is Nick Petalas, PR Bridge the T74 root to the outer dimension and signed format #3493. The K = floor(selected^{1/500}) law and normalized H0 are Jose Manuel, PR Prove and bind T74 cost in the actual selected width #3599. Sagar Thakrar, PR Execute complete encoded twiddle diagonal passes on fixed tapes #3617, is credited and not replayed.

Ordered eligible handoffs, chronological on #1166, after dropping tasks already delivered, matched by an open PR title, or refined by this round's PRs #4107 and #4110–#4120:

1. Typed actual-width H and one rounded return (comment 19016699). This session.
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 PR Prove minimum binary carrier and native XOR bill for the integer-shear target #3947 (comment 19031016).

Set aside before that cut: comment 19016211, because open PR #3681 already replays the PR #3605 fresh bootstrap; the L22 census handoff, refined by #4120; the QQ16 six-tape handoff, refined by the #4114 F0 ledger. Overlap refresh at 2026-10-20T02:26:38Z covered 814 open PR bodies and 2642 discussion records. No open PR builds this one-round wrapper.

Validation

CPython 3.12.3. Registered replay:

python3.12 -B scripts/research_package.py --package research/typed-actual-width-h

Passed locally as PASS_REGISTERED_RESEARCH_PACKAGE research/typed-actual-width-h outcome=partial. Both the normal command and -O print PASS_TYPED_ACTUAL_WIDTH_H and the same receipt. check_research_commit.py --check-source-history passed for base bb4bccbf42844d2801487450727c2f63e10d1d0f.

• Exhaustive g = 1, p = 3 disk: 197 points, 38,809 pairs, each checked against the explicit two-point matrix.
• Eighteen common-round bit identities on the ten small layouts.
• Exact k = 2, 3 floors, including six-multiple offsets +2 and +5 at k = 2.
• Frozen source SHA-256 checks. Those modules are not imported.

Not run: a production T74 root, the PR #3617 diagonal, the multi-round lifecycle, Lean, Docker, or the selected cold replay. Finite checks corroborate the written width and error argument; they are not an all-size theorem.

Credit

Nick Petalas for the outer guard. Jose Manuel for the actual-width plan and H0, retaining Hiroki Kumazaki's factory work. Clear1nsight for the matched layout, the address-2 witness, and the stop rule. Sagar Thakrar for the unexecuted diagonal. The transform lineage named in PR #3623 stays with its authors, including Douglas Colkitt and Philipp.

unpaidmoon, with substantial AI assistance. No independent human review or formal-kernel proof is claimed. Apache-2.0.

Open next questions

1. Can one fixed cube and packet inventory carry every round, with descriptor updates, previous-output reset, borrowed restoration, and an explicit buffer count? Blocked because Q <= C M R (r w + A + p^3) + poly(p) is still only a sufficient criterion.
2. Does this 6 N (p + 2) bit invoice plug into PR Execute complete encoded twiddle diagonal passes on fixed tapes #3617 without a second producer charge? Blocked because that pass was not replayed.
3. The k = 2 cube is not executable: the source band forces ell >= 45,812,984,491.

Workflow feedback (optional)

The handoff's stop rule, once the multi-round allowance is unnamed, is enough to keep the session on one round. #1200