=== Part G: the base schemes B_A, B_B and their spans (U)
  ok   B_A: length 4, dim nu_k = [4, 4, 4, 4] for k = 3..6, and nu_5, nu_6 are the prolongations of nu_4, nu_5
  ok   B_B: length 4, dim nu_k = [4, 4, 4, 4] for k = 3..6, and nu_5, nu_6 are the prolongations of nu_4, nu_5
  ok   nu_3(B_A) = <X^3, X^2Y, X^2Z, XZ^2>; nu_5(B_A) = [X^3*Z^2, X^4*Z, X^4*Y, X^5]
  ok   nu_3(B_B) = <X^3, X^2Y, X^2Z, Y^3>; nu_5(B_B) = [X^4*Z, Y^5, X^4*Y, X^5]
=== Part X: representative graded data, Hilbert functions and excess bounds (U on the representatives)
  ok   B_A  X^5 not in D_5                  F = X^4*Z^2      H = [1, 3, 6, 4, 4, 2, 1], k(p) = [5], 27 + sum (k-3) e_p = 37
  ok   B_A  F = X^6                         F = X^6          H = [1, 3, 6, 4, 4, 2, 1], k(p) = [6], 27 + sum (k-3) e_p = 42
  ok   B_A  F = X^5 Y (transverse)          F = X^5*Y        H = [1, 3, 6, 4, 4, 2, 1], k(p) = [6], 27 + sum (k-3) e_p = 42
  ok   B_A  F = X^5 Z (aligned)             F = X^5*Z        H = [1, 3, 6, 4, 4, 2, 1], k(p) = [6], 27 + sum (k-3) e_p = 42
  ok   B_B  X^5 not in D_5                  F = 7*Y^6        H = [1, 3, 6, 4, 4, 2, 1], k(p) = [5, 6], 27 + sum (k-3) e_p = 38
  ok   B_B  X^5 in, Y^5 not in D_5          F = X^5*Z        H = [1, 3, 6, 4, 4, 2, 1], k(p) = [6, 5], 27 + sum (k-3) e_p = 41
  ok   B_B  D_5 = <X^5,Y^5>, F = X^6        F = X^6          H = [1, 3, 6, 4, 4, 2, 1], k(p) = [6, 6], 27 + sum (k-3) e_p = 42
  ok   B_B  D_5 = <X^5,Y^5>, F = X^6+Y^6    F = X^6 + Y^6    H = [1, 3, 6, 4, 4, 2, 1], k(p) = [6, 6], 27 + sum (k-3) e_p = 42
  ok   B_B  D_5 = <X^5,Y^5>, F = Y^6        F = Y^6          H = [1, 3, 6, 4, 4, 2, 1], k(p) = [6, 6], 27 + sum (k-3) e_p = 42
=== Part D3: rank condition of [15, Lemma D.3] at [X] (U)
  ok   D_A: dim K = 7, rank of x o K modulo D_3 = 3
  ok   D_B: dim K = 7, rank of x o K modulo D_3 = 3
=== Part D4: the transverse top X^5 Y on B_A (U)
  ok   pencil (xy^2+yz^2+z^3, xyz+y^2z) in I*_3 without y^3, local length 5 = e_X at [X]; R = Y^4 has x o R = z o R = 0
=== Part F: families with anchors (U + A)
  ok   D_4 = nu_4(B) is the monomial space <X^4,X^3Y,X^3Z,X^2Z^2> (B_A), <X^4,X^3Y,X^3Z,Y^4> (B_B)
  ok   B_A aligned F = X^5Z: 35 closedness conditions in 48 tail coefficients, solved by elimination with constant pivots; 30 free: one family over an affine space
  ok   B_A aligned F = X^5Z anchor (A): generators [X^5*Z + X^2*Y*Z^2, X^3*Y, X^2*Z^2]; the triple [-2*x^4 - 2*x^3*z + 4*x*y^2 + y^3 + 60*x*y*z + 4*y^2*z + 60*y*z^2 + 4*z^3, -2*x^4 - 2*x^3*z + x*y^2 + 3*y^3 + 60*x*y*z + 4*y^2*z + 60*y*z^2 + 2*z^3, -2*x^4 - 2*x^3*z + x*y^2 + y^3 + 60*x*y*z + 4*y^2*z + 60*y*z^2 + z^3] of I has local length 33 (local standard basis); stabilization (33, 9); e(I) <= 33
  ok   B_B secant F = X^6+Y^6: 41 closedness conditions in 48 tail coefficients, solved by elimination with constant pivots; 30 free: one family over an affine space
  ok   B_B secant F = X^6+Y^6 anchor (A): generators [X^6 + Y^6 + X^3*Y*Z, X^3*Y, X^3*Z]; the triple [3*x^6 - 3*y^6 - 4*x^4 + 2*x*y^2 + 480*x*y*z + y^2*z + 4*x*z^2 + 4*y*z^2 + z^3, x^6 - y^6 - 3*x^4 + 3*x*y^2 + 360*x*y*z + 4*y^2*z + 4*x*z^2 + y*z^2 + 4*z^3, 2*x^6 - 2*y^6 - x^4 + 4*x*y^2 + 120*x*y*z + 4*y^2*z + 2*x*z^2 + 2*y*z^2 + 2*z^3] of I has local length 34 (local standard basis); stabilization (34, 10); e(I) <= 34
  ok   B_B secant F = Y^6: elimination with constant pivots leaves exactly the three equations (2/5) p G_i = 4i (q H1_i + p H2_i), p = coefficient of X^4Z, q = of X^4Y in the quintic tail of the lift of Y^6 (none of these variables was eliminated)
   components: closure of {p != 0} (a graph over an open set) and {p = q = 0} (linear); {p = 0, H1 = 0} lies in the closure of the first
  ok   B_B secant F = Y^6, component p = q = 0 anchor (A): generators [Y^6, X^5 + X^2*Y*Z, X^3*Y, X^3*Z]; the triple [2*y^7 + x^6 - 2*x^4 + 3*x*y^2 + 120*x*y*z + 3*y^2*z + x*z^2 + 3*y*z^2 + z^3, 4*y^7 + 3*x^6 - 4*x^4 + x*y^2 + 240*x*y*z + 2*y^2*z + 2*x*z^2 + 2*y*z^2 + 3*z^3, y^7 + 2*x^6 - 4*x^4 + x*y^2 + 240*x*y*z + 3*y^2*z + 3*x*z^2 + 2*y*z^2 + 3*z^3] of I has local length 35 (local standard basis); stabilization (35, 11); e(I) <= 35
  ok   the chosen point (p = 1) satisfies the residual equations
  ok   B_B secant F = Y^6, component closure{p != 0} anchor (A): generators [Y^6 - X*Y^4 + X^4*Z + 2*Y^4*Z + 1/5*X^2*Y^2 + 2*X*Y^3 + 2*X^2*Y*Z + 36/5*X*Y^2*Z - 4*X^2*Z^2 - 4*X*Y*Z^2 + 4/5*Y^2*Z^2 - 8*X*Z^3 - X*Y*Z - Y^2*Z + 2*X*Z^2 - Y*Z^2 + Z^3, X^5 + 20*X^2*Y^2 - 20*X^2*Y*Z - 60*X^2*Z^2 + 2*X*Y^2 - 2*X*Y*Z + Y^2*Z - X*Z^2 - Y*Z^2, X^3*Y + Y^2*Z + Y*Z^2 - Z^3, X^3*Z + X*Y*Z + 2*Y^2*Z - 2*X*Z^2 - Y*Z^2 - 2*Z^3]; the triple [18*y^6 + 160*x^5 - 157*y^5 + 3080*x^4 - 1200*x^3*y - 94*y^4 + 240*x^3*z + 60*x*y^2 + 7200*x*y*z + 1440*y^2*z + 360*x*z^2 + 3600*y*z^2 + 480*z^3, 24*y^6 + 253*x^5 - 295*y^5 + 5120*x^4 - 600*x^3*y - 70*y^4 - 1680*x^3*z + 60*x*y^2 + 14400*x*y*z + 1080*y^2*z + 180*x*z^2 + 1800*y*z^2 + 360*z^3, 34*y^6 + 317*x^5 - 308*y^5 + 5800*x^4 - 2040*x^3*y - 68*y^4 - 720*x^3*z + 120*x*y^2 + 14400*x*y*z + 1080*y^2*z + 540*x*z^2 + 5400*y*z^2 + 120*z^3] of I has local length 33 (local standard basis); stabilization (33, 9); e(I) <= 33
=== Part T: the transverse family (S)
  ok   B_A transverse F = X^5Y: 35 conditions, elimination with constant pivots solves all (29 free)
  ok     (S) random transverse member: H ok, e(I) <= 33 <= 37
  ok     (S) random transverse member: H ok, e(I) <= 34 <= 37
  ok     (S) random transverse member: H ok, e(I) <= 33 <= 37
=== Part D4s: the deformation M_lambda of the Cor D.4 rerun on the graded transverse member (S, discriminating)
  ok   lambda = 0: H ok, e(I_lambda) = 42 (graded member 42; the bound of the deformation is 37)
  ok   lambda = 1: H ok, e(I_lambda) = 37 (graded member 42; the bound of the deformation is 37)
  ok   lambda = 3: H ok, e(I_lambda) = 37 (graded member 42; the bound of the deformation is 37)
time 6.9 s
ALL CHECKS PASSED
