Part S1: (O2) binary branch, W_2 = z<y,z>, gamma_f = k z^3
  ok   O2 [Y^7, Y^4 Z, Y^2Z^2 + Z^4] (seed 1): H ok; f = x^2 + 1 z^3 in I: True; gamma = z^3; D_7 = C Y^7: True; XY^2 in D_3: True; y^5 o M in T_{<=1} + C Y^2: True; Lemma F family (dim I/m^8 = 99, |g_w| = 2, dim ker eps = 92): colength 21 at t = 0, 3, -2, I_0 = I, f + t y^5 in I_t: True; split jets: z^3: 1, y^4: 0, y^3 z: 0, y^5: 0 -> E8 for t != 0
  ok   O2 [Y^7, Y^4 Z, Y^2Z^2 + Z^4] (seed 2): H ok; f = x^2 + 1 z^3 in I: True; gamma = z^3; D_7 = C Y^7: True; XY^2 in D_3: True; y^5 o M in T_{<=1} + C Y^2: True; Lemma F family (dim I/m^8 = 99, |g_w| = 2, dim ker eps = 92): colength 21 at t = 0, 3, -2, I_0 = I, f + t y^5 in I_t: True; split jets: z^3: 1, y^4: 0, y^3 z: 0, y^5: 0 -> E8 for t != 0
  ok   O2 [Y^7, Y^4 Z, Y^2Z^2 + Z^4]: 2 accepted samples
  ok   O2 [Y^7, Y^3 Z^2] (seed 1): H ok; f = x^2 + 2 z^3 in I: True; gamma = 2*z^3; D_7 = C Y^7: True; XY^2 in D_3: True; y^5 o M in T_{<=1} + C Y^2: True; Lemma F family (dim I/m^8 = 99, |g_w| = 2, dim ker eps = 92): colength 21 at t = 0, 3, -2, I_0 = I, f + t y^5 in I_t: True; split jets: z^3: 2, y^4: 0, y^3 z: 0, y^5: 0 -> E8 for t != 0
  ok   O2 [Y^7, Y^3 Z^2] (seed 2): H ok; f = x^2 + 2 z^3 in I: True; gamma = 2*z^3; D_7 = C Y^7: True; XY^2 in D_3: True; y^5 o M in T_{<=1} + C Y^2: True; Lemma F family (dim I/m^8 = 99, |g_w| = 2, dim ker eps = 92): colength 21 at t = 0, 3, -2, I_0 = I, f + t y^5 in I_t: True; split jets: z^3: 2, y^4: 0, y^3 z: 0, y^5: 0 -> E8 for t != 0
  ok   O2 [Y^7, Y^3 Z^2]: 2 accepted samples
  ok   O2 [Y^7 + Y^4 Z^2, Y^2 Z^2 + Z^4] (seed 1): H ok; f = x^2 + -1 z^3 in I: True; gamma = -z^3; D_7 = C Y^7: True; XY^2 in D_3: True; y^5 o M in T_{<=1} + C Y^2: True; Lemma F family (dim I/m^8 = 99, |g_w| = 2, dim ker eps = 92): colength 21 at t = 0, 3, -2, I_0 = I, f + t y^5 in I_t: True; split jets: z^3: -1, y^4: 0, y^3 z: 0, y^5: -2/35 -> E8 for t != 2/35
  ok   O2 [Y^7 + Y^4 Z^2, Y^2 Z^2 + Z^4]: 1 accepted samples
Part S2: (O3) cube branch, I*_3 = x^2 S_1 + C x z^2 + C z^3
  ok   O3 [Y^7, Y^4 Z, X Y^3 + Y^2 Z^2] (seed 1): H ok; f = x^2 + 1 z^3 in I: True; gamma = z^3; D_7 = C Y^7: True; XY^2 in D_3: True; y^5 o M in T_{<=1} + C Y^2: True; Lemma F family (dim I/m^8 = 99, |g_w| = 2, dim ker eps = 92): colength 21 at t = 0, 3, -2, I_0 = I, f + t y^5 in I_t: True; split jets: z^3: 1, y^4: 0, y^3 z: 0, y^5: 0 -> E8 for t != 0
  ok   O3 [Y^7, Y^4 Z, X Y^3 + Y^2 Z^2]: 1 accepted samples
  ok   O3 [Y^7, Y^4 Z, X Y^2 Z] (seed 1): H ok; f = x^2 + 1 z^3 in I: True; gamma = z^3; D_7 = C Y^7: True; XY^2 in D_3: True; y^5 o M in T_{<=1} + C Y^2: True; Lemma F family (dim I/m^8 = 99, |g_w| = 2, dim ker eps = 92): colength 21 at t = 0, 3, -2, I_0 = I, f + t y^5 in I_t: True; split jets: z^3: 1, y^4: 0, y^3 z: 0, y^5: 0 -> E8 for t != 0
  ok   O3 [Y^7, Y^4 Z, X Y^2 Z]: 1 accepted samples
Part S3: (O1), delta = 3, z^2 | g: Lemma F operators from the actual inverse system
  ok   O1 g = z^3, e != 0 (seed 1): gcd J_4 = z^3, gamma_f = 0; admissible u (R = Y^2): dim 40, dim of pi(u_3)-image 1; example pi(u_3) = z^3; Lemma F family for u + y^5: colength 21 at t = 0, 3, contains f + 3(u + y^5): True
  ok   O1 g = z^3, e != 0: 1 accepted samples
  ok   O1 g = z^3, e = 0 != d (seed 1): gcd J_4 = z^3, gamma_f = 0; admissible u (R = Y^2): dim 41, dim of pi(u_3)-image 1; example pi(u_3) = z^3; Lemma F family for u + y^5: colength 21 at t = 0, 3, contains f + 3(u + y^5): True
  ok   O1 g = z^3, e = 0 != d: 1 accepted samples
  ok   O1 g = z^3, B6 = 0 (seed 3): gcd J_4 = z^3, gamma_f = 0; admissible u (R = Y^2): dim 44, dim of pi(u_3)-image 3; example pi(u_3) = z^3; Lemma F family for u + y^5: colength 21 at t = 0, 3, contains f + 3(u + y^5): True
  ok   O1 g = z^3, B6 = 0: 1 accepted samples
  ok   O1 g = z^2 y, A, B != 0 (seed 1): gcd J_4 = y*z^2, gamma_f = 0; admissible u (R = Y^2): dim 40, dim of pi(u_3)-image 1; example pi(u_3) = y * z^2; Lemma F family for u + y^5: colength 21 at t = 0, 3, contains f + 3(u + y^5): True
  ok   O1 g = z^2 y, A, B != 0: 1 accepted samples
  ok   O1 g = z^2 y, A != 0 = B (seed 1): gcd J_4 = y*z^2, gamma_f = 0; admissible u (R = Y^2): dim 41, dim of pi(u_3)-image 1; example pi(u_3) = z^3; Lemma F family for u + y^5: colength 21 at t = 0, 3, contains f + 3(u + y^5): True
  ok   O1 g = z^2 y, A != 0 = B: 1 accepted samples
  ok   O1 g = z^2 y, A = 0 != B (seed 1): gcd J_4 = y*z^2, gamma_f = 0; admissible u (R = Y^2): dim 42, dim of pi(u_3)-image 2; example pi(u_3) = y * z^2; Lemma F family for u + y^5: colength 21 at t = 0, 3, contains f + 3(u + y^5): True
  ok   O1 g = z^2 y, A = 0 != B: 1 accepted samples
  ok   O1 g = z^2 y, B6 = 0 (seed 3): gcd J_4 = y*z^2, gamma_f = 0; admissible u (R = Y^2): dim 43, dim of pi(u_3)-image 2; example pi(u_3) = y * z^2; Lemma F family for u + y^5: colength 21 at t = 0, 3, contains f + 3(u + y^5): True
  ok   O1 g = z^2 y, B6 = 0: 1 accepted samples
  ok   Part S3: 7 accepted samples in total
time 15.8 s
ALL CHECKS PASSED
