Negative / threshold tests for Theorem 4 (seed 7; 40 lpv, all 13 ncA, 40 ms5, 40 cells)
  cell a1 lowered to 0 rejected                         36/36
  cell c0 + 0.5*minh passes                             40/40
  cell c0 + 1.01*minh rejected (V4)                     40/40
  cell c0 + 1e-2|c0| rejected                           4/40
  cell c0 - 0.5*min(V1-V3) passes V1-V3                 40/40
  cell c0 - 1.01*min(V1-V3) rejected by V1-V3           40/40
  cell c0 - 1e-3|c0| rejected                           40/40
  cell c10 - 0.1|c10| rejected                          40/40
  cell orig passes                                      40/40
  lpv  W1 margin > 0 (float)                            40/40
  lpv  W2 margin > 0 (float)                            40/40
  lpv  c0 + 0.5*minh passes W2                          40/40
  lpv  c0 + 1.01*minh rejected (W2)                     40/40
  lpv  c0 - 0.5*minW1 passes W1                         40/40
  lpv  c0 - 1.01*minW1 rejected (W1)                    40/40
  lpv  one vertex moved by 1e-6 rejected (P1/P2)        40/40
  lpv  orig passes                                      40/40
  lpv  polytope enlarged 10% still passes P1-P4         40/40
  lpv  polytope shrunk 10% rejected (P4)                40/40
  lpv  r1 - 1/20 rejected                               40/40
  lpv  sign flipped rejected                            33/40
  lpv  t1 - 1/20 rejected                               27/27
  lpv  vertex c0 + 1 rejected                           40/40
  lpv  vertex c0 - 1/20 rejected                        40/40
  lpv  vertex set passes                                40/40
  ms   A3 piece widened to [m1/2, m2] rejected          16/16
  ms   A4 sub-piece widened to [n1/2, n2] rejected      13/13
  ms   SG m1 set to 7/10 (2 m1^2 < 1) rejected          18/18
  ms   SG p* set to phi(m1) - 1/100 rejected            18/18
  ms   TC cap replaced by m2 rejected                   40/40
  ms   TC polytope shrunk 10% rejected                  40/40
  ms   TC vertex c0 - 1/20 rejected                     40/40
  ms   TCS C'' vertex c0 - 1/20 rejected                13/13
  ms   TCS cap replaced by m2 rejected                  13/13
  ms   TCS first sub-piece removed rejected             13/13
  ms   TCS p'* below phi(mc') rejected                  13/13
  ms   W1 margin > 0 (float)                            40/40
  ms   W2 margin > 0 (float)                            40/40
  ms   c0 + 0.5*minh passes W2                          40/40
  ms   c0 + 1.01*minh rejected (W2)                     40/40
  ms   c0 - 0.5*minW1 passes W1                         40/40
  ms   c0 - 1.01*minW1 rejected (W1)                    40/40
  ms   mc - 1/10 rejected                               40/40
  ms   one minus piece removed rejected                 40/40
  ms   one plus piece removed rejected                  40/40
  ms   orig passes                                      40/40
  ms   r1 - 1/20 rejected                               40/40
  ms   sign flipped rejected                            40/40
  ms   top of minus pieces lowered by 1/100 rejected    40/40
  ms   top of plus pieces lowered by 1/100 rejected     40/40
  nc   orig passes                                      13/13
  nc   r1 - 1/10 rejected                               13/13
  nc   t1 - 1/20 rejected                               12/13
  sampled vertex margins: min_y(P - Phi_V) in [4.674e-05, 1.510e-03], min_rho h in [5.989e-04, 2.793e-01]
  cover sign +1: deleting one random box breaks coverage 10/10
  cover sign -1: deleting one random box breaks coverage 10/10
  cells: deleting one random cell breaks coverage 10/10
  four-bit: H_OR1234(q4 - 1e-3) = -1.567e-02 < 0 (the lemma fails below q4): True
  four-bit: H_OR123 identically 0 (equality case, so a target 12 - 1e-3 fails): True
