N2: H = (1,3,5,6,4,2), W_2 = C y^2 (double zero at [Z]), g = y z^3 in J_4 (roots [Z] (mu 1), [Y] (mu 3))
  ok   graded member: closed True, H = [1, 3, 5, 6, 4, 2]
  ok   W_2 = C y^2: x y^2 in I and the cubic parts of I are x^2 S_1 + C x y^2 (dim I*_3 = 4)
  ok   g = y z^3 lies in I*_4 (graded member) and is binary: True
  ok   graded, t = 3, nu = 5: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 13) <= 40
  ok   graded, t = -2, nu = 5: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 13) <= 40
  ok   negative control (graded, t = 0, no turning): local length -1 > 41 or infinite
  (the bound 24 + 2*3 + 2*1 + 4*2 = 40)
  seed 1: theta_m = -1/6*y^4 - y^3*z + 3/2*y^2*z^2 + 1/12*y*z^3 - 8/9*z^4, theta_m([Z]) = -8/9 (bad point at [Z]: False)
  ok   seed 1: closed, H = [1, 3, 5, 6, 4, 2], gamma_f = 0
  ok   seed 1, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 32 (ds) = 32 (stable N = 9) <= 40
  seed 2: theta_m = 1/12*y^4 - 1/2*y^2*z^2 - 1/3*y*z^3 + 5/3*z^4, theta_m([Z]) = 5/3 (bad point at [Z]: False)
  ok   seed 2: closed, H = [1, 3, 5, 6, 4, 2], gamma_f = 0
  ok   seed 2, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 32 (ds) = 32 (stable N = 9) <= 40
  seed 3: theta_m = -1/4*y^4 + y^3*z - 1/2*y^2*z^2 + 22/3*y*z^3 + z^4, theta_m([Z]) = 1 (bad point at [Z]: False)
  ok   seed 3: closed, H = [1, 3, 5, 6, 4, 2], gamma_f = 0
  ok   seed 3, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 32 (ds) = 32 (stable N = 9) <= 40
  bad stratum among actual algebras: K_m + c P_4 with c chosen so that theta([Z]) = 0, where P_4 in I has order 4 and pi(P_4)([Z]) != 0
  ok   seed 4: theta_m([Z]) = 0 after the change (bad point)
  ok   seed 4 (bad), t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 36 (ds) = 36 (stable N = 11) <= 40
  ok   seed 5: theta_m([Z]) = 0 after the change (bad point)
  ok   seed 5 (bad), t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 34 (ds) = 34 (stable N = 10) <= 40
N3: H = (1,3,5,4,5,3), D_4 = C[Y,Z]_4 (J_4 = 0), g in J_5, m nonzero at the roots of g
  ok   D_5 = <Y^5, Y^4 Z, Y^3 Z^2>, g = z^5 (root [Y], mult 5), m = y^2: graded member closed, H = [1, 3, 5, 4, 5, 3], g in I*_5
  ok   graded, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok     seed 1: closed, H = [1, 3, 5, 4, 5, 3], gamma_f = 0
  ok     seed 1, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok     seed 2: closed, H = [1, 3, 5, 4, 5, 3], gamma_f = 0
  ok     seed 2, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok   D_5 = <Y^5, Z^5, Y^2 Z^3>, g = y z^4 (roots [Y], [Z]), m = y^2 + z^2: graded member closed, H = [1, 3, 5, 4, 5, 3], g in I*_5
  ok   graded, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok     seed 1: closed, H = [1, 3, 5, 4, 5, 3], gamma_f = 0
  ok     seed 1, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok     seed 2: closed, H = [1, 3, 5, 4, 5, 3], gamma_f = 0
  ok     seed 2, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok   D_5 = <Y^5, Z^5, (Y+Z)^5>, g = y z (y - z)(y + z)(y + 2z), m = y^2 + y z + 2 z^2: graded member closed, H = [1, 3, 5, 4, 5, 3], g in I*_5
  ok   graded, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok     seed 1: closed, H = [1, 3, 5, 4, 5, 3], gamma_f = 0
  ok     seed 1, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok     seed 2: closed, H = [1, 3, 5, 4, 5, 3], gamma_f = 0
  ok     seed 2, t = 3: u in (I:m^2); colength I_0 = 21, I_t = 21 (d = 21); triple in I_t: True; local length 40 (ds) = 40 (stable N = 10) <= 40
  ok   negative control: m = z^2 vanishes at the root [Y] of g = z^5 (theta = 0): local length -1 > 40
N4: H = (1,3,5,7,3,2), I*_3 = x^2 S_1, J_4 a pencil with gcd of degree delta; N = l^6 (k = 6): bound 32 + 4 delta
  ok   delta = 0: graded member closed, H = [1, 3, 5, 7, 3, 2], p = y^4, q = z^4 in I*_4
  ok   delta = 0: graded, (x^2, P, Q + 3 l^6): local length 32 = 32 (stable N = 8) <= 32
  ok     delta = 0 seed 1 (random tails, any gamma_f; gamma_f = -4/3*y^3 - 4*y^2*z - 11/52*y*z^2 + 4/3*z^3): local length 32 = 32 (stable N = 8) <= 32
  ok     delta = 0 seed 2 (random tails, any gamma_f; gamma_f = -4/3*y^3 - 17/12*y^2*z - 3*y*z^2 - 1/3*z^3): local length 32 = 32 (stable N = 8) <= 32
  ok   delta = 1: graded member closed, H = [1, 3, 5, 7, 3, 2], p = y^3*z, q = z^4 in I*_4
  ok   delta = 1: graded, (x^2, P, Q + 3 l^6): local length 36 = 36 (stable N = 10) <= 36
  ok     delta = 1 seed 1 (random tails, any gamma_f; gamma_f = -4/3*y^3 - 4*y^2*z + 1108/9*y*z^2 + 4/3*z^3): local length 35 = 35 (stable N = 10) <= 36
  ok     delta = 1 seed 2 (random tails, any gamma_f; gamma_f = y^3 - 73/2*y^2*z - 3*y*z^2 - 1/3*z^3): local length 34 = 34 (stable N = 9) <= 36
  ok   delta = 2: graded member closed, H = [1, 3, 5, 7, 3, 2], p = y^2*z^2, q = z^4 in I*_4
  ok   delta = 2: graded, (x^2, P, Q + 3 l^6): local length 40 = 40 (stable N = 10) <= 40
  ok     delta = 2 seed 1 (random tails, any gamma_f; gamma_f = -4/3*y^3 - 4*y^2*z - 177/2*y*z^2 + 4/3*z^3): local length 36 = 36 (stable N = 9) <= 40
  ok     delta = 2 seed 2 (random tails, any gamma_f; gamma_f = 1/3*y^3 + 13/14*y^2*z - 3*y*z^2 - 1/3*z^3): local length 38 = 38 (stable N = 10) <= 40
Part D3: delta = 3: J_4 = z^3 C[y,z]_1, D_4 = nu_4(Gamma), Gamma = V(x, z^3); h = z^3 kills D_5 and lies in (I : m^2)
  ok   graded member closed, H = [1, 3, 5, 7, 3, 2]
  ok   seed 1: closed, H = [1, 3, 5, 7, 3, 2]; z^3 o M in T_{<=1} (z^3 in (I : m^2), pi(z^3) = z^3 != 0)
  ok   seed 2: closed, H = [1, 3, 5, 7, 3, 2]; z^3 o M in T_{<=1} (z^3 in (I : m^2), pi(z^3) = z^3 != 0)
  ok   seed 3: closed, H = [1, 3, 5, 7, 3, 2]; z^3 o M in T_{<=1} (z^3 in (I : m^2), pi(z^3) = z^3 != 0)
accepted samples: 22
total time 942.7 s
ALL CHECKS PASSED
