Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 136Settled

On one row of nine blocks (a 9 × 81 strip) of each of H, U, U′, T and X, how far does chaining the block's left-to-right operator T fall short of the strip's exact crossing? T misses only clusters that cross a seam and come back, so this measures the back-flow. How far short does truncating T to rank k fall?

slug
the-strip-and-its-back-flow
state
run
created
2026-10-24
sealed
2026-10-24 (committed before any strip of H, U, U′, T or X at B = 9 is computed)
builds_on
[051, 131, 132]
concepts
[transfer-operator, renormalisation, percolation, finite-boundary-state]
programme
PG-01
line
the block as an operator (composition; the symmetry theme's E4, strip form)
from
the design reader on composing the block operator (R28_reading/design-e4/REPORT.md). "Rank-k T composed on the 9 × 9 arrangement" is not defined, because T is one-sided and does not compose exactly past height 2. The strip, one row of nine blocks, is exact and cheap. The tool chat built strip.py and its premises (strip_premises.log); the codifier added a grid premise (strip_grid_premise.py and .log). The owner's ruling, 2026-10-24 14:35 ("yes to all", PG-01's composition). Guesses: docs/walk/2026-10-21-pre-registration-symmetry.md, section of 2026-10-24 17:15.
statistic
R27_tree_groups/carpet_perc/strip.py, called with --sealed 136 and not edited; transfer_op.build_T and generator are imported, not edited. Hosts: census patterns 17 (H), 11 (U), 12 (U′), 26 (T) and 6 (X), B = 9. Per host:
outcome
confirmed
scored
2026-10-24 (blind scorer, R28_reading/136/scorer136.md)
pre_seal_glimpse
the tool chat's premises (toys, U′ at B ≤ 3, the square at B = 9). No carpet strip at B = 9 has been computed, by anyone, at any p.
source
../lab/runs/136-the-strip-and-its-back-flow.md

Prediction(A) T's chain falls short of the strip by at most a quarter of the strip's width on every host: 0 ≤ g_bf ≤ 0.25 for H, U, U′, T and X. Confirmed if (A) holds.

Falsifier(A) g_bf ≥ 0.5 on any host: clusters that cross a seam and come back move the strip's threshold by half a width or more, so the left-to-right operator is not the strip's boundary state. Between 0.25 and 0.5 on any host, with none at 0.5 or more, unanticipated. Withdrawn if a control fails.

Statement

The block operator T says which right-edge cells get wet given which left-edge cells are wet. Chain nine copies along a row and you have a prediction for a 9 × 81 strip, missing exactly one thing: a cluster that crosses a seam into the next block and comes back through another seam bond. T can't see that, because each block's T only looks left to right through itself. The strip's exact crossing can be computed directly. The gap between the two is the back-flow, the first dropped term in PG-01's ledger after truncation. This run measures it on the five census patterns, and also how far truncating T to rank 1, 2, 4, 8 or 16 falls short.

Sealed 2026-10-24

As written. The lab chat reads this file above "Not part of the seal" only, plus strip.py, transfer_op.py and strip_grid_premise.py.

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 RESULTS136.md (e48c6a2) and nothing else (R28_reading/136/scorer136.md). Scores are as written.

Controls. - (i): the square strip's p* is off by +1.25e−9, against a line of 1e−7. - (ii): max(R_T − R_strip) ≤ −1.58e−4 on every host. - (iii): H, U and U′ clear run 051's bounds by 0.07 or more.

(A) Confirmed. g_bf is 0.0132 (H), 0.0212 (U), 0.0142 (U′), 0.0218 (T) and 0.0341 (X), all inside [0, 0.25]. Chaining the left-to-right operator T falls short of the exact 9 × 81 strip by 1.3% to 3.4% of a strip width. So clusters that cross a seam and come back barely move the strip's threshold, and T is close to the strip's boundary state.

Beyond the clause. 1. Rank 1 suffices. Rank k = 1 is within 0.1·w_s on all five hosts (the largest gap is −0.051 w_s, on U′), rank 4 is exact to 1e−4, and k = 1 and k = 2 agree exactly. With run 135's "rank 1 carries no curvature", the strip's threshold is set by one mode. 2. The vertical gap in run 051's widths at L = 81 was converted by the codifier from RESULTS051.md:36, which the lab did not open. H is 2.779, U 2.264 and U′ 1.172. The pre-registered guess, at least 2 widths for all three (2026-10-24, 17:15; P = 0.75), missed on U′. The strip sits above the lattice at L = 81, as an upper bound should, but by very different amounts.

Deviations, none of which changes the score. - U′'s pass-1 bracket was extended with the exact point R_strip(1) = 1, as ruled at 20:52. - Build children peaked at 5.84 GiB, against the sealed 3.0 GiB, and the builds took 18,201 s against 16,300 s. This is a sizing miss for OPS. - The lab chat called strip.block_T, which by_lab does not list; strip.py's md5 was unchanged. - The lab chat committed with --no-verify and ran with memory on, the second lab chat in a row to do both. This goes to OPS, and the lab prompt template needs a line forbidding both.

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

generated 2026-10-25the lab index