On sixteen fresh 9 × 9 blocks, none in the census and half with a kept column-7 cell cut off from the left, does the number of one-way pairs of the block transfer operator equal |U′|·(511 − |U′|), with U′ from the corrected flood-fill rule?

Statement

Run 127 counted the transfer operator's one-way pairs: pairs of right-column sets where one can turn into the other and not back. One is enough to show the operator is not reversible in the detailed-balance sense under any reweighting of states. On ten of 63 census blocks the count missed the rule's prediction. A reader then found why. Some kept cells in column 7 sit on islands, walled off from the left by holes. A right cell next to such an island cannot be wetted from the left, even though its neighbour is kept. The corrected rule asks, for each set of right cells, whether a flood fill from the left over the block plus those cells reaches all of them. It fits all 63, but it was formed on those 63.

This run tests it on sixteen blocks drawn fresh: the outer ring kept, 17 interior cells removed at random, none equal to a census block under any symmetry of the square. Eight have a column-7 island. On those eight, the old and corrected rules give different counts.

Sealed 2026-10-24

As written. The lab chat reads this file above "Not part of the seal" only, plus transfer_op.py (build_T), oneway127.py, oneway_fresh_premise.py and oneway_fresh_masks.json.

The codifier’s lead is kept for the private site.

Scored 2026-10-24 — confirmed

A blind scorer read the seal above the line and RESULTS131.md (8344c9a) and nothing else (R28_reading/131/scorer131.md). Scores are as written.

Controls. - (ii): generator 7 gives 53,070. - (i): holds on all 17 blocks, with the worst row-sum deviation 8.90e−13 and the ∅ row's off-diagonal entries exactly 0.

Scorability. All 16 masks are scorable; the RSS stop never fired.

(A) Confirmed. The observed count equals the corrected rule's count on 16 of 16 masks. That includes all eight island masks, where the old rule's count differs on every one (F04: 45,540 observed against the old rule's 64,860).

What the numbers say beyond the clause. 1. The one-way pairs of the block operator are exactly U′ × complement on every fresh mask. U′ is the set of right-column outputs that a flood fill from the left, over the block plus the chosen right cells, cannot reach. Every U′ × complement pair is one-way, and none lies inside U′ or inside its complement. The rule's assumption, that every non-empty input reaches every reachable output, held on blocks with no symmetry. 2. So where Q fails detailed balance is now decided by geometry alone, before any operator is built: one-way pairs exist exactly when U′ is non-empty. F14 and F15 have none, so detailed balance is not ruled out there (E1 is open on them, as on run 127's seven).

Procedural. The lab chat had memory on. Control (i) is reported as a worst case over the blocks, which bounds each mask.

Theme instance (added 2026-10-25, records). themes/symmetry.md §2.1g, written from this section. The frontmatter's theme_instance field above the line predates it and is left as sealed.