(U0) case (d): D_3 = C[X,Y]_3 + C G with rank(z o G) >= 2 has D_3' = C[X,Y]_4
  [ok] (U0) G = sum g_i (Z-monomials), g symbolic: dim D_3' = 5 = dim C[X,Y]_4 over Q(g)
  [ok] (U0) G = X^2Z + Y^2Z (rank 2, Z(X^2+Y^2)): rank z o G = 2, dim D_3' = 5 (5 iff rank >= 2)
  [ok] (U0) G = XYZ: rank z o G = 2, dim D_3' = 5 (5 iff rank >= 2)
  [ok] (U0) G = XZ^2 (= Z.XZ, rank 2): rank z o G = 2, dim D_3' = 5 (5 iff rank >= 2)
  [ok] (U0) G = Z^3 + X^2Z: rank z o G = 2, dim D_3' = 5 (5 iff rank >= 2)
  [ok] (U0) G = X^2Z (rank 1, embedded point at [X]): rank z o G = 1, dim D_3' = 6 (5 iff rank >= 2)
  [ok] (U0) G = Z^3 (rank 1, the point [Z]): rank z o G = 1, dim D_3' = 6 (5 iff rank >= 2)
  [ok] (U0) G = (X+Y)^2 Z (rank 1): rank z o G = 1, dim D_3' = 6 (5 iff rank >= 2)
(U1) the A-space and q for binary R (symbolic a, b, c)
  [ok] (U1) over Q(a,b,c): {A : A_Y, A_Z in D_4} = <X^5, X^4Y, X^3Y^2, X^4Z, A_R>, A_R = aX^2Y^3/3 + bXY^4/4 + cY^5/5, y o A_R = R
  [ok] (U1) R = X*Y^3: A-space = <X^5, X^4Y, X^3Y^2, X^4Z, A_R>, q = 2 (8ac - 3b^2 = -3)
  [ok] (U1) R = X^2*Y^2 + Y^4: A-space = <X^5, X^4Y, X^3Y^2, X^4Z, A_R>, q = 2 (8ac - 3b^2 = 8)
  [ok] (U1) R = Y^4: A-space = <X^5, X^4Y, X^3Y^2, X^4Z, A_R>, q = 1 (8ac - 3b^2 = 0)
  [ok] (U1) R = X^2*Y^2: A-space = <X^5, X^4Y, X^3Y^2, X^4Z, A_R>, q = 1 (8ac - 3b^2 = 0)
  [ok] (U1) R = 17/5*X^2*Y^2 + X*Y^3 + Y^4: A-space = <X^5, X^4Y, X^3Y^2, X^4Z, A_R>, q = 2 (8ac - 3b^2 = 121/5)
  [ok] (U1) R = 3*X^2*Y^2 + X*Y^3: A-space = <X^5, X^4Y, X^3Y^2, X^4Z, A_R>, q = 2 (8ac - 3b^2 = -3)
(U2) the reduction to rank(pi(P'_Y), pi(P'_Z), pi(B'_Z)) (symbolic tails)
  [ok] (U2) A_Y = a1 X^4 + a2 X^3Y + e R and A_Z = b1 X^4 identically (so a3 = e, b2 = b3 = 0)
  [ok] (U2) pi(x^2 o A/30) = (2/5) b1 X^2Z identically on the A-space
  [ok] (U2) pi(h3) = pi(P'_Y), pi(h4) = pi(P'_Z), pi(h2) = pi(B'_Z) identically (P' = P4 - (2/5)b1 X^2YZ, B' = B - (1/5)b1^2 X^2Z^2)
  [ok] (U2) B -> pi(B_Z) has rank 6 = dim T_3/C[X,Y]_3 (onto); the non-free algebras with D_3 in N are e != 0 and rank(pi(P'_Y), pi(P'_Z), pi(B'_Z)) = 1
(U3) the rank <= 1 locus of beta : P -> (pi(P_Y), pi(P_Z))
  [ok] (U3) rank <= 1 locus of beta: 1 minimal prime(s) of dimension [10] (in T_4, dim 15); ker beta = C[X,Y]_4 + C X^3Z has dimension 6 and lies in it
  [ok] (U3) cross-check: the minimal prime is prime and equals rad(I_u) (2x2 minors of (pi(P_Y), pi(P_Z)))
(A4) certificates: actual non-free members of (alpha'), (beta')
  [ok] (A4) R = XY^3, piece (alpha'): actual non-free member (closed, H, D_3 = N_3; 1 accepted of 1 tries) with an S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      D_3 = [1/6*X^2*Z - X*Y*Z + X*Z^2, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [4*x^6 - 3*x^5*y + 7*x^4*y^2 + 4*x^3*y^3 + 4*x^2*y^4 + 7*x*y^5 + 7*y^6 - x^5*z + x^4*y*z - 4*x^3*y^2*z + 6*x^2*y^3*z + 4*x*y^4*z + 7*y^5*z - x^4*z^2 - 7*x^3*y*z^2 + 6*x^2*y^2*z^2 - 4*x*y^3*z^2 - y^4*z^2 + 6*x^3*z^3 + 2*x^2*y*z^3 - 2*x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 + 2*x*y*z^4 - 5*y^2*z^4 - 2*x*z^5 - 5*y*z^5 - 2*z^6 + 2*x^3*y^2 - 2*x^2*y^3 - 6*x*y^4 + 5*y^5 - 3*x^4*z + 3*x^3*y*z - 7*x^2*y^2*z - 6*x*y^3*z - 3*y^4*z + 4*x^3*z^2 - 3*x^2*y*z^2 - 7*x*y^2*z^2 + y^3*z^2 + 2*x^2*z^3 + 5*y^2*z^3 - 3*x*z^4 + 3*y*z^4 - 5*z^5 + 2*x^2*y^2 + 6*y^4 - 7*x^3*z + x^2*y*z - 4*x*y^2*z - 3*y^3*z - 2*x^2*z^2 + 6*x*y*z^2 + 7*y^2*z^2 + 5*x*z^3 - 2*y*z^3 + 4*z^4 - 15*x^2*z - x*y*z + 2*y^2*z + 2*x*z^2 + 7*y*z^2 + 6*z^3, -4*x^6 + 3*x^5*y + 7*x^4*y^2 - 3*x^3*y^3 + x^2*y^4 + 2*x*y^5 - 2*y^6 + 5*x^5*z + 7*x^4*y*z + 6*x^2*y^3*z + 4*x*y^4*z - 6*y^5*z + 5*x^4*z^2 + x^3*y*z^2 + 2*x^2*y^2*z^2 + 5*x*y^3*z^2 - 5*y^4*z^2 - x^3*z^3 - 6*x^2*y*z^3 - 3*x*y^2*z^3 + y^3*z^3 + 2*x^2*z^4 - 2*x*y*z^4 - 2*y^2*z^4 + 2*x*z^5 - 7*y*z^5 + z^6 + 4*x^3*y^2 - 7*x^2*y^3 + 7*x*y^4 + 7*y^5 - 7*x^4*z - 2*x^3*y*z - 4*x^2*y^2*z + 2*x*y^3*z + 5*y^4*z - 5*x^3*z^2 - 3*x^2*y*z^2 - x*y^2*z^2 - y^3*z^2 + 4*x^2*z^3 - 2*x*y*z^3 - 3*y^2*z^3 + 5*x*z^4 - y*z^4 - z^5 - 3*x^2*y^2 + 6*y^4 + 4*x^3*z - 2*x^2*y*z - 7*x*y^2*z + y^3*z - x^2*z^2 + 2*x*y*z^2 - 5*y^2*z^2 + x*z^3 - 2*y*z^3 - z^4 + 21*x^2*z + 3*x*y*z + 7*y^2*z - 2*x*z^2 + 7*y*z^2, 4*x^6 - 5*x^5*y + 6*x^4*y^2 + x^3*y^3 + x^2*y^4 - 4*x*y^5 + 3*x^5*z - 2*x^4*y*z + 3*x^3*y^2*z - 6*x*y^4*z + 6*y^5*z - 7*x^4*z^2 - x^3*y*z^2 + 3*x^2*y^2*z^2 + 5*x*y^3*z^2 + 6*y^4*z^2 - 3*x^3*z^3 + 3*x^2*y*z^3 + x*y^2*z^3 - 2*y^3*z^3 + 2*x^2*z^4 + 4*x*y*z^4 - 7*y*z^5 - 3*z^6 - 5*x^3*y^2 + 6*x^2*y^3 - 2*x*y^4 - 4*y^5 - 3*x^4*z + 7*x^3*y*z - 4*x^2*y^2*z + x*y^3*z - 6*y^4*z - x^3*z^2 + 4*x*y^2*z^2 + 5*y^3*z^2 - x^2*z^3 + 6*x*y*z^3 + 4*y^2*z^3 + 3*x*z^4 - 5*y*z^4 + 3*z^5 + 3*x^2*y^2 - 4*y^4 + 5*x^3*z - x^2*y*z + x*y^2*z + 3*y^3*z + 3*x^2*z^2 - x*y*z^2 - 7*y^2*z^2 + 6*x*z^3 + 3*y*z^3 + 15*x^2*z - x*y*z - 3*x*z^2 - 4*y*z^2 + 2*z^3]
  [ok] (A4) R = XY^3, piece (beta'): actual non-free member (closed, H, D_3 = N_3; 1 accepted of 1 tries) with an S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      D_3 = [1/8*X^2*Z - 3/8*X*Y*Z - 3/8*Y^2*Z - 3/16*X*Z^2 + 3/8*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-x^6 - 2*x^5*y - 6*x^4*y^2 - 7*x^3*y^3 - 6*x*y^5 + 7*y^6 - 3*x^5*z - 6*x^4*y*z + 6*x^3*y^2*z + 4*x^2*y^3*z + 4*x*y^4*z - 2*y^5*z + 4*x^4*z^2 + 6*x^3*y*z^2 + 3*x^2*y^2*z^2 + 6*x*y^3*z^2 + 6*y^4*z^2 + 5*x^2*y*z^3 - 2*x*y^2*z^3 - 4*y^3*z^3 - 7*x^2*z^4 - x*y*z^4 - 7*y^2*z^4 + x*z^5 + 7*z^6 - 2*x^3*y^2 + 4*x*y^4 - 4*y^5 + 4*x^4*z - 4*x^3*y*z - x^2*y^2*z - 7*x*y^3*z + 6*y^4*z - 2*x^3*z^2 - 4*x^2*y*z^2 + 7*x*y^2*z^2 - y^3*z^2 + x^2*z^3 + x*y*z^3 - 5*y^2*z^3 - 7*x*z^4 + 7*y*z^4 + 6*z^5 - 5*y^4 - x^3*z + 5*x^2*y*z + 2*x*y^2*z - 3*y^3*z - x^2*z^2 + 6*x*y*z^2 + 4*y^2*z^2 + 6*x*z^3 - 7*y*z^3 + z^4 - 165/2*x^2*z + x*y*z + 6*y^2*z + 2*x*z^2 - 5*y*z^2 + 5*z^3, -5*x^6 - 6*x^5*y - 2*x^4*y^2 - x^3*y^3 - 5*x^2*y^4 + 7*x*y^5 - 7*y^6 - 5*x^5*z + 6*x^4*y*z + 3*x^3*y^2*z + 2*x^2*y^3*z - 3*x*y^4*z - 6*y^5*z - 6*x^4*z^2 - 3*x^3*y*z^2 - 5*x^2*y^2*z^2 + 3*x*y^3*z^2 - 6*y^4*z^2 + x^3*z^3 - 7*x^2*y*z^3 - 5*x*y^2*z^3 - y^3*z^3 - 7*x^2*z^4 - 3*x*y*z^4 - 3*y^2*z^4 + 4*x*z^5 + 2*y*z^5 + 7*z^6 + 3*x^3*y^2 + 5*x^2*y^3 + 4*x*y^4 + y^5 + 7*x^4*z - 2*x^3*y*z - 5*x^2*y^2*z - 6*x*y^3*z + 6*y^4*z + 3*x^3*z^2 - 3*x^2*y*z^2 + 4*x*y^2*z^2 + 5*y^3*z^2 + 3*x^2*z^3 - 6*x*y*z^3 - y^2*z^3 + 7*x*z^4 + 5*z^5 + 4*x^2*y^2 + 5*y^4 + 5*x^3*z - 2*x^2*y*z - 5*x*y^2*z - 6*y^3*z + 2*x^2*z^2 + 3*x*y*z^2 - y^2*z^2 + 7*x*z^3 + 5*y*z^3 + 6*z^4 - 81*x^2*z - 2*y^2*z - 6*x*z^2 + 6*y*z^2 + 2*z^3, -4*x^6 + 6*x^5*y - 7*x^4*y^2 - 7*x^3*y^3 - 6*x^2*y^4 + 3*x*y^5 + 7*y^6 - 4*x^5*z + 5*x^4*y*z + 3*x^3*y^2*z - 4*x^2*y^3*z + 5*x*y^4*z + 7*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 + 2*y^4*z^2 - 7*x^3*z^3 - 7*x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 - x*y*z^4 + 5*y^2*z^4 - 2*x*z^5 + 5*y*z^5 - z^6 + 6*x^3*y^2 + 6*x^2*y^3 + 4*x*y^4 + 3*y^5 + 6*x^4*z + 4*x^3*y*z - 3*x^2*y^2*z - 2*x*y^3*z - 4*y^4*z - 3*x^3*z^2 + 6*x^2*y*z^2 + 7*y^3*z^2 - 4*x^2*z^3 - 4*x*y*z^3 + 7*x*z^4 - 3*y*z^4 - z^5 + 5*x^2*y^2 - 5*y^4 + 6*x^3*z - 4*x^2*y*z - 3*x*y^2*z + 5*y^3*z - 5*x^2*z^2 - x*y*z^2 - 4*y^2*z^2 - 3*x*z^3 - 2*y*z^3 - 2*z^4 + 99*x^2*z - 4*x*z^2 - 3*y*z^2 - 4*z^3]
  [ok] (A4) R = X^2Y^2 + Y^4, piece (alpha'): actual non-free member (closed, H, D_3 = N_3; 1 accepted of 1 tries) with an S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      D_3 = [3/2*X^2*Z + 3*Y^2*Z + 3*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-7*x^6 + x^5*y + 4*x^4*y^2 - 2*x^3*y^3 + 4*x^2*y^4 + 3*x*y^5 - x^5*z - 4*x^4*y*z + 6*x^3*y^2*z - x^2*y^3*z + 5*x*y^4*z + 4*y^5*z + 2*x^4*z^2 - 5*x^3*y*z^2 + 6*x^2*y^2*z^2 - 3*x*y^3*z^2 + 3*y^4*z^2 + 5*x^3*z^3 - 3*x^2*y*z^3 + x*y^2*z^3 + y^3*z^3 + 4*x^2*z^4 - x*y*z^4 - x*z^5 - 2*y*z^5 + 7*z^6 - 2*x^2*y^3 - y^5 - x^4*z + 5*x^3*y*z + x^2*y^2*z + 3*x*y^3*z - 7*y^4*z + 6*x^3*z^2 + 5*x^2*y*z^2 + 3*x*y^2*z^2 - 3*y^3*z^2 - 4*x^2*z^3 - 6*x*y*z^3 - y^2*z^3 - 2*x*z^4 - 7*y*z^4 + 7*z^5 + 36*x^2*y^2 + 7*x*y^3 - 6*y^4 + 4*x^3*z - 3*x^2*y*z - 4*x*y^2*z - 5*y^3*z - 4*x^2*z^2 + 6*x*y*z^2 + y^2*z^2 - x*z^3 + y*z^3 - 4*z^4 + 10*x^2*z + 6*x*y*z + 4*y^2*z - 2*x*z^2 - 7*y*z^2 - 2*z^3, -4*x^6 + 4*x^5*y + 6*x^4*y^2 + 4*x^3*y^3 + 5*x^2*y^4 - 2*x*y^5 + 6*y^6 - 2*x^5*z + 6*x^4*y*z + 5*x^3*y^2*z + 7*x^2*y^3*z - 7*x*y^4*z - 5*y^5*z - 6*x^4*z^2 - 4*x^3*y*z^2 + 2*x^2*y^2*z^2 + x*y^3*z^2 + 2*y^4*z^2 + 2*x^2*y*z^3 + 3*x*y^2*z^3 + 6*y^3*z^3 + x^2*z^4 - 3*x*y*z^4 + 7*y^2*z^4 + 2*x^3*y^2 - 5*x^2*y^3 - 4*x*y^4 + 4*y^5 + 2*x^4*z + 7*x^3*y*z - 4*x^2*y^2*z - 6*x*y^3*z - 3*y^4*z + 3*x^3*z^2 - 7*x^2*y*z^2 + 3*x*y^2*z^2 - 2*y^3*z^2 - 7*x^2*z^3 + 5*x*y*z^3 + y^2*z^3 - 6*x*z^4 + 3*y*z^4 - 5*z^5 - 6*x^2*y^2 - 2*x*y^3 + y^4 + 2*x^3*z + 3*x^2*y*z + 7*x*y^2*z - 7*y^3*z - x^2*z^2 - 6*x*y*z^2 + 3*x*z^3 - 3*y*z^3 + 4*z^4 + 14*x^2*z - 7*x*y*z - 7*y^2*z - 3*x*z^2 + 6*y*z^2 - 6*z^3, 4*x^6 - 4*x^5*y - 7*x^4*y^2 + 7*x^3*y^3 + x^2*y^4 - 4*x*y^5 + 3*y^6 + x^5*z + 6*x^4*y*z - 3*x^3*y^2*z + 7*x^2*y^3*z + 6*x*y^4*z + 6*y^5*z - 3*x^4*z^2 + 5*x^3*y*z^2 + 6*x^2*y^2*z^2 + 3*x*y^3*z^2 + 2*y^4*z^2 + x^3*z^3 - 3*x^2*y*z^3 + 3*x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 - 2*x*y*z^4 + y^2*z^4 + 2*x*z^5 + 6*y*z^5 - 5*z^6 - 4*x^3*y^2 + 7*x^2*y^3 + 4*x*y^4 - 6*y^5 + 4*x^4*z - 6*x^3*y*z - 5*x^2*y^2*z - 5*x*y^3*z - 5*y^4*z - x^2*y*z^2 + 4*x*y^2*z^2 - 5*y^3*z^2 - 2*x^2*z^3 - 4*x*y*z^3 - 6*y^2*z^3 - 7*x*z^4 - 2*y*z^4 + 4*z^5 + 6*x^2*y^2 - 5*x*y^3 - y^4 - 4*x^3*z + 3*x^2*y*z - 5*x*y^2*z + 6*y^3*z - 7*x^2*z^2 - 4*x*y*z^2 + y^2*z^2 - 7*x*z^3 - 7*y*z^3 - z^4 - 28*x^2*z + x*y*z + 6*y^2*z - 3*x*z^2 + 6*y*z^2 + 2*z^3]
  [ok] (A4) R = X^2Y^2 + Y^4, piece (beta'): actual non-free member (closed, H, D_3 = N_3; 1 accepted of 1 tries) with an S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      D_3 = [-5/8*X^2*Z + 1/2*X*Y*Z + 5/8*Y^2*Z + 9/16*X*Z^2 - 3/4*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-4*x^6 - 7*x^5*y + 2*x^4*y^2 - 4*x^3*y^3 - 7*x^2*y^4 - 2*x*y^5 + y^6 - 5*x^5*z - 7*x^4*y*z + 2*x^3*y^2*z - 4*x^2*y^3*z - 4*x*y^4*z + 7*y^5*z + 3*x^4*z^2 + 6*x^3*y*z^2 - x^2*y^2*z^2 - 3*x*y^3*z^2 + 2*y^4*z^2 - 5*x^3*z^3 + 2*x^2*y*z^3 + 5*x*y^2*z^3 + 4*y^3*z^3 - 4*x^2*z^4 - x*y*z^4 - 7*y^2*z^4 - 5*x*z^5 - 5*y*z^5 + 4*z^6 + 2*x^3*y^2 + 2*x^2*y^3 + 7*x*y^4 - y^5 + 6*x^4*z + x^3*y*z + 6*x^2*y^2*z - 7*x*y^3*z - 3*y^4*z - 3*x^3*z^2 + 7*x^2*y*z^2 - 7*x*y^2*z^2 - 7*y^3*z^2 - 5*x^2*z^3 - 7*y^2*z^3 - 7*x*z^4 + 3*y*z^4 + 6*z^5 + 30*x^2*y^2 + 7*x*y^3 - 5*y^4 - 2*x^3*z - x^2*y*z + 2*x*y^2*z + 5*y^3*z + 2*x^2*z^2 - 6*x*y*z^2 - 6*x*z^3 + 4*y*z^3 + 4*z^4 + 307/10*x^2*z - 4*x*y*z - y^2*z - 7*x*z^2 - 5*y*z^2 + 7*z^3, -3*x^6 + 4*x^5*y + 3*x^4*y^2 - 5*x^3*y^3 + 3*x^2*y^4 + 5*x*y^5 - 2*x^5*z - 4*x^4*y*z - 5*x^3*y^2*z + x*y^4*z - 3*y^5*z + 4*x^4*z^2 + 4*x^3*y*z^2 - 7*x^2*y^2*z^2 + 7*y^4*z^2 - 6*x^3*z^3 + 2*x^2*y*z^3 - 5*x*y^2*z^3 - 6*y^3*z^3 - 6*x^2*z^4 + 6*x*y*z^4 + 7*y^2*z^4 - 5*x*z^5 - y*z^5 + 4*z^6 - 6*x^3*y^2 + 2*x^2*y^3 - 6*x*y^4 + 6*y^5 - 5*x^4*z - x^3*y*z + x^2*y^2*z - 7*x*y^3*z + 4*y^4*z + 7*x^3*z^2 - 5*x^2*y*z^2 + 4*x*y^2*z^2 - y^3*z^2 - 4*x*y*z^3 - 4*y^2*z^3 + 4*x*z^4 - 6*y*z^4 + 2*z^5 + 6*x^2*y^2 + 7*x*y^3 - y^4 - 4*x^3*z - 3*x^2*y*z + 3*x*y^2*z - y^3*z - 5*x^2*z^2 - 7*x*y*z^2 - y^2*z^2 + 5*x*z^3 + 7*y*z^3 - 6*z^4 - 33/5*x^2*z - x*y*z - 5*y^2*z + y*z^2, -2*x^6 - 2*x^5*y + 7*x^4*y^2 - 5*x^3*y^3 + x^2*y^4 - y^6 + 6*x^5*z - 7*x^4*y*z + 4*x^3*y^2*z + 5*x*y^4*z + y^5*z + 3*x^4*z^2 + 6*x^3*y*z^2 + 6*x*y^3*z^2 + 3*y^4*z^2 + x^3*z^3 + x^2*y*z^3 + 7*x*y^2*z^3 - 6*y^3*z^3 + 4*x^2*z^4 - 7*x*y*z^4 - 6*y^2*z^4 - 7*x*z^5 + 3*z^6 - 2*x^3*y^2 - 6*x^2*y^3 + 4*x*y^4 - 6*y^5 - 4*x^4*z + 4*x^3*y*z - 4*x^2*y^2*z + 7*x*y^3*z + 5*y^4*z + 2*x^3*z^2 + 6*x^2*y*z^2 - 4*x*y^2*z^2 - 3*y^3*z^2 + x^2*z^3 - x*y*z^3 + 5*y^2*z^3 + 3*x*z^4 + 6*y*z^4 - 6*z^5 + 6*x^2*y^2 - 5*x*y^3 - y^4 + x^3*z + 7*x^2*y*z + 7*x*y^2*z - 3*x^2*z^2 - 5*x*y*z^2 - 3*y^2*z^2 - 5*y*z^3 + 6*z^4 - 71/5*x^2*z - x*y*z - 6*y^2*z - 2*x*z^2 + 5*y*z^2]
  [ok] (A4) R = R_l, l = 17/5, piece (alpha'): actual non-free member (closed, H, D_3 = N_3; 1 accepted of 1 tries) with an S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      D_3 = [-1/9*X^2*Z + 10/9*X*Y*Z + Y^2*Z, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-x^6 - 2*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 3*x^2*y^4 + x*y^5 - 3*y^6 - 5*x^5*z + 3*x^4*y*z - 6*x^3*y^2*z - x^2*y^3*z - 2*x*y^4*z - 3*y^5*z - 5*x^4*z^2 - 5*x^2*y^2*z^2 - 7*x*y^3*z^2 + 7*x^2*y*z^3 + 3*x*y^2*z^3 + 2*y^3*z^3 - 5*x^2*z^4 - 4*x*y*z^4 - y^2*z^4 + 6*x*z^5 + 3*y*z^5 + 6*z^6 + 5*x^3*y^2 + x^2*y^3 - 7*x*y^4 + 4*y^5 + 5*x^4*z - 2*x^3*y*z - 2*x^2*y^2*z - 4*x*y^3*z + 3*y^4*z + x^3*z^2 + 4*x^2*y*z^2 + 7*x*y^2*z^2 - 7*y^3*z^2 + 5*x^2*z^3 - 6*x*y*z^3 - 2*y^2*z^3 + x*z^4 - 5*y*z^4 + 3*z^5 + 15/34*x^2*y^2 - 5*x*y^3 + y^4 - 3*x^3*z + 5*x^2*y*z + 5*x*y^2*z - 2*y^3*z - 3*x^2*z^2 - 7*x*y*z^2 - 2*y^2*z^2 + 6*x*z^3 + 3*y*z^3 - 2*z^4 - 39*x^2*z - 6*x*y*z - y^2*z + 5*x*z^2 - 6*y*z^2 + 7*z^3, 4*x^6 - 2*x^5*y + 4*x^4*y^2 - 6*x^3*y^3 + 6*x^2*y^4 - 2*x*y^5 + 3*y^6 - 2*x^5*z + 5*x^3*y^2*z - 2*x^2*y^3*z - x*y^4*z + 3*y^5*z + 4*x^4*z^2 + 3*x^3*y*z^2 + 6*x^2*y^2*z^2 + 3*x*y^3*z^2 + 4*y^4*z^2 - 3*x^3*z^3 - 7*x^2*y*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 + 3*x*y*z^4 - 6*y^2*z^4 + 4*x*z^5 - 4*y*z^5 - 2*z^6 + 2*x^3*y^2 - 2*x^2*y^3 - 3*x*y^4 - 5*y^5 + 3*x^3*y*z - 5*x^2*y^2*z + 2*x*y^3*z - 5*y^4*z - 2*x^3*z^2 - 7*x^2*y*z^2 + 2*x*y^2*z^2 - 2*y^3*z^2 - 7*x^2*z^3 - 2*x*y*z^3 + 4*y^2*z^3 + 3*z^5 - 15/17*x^2*y^2 - 2*x*y^3 + y^4 - 5*x^3*z - 7*x^2*y*z + 4*x*y^2*z + y^3*z - 2*x^2*z^2 + 4*x*y*z^2 - 2*y^2*z^2 + 6*x*z^3 + 4*y*z^3 + 4*z^4 + 37*x^2*z + 2*x*y*z + 3*y^2*z - 7*x*z^2 - 7*y*z^2 + z^3, x^6 + 4*x^5*y - 3*x^4*y^2 - 3*x^3*y^3 + 4*x^2*y^4 + 4*y^6 - 6*x^5*z - 6*x^4*y*z + 6*x^3*y^2*z - 2*x^2*y^3*z + 7*x*y^4*z - 3*y^5*z + 7*x^4*z^2 + x^3*y*z^2 - 4*x^2*y^2*z^2 + 7*x*y^3*z^2 + 4*y^4*z^2 + 6*x^3*z^3 + 2*x^2*y*z^3 - 5*x*y^2*z^3 + 2*y^3*z^3 - 2*x*y*z^4 - x*z^5 - 6*y*z^5 - z^6 - 2*x^3*y^2 - 4*x^2*y^3 + 6*x*y^4 + 6*y^5 + 7*x^4*z + 4*x^2*y^2*z + y^4*z - 3*x^3*z^2 - 6*x^2*y*z^2 + 4*x*y^2*z^2 - 4*x^2*z^3 + 3*x*y*z^3 - 5*x*z^4 + 5*y*z^4 + 7*z^5 - 90/17*x^2*y^2 - 4*x*y^3 + 4*y^4 + 6*x^3*z - 5*x*y^2*z + 2*y^3*z + 5*x^2*z^2 - 6*x*y*z^2 + 2*y^2*z^2 - 5*x*z^3 + 2*y*z^3 - 6*z^4 - 17*x^2*z - 7*x*y*z + 2*y^2*z + 6*x*z^2 - 3*y*z^2 + 5*z^3]
  [ok] (A4) R = R_l, l = 17/5, piece (beta'): actual non-free member (closed, H, D_3 = N_3; 1 accepted of 1 tries) with an S6.2 triple of local length 40 (ds = stabilization) in [21, 41]
      D_3 = [1/2*X*Y*Z + 1/6*Y^2*Z + 5/4*X*Z^2 + 1/4*Y*Z^2 + Z^3, Y^3, X*Y^2, X^2*Y, X^3]
      triple: [-3*x^6 - 3*x^5*y + 4*x^4*y^2 + 5*x^3*y^3 + 5*x^2*y^4 - 4*x*y^5 - 3*y^6 - 4*x^4*y*z + 2*x^3*y^2*z + 3*x^2*y^3*z - 3*x*y^4*z - 7*y^5*z + 6*x^4*z^2 + 7*x^3*y*z^2 + 3*x^2*y^2*z^2 - 5*x*y^3*z^2 + 2*y^4*z^2 + 2*x^3*z^3 + 3*x^2*y*z^3 + 6*x*y^2*z^3 - y^3*z^3 - 6*x^2*z^4 - 5*y^2*z^4 - 7*x*z^5 - 5*y*z^5 - 5*z^6 - 5*x^3*y^2 + 3*x^2*y^3 - y^5 + 7*x^3*y*z - 4*x^2*y^2*z + 2*x*y^3*z + y^4*z - 6*x^3*z^2 - 4*x^2*y*z^2 - 2*x*y^2*z^2 + 7*y^3*z^2 - 7*x^2*z^3 + 2*x*y*z^3 - 2*y^2*z^3 - 3*y*z^4 + 4*z^5 - 240/17*x^2*y^2 + 4*x*y^3 + 7*y^4 + 2*x^3*z - x^2*y*z - 2*x*y^2*z - 5*y^3*z - 7*x^2*z^2 + 7*x*y*z^2 - 4*y^2*z^2 - 5*x*z^3 - 3*y*z^3 - z^4 - 6*x^2*z - 250/3*x*y*z + 2*y^2*z + 7*x*z^2 - y*z^2 + 4*z^3, -3*x^6 + 6*x^5*y - x^4*y^2 - 4*x^3*y^3 + 4*x^2*y^4 - x*y^5 - 6*x^5*z + 3*x^4*y*z - 5*x^3*y^2*z - 2*x*y^4*z + 2*y^5*z - 2*x^4*z^2 + 2*x^3*y*z^2 + 5*x^2*y^2*z^2 + 2*x*y^3*z^2 + y^4*z^2 - 7*x^3*z^3 - 2*x^2*y*z^3 + 5*x*y^2*z^3 + 6*y^3*z^3 - 5*x*y*z^4 - y^2*z^4 + 5*x*z^5 - 6*y*z^5 + 7*z^6 + 6*x^3*y^2 + 6*x^2*y^3 + 7*x*y^4 - 6*y^5 - 5*x^4*z + 3*x^3*y*z - 5*x^2*y^2*z - 4*x*y^3*z - 4*y^4*z + 6*x^3*z^2 + x*y^2*z^2 - 7*y^3*z^2 - 6*x^2*z^3 - 6*x*y*z^3 + 7*y^2*z^3 - x*z^4 + 7*y*z^4 - 2*z^5 - 165/17*x^2*y^2 + 6*x*y^3 + 4*y^4 + 6*x^3*z + 3*x^2*y*z - 7*x*y^2*z - 3*y^3*z + 3*x^2*z^2 + 3*x*y*z^2 + 3*y^2*z^2 - 2*x*z^3 + 4*y*z^3 + 3*z^4 + 7*x^2*z + 32/3*x*y*z - 4*y^2*z + 3*x*z^2 + y*z^2 - 2*z^3, 4*x^6 + 5*x^5*y - 6*x^4*y^2 + 7*x^3*y^3 + 5*x^2*y^4 + 7*x*y^5 - 5*y^6 - 4*x^5*z - 7*x^4*y*z + 3*x^3*y^2*z + 2*x^2*y^3*z - 7*x*y^4*z + 4*y^5*z + 3*x^4*z^2 + 5*x^3*y*z^2 - 7*x^2*y^2*z^2 - x*y^3*z^2 - 6*y^4*z^2 - 3*x^2*y*z^3 - 5*x*y^2*z^3 - 4*y^3*z^3 - 2*x^2*z^4 - 7*x*y*z^4 + 5*y^2*z^4 + 7*x*z^5 + x^3*y^2 + x^2*y^3 - 4*x*y^4 + 4*y^5 - x^4*z + 6*x^3*y*z + 6*x^2*y^2*z + 6*x*y^3*z + 2*y^4*z - 6*x^3*z^2 + 3*x^2*y*z^2 + 5*x*y^2*z^2 + 5*y^3*z^2 - 6*x^2*z^3 - 7*x*y*z^3 - y^2*z^3 + 2*x*z^4 - 3*y*z^4 - 2*z^5 + 195/34*x^2*y^2 - x*y^3 - 3*y^4 - 7*x^3*z + x^2*y*z + 4*x*y^2*z - 5*y^3*z - x^2*z^2 + 3*x*y*z^2 + 6*y^2*z^2 - 7*x*z^3 - 7*y*z^3 + 6*z^4 - 4*x^2*z + 11*x*y*z + 3*y^2*z - 6*x*z^2 + 5*y*z^2 + z^3]
27 checks, 0 failed
ALL CHECKS PASSED
