Part M1: gcd(P) = z^2, delta = 0: Phi = Y^7 + alpha X Y^4 + B5 + Q4 + Q3 (all symbolic)
  ok   v o Phi: degree 3 part = 0 (p4 kills Y^7)
  ok   v o Phi: degree 2 part = 2520 [y^5]v5 Y^2
  ok   v o Phi: degree 1 part = 5040 [y^6]v6 Y + p4 o B5 (the X Y^4 term is killed, p4(Y) = 0)
  ok   u o Phi (u = x r2 + u4 + u5): degree 3 part = 840 [y^4]u4 Y^3
  ok   B5 in V = <Y^5, Y^4 Z>: y z^2 o Phi in T_{<=1}
  ok   y^5 z o Phi = 0
  ok   [y^4]u4 = 0 gives A(t,0) = O(t^2) on the strict transform of u
  ok   [y^5]v5 = 0, p4 in z^2 beta_2: V(t,0) = [y^6]v6 t^2 + O(t^3)
  ( 0.5 s)
Part M2-secant: V = <Y^5, Z^5>, B6 = sigma Z^6, P5 = X(A Y^4 + B Z^4) + sum d_i Y^(5-i) Z^i, Q4, Q3 symbolic
   high part of U0 o Phi: (12*p1*d3 + 12*p2*d2)*Y*Z
  ok   secant identity: high part of U0 o Phi = 12 (d3 p1 + d2 p2) Y Z
  ok   cubic part y z (p1 z + p2 y) kills V = <Y^5, Z^5> and Y^7
   YZ-coefficient of r2 o A4: 6*a1*r0 + 4*a2*r1 + 6*a3*r2 ; matrix [(0, 6, 0, 0, 0), (0, 0, 4, 0, 0), (0, 0, 0, 6, 0)]
  ok   the YZ-coefficient of r2 o A4 vanishes for all r2 iff a1 = a2 = a3 = 0 (A4 in <Y^4, Z^4>)
  ok   quartic monomials m != y^4: m o (sigma Z^6) in C Z^2 (so u4 o B6 in <Z^2> when [y^4]u4 = 0)
  ( 0.7 s)
Part M1-E: actual algebras with gcd(P) = z^2, P6 = 0 (delta = 0), B5 not in V; exact local lengths
  ok   tight example: Phi = Y^7 + B5, Y^4 Z, X Y, X Z; P'' = <z^2 - 20y^2, yz>: H = [1, 3, 5, 4, 3, 2, 1, 1], I*_2 = <x^2>, I*_3 = x S_2, gcd P = z^2, gamma_f = 0, P6 = 0, p4, p' in P, lambda(p4) = -684*Y
  ok      y^5 z in I; [y^4]u4 = 0, [y^5]v5 = 0, [y^6]v6 = -lambda/5040
  ok      form A (f + nu y^5 z, K + mu Q, v): (sigma, mu, rho, nu) = (0, 2, 3, 5): local length 38 (stable at N = 13), 20 <= L <= 38
  ok      form A: (sigma, mu, rho, nu) = (0, 1, -2, 7): local length 38 (stable at N = 13), 20 <= L <= 38
  ok      radial form (f + sigma Q + nu y^5 z, K, v): (sigma, mu, rho, nu) = (2, 0, 3, 5): local length 36 (stable at N = 11), 20 <= L <= 38
   [info] without y^5 z: (sigma, mu, rho, nu) = (0, 2, 3, 0): local length 40 (stable at N = 13)
  ok   random tails: Phi = Y^7 + 2XY^4 + B5 + ..., Y^4 Z + ...: H = [1, 3, 5, 4, 3, 2, 1, 1], I*_2 = <x^2>, I*_3 = x S_2, gcd P = z^2, gamma_f = 0, P6 = 0, p4, p' in P, lambda(p4) = -684*Y
  ok      y^5 z in I; [y^4]u4 = 0, [y^5]v5 = 0, [y^6]v6 = -lambda/5040
  ok      form A: (sigma, mu, rho, nu) = (0, 2, 3, 5): local length 36 (local standard basis, ring ds), 20 <= L <= 38
  ok      radial form: (sigma, mu, rho, nu) = (2, 0, 3, 5): local length 36 (local standard basis, ring ds), 20 <= L <= 38
   [info] without y^5 z: (sigma, mu, rho, nu) = (0, 2, 3, 0): local length 36 (local standard basis, ring ds)
  ok   second orbit: B5 = Y Z^4 + Y^3 Z^2 (P'' = <y^2, z^2>): H = [1, 3, 5, 4, 3, 2, 1, 1], I*_2 = <x^2>, I*_3 = x S_2, gcd P = z^2, gamma_f = 0, P6 = 0, p4, p' in P, lambda(p4) = -48*Y
  ok      y^5 z in I; [y^4]u4 = 0, [y^5]v5 = 0, [y^6]v6 = -lambda/5040
  ok      form A: (sigma, mu, rho, nu) = (0, 2, 3, 5): local length 36 (local standard basis, ring ds), 20 <= L <= 38
  ok      radial form: (sigma, mu, rho, nu) = (2, 0, 3, 5): local length 36 (local standard basis, ring ds), 20 <= L <= 38
   [info] without y^5 z: (sigma, mu, rho, nu) = (0, 2, 3, 0): local length 36 (local standard basis, ring ds)
Part M2-E: actual algebras with deg gcd(P) = 3; the element U0 of the proof lies in (I : m^2)
  ok   h = y z^2, W full                A4 = 0; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [2/3*Y^4*Z + Z^5, Y^5]: True;  case A: U0 in (I:m^2): True, cubic part y * z^2
  ok   h = y z^2, W full      (sheared) A4 = 418*Y^4 + 24*Y^3*Z - 15*Z^4; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [2/3*Y^4*Z + Z^5, Y^5]: True;  case A: U0 in (I:m^2): True, cubic part y * z^2
  ok   h = y z^2, secant                A4 = 0; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [Z^5, Y^5]: True;  case B-secant: U0 in (I:m^2): True, cubic part (-1) * z * y * (2*y + z)
  ok   h = y z^2, secant      (sheared) A4 = 420*Y^4 - 60*Z^4; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [Z^5, Y^5]: True;  case B-secant: U0 in (I:m^2): True, cubic part (-1) * z * y * (2*y + z)
  ok   h = y z^2, tangent               A4 = 0; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [Y^4*Z, Y^5]: True;  case B-tangent: U0 in (I:m^2): True, cubic part (12/35) * (175*y - 76*z) * z^2
  ok   h = y z^2, tangent     (sheared) A4 = 414*Y^4 + 72*Y^3*Z; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [Y^4*Z, Y^5]: True;  case B-tangent: U0 in (I:m^2): True, cubic part (12/35) * (175*y - 76*z) * z^2
  ok   h = y z^2, G = 0 (C)             A4 = 0; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [2/3*Y^4*Z + Z^5, Y^5]: True;  case C: U0 in (I:m^2): True, cubic part y * z^2
  ok   h = y z^2, G = 0 (C)   (sheared) A4 = 420*Y^4; H, tangent cone, gcd P = y*z^2, gamma_f = 0, V = [2/3*Y^4*Z + Z^5, Y^5]: True;  case C: U0 in (I:m^2): True, cubic part y * z^2
  ok   h = z^3,   W full                A4 = 0; H, tangent cone, gcd P = z^3, gamma_f = 0, V = [2/3*Y^4*Z + Y^3*Z^2, Y^5]: True;  case A: U0 in (I:m^2): True, cubic part z^3
  ok      cube case: an element x y^2 + ... of I exists
  ok   h = z^3,   W full      (sheared) A4 = 418*Y^4 + 18*Y^3*Z + 27*Y^2*Z^2; H, tangent cone, gcd P = z^3, gamma_f = 0, V = [2/3*Y^4*Z + Y^3*Z^2, Y^5]: True;  case A: U0 in (I:m^2): True, cubic part z^3
  ok      cube case: an element x y^2 + ... of I exists
  ok   h = z^3,   tangent               A4 = 0; H, tangent cone, gcd P = z^3, gamma_f = 0, V = [Y^4*Z, Y^5]: True;  case B-tangent: U0 in (I:m^2): True, cubic part (24/35) * (175*y - 38*z) * z^2
  ok   h = z^3,   tangent     (sheared) A4 = 414*Y^4 + 72*Y^3*Z; H, tangent cone, gcd P = z^3, gamma_f = 0, V = [Y^4*Z, Y^5]: True;  case B-tangent: U0 in (I:m^2): True, cubic part (24/35) * (175*y - 38*z) * z^2
  ok   h = z^3,   G = 0 (C)             A4 = 0; H, tangent cone, gcd P = z^3, gamma_f = 0, V = [Y^4*Z + Y^3*Z^2, Y^5]: True;  case C: U0 in (I:m^2): True, cubic part z^3
  ok      cube case: an element x y^2 + ... of I exists
  ok   h = z^3,   G = 0 (C)   (sheared) A4 = 420*Y^4; H, tangent cone, gcd P = z^3, gamma_f = 0, V = [Y^4*Z + Y^3*Z^2, Y^5]: True;  case C: U0 in (I:m^2): True, cubic part z^3
  ok      cube case: an element x y^2 + ... of I exists
  ok   h = y^2 z, secant                A4 = 0; H, tangent cone, gcd P = y^2*z, gamma_f = 0, V = [Z^5, Y^5]: True;  case B-secant: U0 in (I:m^2): True, cubic part (-1) * z * (y - 3*z) * y
  ok   h = y^2 z, secant      (sheared) A4 = 420*Y^4 - 60*Z^4; H, tangent cone, gcd P = y^2*z, gamma_f = 0, V = [Z^5, Y^5]: True;  case B-secant: U0 in (I:m^2): True, cubic part (-1) * z * (y - 3*z) * y
  ok   h = y^2 z, G = 0 (C)             A4 = 0; H, tangent cone, gcd P = y^2*z, gamma_f = 0, V = [Z^5, Y^5]: True;  case C: U0 in (I:m^2): True, cubic part z * y^2
  ok   h = y^2 z, G = 0 (C)   (sheared) A4 = 420*Y^4; H, tangent cone, gcd P = y^2*z, gamma_f = 0, V = [Z^5, Y^5]: True;  case C: U0 in (I:m^2): True, cubic part z * y^2
total time 32.7 s
ALL CHECKS PASSED
