Math Mapgenerated 2026-10-25 · a354234
PUBLIC BUILD

run 122Settled

On census generators run 119 did not build, does a single-cell flip change the block transfer operator T exactly when the cell lies on a simple path between two distinct cells of the left and right columns?

slug
what-the-block-operator-sees
state
run
created
2026-10-23
sealed
2026-10-23 (committed before any flipped operator is built on the nine generators)
builds_on
[119]
concepts
[transfer-operator, symmetry, percolation, biconnectivity]
programme
PG-01
line
the block as an operator
from
run 119, read again by the codifier (themes/symmetry.md §2.1d; OPS instance 13). Its isotropy was forced, and 37 of its 38 remaining coincidences were flips T could not see. A criterion for which single cells T sees was formed on run 119's data and classifies all of it (R27_tree_groups/carpet_perc/vis_premise122.py and .log). This run tests it on generators run 119 never built. The owner's ruling, 2026-10-23 14:47: "yes to new seal". The guesses are in docs/walk/2026-10-21-pre-registration-symmetry.md (section of 15:00), committed before this seal.
statistic
R27_tree_groups/carpet_perc/, transfer_op.build_T and generator imported, and e3_119.flipped and pstar imported; none of them edited. Generators are the nine non-cut-2 census patterns (C₁ ≠ 2) whose p* build in transfer_op.json took wall_s ≤ 140 and peak_rss_bytes ≤ 3000·2²⁰. These are indices 4, 5, 6, 7, 8, 9, 13, 18, 26. Cells are the 15 wedge cells (i, j), 0 ≤ i ≤ j ≤ 4, each toggled alone (hole ↔ kept), as in run 119; there are 135 flips. For each flip build T(K′) = build_T(K′, p*), with p* the unflipped pattern's census value, and T(K) for the unflipped pattern. The flip's change is Δ = max over all 512² entries of |T(K′) − T(K)|, in the same basis. The criterion: let G be the pattern in which the cell is kept (K if it is kept in K, else K′), with 4-adjacent kept cells as vertices, plus a super-vertex s joined to every kept cell of column 0 or column 8. The cell is predicted visible iff it and s lie in one biconnected component of G + s with more than two vertices. This is vis_premise122.on_boundary_cycle, which the lab imports rather than re-implements. Observed visible means Δ > 1e−11. Scored: the 2 × 2 table (predicted visible or not) × (observed visible or not) over the 135 flips. Reported beside: Δ for every flip. Also reported beside: for each non-diagonal flip (90 of them), T(K′ᵀ) = build_T(K′ᵀ, p*) and max|T(K′) − T(K′ᵀ)|, with every flip where that is ≤ 1e−11 and Δ > 1e−11 listed. And σ₁…σ₄ of every T, wall time and peak memory per build.
scored
2026-10-23
outcome
confirmed
outcome_note
by the prediction as written — 129 / 0 / 0 / 6 over 135 flips on nine generators run 119 never built: T changes exactly when the flipped cell lies on a boundary-to-boundary path. The classes are about 13 orders of magnitude apart. Controls pass. The p* deviation (census expression in place of e3_119.pstar, which refuses C₁ ≠ 2) built the same operators, checked bit for bit. Side list: generator 8, cell (2,3), has T(K′) = T(K′ᵀ) while T moved, the same cell as run 119's pattern 51. M5, 17,588 s.
pre_seal_glimpse
run 119's 285 flips on the 19 cut-2 patterns (where the criterion was formed); transfer_op.json's unflipped spectra and costs for the nine; the predicted split from the criterion. Nothing built on any flipped pattern of the nine.
source
../lab/runs/122-what-the-block-operator-sees.md

Prediction(A) both off-diagonal cells of the table are 0, with 6 flips predicted invisible and 129 visible (vis_premise122.log). Confirmed if (A) holds.

Falsifier(A) any flip off the diagonal of the table. Unanticipated, recorded and not scored: any predicted-invisible flip with Δ between 1e−14 and 1e−11 (a change at the instrument's edge). Withdrawn if any control fails.

Statement

Run 119 showed that the block transfer operator T cannot see many single cells. Its leading singular values are unchanged when the cell flips, and there is no symmetry behind it. T records which right-column cells connect to which left-column cells through open bonds. A cell that lies on no route between two boundary cells cannot change those connections, so its flip should be invisible. A cell on such a route should change them, unless something cancels. Both halves are tested here, on nine census generators no flipped operator has been built for.

Sealed 2026-10-23

As written. The lab chat reads this file above "Not part of the seal" only, and transfer_op.py, e3_119.py, vis_premise122.py, census068.json and transfer_op.json for the method.

Scored 2026-10-23 — confirmed

Scored by a blind reader given the seal above the "Not part of the seal" heading and RESULTS122.md (R28_reading/122/scorer122.md). The criterion classified all 135 flips, and visible and invisible differ by about 13 orders of magnitude (2.7e−3 against 2.2e−16). The reader's fault 1 is the codifier's: the seal named a function that refuses these generators. The lab chat should have raised it as an RFI. It documented the change instead, and the result does not depend on it.

Reading (the codifier's, after the score). What T can see is now a criterion, out of sample: a cell matters exactly when it lies on a route between two boundary cells. The side list repeats something. The one flip where T cannot tell K′ from its transpose while still changing is again cell (2,3), as in run 119 (pattern 51). The reader also lists pairs of different cells that give identical spectra on one generator. Both say that T's blindness is structured beyond the criterion: it sees routes, not cells. That is case material for the formation step. Which cells give the same T, and is it the block-cut tree of the route graph?

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

generated 2026-10-25the lab index