(1) general C_5: the remainders of degree >= 4 say exactly z o C_5 in <X^4, X^3 Y>
  ok   the 19-vector reduction frame (7 lifts, T_{<=2}, E_0) has distinct pivots with coefficient 1; adjoining G not in E_0 gives dimension 20
  ok   ideal of the degree >= 4 remainders = ideal of the coefficients of z o C_5 outside <X^4, X^3 Y> (13 conditions)
(2) C_5 = c_1 X^3 Y Z + c_2 X^3 Y^2 + c_3 X^2 Y^3 + c_4 X Y^4 + c_5 Y^5: the remainders
     nonzero remainders: [('Phi', 'z'), ('y o Phi/(2 delta)', 'y')]
     rho_2 = (-3/(4*d)*c1*c3 + h_XY2Z)*X*Y^2 + (-1/(2*d)*c1*c4 + h_Y3Z)*Y^3 + (-3/(8*d)*c1^2 + 2*h_X2Z2)*X^2*Z + 2*h_XYZ2*X*Y*Z + 2*h_Y2Z2*Y^2*Z + 3*h_XZ3*X*Z^2 + 3*h_YZ3*Y*Z^2 + 4*h_Z4*Z^3
  ok   exactly two nonzero remainders: (Phi, z) and (y o Phi/(2 delta), y)
  ok   the remainder of y o (y o Phi/(2 delta)) is rho_1 = (6 c_4 X Y^2 + 10 c_5 Y^3)/delta
  ok   the remainder of z o Phi is rho_2 = z o H_4 - c_1 (xy o C_5)/(8 delta) modulo E_0 (a cubic)
  ok   J_3 occurs in no remainder
     explicitly: -c_1 (xy o C_5)/(8 delta) = (-3/(4*d)*c1*c2)*X^2*Y + (-3/(4*d)*c1*c3)*X*Y^2 + (-1/(2*d)*c1*c4)*Y^3 + (-3/(8*d)*c1^2)*X^2*Z
(3) dimension of the admissible H_4
  ok   z o : T_4 -> T_3 has rank 10 (onto), kernel of dimension 5 = 5; admissible H_4: affine space of dimension 15 - 7 = 8
(4) x^3 o Phi/336 = X^5 + (delta/14) X Y^2 modulo T_{<=2}
  ok   x^3 o Phi/336 - X^5 - (delta/14) X Y^2 has degree <= 2
(5) family (U_1): I*_3 and the cubic base scheme
  ok   z S_2 + C y^2 (5 c_5 x - c_4 y) kills D_3 (symbolic c_4, c_5)
  ok   these seven cubics are independent (dim I*_3 = 10 - 3 = 7)
     c_4, c_5 = 2, 3: V(z, y^2 (5 c_5 x - c_4 y)) has components [[z, y^2], [z, 15*x - 2*y]]
     c_4, c_5 = 1, 0: V(z, y^2 (5 c_5 x - c_4 y)) has components [[z, y^3]]
     c_4, c_5 = 0, 1: V(z, y^2 (5 c_5 x - c_4 y)) has components [[z, y^2], [z, x]]
  ok   for c_5 != 0 the zero set is Gamma_0 = V(z, y^2) together with [c_4 : 5 c_5 : 0] (checked symbolically: y^2 (5 c_5 x - c_4 y) vanishes there)
(6) the same conditions in other names: C_5 = b X^3 Y Z + c X^3 Y^2 + d X^2 Y^3 + e X Y^4 + f Y^5
  ok   w = z o H_4 - (b/(8 delta)) (3 b X^2 Z + 6 c X^2 Y + 6 d X Y^2 + 4 e Y^3) mod E_0 equals rho_2; v = (6 e X Y^2 + 10 f Y^3)/delta equals rho_1
(7) the member Phi = X^8 + X^4 Y^2 + X Y^4, D_3 = <X^3, X^2 Y, X Y^2> (family (U_1), c_5 = 0) and a complete intersection in its ideal
  ok   closure of Phi, D_3 and T_{<=2}: dimension 20, H = [1, 3, 6, 3, 2, 2, 1, 1, 1]
     triple: [x^2*z, -1/2*x^3*y + y^3, -1/840*x^6 + x^2*y^2 + z^3]
  ok   the triple annihilates M (every element of the triple kills every basis element of M)
     dim S/(J + m^N), N = 8..15: [31, 32, 33, 34, 34, 34, 34, 34]
  ok   local length 34 (local standard basis) = dim S/(J + m^N) for the stable pair N = 11, 12
total time 2.3 s
ALL CHECKS PASSED
