Part V: avoidance (some p in J_4 avoids P): N1 with the turning power b^6
  ok   [Y^7, Y^4*Z], Q_D = Z^2: H = [1, 3, 5, 5, 3, 2, 1, 1], gcd(p, r) = 1: True; CI(f + 3 b^6, K_r, P) = 34 <= 40
  ok   [Y^7, Y^3*Z^2 + Z^5], Q_D = Y*Z: H = [1, 3, 5, 5, 3, 2, 1, 1], gcd(p, r) = 1: True; CI(f + 3 b^6, K_r, P) = 34 <= 40
  ok   [Y^7, Y^4*Z], Q_D = Y^2 + 2*Y*Z + Z^2: H = [1, 3, 5, 5, 3, 2, 1, 1], gcd(p, r) = 1: True; CI(f + 3 b^6, K_r, P) = 36 <= 40
  ok   Part V: 3 accepted samples
  ok   Part V anchor certified: 36 (N = 11) <= 40
Part S: bad configuration with P != [Y]: slide at P, turning elsewhere
  ok   [Y^7, Z^5], c_0 = y: H = [1, 3, 5, 5, 3, 2, 1, 1]; every p in J_4 vanishes at P and at [Y]: True; x phi, m^2 phi in I: True; m meets p only at P and phi(m) only at [Y]: True
  ok     tau = 1: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     tau = -2: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     CI(f + 3 b^6, K_m, P) = 37 (no slide; information, -1 = infinite); CI(f + 3 b^6, K_m + 2 phi(m), P) = 36 <= 40
  ok   [Y^7, Y^4*Z + Z^5], c_0 = y: H = [1, 3, 5, 5, 3, 2, 1, 1]; every p in J_4 vanishes at P and at [Y]: True; x phi, m^2 phi in I: True; m meets p only at P and phi(m) only at [Y]: True
  ok     tau = 1: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     tau = -2: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     CI(f + 3 b^6, K_m, P) = -1 (no slide; information, -1 = infinite); CI(f + 3 b^6, K_m + 2 phi(m), P) = 38 <= 40
  ok   [Y^7, Y^5 + 5*Y^4*Z + 10*Y^3*Z^2 + 10*Y^2*Z^3 + 5*Y*Z^4 + Z^5], c_0 = y - z: H = [1, 3, 5, 5, 3, 2, 1, 1]; every p in J_4 vanishes at P and at [Y]: True; x phi, m^2 phi in I: True; m meets p only at P and phi(m) only at [Y]: True
  ok     tau = 1: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     tau = -2: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     CI(f + 3 b^6, K_m, P) = 40 (no slide; information, -1 = infinite); CI(f + 3 b^6, K_m + 2 phi(m), P) = 40 <= 40
  ok   [Y^7, Z^5], c_0 = y: H = [1, 3, 5, 5, 3, 2, 1, 1]; every p in J_4 vanishes at P and at [Y]: True; x phi, m^2 phi in I: True; m meets p only at P and phi(m) only at [Y]: True
  ok     tau = 1: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     tau = -2: slid ideal colength 21, H = [1, 3, 5, 5, 3, 2, 1, 1], contains f, P, K + tau phi: True
  ok     CI(f + 3 b^6, K_m, P) = -1 (no slide; information, -1 = infinite); CI(f + 3 b^6, K_m + 2 phi(m), P) = 38 <= 40
  ok   Part S: 4 accepted samples
  ok   Part S anchor (first sample) certified by a local standard basis (Singular, ordering ds): 36 <= 40
ALL CHECKS PASSED
