=== (1) face structure on columns [-3000,3000] ===
edges lying twice on the same face / vertical edges without a square: 0

=== (2) Lemma dual: exhaustive bonds of the windows ===
R=2: |V|=14, bonds=24, bonds whose boundary is a simple dual BOT->squares->TOP path: 24
R=3: |V|=20, bonds=100, bonds whose boundary is a simple dual BOT->squares->TOP path: 100
R=4: |V|=26, bonds=374, bonds whose boundary is a simple dual BOT->squares->TOP path: 374
exact min |dS| over window bonds touching column x vs L_x: {(2, 1): (2, 2), (3, 1): (2, 2), (3, 2): (3, 2), (4, 1): (2, 2), (4, 2): (3, 2), (4, 3): (3, 3)}

=== (3) Lemma len via BFS lower bound, 1 <= x <= 3000 ===
violations of min-walk-length >= L_x: [] count 0
x with equality: 28 ; sample (x, BFS lower bound, L_x): [(1, 2, 2), (2, 3, 2), (3, 3, 3), (5, 3, 3), (10, 4, 3), (50, 5, 4), (100, 6, 5), (500, 9, 8), (1000, 11, 9), (2000, 14, 11), (3000, 16, 13)]
unconstrained walks (through TOP/BOT allowed): violations of >= L_x: 2946

=== (4) EVIDENCE: minimal end-separating cut in windows, one U(0,1) sample ===
R=25: min cut weight 0.6005, |cut|=5, columns 4..6
R=50: min cut weight 0.6005, |cut|=5, columns 4..6
R=100: min cut weight 0.6005, |cut|=5, columns 4..6
R=200: min cut weight 0.6005, |cut|=5, columns 4..6
R=400: min cut weight 0.6005, |cut|=5, columns 4..6
R=800: min cut weight 0.6005, |cut|=5, columns 4..6
R=1600: min cut weight 0.6005, |cut|=5, columns 4..6
cut identical for R=200..1600: True
