(U5) the reduction for free algebras (symbolic tails)
  [ok] (U5) A_Y = al1 X^4 + 2 al2 X^3Y, A_Z = be X^4 (a3 = b3 = 0, free)
  [ok] (U5) pi(h1) = pi(B'_Y) - (al2/2) pi(P'_X), pi(h2) = pi(B'_Z), pi(h3) = pi(P'_Y), pi(h4) = pi(P'_Z) identically
(U6) the incidence I: (g, lam, al2) -> tails, and back
  [ok] (U6) for (g, lam, al2) in I the tails P', B' built from (g, lam) give u_1..u_4 = lam_1 g, .., lam_4 g (differences lie in the ideal of I)
  [ok] (U6) conversely the components of u_i - lam_i g generate the ideal of I (so every free algebra with rank exactly 1 arises)
(U7) minimal primes of the incidence
   prime: dim 7, contains lam = 0: True, contains g = 0: False, gens [l4, l3, l2, l1]
   prime: dim 6, contains lam = 0: False, contains g = 0: False, gens [l4, l2, f, e, d]
   prime: dim 6, contains lam = 0: False, contains g = 0: False, gens [-3*f*l1 + e*l2, -e*l1 + c*l2, -e^2 + 3*c*f, -2*d*l1 + b*l2, -2*d*e + 3*b*f, -2*c*d + b*e, l4, l3]
   prime: dim 6, contains lam = 0: False, contains g = 0: False, gens [-3*f*l3 + e*l4, d*l4*al + 6*f*l1 - 2*e*l2, 3*d*f*l3*al + 6*e*f*l1 - 2*e^2*l2, -e*l3 + c*l4, -d*l3*al - 2*e*l1 + 2*c*l2, -e^2 + 3*c*f, -2*d*l3 + b*l4, -2*d*e + 3*b*f, -2*c*d + b*e, a*l4*al + 2*d*l1 - b*l2, -d^2 + 3*a*f, -b*d + 2*a*e, 2*a*d*l3*al + 2*b*d*l1 - b^2*l2, -b^2 + 4*a*c]
   prime: dim 6, contains lam = 0: False, contains g = 0: False, gens [-l2*l3 + l1*l4, -3*f*l3 + e*l4, -3*f*l1 + e*l2, -e*l3 + c*l4, -e*l1 + c*l2, -e^2 + 3*c*f, -2*d*l3 + b*l4, -2*d*l1 + b*l2, -2*d*e + 3*b*f, -2*c*d + b*e, al]
  [ok] (U7) I has 5 minimal primes: lam = 0 and 4 components of dimension [6, 6, 6, 6], none containing g = 0 or lam = 0
  [ok] (U7) cross-check: each of the five ideals is prime and contains I, and their intersection equals rad(I)
(U7b) geometric irreducibility: each component is the closure of the image of a rational map from an affine 6-space over Q
  [ok] (U7b) K_1: the parametrization lies in component(s) [3] (one of dimension 6) and has Jacobian rank 6
  [ok] (U7b) K_2: the parametrization lies in component(s) [1] (one of dimension 6) and has Jacobian rank 6
  [ok] (U7b) K_3: the parametrization lies in component(s) [2] (one of dimension 6) and has Jacobian rank 6
  [ok] (U7b) K_D is the linear space {d = e = f = lam2 = lam4 = 0}
(A8) certificates: one actual free member per component
  [ok] (A8) K_1 (point in component(s) [3]), R = XY^3: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (-4, -30, -27, 5, 9, -1, 3, -1, -12, 4, 0)
      D_3 = [4*X^2*Z + 30*X*Y*Z + 27*Y^2*Z - 5*X*Z^2 - 9*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-x^6 - 6*x^5*y - 7*x^4*y^2 - 2*x^3*y^3 + 3*x^2*y^4 - x*y^5 - y^6 - 7*x^5*z - 2*x^4*y*z + 7*x^3*y^2*z - 3*x^2*y^3*z - x*y^4*z + y^5*z - 3*x^4*z^2 + 7*x^3*y*z^2 + 2*x^2*y^2*z^2 + 2*x*y^3*z^2 - 6*y^4*z^2 + 3*x^3*z^3 + x^2*y*z^3 + 3*x*y^2*z^3 + 3*y^3*z^3 - 6*x^2*z^4 + 4*x*y*z^4 - 3*y^2*z^4 - 7*x*z^5 - 5*y*z^5 - 3*z^6 - 4*x^3*y^2 + 2*x^2*y^3 + 6*x*y^4 - y^5 + 7*x^4*z + 2*x^3*y*z - x^2*y^2*z - 7*x*y^3*z - 5*y^4*z - 6*x^3*z^2 - 3*x^2*y*z^2 - 7*x*y^2*z^2 - 3*y^3*z^2 + 5*x^2*z^3 + 7*x*y*z^3 + 5*y^2*z^3 + 2*x*z^4 - 7*y*z^4 - 5*z^5 - 2*x^2*y^2 - 4*y^4 + 5*x^3*z + 7*x^2*y*z + 3*x*y^2*z + 7*y^3*z - 6*x^2*z^2 - 6*x*y*z^2 - 3*y^2*z^2 - 6*x*z^3 + 4*y*z^3 + 2*z^4 + 95/4*x^2*z - 5*x*y*z - y^2*z - 5*x*z^2 + 4*y*z^2 + 6*z^3, 2*x^6 + x^5*y + 7*x^4*y^2 - x^3*y^3 + 7*x*y^5 + 4*x^5*z + 2*x^4*y*z + x^3*y^2*z - 3*x^2*y^3*z - 6*x*y^4*z + y^5*z + 7*x^4*z^2 - 5*x^3*y*z^2 + 6*x^2*y^2*z^2 + 6*x*y^3*z^2 - 7*x^3*z^3 + 6*x^2*y*z^3 - 4*x*y^2*z^3 - 2*y^3*z^3 + 6*x^2*z^4 + 6*x*y*z^4 + 6*y^2*z^4 - 6*x*z^5 - 5*y*z^5 - x^3*y^2 + 5*x^2*y^3 - 3*x*y^4 - 7*y^5 - x^4*z + 5*x^2*y^2*z - 6*x*y^3*z + 6*y^4*z - 6*x^3*z^2 - 6*x^2*y*z^2 + 3*x*y^2*z^2 + y^3*z^2 + x^2*z^3 - 3*x*y*z^3 - 6*y^2*z^3 - 5*x*z^4 - 4*y*z^4 + z^5 + 5*y^4 + 2*x^3*z + 4*x^2*y*z - x*y^2*z + 4*x^2*z^2 - 6*x*y*z^2 - 6*y^2*z^2 - x*z^3 + 3*y*z^3 - 5*z^4 - 297/4*x^2*z + 7*x*y*z + 6*y^2*z - 4*y*z^2 - 2*z^3, -4*x^6 - 3*x^5*y + 7*x^4*y^2 + 3*x^3*y^3 - 6*x^2*y^4 + 4*x*y^5 - 7*y^6 + 6*x^5*z + 7*x^4*y*z + 6*x^3*y^2*z + 4*x^2*y^3*z - 6*x*y^4*z + 4*y^5*z - 3*x^4*z^2 + x^2*y^2*z^2 - x*y^3*z^2 + 7*y^4*z^2 - x^3*z^3 - 5*x^2*y*z^3 + 3*x*y^2*z^3 - 7*y^3*z^3 + x^2*z^4 + 3*x*y*z^4 + 4*x*z^5 - 3*y*z^5 + 7*z^6 - 4*x^3*y^2 + 2*x^2*y^3 + x*y^4 + y^5 + 5*x^4*z - 2*x^3*y*z + x^2*y^2*z - 4*x*y^3*z + y^4*z - 2*x^3*z^2 + x*y^2*z^2 + 4*y^3*z^2 - 7*x^2*z^3 + x*y*z^3 + 2*y^2*z^3 + 3*x*z^4 - 6*y*z^4 - 6*z^5 + 5*x^2*y^2 - 2*y^4 - 6*x^2*y*z - 4*x*y^2*z - 6*y^3*z - 4*x^2*z^2 + 4*x*y*z^2 + 6*y^2*z^2 + 3*x*z^3 - 7*y*z^3 + 2*z^4 + 47/4*x^2*z - 6*x*y*z + 3*y^2*z + 4*x*z^2 - 6*z^3]
  [ok] (A8) K_1 (point in component(s) [3]), R = X^2Y^2 + Y^4: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (2, 4, -3, -2, 3, -1, -4, 4, 1, -1, 0)
      D_3 = [-2*X^2*Z - 4*X*Y*Z + 3*Y^2*Z + 2*X*Z^2 - 3*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-2*x^6 + 2*x^5*y + 4*x^4*y^2 + 3*x^3*y^3 - 2*x^2*y^4 + 3*x*y^5 + 5*y^6 - 7*x^5*z - x^4*y*z - 4*x^3*y^2*z + 2*x^2*y^3*z + 3*x*y^4*z + 3*y^5*z - 2*x^4*z^2 - 5*x^3*y*z^2 + 2*x^2*y^2*z^2 + 3*x*y^3*z^2 + 2*y^4*z^2 + 5*x^2*y*z^3 + 5*x*y^2*z^3 + 4*y^3*z^3 + 6*x^2*z^4 - 7*x*y*z^4 + 6*y^2*z^4 + 4*x*z^5 + 5*y*z^5 - z^6 + 2*x^3*y^2 - x^2*y^3 - x*y^4 - 7*x^4*z - 2*x^3*y*z + x^2*y^2*z - 7*x*y^3*z + 2*y^4*z + 6*x^3*z^2 - x^2*y*z^2 - 6*x*y^2*z^2 - 6*y^3*z^2 + x^2*z^3 - 3*x*y*z^3 - 5*y^2*z^3 - 7*x*z^4 + 6*y*z^4 - 5*z^5 - 6*x^2*y^2 - x*y^3 + y^4 - 3*x^3*z + 2*x^2*y*z - 6*x*y^2*z + 7*y^3*z - x^2*z^2 + 6*x*y*z^2 + 2*y^2*z^2 - 2*x*z^3 - 7*z^4 + 14*x^2*z - 7*x*y*z + 3*y^2*z + 4*x*z^2 + 6*y*z^2 + 5*z^3, -3*x^6 - 3*x^4*y^2 - 4*x^3*y^3 - x^2*y^4 + 4*x*y^5 - 7*y^6 + 4*x^5*z + 5*x^4*y*z - 6*x^3*y^2*z - 2*x^2*y^3*z - 6*x*y^4*z - 2*y^5*z + x^4*z^2 + 7*x^3*y*z^2 + 7*x^2*y^2*z^2 + x*y^3*z^2 - y^4*z^2 + 6*x^3*z^3 - 3*x^2*y*z^3 - 7*x*y^2*z^3 + 4*y^3*z^3 + 7*x^2*z^4 + 3*x*y*z^4 - y^2*z^4 + 3*x*z^5 + 3*y*z^5 - 6*z^6 + 2*x^3*y^2 - x^2*y^3 - 4*x*y^4 - 7*y^5 + 2*x^4*z - 3*x^3*y*z - x^2*y^2*z + 2*x*y^3*z - 3*y^4*z + 3*x^3*z^2 + 5*x^2*y*z^2 + 5*x*y^2*z^2 - y^3*z^2 + 4*x^2*z^3 + 4*x*y*z^3 - 7*y^2*z^3 + 6*x*z^4 + 7*y*z^4 - 5*z^5 + 18*x^2*y^2 + 5*x*y^3 - 3*y^4 - 7*x^3*z + 5*x^2*y*z - 2*x*y^2*z - 3*y^3*z + 5*x^2*z^2 + 2*x*y*z^2 + 6*y^2*z^2 + 5*x*z^3 + 7*y*z^3 - z^4 - 39/2*x^2*z + 4*x*y*z - 5*y^2*z - 2*x*z^2 + y*z^2 - 3*z^3, -3*x^6 + 5*x^5*y + x^3*y^3 - 4*x^2*y^4 + 7*x*y^5 + 2*y^6 + 3*x^5*z + 7*x^3*y^2*z - x^2*y^3*z - 4*x*y^4*z - 2*y^5*z - 7*x^4*z^2 - x^3*y*z^2 + 3*x^2*y^2*z^2 - x*y^3*z^2 - 6*y^4*z^2 - 6*x^3*z^3 - x^2*y*z^3 - 6*x*y^2*z^3 + 6*y^3*z^3 + 6*x^2*z^4 + 2*x*y*z^4 + 4*y^2*z^4 - 6*x*z^5 + 7*y*z^5 - 3*z^6 + 4*x^3*y^2 - 5*x^2*y^3 + 4*x*y^4 + 6*y^5 - 7*x^4*z + 3*x^3*y*z - 7*x^2*y^2*z + 3*x*y^3*z + 6*y^4*z + x^3*z^2 - x^2*y*z^2 + 6*x*y^2*z^2 - 6*y^3*z^2 + 3*x^2*z^3 - 5*x*y*z^3 - 3*y^2*z^3 - 5*x*z^4 - 6*z^5 - 36*x^2*y^2 + 2*x*y^3 + 6*y^4 + 3*x^3*z - x^2*y*z + x*y^2*z - 7*y^3*z - 4*x^2*z^2 + 4*x*y*z^2 + 3*y^2*z^2 - 5*x*z^3 + 6*y*z^3 - 7*z^4 + 13*x^2*z - 3*x*y*z + 2*y^2*z + 7*x*z^2 - 2*y*z^2 - 2*z^3]
  [ok] (A8) K_2 (point in component(s) [1]), R = XY^3: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (3, -30, -27, -5, -9, -1, -6, -2, 0, 0, 4)
      D_3 = [-3*X^2*Z + 30*X*Y*Z + 27*Y^2*Z + 5*X*Z^2 + 9*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-2*x^6 + x^5*y - 5*x^4*y^2 - 3*x^3*y^3 + 5*x^2*y^4 + 5*x*y^5 + 3*x^5*z - 5*x^4*y*z - 5*x^3*y^2*z - x^2*y^3*z - 5*x*y^4*z - 6*y^5*z + 2*x^4*z^2 + 6*x^3*y*z^2 - 2*x^2*y^2*z^2 - 6*x*y^3*z^2 - 7*y^4*z^2 + 7*x^3*z^3 + 7*x^2*y*z^3 + 3*x*y^2*z^3 - 6*y^3*z^3 - 3*x^2*z^4 + x*y*z^4 + 4*y^2*z^4 - 2*x*z^5 + 2*y*z^5 + 7*z^6 + 7*x^3*y^2 - 2*x^2*y^3 - 3*x*y^4 - 5*x^4*z - 4*x^3*y*z - 7*x^2*y^2*z - 3*x*y^3*z + 6*x^3*z^2 + 7*x^2*y*z^2 - 4*x*y^2*z^2 - 5*y^3*z^2 - 6*x^2*z^3 + 3*x*y*z^3 - 7*y^2*z^3 - 2*x*z^4 - 3*z^5 + 2*x^2*y^2 - 7*y^4 + 4*x^3*z - 3*x*y^2*z + 5*y^3*z - 3*x*y*z^2 + y^2*z^2 - 6*x*z^3 + 4*y*z^3 - 5*z^4 - 128/3*x^2*z - 4*x*y*z - 3*y^2*z + 2*x*z^2 + z^3, -5*x^5*y - 7*x^4*y^2 - 6*x^3*y^3 + 7*x^2*y^4 + x*y^5 - y^6 - 5*x^3*y^2*z - 5*x^2*y^3*z - 3*x*y^4*z - 7*y^5*z + 6*x^4*z^2 - 6*x^3*y*z^2 - 5*x^2*y^2*z^2 - 4*y^4*z^2 - 2*x^3*z^3 + x^2*y*z^3 + x*y^2*z^3 + y^3*z^3 - 4*x^2*z^4 + 2*x*y*z^4 + 2*y^2*z^4 + 2*x*z^5 - 2*y*z^5 + 5*z^6 + 3*x^3*y^2 + 3*x^2*y^3 - 7*x*y^4 + 3*y^5 + x^4*z - 4*x^3*y*z - 2*x^2*y^2*z + x*y^3*z + y^4*z - 7*x^3*z^2 + 5*x^2*y*z^2 - x*y^2*z^2 + y^3*z^2 - 5*x^2*z^3 + 5*x*y*z^3 + 4*y^2*z^3 - 5*z^5 + 7*x^2*y^2 - 6*y^4 + x^3*z - 5*x^2*y*z + x*y^2*z - 6*y^3*z + 6*x^2*z^2 + 7*y^2*z^2 + 7*x*z^3 - 2*y*z^3 + 2*z^4 - 133/3*x^2*z + 4*x*y*z - 7*y^2*z + x*z^2 + y*z^2 - 6*z^3, 6*x^6 + 3*x^5*y + x^4*y^2 + 5*x^3*y^3 + 3*x^2*y^4 - y^6 - 5*x^5*z - 5*x^4*y*z - 3*x^3*y^2*z + 5*x^2*y^3*z + 7*x*y^4*z - 3*y^5*z + 4*x^4*z^2 + x^3*y*z^2 + x^2*y^2*z^2 - 4*x*y^3*z^2 + 4*y^4*z^2 + 4*x^3*z^3 + 4*x^2*y*z^3 + 7*x*y^2*z^3 - 3*y^3*z^3 + x^2*z^4 - x*y*z^4 - 6*y^2*z^4 + 3*y*z^5 + 7*z^6 + 7*x^3*y^2 - 6*x^2*y^3 - 4*x*y^4 + y^5 - 6*x^4*z + 4*x^3*y*z + 7*x^2*y^2*z + 6*x*y^3*z - y^4*z + 7*x^3*z^2 - 6*x^2*y*z^2 - x*y^2*z^2 - 5*y^3*z^2 - 3*x^2*z^3 + 3*x*y*z^3 - 4*y^2*z^3 + 6*y*z^4 - 7*z^5 + 7*x^2*y^2 - 7*y^4 + 7*x^3*z - 3*x^2*y*z + 4*x*y^2*z + y^3*z + 7*x^2*z^2 - x*y*z^2 + 5*y^2*z^2 - x*z^3 + 6*y*z^3 + 6*z^4 + 49*x^2*z + 2*x*y*z + 5*y^2*z + 3*x*z^2 - 6*y*z^2 + 7*z^3]
  [ok] (A8) K_2 (point in component(s) [1]), R = X^2Y^2 + Y^4: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (-2, -8, 12, -2, 6, 1, 4, 2, 0, 0, 5)
      D_3 = [-2*X^2*Z - 8*X*Y*Z + 12*Y^2*Z - 2*X*Z^2 + 6*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [2*x^6 + 7*x^5*y - 7*x^4*y^2 + 7*x^3*y^3 - 4*x^2*y^4 + 2*x*y^5 + y^6 - 3*x^5*z - 5*x^4*y*z + x^3*y^2*z + 3*x^2*y^3*z - 2*x*y^4*z + y^5*z + 7*x^4*z^2 - 2*x^3*y*z^2 - 2*x^2*y^2*z^2 - x*y^3*z^2 - 5*y^4*z^2 - 5*x^3*z^3 - 5*x^2*y*z^3 + 3*x*y^2*z^3 - 5*y^3*z^3 + x^2*z^4 + 7*x*y*z^4 - 5*y^2*z^4 + 2*x*z^5 - 3*y*z^5 - 6*z^6 - 2*x^3*y^2 - 5*x^2*y^3 + 4*x*y^4 + 6*y^5 + 5*x^4*z + 4*x^3*y*z + 3*x^2*y^2*z + 5*x*y^3*z + 5*y^4*z - x^3*z^2 + x^2*y*z^2 + 6*x*y^2*z^2 - 6*y^3*z^2 + 7*x^2*z^3 + 4*x*y*z^3 - 6*y^2*z^3 + 7*x*z^4 + 4*y*z^4 + z^5 + 30*x^2*y^2 + 7*x*y^3 - 5*y^4 + 6*x^3*z - 2*x^2*y*z + 6*x*y^2*z + 7*y^3*z - 6*y^2*z^2 - 3*x*z^3 + 5*y*z^3 - 6*z^4 - 49/2*x^2*z - 3*y^2*z + 2*x*z^2 - 2*y*z^2 + z^3, -3*x^6 + x^5*y + 2*x^4*y^2 - 3*x^3*y^3 + 4*x^2*y^4 + 7*x*y^5 - 6*y^6 - 2*x^5*z - 3*x^4*y*z + 6*x^3*y^2*z - 7*x^2*y^3*z + 3*x*y^4*z + 2*y^5*z - 7*x^4*z^2 - 3*x^3*y*z^2 + 7*x^2*y^2*z^2 + x*y^3*z^2 + 3*y^4*z^2 + 3*x^3*z^3 + 5*x^2*y*z^3 - 5*x*y^2*z^3 - 6*y^3*z^3 - 6*x^2*z^4 + 2*x*y*z^4 - 6*y^2*z^4 + x*z^5 + 7*y*z^5 + z^6 - 2*x^3*y^2 + 2*x^2*y^3 - 4*x*y^4 - y^5 - 4*x^4*z + 6*x^3*y*z + 6*x^2*y^2*z - x*y^3*z - 2*y^4*z + 7*x^3*z^2 + x^2*y*z^2 - 6*x*y^2*z^2 + 2*y^3*z^2 + 3*x^2*z^3 + x*y*z^3 + 4*y^2*z^3 - x*z^4 - 6*y*z^4 - 4*z^5 + 30*x^2*y^2 - 6*x*y^3 - 5*y^4 - 6*x^3*z + x^2*y*z + 7*x*y^2*z + 2*y^3*z + 2*x*y*z^2 + 3*y^2*z^2 - 5*x*z^3 + 3*y*z^3 - 4*z^4 - 121/2*x^2*z + 5*x*y*z - 4*y^2*z + 7*x*z^2 - 4*y*z^2 - 5*z^3, 4*x^6 - 4*x^4*y^2 + 7*x^3*y^3 - 4*x^2*y^4 - 6*x*y^5 - 6*y^6 + 4*x^5*z - 5*x^3*y^2*z + 5*x^2*y^3*z + 5*x*y^4*z - 5*y^5*z - x^4*z^2 - 7*x^3*y*z^2 - 3*x^2*y^2*z^2 + 2*x*y^3*z^2 + 7*y^4*z^2 - 3*x^3*z^3 - 7*x^2*y*z^3 - 5*x*y^2*z^3 + 2*y^3*z^3 - 3*x*y*z^4 + 6*y^2*z^4 - 2*x*z^5 + 5*y*z^5 - 7*z^6 - 2*x^3*y^2 - 7*x^2*y^3 + 2*x*y^4 + 6*y^5 + 3*x^4*z - 5*x^3*y*z - x^2*y^2*z + 4*x*y^3*z + 3*y^4*z + 7*x^3*z^2 + 6*x^2*y*z^2 - 3*x*y^2*z^2 + 5*y^3*z^2 + 2*x*y*z^3 - y^2*z^3 - 2*x*z^4 + 5*y*z^4 + 4*z^5 + 7*x*y^3 + 4*x^3*z + 4*x^2*y*z - 3*x*y^2*z + 4*y^3*z + 7*x^2*z^2 - 3*x*y*z^2 - 4*y^2*z^2 - 7*x*z^3 + 4*y*z^3 - 6*z^4 - 47/2*x^2*z + 7*x*y*z + 3*y^2*z + 2*x*z^2 - 7*y*z^2 - 3*z^3]
  [ok] (A8) K_3 (point in component(s) [2]), R = XY^3: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 41 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (-4/15, 12, -135, -2, 45, -5, -2, 1, 15, -5, 3)
      D_3 = [4/75*X^2*Z - 12/5*X*Y*Z + 27*Y^2*Z + 2/5*X*Z^2 - 9*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-6*x^6 + 2*x^5*y - 3*x^3*y^3 + 7*x^2*y^4 + 6*x*y^5 + 2*y^6 + x^5*z + 2*x^4*y*z + x^3*y^2*z + 3*x^2*y^3*z + 4*x*y^4*z - 6*y^5*z + 7*x^4*z^2 + 6*x^3*y*z^2 - x^2*y^2*z^2 + 2*x*y^3*z^2 + y^4*z^2 + 4*x^3*z^3 + 2*x^2*y*z^3 + x*y^2*z^3 - y^3*z^3 + 2*x^2*z^4 + 2*x*y*z^4 - 6*y^2*z^4 + 7*x*z^5 + 2*y*z^5 - 7*z^6 + 2*x^3*y^2 - x^2*y^3 - 2*x*y^4 - y^5 + 5*x^4*z - 7*x^3*y*z + 7*x^2*y^2*z + 7*x*y^3*z + 2*y^4*z + 6*x^3*z^2 - 2*x^2*y*z^2 + 3*x*y^2*z^2 - 5*y^3*z^2 + 2*x^2*z^3 - 4*x*y*z^3 + 7*y^2*z^3 + 5*x*z^4 + 7*y*z^4 - z^5 + 7*x^2*y^2 + y^4 - 5*x^2*y*z + 6*x*y^2*z + 2*y^3*z - 2*x^2*z^2 + 4*x*y*z^2 + 3*y^2*z^2 - 7*x*z^3 + 5*y*z^3 - 6*z^4 + 6315/4*x^2*z + 6*x*y*z - 5*y^2*z - 5*x*z^2 - 7*y*z^2 - z^3, -2*x^6 - 7*x^5*y + 7*x^4*y^2 + 2*x^3*y^3 - 5*x^2*y^4 - 7*x*y^5 + 3*y^6 + 6*x^5*z - 3*x^4*y*z + 6*x^3*y^2*z - 5*x*y^4*z - y^5*z + 3*x^4*z^2 - 3*x^3*y*z^2 - 2*x^2*y^2*z^2 - 4*x*y^3*z^2 - 5*y^4*z^2 - 7*x^3*z^3 + x^2*y*z^3 - 4*x*y^2*z^3 - 3*y^3*z^3 - 7*x^2*z^4 + 5*y^2*z^4 - 3*x*z^5 + 5*y*z^5 + 6*z^6 - 7*x^3*y^2 - 7*x*y^4 + 2*y^5 + 5*x^4*z + 2*x^3*y*z - 3*x^2*y^2*z + 3*x*y^3*z + 7*y^4*z - 5*x^3*z^2 + 6*x^2*y*z^2 + 2*x*y^2*z^2 + 6*y^3*z^2 - 4*x^2*z^3 + 6*x*y*z^3 - 5*y^2*z^3 - 6*x*z^4 + 6*y*z^4 + 2*z^5 + 4*x^2*y^2 + 7*y^4 - 6*x^3*z - 4*x^2*y*z - 7*x*y^2*z - 4*y^3*z - 7*x^2*z^2 - 5*x*y*z^2 + 6*y^2*z^2 + 5*x*z^3 + 7*y*z^3 - 4*z^4 + 11325/4*x^2*z - 5*x*y*z - 5*y^2*z + 5*x*z^2 + 5*y*z^2 + 7*z^3, -2*x^6 + 3*x^4*y^2 + 2*x^2*y^4 - 5*x*y^5 + 2*y^6 + 3*x^5*z + 7*x^4*y*z - 6*x^3*y^2*z + 6*x^2*y^3*z - 4*x*y^4*z + 4*y^5*z + 7*x^4*z^2 - 5*x^3*y*z^2 - 3*x^2*y^2*z^2 - 6*x*y^3*z^2 + 3*y^4*z^2 + 6*x^3*z^3 - 6*x^2*y*z^3 - 3*x*y^2*z^3 - x^2*z^4 + 2*x*y*z^4 - 5*y^2*z^4 + 4*x*z^5 - 6*y*z^5 - 6*z^6 + 6*x^3*y^2 - 7*x^2*y^3 + 4*y^5 + 4*x^4*z - x^3*y*z - 7*x^2*y^2*z - 5*x*y^3*z - 5*y^4*z - 4*x^3*z^2 - 7*x^2*y*z^2 + 3*x*y^2*z^2 + 6*y^3*z^2 + 2*x^2*z^3 + 4*x*y*z^3 + 2*y^2*z^3 + 4*x*z^4 - 4*y*z^4 + 4*z^5 - 3*x^2*y^2 - 2*x^3*z - 6*x^2*y*z + 2*x*y^2*z + 3*y^3*z + 2*x^2*z^2 - 3*x*y*z^2 - 2*y^2*z^2 + 7*x*z^3 - 2*y*z^3 + 4*z^4 - 5145/2*x^2*z - 2*x*y*z + 5*y^2*z + 7*x*z^2 - 2*y*z^2 - 7*z^3]
  [ok] (A8) K_3 (point in component(s) [2]), R = X^2Y^2 + Y^4: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 41 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (-4/9, 4, -9, -2, 9, -3, 10/3, -5, -5, 5, 3)
      D_3 = [4/27*X^2*Z - 4/3*X*Y*Z + 3*Y^2*Z + 2/3*X*Z^2 - 3*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [2*x^5*y - 2*x^4*y^2 + 5*x^2*y^4 - 5*x*y^5 + 4*y^6 + 3*x^5*z + x^4*y*z + 2*x^3*y^2*z + 4*x^2*y^3*z + 6*x*y^4*z - 6*y^5*z + 7*x^4*z^2 + 5*x^3*y*z^2 - 2*x^2*y^2*z^2 - 4*x*y^3*z^2 - 4*y^4*z^2 + 4*x^3*z^3 + 5*x^2*y*z^3 + 7*x*y^2*z^3 + 4*y^3*z^3 + 3*x^2*z^4 + 2*x*y*z^4 - 3*y^2*z^4 - 4*x*z^5 + 3*y*z^5 + z^6 - 4*x^3*y^2 - 7*x^2*y^3 - 4*x*y^4 + y^5 + 3*x^4*z + 3*x^3*y*z + 2*x^2*y^2*z + 4*y^4*z + 5*x^3*z^2 - 7*x^2*y*z^2 - 3*x*y^2*z^2 + 4*y^3*z^2 + 6*x^2*z^3 + 4*x*y*z^3 + 3*y^2*z^3 - 6*y*z^4 + 2*z^5 - 18*x^2*y^2 - 4*x*y^3 + 3*y^4 - x^3*z - 7*x^2*y*z - 2*x*y^2*z - y^3*z - 4*x^2*z^2 + 3*x*y*z^2 + 7*y^2*z^2 - 6*x*z^3 - 7*y*z^3 + 5*z^4 - 135*x^2*z - 6*x*y*z + 3*y^2*z - 3*x*z^2 - 4*y*z^2 - z^3, 3*x^6 + x^5*y - 5*x^4*y^2 - x^3*y^3 + 4*x^2*y^4 + x*y^5 + 7*y^6 + 2*x^5*z + 2*x^4*y*z + 7*x^3*y^2*z - 3*x^2*y^3*z + 5*x*y^4*z + 2*y^5*z + x^4*z^2 - 5*x^3*y*z^2 - 6*x^2*y^2*z^2 - x*y^3*z^2 - 5*y^4*z^2 + 3*x^3*z^3 - 3*x^2*y*z^3 - 2*x*y^2*z^3 - 7*y^3*z^3 + 5*x^2*z^4 - 5*x*y*z^4 + x*z^5 - 5*y*z^5 + 2*z^6 - 3*x^3*y^2 + 7*x^2*y^3 - 3*x*y^4 - 7*y^5 + 3*x^4*z - 5*x^3*y*z + x^2*y^2*z + 6*x*y^3*z + y^4*z - 3*x^3*z^2 - 2*x^2*y*z^2 + 4*x*y^2*z^2 - 2*y^3*z^2 + 4*x^2*z^3 - 5*x*y*z^3 - y^2*z^3 + x*z^4 + 6*y*z^4 + 4*z^5 - 12*x^2*y^2 - 4*x*y^3 + 2*y^4 - 7*x^3*z + 7*x^2*y*z + 6*x*y^2*z - 2*y^3*z + 3*x^2*z^2 + 4*x*y*z^2 + 6*x*z^3 + 3*y*z^3 + 2*z^4 + 189/2*x^2*z + 7*x*y*z + y^2*z - 5*x*z^2 + 6*y*z^2 + 3*z^3, -4*x^6 - 7*x^5*y - 4*x^4*y^2 - 4*x^3*y^3 - 5*x*y^5 + y^6 + x^5*z - 6*x^4*y*z + 2*x^3*y^2*z + 4*x^2*y^3*z + 7*x*y^4*z + 6*y^5*z - 7*x^4*z^2 - 5*x^3*y*z^2 + 5*x^2*y^2*z^2 + x*y^3*z^2 - 5*y^4*z^2 - 6*x^3*z^3 + x^2*y*z^3 + 5*x*y^2*z^3 - 5*y^3*z^3 + 5*x^2*z^4 - x*y*z^4 + 3*y^2*z^4 + 2*x*z^5 + 6*y*z^5 - 3*z^6 - x^2*y^3 - 7*x*y^4 + 2*y^5 - 6*x^4*z - 4*x^3*y*z + 3*x^2*y^2*z + 2*x*y^3*z - 5*y^4*z + 2*x^3*z^2 + 4*x^2*y*z^2 + 2*x*y^2*z^2 - 5*x^2*z^3 - x*y*z^3 - 5*y^2*z^3 - 4*x*z^4 - 7*y*z^4 + 2*z^5 + 42*x^2*y^2 + 2*x*y^3 - 7*y^4 + 7*x^3*z - 6*x^2*y*z - x*y^2*z - 3*y^3*z + 7*x^2*z^2 - 5*x*y*z^2 - 5*y^2*z^2 - 2*x*z^3 + 3*z^4 - 27*x^2*z - x*y*z + 6*y^2*z - 4*x*z^2 + 3*y*z^2 - z^3]
  [ok] (A8) K_D (point in component(s) [0]), R = XY^3: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (-3, 2, -4, 0, 0, 0, -5, 0, 5, 0, 3)
      D_3 = [3/4*X^2*Z - 1/2*X*Y*Z + Y^2*Z, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-2*x^6 - 7*x^5*y - 3*x^4*y^2 + 6*x^3*y^3 + 4*x^2*y^4 - 6*x*y^5 + 6*y^6 - x^5*z + 6*x^4*y*z + 7*x^3*y^2*z - 2*x^2*y^3*z + 6*x*y^4*z + 4*y^5*z - 6*x^4*z^2 + 5*x^3*y*z^2 + x*y^3*z^2 + 2*y^4*z^2 + 2*x^3*z^3 + 5*x^2*y*z^3 - 3*x*y^2*z^3 + 5*y^3*z^3 + 2*x^2*z^4 - 4*x*y*z^4 + 6*y^2*z^4 - x*z^5 - 5*y*z^5 - 7*z^6 - x^3*y^2 + 3*x^2*y^3 + x*y^4 + 4*y^5 - x^4*z - 4*x^3*y*z + 5*x^2*y^2*z + 4*x*y^3*z - 2*y^4*z + 5*x^3*z^2 + x^2*y*z^2 - 7*x*y^2*z^2 + 4*y^3*z^2 - 5*x^2*z^3 - y^2*z^3 + 7*x*z^4 + 4*z^5 - 5*x^2*y^2 - 7*y^4 - 5*x^3*z - 5*x^2*y*z + 7*x*y^2*z + 5*y^3*z + x^2*z^2 - 6*x*y*z^2 - 7*x*z^3 + 4*z^4 - 7*x^2*z - 5*x*y*z + 4*y^2*z + 6*x*z^2 - y*z^2, 3*x^6 + 3*x^5*y - 5*x^4*y^2 - 5*x^3*y^3 - x^2*y^4 - 3*x*y^5 + 2*y^6 + x^5*z + x^4*y*z + x^3*y^2*z + 4*x*y^4*z + 6*y^5*z + 4*x^4*z^2 - 5*x^3*y*z^2 + x^2*y^2*z^2 - 7*x*y^3*z^2 + y^4*z^2 - 6*x^3*z^3 + 7*x^2*y*z^3 + 3*x*y^2*z^3 - 7*y^3*z^3 + 5*x^2*z^4 - 6*x*y*z^4 - 6*y^2*z^4 - x*z^5 + 4*y*z^5 + x^3*y^2 + 7*x^2*y^3 - 7*x*y^4 - 7*x^4*z + 2*x^3*y*z + 6*x^2*y^2*z - 2*x*y^3*z + 3*y^4*z + 4*x^3*z^2 + 2*x^2*y*z^2 + 5*x*y^2*z^2 + y^3*z^2 + 2*x^2*z^3 - 6*x*y*z^3 - 7*y^2*z^3 + 3*x*z^4 - 4*y*z^4 + 4*z^5 - 7*x^2*y^2 + 3*y^4 + x^3*z + 2*x*y^2*z - 5*x^2*z^2 + 4*x*y*z^2 + 2*y^2*z^2 + x*z^3 - 2*y*z^3 - 7*z^4 + 14/3*x^2*z + 6*x*y*z - 2*y^2*z - 2*x*z^2 - 4*y*z^2 - 3*z^3, 7*x^6 - 7*x^5*y - 6*x^4*y^2 - 4*x^3*y^3 + 4*x^2*y^4 - 7*x*y^5 - 5*y^6 + 7*x^5*z - 3*x^4*y*z + x^3*y^2*z - x^2*y^3*z + x*y^4*z + 6*y^5*z + x^4*z^2 + 5*x^3*y*z^2 + 3*x^2*y^2*z^2 + 6*x*y^3*z^2 + 7*y^4*z^2 - 3*x^3*z^3 + 2*x^2*y*z^3 + y^3*z^3 - 2*x^2*z^4 + 7*x*y*z^4 - 4*y^2*z^4 + 5*x*z^5 + 6*y*z^5 + 2*z^6 + 6*x^3*y^2 + 2*x^2*y^3 + 5*y^5 + x^4*z + 4*x^3*y*z - 2*x^2*y^2*z + 5*x*y^3*z + 5*y^4*z + 4*x^3*z^2 + 7*x^2*y*z^2 - 4*x*y^2*z^2 + 5*y^3*z^2 - 2*x^2*z^3 + 5*x*y*z^3 - 6*x*z^4 + 4*y*z^4 - 6*z^5 - 4*x^2*y^2 - 2*y^4 + 5*x^3*z - 5*x^2*y*z + 2*x*y^2*z - 3*x^2*z^2 - 6*x*y*z^2 - 7*x*z^3 - 5*z^4 - 23/3*x^2*z + 5*x*y*z + 7*y^2*z - 3*x*z^2 - 5*y*z^2 + z^3]
  [ok] (A8) K_D (point in component(s) [0]), R = X^2Y^2 + Y^4: actual free member, closed with H, D_3 = N_3 = C[X,Y]_3 + C G (1 accepted of 1); S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      point (a,b,c,d,e,f,l1,l2,l3,l4,al2) = (-1, 4, 3, 0, 0, 0, -4, 0, 2, 0, 1)
      D_3 = [-1/3*X^2*Z + 4/3*X*Y*Z + Y^2*Z, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [3*x^6 - 6*x^5*y + 6*x^4*y^2 + 7*x^3*y^3 + 3*x^2*y^4 - 2*x*y^5 + 6*y^6 + 5*x^5*z + 3*x^4*y*z + 5*x^3*y^2*z - 2*x^2*y^3*z - 2*y^5*z + 7*x^2*y^2*z^2 + x*y^3*z^2 + 7*y^4*z^2 - 6*x^3*z^3 - 6*x^2*y*z^3 - 7*x*y^2*z^3 + 4*y^3*z^3 + 7*x^2*z^4 - 7*y^2*z^4 + 7*x*z^5 + y*z^5 + 5*z^6 + 2*x^3*y^2 + 4*x^2*y^3 + 2*x*y^4 + 6*y^5 - 5*x^4*z + x^3*y*z + 4*x^2*y^2*z - 5*x*y^3*z + 2*y^4*z + 3*x^3*z^2 + 6*x^2*y*z^2 - x*y^2*z^2 - 6*y^3*z^2 - 3*x^2*z^3 + 2*x*y*z^3 - 4*y^2*z^3 + 3*x*z^4 + 2*y*z^4 - 5*z^5 + 24*x^2*y^2 + 3*x*y^3 - 4*y^4 - 6*x^3*z - 5*x^2*y*z - 3*x*y^2*z + 3*y^3*z + 4*x^2*z^2 + 5*x*y*z^2 - 4*y^2*z^2 + 3*x*z^3 - 2*y*z^3 + 6*z^4 + 3*x^2*z + y^2*z + 7*y*z^2 + 6*z^3, 3*x^6 - 3*x^5*y - 5*x^4*y^2 + 2*x^3*y^3 + 6*x^2*y^4 - 3*y^6 - 3*x^4*y*z - 5*x^3*y^2*z + 7*x^2*y^3*z + 2*x*y^4*z - 3*y^5*z - 2*x^4*z^2 + x^3*y*z^2 - 4*x^2*y^2*z^2 + 3*x*y^3*z^2 + y^4*z^2 - 4*x^3*z^3 - 3*x^2*y*z^3 - 2*x*y^2*z^3 + 4*y^3*z^3 - 4*x^2*z^4 + 4*x*y*z^4 - 5*y^2*z^4 + x*z^5 + 3*y*z^5 + 4*z^6 + 6*x^3*y^2 + 2*x*y^4 + 3*y^5 + x^4*z - 5*x^3*y*z - 6*x^2*y^2*z + 2*x*y^3*z - 2*y^4*z - 7*x^3*z^2 - 7*x^2*y*z^2 - 6*x*y^2*z^2 - 4*y^3*z^2 + 3*x^2*z^3 + 7*x*y*z^3 + 6*y^2*z^3 + 4*x*z^4 - 3*y*z^4 - 3*z^5 - 24*x^2*y^2 + 2*x*y^3 + 4*y^4 + 4*x^3*z - x^2*y*z - 5*x*y^2*z - 5*y^3*z + 2*x^2*z^2 - 2*x*y*z^2 - 3*y^2*z^2 - 4*x*z^3 - 7*y*z^3 - 2*z^4 + 19*x^2*z + 2*x*y*z + 5*y^2*z - 5*x*z^2 - z^3, -5*x^6 - 5*x^5*y + 7*x^4*y^2 + 6*x^3*y^3 - 3*x^2*y^4 - 2*x*y^5 + 4*y^6 - 4*x^5*z - 2*x^4*y*z + 6*x^2*y^3*z - x*y^4*z - 6*y^5*z - 7*x^4*z^2 + 3*x^3*y*z^2 - 6*x^2*y^2*z^2 - 5*x*y^3*z^2 + y^4*z^2 - 4*x^3*z^3 - 5*x^2*y*z^3 + 4*x*y^2*z^3 - 4*y^3*z^3 - 2*x^2*z^4 + 2*x*y*z^4 - 6*y^2*z^4 - 7*x*z^5 + 5*y*z^5 - 4*z^6 + 6*x^3*y^2 + 3*x^2*y^3 + 3*x*y^4 - 2*y^5 + 7*x^4*z - 3*x^3*y*z - 3*x^2*y^2*z + 4*x*y^3*z - y^4*z - 3*x^3*z^2 + 2*x^2*y*z^2 - 5*x*y^2*z^2 + y^3*z^2 + 2*x^2*z^3 + 3*x*y*z^3 + 6*y^2*z^3 - x*z^4 - y*z^4 + z^5 - 12*x^2*y^2 + 5*x*y^3 + 2*y^4 - 5*x^3*z - 2*x^2*y*z - 7*x*y^2*z - 7*y^3*z + 7*x^2*z^2 + 7*x*y*z^2 - 4*y^2*z^2 - 5*x*z^3 - y*z^3 + 7*z^4 - 29*x^2*z - 4*x*y*z - 7*y^2*z - 6*x*z^2 + 4*y*z^2 + 4*z^3]
accepted actual free samples: 8
  [ok] (A8) every one of the 4 components of I (other than lam = 0) contains a certified member (covered: [0, 1, 2, 3])
(U9) Lemma FQ for the free algebras with q(R) = 1
  [ok] (U9) R = Y^4 (q = 1), W = X^2*Y^2 (binary, so S_1 o W in C[X,Y]_3 in D_3): 8ac - 3b^2 of R + tW is 8*t, nonzero for t != 0; q(R + tW) = 2 at t = 1, 2, -3, 1/7
  [ok] (U9) R = X^2Y^2 (q = 1), W = Y^4 (binary, so S_1 o W in C[X,Y]_3 in D_3): 8ac - 3b^2 of R + tW is 8*t, nonzero for t != 0; q(R + tW) = 2 at t = 1, 2, -3, 1/7
  [ok] (U9) R = R_{3/8} = (3/8)X^2Y^2 + XY^3 + Y^4 (q = 1), W = X^2*Y^2 (binary, so S_1 o W in C[X,Y]_3 in D_3): 8ac - 3b^2 of R + tW is 8*t, nonzero for t != 0; q(R + tW) = 2 at t = 1, 2, -3, 1/7
(S9) Lemma FQ on actual free algebras of F1 data (G = Z(X^2 + Y^2) + Z^3, rank z o G = 2): M_t = M' + C(Xi + tW) closed with H and D_4(t)
  [ok] (S9) R = Y^4, G = X^2*Z + Y^2*Z: 2 actual free algebras (dim M' = 20) of 2 tries; M_t closed with H and D_4(t) for t = 1, 2 on 2 of 2
  [ok] (S9) R = X^2Y^2, G = X*Y*Z: 2 actual free algebras (dim M' = 20) of 2 tries; M_t closed with H and D_4(t) for t = 1, 2 on 2 of 2
accepted FQ samples: 4
24 checks, 0 failed
ALL CHECKS PASSED
