On the 19 cut-2 census patterns, does flipping one cell of the 15-cell wedge break boundary isotropy visibly — the left–right and top–bottom transfer operators acquiring different singular values — for every flip that removes the pattern's D4 symmetry, and does the size of the break order by the flipped cell's symmetry class?

Statement

The symmetry theme's E3 asks whether the block reads the same from its left–right sides as from its top–bottom sides. On the census it does by construction (every pattern is D4-symmetric). A single flipped cell breaks the symmetry in one of three ways: on a diagonal (the pattern keeps its diagonal reflections), on a midline (it keeps one axis reflection), or off-axis (it keeps nothing). Comparing σ(T_LR) with σ(T_TB) in the same 512-state basis and the same (standard) metric measures the break in the per-side reading. The centre flip keeps D4 and is the control.

Sealed 2026-10-23

As written. The lab chat reads this file above "Not part of the seal" only, and transfer_op.py, wedge_op.json's "cells" list, census068.json and RESULTS-transfer-operator.md §4.4 for the method.

Scored 2026-10-23 — falsified

Scored by a blind reader given the seal above the "Not part of the seal" heading and R27_tree_groups/carpet_perc/RESULTS119.md. Controls held (centre flips g = 0; unflipped σ to 4.6e−16). The falsifier fired: 114 of 266 non-centre flips at g ≤ 1e−12. The unanticipated clause also fired, on the diagonal class (76 of 76). The seal gave no precedence between them; the reader chose falsified because 38 isospectral flips lie outside the diagonal class, and recorded the diagonal class as the theme's hidden symmetry beside the verdict, as the seal says.

The 38 non-diagonal isospectral flips fall on four cells: (1,4) 5, (2,3) 9, (2,4) 8, (3,4) 16.

Faults in the seal's wording (the reader's seven, kept): no precedence between the falsifier and the unanticipated clause, which cannot fire apart; the question ("every flip that removes D4") and the test ("every non-centre flip") assume breaking D4 breaks the left–right/top–bottom equivalence, which the diagonal class shows it need not; two zeros (control (i)'s 1e−15 and (A)'s 1e−12) not reconciled (the diagonal flips sit at 1e−15 to 7.6e−15); the class's scope (pooled or per pattern) unsaid; no tie rule for (B); the sizing (under 2 GB) was wrong by 2.5× (peak 4.9 GB) with no consequence attached. On the reader's last point: the pattern-12 test that showed a diagonal flip at 1.5e−15 was the lab's pre-launch test, run after this seal was committed; it was not a pre-seal glimpse.

Reading (the codifier's, after the score): the symmetry theme's first carpet result is a hidden symmetry, and it is exact. A single cell flipped on the diagonal leaves the pattern symmetric under transposition only, and transposition swaps rows with columns — so the top–bottom operator of the flipped pattern is the left–right operator of its transpose, which is itself; the two operators are the same object read with left and right relabelled, and their singular values agree to the instrument's zero. Boundary isotropy, as the per-side reading sees it, needs only the transpose, not all of D4. That is the forced-situation ladder's first rung measured on the carpets, and it says the ladder's E3 row was written too coarsely: the symmetry that controls the per-side reading is one reflection, and the theme's table should say which. The 38 flips off the diagonal that are also isospectral sit on four cells, which suggests flips that do not change what the boundary sees at all (a cell whose toggle leaves the crossing structure untouched); that is a question with a cheap test — compare T itself, not its spectrum — and it goes to the theme before any further seal.

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