(L5) Lemma lem:L5: parameter ideals of colength 6 inside the three monomial ideals
  ok   (u^2, v^3): supported at the origin, colength 6, contained in (u^2, u*v^2, v^3) (colength 5)
  ok   (u*v, u^3 + v^3): supported at the origin, colength 6, contained in (u*v, u^3, v^3) (colength 5)
  ok   (u*v, v^4 + u^2): supported at the origin, colength 6, contained in (u^2, u*v, v^4) (colength 5)
(R) Lemma lem:realize: dual tails on the data of prop:C, gamma = y^2 z, F = Y^5 + Y Z^4 + Z^5
  ok   <w_i, C_j> = delta_ij
  ok   p_i o F = 0 (the quartic parts annihilate D_5 = C F)
  ok   closed under differentiation, dim 20, H = (1,3,6,4,5,1)
  ok   f_1, f_2, f_3 lie in the ideal (orthogonal to M)
(S4) Theorem thm:S4 (a): the anchor of (1,3,6,5,5)
  ok   closed, H = (1,3,6,5,5)
  ok   x^3 + y^4 + z^4, x y^2 - z^4, x z^2 - y^4 lie in the ideal
  ok   local length 36 (stable at N = 9, 10); residual 16
  ok   the same ideal is generated by x(x^2 + y^2 + z^2), x y^2 - z^4, x z^2 - y^4
  ok   factor x: local length 16 (N = 7); factor x^2+y^2+z^2: local length 20 (N = 7); sum 36
  ok   the tangent cones x^2+y^2+z^2, x y^2, x z^2 meet in P^2 exactly in V(x, y^2 + z^2) = {[0:1:i], [0:1:-i]}
  ok   proper transforms: [u^2 + w^2 + 1, -w^4*s + u, u*w^2 - s]
     at (u,w,s) = (0,ii,0): Jacobian determinant 4*ii
     at (u,w,s) = (0,-ii,0): Jacobian determinant -4*ii
  ok   they meet transversally at (0, +-i, 0): local length 1 each; 16 + 2*3*3 + 1 + 1 = 36
(U1) Theorem thm:136322111: the member X^8 + X^4 Y^2 + X Y^4 of (U_1)
  ok   H = (1,3,6,3,2,2,1,1,1)
  ok   x^2 z, q, r lie in the ideal
  ok   local length of (x^2 z, q, r): 34 (stable at N = 11, 12)
  ok   l(R/x) = 9 (N = 5), l(R/z) = 16 (N = 9), 2*l(R/x) + l(R/z) = 34
  ok   R/x = C[[y,z]]/(y^3, z^3): 9
  ok   l(R/z) = i(y(y^2 - x^3/2), x^2(y^2 - x^4/840)) = i(y,x^2) + i(y, y^2 - x^4/840) + i(y^2 - x^3/2, x^2) + i(y^2 - x^3/2, y^2 - x^4/840) = 2 + 4 + 4 + 6
  ok   the difference of the last pair is x^3 (1/2 - x/840), and i(y^2 - x^3/2, x^3) = 6
(A14) Theorem thm:136631, concise top form: zero-tail anchors in (x^3, y^3, z^3)
  ok   F = X^2*Y^2*Z + X^2*Y*Z^2 + X*Y^2*Z^2, D_3 = S_2 o F + []: concise True, h = 6, closed True, H = [1, 3, 6, 6, 3, 1]
  ok      x^3, y^3, z^3 annihilate M: a complete intersection of length 27, residual 7
  ok   F = X^2*Y^2*Z, D_3 = S_2 o F + [X*Z^2]: concise True, h = 5, closed True, H = [1, 3, 6, 6, 3, 1]
  ok      x^3, y^3, z^3 annihilate M: a complete intersection of length 27, residual 7
(B12) Theorem thm:1355321, case (R3): an algebra outside the subcase of the operator u
  ok   M = N + <Y^4, X Y^3, R_4, y o Phi, X Y^4, Phi>: closed, dim 20, H = [1, 3, 5, 5, 3, 2, 1], x^2 in I
  ok   D_5 = <Y^5, X Y^4>: T_5 = X Y^4, G_5 = 0; Phi has no term with X, so A_4 = 0; z o (Y^2 Z^3/3 + Z^5/5) = R_4
  ok   (y^2 z) o Phi = 2*Z^2, (x y^2) o Phi = 0, y^4 o Phi = 360*Y^2: y^2 z + c' x y^2 + c y^4 sends Phi to 2 Z^2 + 360 c Y^2, never into T_{<=1}
  ok   p = z^3 - 6 y^2 z: p o R_4 = 0, p o Phi = 2*Y^2; u = p - y^4/180: u o M in T_{<=1}, u m^2 in I, binary cubic part -6*y^2*z + z^3
total time 3.2 s
ALL CHECKS PASSED
