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PART A (non-free algebras, every D_3 subset N, every m)
(U0) the graded datum for every m
  [ok] (U0) S_1 o R_m lies in C[X,Z]_3 + C X^2Y identically in m (so (D_3 = C[X,Z]_3 + C X^2Y, <X^4, X^3Y, R_m>, <X^5, X^4Y>, X^6) is a graded datum for every m)
  [ok] (U0) dim D_4(m) = 3 for every m (gcd of the 3x3 minors = 1)
(U1) the A-space {A in T_5 : A_Y, A_Z in D_4(m)} for every m
  [ok] (U1) D_4(m)^perp = <10 monomials, 2x^3z - 3m x^2z^2, 6x^2z^2 - z^4> for every m (orthogonal identically; gcd of the 12x12 minors = 1)
  [ok] (U1) X^5, X^4Y, X^3Y^2, X^4Z, A_R = (m/2)X^3Z^2 + X^2Z^3/3 + Z^5/5 lie in the A-space identically and are independent for every m (gcd 1)
  [ok] (U1) the 21 x 24 condition matrix has rank 16 over Q(m); a pivot 16x16 minor has determinant (-49302469038676377600) * m; the rank is < 16 at []: the A-space is 5-dimensional, spanned by these five, for EVERY m
  [ok] (U1) y o A_R = 0, z o A_R = R_m: for A = c50 X^5 + c41 X^4Y + c32 X^3Y^2 + c4z X^4Z + e A_R, A_Y = c41 X^4 + 2c32 X^3Y (a3 = 0) and A_Z = c4z X^4 + e R_m (b2 = 0, b3 = e)
  [ok] (U1) y o R_m = 0 and P(m) = <X^3, X^2Y> + S_1 o R_m = <X^3, X^2Y, x o R_m, z o R_m> has dimension 4 for every m (q(R_m) = 2; gcd of the 4x4 minors = 1)
(U2) a basis of P(m)^perp polynomial in m
  [ok] (U2) P(m)^perp = <xy^2, y^3, y^2z, xyz, yz^2, 6x^2z - 9m xz^2 - z^3> for every m (orthogonal identically; gcd of the 6x6 minors = 1)
(U3) the A-shift p(A, m)
  [ok] (U3) (x^2 o A/30, 0) = (p_Y, p_Z) modulo P(m), identically in m, for p = c32 XY^3/15 + (2c4z/5 - e(3m^2/20 + 1/90)) X^2YZ
(U4) the linear map L_m : P_4' -> ((P_4')_Y, (P_4')_Z) mod P(m)
  [ok] (U4) rank 9 over Q(m); a pivot 9x9 minor has determinant 23887872; the rank is < 9 at []: ker L_m is a rank-6 bundle over the whole m-line
  [ok] (U4) B -> B_Y maps T_4 onto T_3 (rank 10), with a 5-dimensional kernel
(U5) the rank <= 1 locus Z = {(m, P_4') : rank L_m(P_4') <= 1}
  [ok] (U5a) the 15 x 6 matrix of P -> (mu y - lam z) o P mod P(m) has rank 6 off lam = mu = 0 (lam^6, mu^6 lie in the ideal of its 6x6 minors over Q[m, lam, mu]); so Z, the image of a rank-9 bundle over A^1 x P^1, is irreducible of dimension <= 11 and dominates the m-line
  [ok] (U5b) cross-check over Q: the ideal of 2x2 minors of (u, w) in Q[m, p_0..p_14] has 1 minimal prime(s), of dimension [11], meeting Q[m] in 0
   [time 8.4 s]
(A) certificates: one actual non-free member of each piece, closed with H and D_3 = N_3, with an S6.2 triple <= 41
   member (m = 17/5, piece alpha): Phi = X^6 + A + B + C, Psi = X^4Y + P_4 + P_3 with
      A = 4*X^5 + 3*X^4*Y + 2*X^4*Z + 17/2*X^3*Z^2 + 5/3*X^2*Z^3 + Z^5
      B = 2*X^4 - 88/3*X^3*Y - 37163/900*X^2*Y^2 - 22/27*X*Y^3 - 11/27*Y^4 - 2*X^3*Z - 31691/180*X^2*Y*Z - 22/3*X*Y^2*Z - 44/9*Y^3*Z + 4*X^2*Z^2 + 48/5*X*Y*Z^2 - 22*Y^2*Z^2 + 5*X*Z^3 - 65/9*Y*Z^3 + 3*Z^4
      C = 2*X^2*Y - 2*X*Y^2 + 3*Y^3 - 4*X^2*Z - 5*X*Y*Z + 4*Y^2*Z - 3*X*Z^2 + 3*Y*Z^2 + 2*Z^3
      P4 = -3*X^4 - 4*X^3*Y - 2/27*X*Y^3 - 1/27*Y^4 - 2*X^3*Z - 34283/900*X^2*Y*Z - 2/3*X*Y^2*Z - 4/9*Y^3*Z - 3*X^2*Z^2 - 2*X*Y*Z^2 - 2*Y^2*Z^2 + 3*X*Z^3 - 4*Y*Z^3 + Z^4
      P3 = 5*X^3 + 5*X^2*Y - 3*X*Y^2 + 4*Y^3 + X^2*Z - 5*X*Y*Z - 4*Y^2*Z - 3*X*Z^2 + 2*Y*Z^2 + 4*Z^3
      D_3 = N_3 = [1/2*X^2*Z + Z^3, 1/18*X*Y^2 + 1/27*Y^3 + 539/120*X^2*Z + 1/3*X*Y*Z + 1/3*Y^2*Z + Y*Z^2, 51/10*X^2*Z + X*Z^2, X^2*Y, X^3]
      S6.2 triple: [-x^6 + 5*x^5*y + x^4*y^2 - 4*x^3*y^3 + 7*x^2*y^4 + 2*x*y^5 - 4*y^6 - 5*x^5*z - 5*x^4*y*z - 4*x^3*y^2*z - 2*x^2*y^3*z - x*y^4*z + 2*y^5*z - 4*x^4*z^2 - 5*x^3*y*z^2 + x^2*y^2*z^2 + 4*x*y^3*z^2 + 3*y^4*z^2 + x^3*z^3 - 4*x^2*y*z^3 + x*y^2*z^3 + y^3*z^3 + 2*x^2*z^4 + x*y*z^4 - 5*y^2*z^4 + 7*x*z^5 - 4*y*z^5 - 2*z^6 + 5*x^3*y^2 - 4*x^2*y^3 + 2*x*y^4 - 6*y^5 + x^4*z + 7*x^3*y*z - 5*x^2*y^2*z - 3*y^4*z + x^3*z^2 - 5*x^2*y*z^2 - 5*x*y^2*z^2 + 4*y^3*z^2 - 2*x^2*z^3 - 3*x*y*z^3 - 3*y^2*z^3 + 2*x*z^4 + 4*y*z^4 + 3*z^5 + 7*x^2*y^2 + 6*x*y^3 + 3*y^4 + 380/51*x^3*z + 5*x*y^2*z - 7*y^3*z - 2*x^2*z^2 - 7*x*y*z^2 + 7*y^2*z^2 - 7*x*z^3 + y*z^3 - 6*z^4 + 5036/5*x*y^2 + 4*y^3 - 12*x^2*z - x*y*z + 5*y^2*z + 306/5*x*z^2 - 4*y*z^2 + 2*z^3, -4*x^6 - 4*x^5*y + 5*x^4*y^2 - 7*x^3*y^3 + 4*x^2*y^4 + 5*x*y^5 + y^6 + 2*x^5*z - 7*x^4*y*z - 6*x^3*y^2*z + 4*x^2*y^3*z - 2*x*y^4*z + 2*y^5*z + 7*x^4*z^2 + 7*x^3*y*z^2 + 5*x^2*y^2*z^2 - 4*x*y^3*z^2 - 3*y^4*z^2 + 5*x^3*z^3 + x^2*y*z^3 + 5*x*y^2*z^3 - 3*y^3*z^3 + 7*x^2*z^4 + x*y*z^4 - 3*y^2*z^4 - x*z^5 - 5*y*z^5 + 4*z^6 - 2*x^3*y^2 - 5*x^2*y^3 + 5*x*y^4 - 2*y^5 + x^3*y*z - 6*x^2*y^2*z - 2*x*y^3*z + 5*y^4*z - 3*x^3*z^2 - 4*x^2*y*z^2 - 3*x*y^2*z^2 - 3*y^3*z^2 - x^2*z^3 + 4*x*y*z^3 - 2*y^2*z^3 - x*z^4 - 4*y*z^4 + z^5 + 6*x^2*y^2 - 6*x*y^3 + 6*y^4 + 140/51*x^3*z - 5*x^2*y*z + x*y^2*z + 3*y^3*z - 2*x^2*z^2 + 3*x*y*z^2 - 7*y^2*z^2 + x*z^3 + 2*y*z^3 - 2*z^4 + 33147/10*x*y^2 - 42*x^2*z + 7*x*y*z + 7*y^2*z + 1071/5*x*z^2 + y*z^2 + 7*z^3, 2*x^6 - 7*x^5*y - x^4*y^2 - 6*x^3*y^3 - 6*x^2*y^4 - 4*x*y^5 - 4*x^5*z + 5*x^4*y*z - 2*x^3*y^2*z + 2*x^2*y^3*z + 7*x*y^4*z + 6*y^5*z - 7*x^3*y*z^2 - 3*x^2*y^2*z^2 + 6*x*y^3*z^2 - 6*y^4*z^2 + 7*x^3*z^3 - x^2*y*z^3 + x*y^2*z^3 - 5*y^3*z^3 + 7*x^2*z^4 - 3*x*y*z^4 + 4*y^2*z^4 - 2*x*z^5 + y*z^5 - 5*z^6 - 6*x^3*y^2 - 3*x^2*y^3 - x*y^4 + 2*y^5 + 5*x^4*z + 5*x^3*y*z + 2*x^2*y^2*z - 3*x*y^3*z + 2*y^4*z - 5*x^3*z^2 + 4*x^2*y*z^2 - 3*x*y^2*z^2 - 6*y^3*z^2 - 3*x^2*z^3 + x*y*z^3 - 7*y^2*z^3 + 5*x*z^4 - 6*z^5 - 6*x^2*y^2 + 2*x*y^3 - 2*y^4 - 50/51*x^3*z + x^2*y*z + 6*x*y^2*z + 3*y^3*z - 7*x^2*z^2 - 3*x*y*z^2 - y^2*z^2 - 5*x*z^3 - y*z^3 + 2*z^4 + 5023/2*x*y^2 + 5*y^3 - 30*x^2*z - 6*x*y*z + 2*y^2*z + 153*x*z^2 - 5*y*z^2 + 5*z^3]
  [ok] (A, class A) m = 17/5, piece alpha': actual member closed with H, non-free (e = 5, dim M' = 21), D_3 = N_3 of dim 5 computed from S_1 o M, rank(h3, h4 mod P) = 1; S6.2 triple (cubic parts independent, in L_def) of local length ds = 35, stabilization = 35, in [21, 41]
   member (m = 17/5, piece beta): Phi = X^6 + A + B + C, Psi = X^4Y + P_4 + P_3 with
      A = -4*X^5 - 4*X^4*Y - 3*X^3*Y^2 + 3*X^4*Z - 17/10*X^3*Z^2 - 1/3*X^2*Z^3 - 1/5*Z^5
      B = -5*X^4 + 4*X^2*Y^2 + 2*X*Y^3 - 5*Y^4 + 3*X^3*Z + 4*X^2*Y*Z - 4*X*Y^2*Z + 2*Y^3*Z + X^2*Z^2 + 3*X*Y*Z^2 - Y^2*Z^2 - 4*X*Z^3 + Y*Z^3 - Z^4
      C = X^3 + X*Y^2 + 5*X^2*Z + 5*X*Y*Z - 5*Y^2*Z + X*Z^2 + 2*Y*Z^2 - 2*Z^3
      P4 = 3*X^2*Y^2 - 1/5*X*Y^3 + 4*X^3*Z + 13253/4500*X^2*Y*Z - 233/20*X^2*Z^2 - X*Z^3 - 4*Z^4
      P3 = 5*X^3 + 3*X^2*Y - 4*X*Y^2 - 3*Y^3 - 4*X^2*Z - X*Y*Z - 3*Y^2*Z + X*Z^2 + 5*Z^3
      D_3 = N_3 = [1/2*X^2*Z + Z^3, -63/10*X*Y^2 + 203/20*Y^3 - 1772779/18000*X^2*Z - 1253/3000*X*Y*Z - 3*Y^2*Z + Y*Z^2, 51/10*X^2*Z + X*Z^2, X^2*Y, X^3]
      S6.2 triple: [6*x^6 + 7*x^4*y^2 - x^3*y^3 + 6*x^2*y^4 + 4*x*y^5 - 3*y^6 - 7*x^5*z - 4*x^4*y*z + 7*x^2*y^3*z + 3*x*y^4*z - 3*y^5*z + 3*x^4*z^2 + 5*x^3*y*z^2 - 7*x^2*y^2*z^2 + 7*x*y^3*z^2 + 4*y^4*z^2 + 6*x^3*z^3 - 6*x^2*y*z^3 - 3*x*y^2*z^3 + 2*y^3*z^3 + 5*x^2*z^4 + 7*x*y*z^4 + 6*y^2*z^4 + 2*x*z^5 + 4*y*z^5 + 3*z^6 + 6*x^3*y^2 - 4*x^2*y^3 + 7*x*y^4 - 2*y^5 + 5*x^4*z - 5*x^3*y*z - 3*x^2*y^2*z + 7*x*y^3*z + 7*y^4*z + 6*x^3*z^2 + 3*x^2*y*z^2 - 3*x*y^2*z^2 + 7*y^3*z^2 + 6*x*y*z^3 + 7*y^2*z^3 + 3*x*z^4 + 3*y*z^4 - 3*z^5 + 4*x^2*y^2 + 3*x*y^3 - 3*y^4 - 10/51*x^3*z + 2*x^2*y*z + 2*x*y^2*z - 2*y^3*z + 7*x^2*z^2 + 5*x*y*z^2 + x*z^3 - y*z^3 - z^4 + 239917/420*x*y^2 + y^3 - 36*x^2*z + 6*x*y*z - 6*y^2*z + 918/5*x*z^2 + 6*y*z^2 + 6*z^3, 7*x^6 + 4*x^5*y + 3*x^4*y^2 + 3*x^3*y^3 - 4*x^2*y^4 + 4*x*y^5 - 5*y^6 - 2*x^5*z - 6*x^4*y*z + 4*x^3*y^2*z + 5*x^2*y^3*z + 7*x*y^4*z - 2*y^5*z + 6*x^4*z^2 - 6*x^3*y*z^2 + 4*x^2*y^2*z^2 + 3*x*y^3*z^2 + 2*y^4*z^2 + 2*x^3*z^3 + 4*x^2*y*z^3 + 6*x*y^2*z^3 + 6*y^3*z^3 + 7*x^2*z^4 + 3*x*y*z^4 + 2*y^2*z^4 + 6*x*z^5 - 2*y*z^5 - 6*z^6 + 5*x^3*y^2 + 3*x^2*y^3 - 7*x*y^4 + 5*y^5 - x^4*z - 6*x^3*y*z + 4*x^2*y^2*z + 5*x*y^3*z - 5*y^4*z - 5*x^3*z^2 + 4*x^2*y*z^2 + 6*x*y^2*z^2 + 3*y^3*z^2 - 5*x^2*z^3 - 5*x*y*z^3 - 3*y^2*z^3 + 6*x*z^4 + y*z^4 - 5*x^2*y^2 - x*y^3 + 6*y^4 - 30/17*x^3*z - 2*x^2*y*z - 6*x*y^2*z - 4*y^3*z + 3*x^2*z^2 + x*y*z^2 + 3*y^2*z^2 - 3*x*z^3 + 3*y*z^3 + z^4 + 1469107/3150*x*y^2 - 30*x^2*z + 2*x*y*z + 5*y^2*z + 153*x*z^2 - y*z^2 + 5*z^3, -4*x^6 - 5*x^5*y - 2*x^4*y^2 + 5*x^3*y^3 + 4*x^2*y^4 - x*y^5 - 6*y^6 - 6*x^5*z - 4*x^4*y*z + 4*x^3*y^2*z - 4*x^2*y^3*z + 7*x*y^4*z - 7*y^5*z - 6*x^4*z^2 + 2*x^3*y*z^2 - 2*x^2*y^2*z^2 + 2*x*y^3*z^2 + 5*y^4*z^2 + x^3*z^3 - 3*x^2*y*z^3 + x*y^2*z^3 - 7*y^3*z^3 - x^2*z^4 - 7*x*y*z^4 + y^2*z^4 - 6*x*z^5 - y*z^5 + 4*z^6 - 7*x^3*y^2 + 4*x*y^4 - 7*y^5 - 5*x^4*z + x^3*y*z + x^2*y^2*z + 2*x*y^3*z + 5*y^4*z + 7*x^3*z^2 - x^2*y*z^2 - x*y^2*z^2 - 4*y^3*z^2 - 5*x^2*z^3 - 5*x*y*z^3 - 5*y^2*z^3 + 3*x*z^4 + 4*y*z^4 + 7*z^5 - x^2*y^2 - 2*x*y^3 - 3*y^4 - 80/17*x^3*z - x^2*y*z - x*y^2*z - 2*y^3*z + 6*x^2*z^2 + 5*x*y*z^2 + y^2*z^2 + y*z^3 + 3*z^4 - 17979719/37800*x*y^2 - y^3 + 30*x^2*z - 7*x*y*z + 2*y^2*z - 153*x*z^2 - 7*y*z^2 - 5*z^3]
  [ok] (A, class A) m = 17/5, piece beta': actual member closed with H, non-free (e = -1, dim M' = 21), D_3 = N_3 of dim 5 computed from S_1 o M, rank(h3, h4 mod P) = 0; S6.2 triple (cubic parts independent, in L_def) of local length ds = 35, stabilization = 35, in [21, 41]
   member (m = 0, piece alpha): Phi = X^6 + A + B + C, Psi = X^4Y + P_4 + P_3 with
      A = 5*X^5 + 3*X^4*Y - 5*X^3*Y^2 + X^4*Z - X^2*Z^3 - 3/5*Z^5
      B = -5*X^4 - 26*X^3*Y + 67/6*X^2*Y^2 + 101/48*X*Y^3 + 17/256*Y^4 - X^3*Z - 112/5*X^2*Y*Z - 61/12*X*Y^2*Z - 109/48*Y^3*Z - X^2*Z^2 + 25*X*Y*Z^2 - 109/8*Y^2*Z^2 + 4*X*Z^3 - 72/5*Y*Z^3 - 2*Z^4
      C = -4*X^3 + 3*X^2*Y - 3*X*Y^2 + 5*Y^3 - 3*X^2*Z + 2*X*Y*Z + 2*Y^2*Z + 4*X*Z^2 + 2*Y*Z^2 - 5*Z^3
      P4 = 4*X^4 - 2*X^2*Y^2 - 5/16*X*Y^3 + 3/256*Y^4 + 3*X^3*Z + 14/15*X^2*Y*Z + 1/4*X*Y^2*Z + 3/16*Y^3*Z + X*Y*Z^2 + 9/8*Y^2*Z^2 - 4*X*Z^3 + 3*Y*Z^3 + 2*Z^4
      P3 = 3*X^3 - X*Y^2 + 2*Y^3 + 3*X^2*Z + 5*X*Y*Z - 5*Y^2*Z + 3*X*Z^2 + 5*Y*Z^2
      D_3 = N_3 = [1/2*X^2*Z + Z^3, 1/36*X*Y^2 + 1/48*Y^3 - 4/9*X^2*Z + 2/9*X*Y*Z + 1/4*Y^2*Z + Y*Z^2, X*Z^2, X^2*Y, X^3]
      S6.2 triple: [6*x^6 - 6*x^5*y - 2*x^4*y^2 + 2*x^3*y^3 - 7*x^2*y^4 - x*y^5 - 3*y^6 + 7*x^5*z - 6*x^4*y*z + 2*x^3*y^2*z - 4*x^2*y^3*z - 2*x*y^4*z - 5*y^5*z + 5*x^4*z^2 - 7*x^3*y*z^2 - 4*x^2*y^2*z^2 + 2*x*y^3*z^2 - 3*y^4*z^2 - x^3*z^3 - 5*x^2*y*z^3 + 5*x*y^2*z^3 - 5*y^3*z^3 + 2*x^2*z^4 - 7*x*y*z^4 + 3*y^2*z^4 - 5*x*z^5 + 4*y*z^5 + 6*z^6 - 7*x^3*y^2 - 2*x^2*y^3 + 6*x*y^4 + 2*y^5 - x^4*z - 6*x^3*y*z + x^2*y^2*z + 5*x*y^3*z + 6*y^4*z - 5*x^3*z^2 + 7*x^2*y*z^2 + x*y^2*z^2 - 5*y^3*z^2 + 3*x^2*z^3 + x*y*z^3 + 7*y^2*z^3 + 7*x*z^4 + 4*y*z^4 - z^5 + 5*x^2*y^2 - 2*x*y^3 - 2*y^4 - 3*x^3*z - x^2*y*z - 3*x*y^2*z - 5*y^3*z + 42*x^2*z^2 + 2*x*y*z^2 + 4*y^2*z^2 - 7*x*z^3 + 2*y*z^3 - 7*z^4 + 845/4*x*y^2 - y^3 + 18*x^2*z - 5*x*y*z + 7*y^2*z + y*z^2 - 3*z^3, -x^6 - 3*x^5*y - 4*x^4*y^2 - 3*x^3*y^3 + 4*x^2*y^4 + x*y^5 + y^6 + 2*x^5*z + x^4*y*z + 4*x^3*y^2*z + 2*x^2*y^3*z + 2*x*y^4*z - 3*x^4*z^2 + 5*x^3*y*z^2 - 7*x^2*y^2*z^2 + 2*x*y^3*z^2 + 5*y^4*z^2 + 2*x^3*z^3 - 2*x^2*y*z^3 + 3*x*y^2*z^3 - y^3*z^3 + 4*x^2*z^4 - 4*y^2*z^4 - x*z^5 + 7*y*z^5 - 6*z^6 + 7*x^3*y^2 - 4*x^2*y^3 + 3*x*y^4 - 4*y^5 - 7*x^3*y*z + 7*x^2*y^2*z - 3*y^4*z + 5*x^3*z^2 - 4*x^2*y*z^2 - 2*x*y^2*z^2 - 7*y^3*z^2 - 2*x^2*z^3 + x*y*z^3 + 2*y^2*z^3 + 6*x*z^4 - y*z^4 - 6*z^5 - 6*x^2*y^2 - 2*x*y^3 + 3*y^4 - 5*x^3*z - x^2*y*z + 2*x*y^2*z + 4*y^3*z + 36*x^2*z^2 - 2*x*y*z^2 - 4*y^2*z^2 + 3*x*z^3 - 3*y*z^3 - 6*z^4 - 1327/2*x*y^2 + 6*y^3 - 30*x^2*z + 2*x*y*z - 6*y^2*z + 6*y*z^2 + 5*z^3, 5*x^6 + 5*x^5*y - x^4*y^2 + 7*x^3*y^3 + 7*x*y^5 + 6*y^6 + 3*x^4*y*z + 2*x^3*y^2*z - x^2*y^3*z - 3*x*y^4*z + 6*y^5*z + 4*x^4*z^2 + 5*x^3*y*z^2 + 3*x^2*y^2*z^2 + 4*x*y^3*z^2 + 2*y^4*z^2 - x^3*z^3 + 6*x^2*y*z^3 - 7*x*y^2*z^3 - 5*y^3*z^3 + 6*x^2*z^4 + 6*x*y*z^4 + 2*y^2*z^4 + 3*x*z^5 - y*z^5 - 2*z^6 - 6*x^3*y^2 + 6*x^2*y^3 - 5*x*y^4 - y^5 + 5*x^4*z - 6*x^3*y*z - 5*x^2*y^2*z - 3*x*y^3*z - y^4*z - 4*x^3*z^2 + 3*x^2*y*z^2 + 2*x*y^2*z^2 + 7*x^2*z^3 - 3*x*y*z^3 + 4*y^2*z^3 + 2*x*z^4 - 4*y*z^4 - 2*z^5 + 2*x^2*y^2 + 7*y^4 - 2*x^3*z + 6*y^3*z + 12*x^2*z^2 + x*y*z^2 + 2*y^2*z^2 - 6*x*z^3 + 2*y*z^3 - 2*z^4 + 163/2*x*y^2 - 6*y^3 + 6*x^2*z + 7*x*y*z - 4*y^2*z + y*z^2 - z^3]
  [ok] (A, class S) m = 0, piece alpha': actual member closed with H, non-free (e = -3, dim M' = 21), D_3 = N_3 of dim 5 computed from S_1 o M, rank(h3, h4 mod P) = 1; S6.2 triple (cubic parts independent, in L_def) of local length ds = 35, stabilization = 35, in [21, 41]
   member (m = 0, piece beta): Phi = X^6 + A + B + C, Psi = X^4Y + P_4 + P_3 with
      A = 2*X^5 - 2*X^4*Y + 2*X^3*Y^2 + 4*X^4*Z - X^2*Z^3 - 3/5*Z^5
      B = -3*X^4 + 3*X^3*Y - 2*X^2*Y^2 + X*Y^3 + 3*Y^4 + 5*X^3*Z + 3*X^2*Y*Z - 4*X*Y^2*Z - 4*Y^3*Z + 5*X^2*Z^2 - 4*X*Y*Z^2 - 3*Y^2*Z^2 - 2*X*Z^3 + Y*Z^3 - 2*Z^4
      C = X^2*Y - 5*X*Y^2 + 4*Y^3 - 3*X^2*Z - 5*X*Y*Z + 3*Y^2*Z - 3*X*Z^2 + 3*Y*Z^2 + 3*Z^3
      P4 = 5*X^4 - 3*X^3*Y + 3*X^2*Y^2 + 2/15*X*Y^3 + X^3*Z + 49/30*X^2*Y*Z + 5*X*Z^3
      P3 = 2*X^3 - X^2*Y - 3*X*Y^2 + 5*Y^3 + 3*X^2*Z - 2*X*Y*Z + 2*Y^2*Z - 5*X*Z^2 - 5*Y*Z^2
      D_3 = N_3 = [1/2*X^2*Z + Z^3, 11/30*X*Y^2 - 89/45*Y^3 - 79/90*X^2*Z + 169/90*X*Y*Z + 2*Y^2*Z + Y*Z^2, X*Z^2, X^2*Y, X^3]
      S6.2 triple: [7*x^6 - x^5*y + 5*x^4*y^2 - x^3*y^3 - 3*x^2*y^4 - 5*x*y^5 + 7*y^6 - 5*x^4*y*z + 4*x^3*y^2*z + x^2*y^3*z - 2*x*y^4*z + 4*y^5*z + 2*x^4*z^2 + 6*x^3*y*z^2 + 6*x^2*y^2*z^2 + x*y^3*z^2 + y^4*z^2 - 6*x^2*y*z^3 - 3*x*y^2*z^3 + 5*y^3*z^3 - 2*x^2*z^4 - 5*x*y*z^4 - 6*y^2*z^4 - 2*x*z^5 - 3*y*z^5 + z^6 - 6*x^3*y^2 - 3*x^2*y^3 - 5*x*y^4 - 3*y^5 - 7*x^4*z - 4*x^3*y*z + 5*x^2*y^2*z + 4*x*y^3*z - 7*y^4*z + 3*x^3*z^2 + x^2*y*z^2 - 6*x*y^2*z^2 - 7*y^3*z^2 + 3*x^2*z^3 + 6*x*y*z^3 - y^2*z^3 + 3*x*z^4 + 5*y*z^4 + 3*x^2*y^2 - 5*x*y^3 + y^4 + 5*x^3*z - 3*x^2*y*z - 2*x*y^2*z + 4*y^3*z - 24*x^2*z^2 + 7*x*y*z^2 + 5*y^2*z^2 + 7*x*z^3 + y*z^3 + 4*z^4 + 19/11*x*y^2 - 3*y^3 + 24*x^2*z + 6*x*y*z - 5*y^2*z + 7*y*z^2 - 4*z^3, -6*x^6 + 5*x^5*y + x^4*y^2 - 5*x^3*y^3 - 2*x^2*y^4 - x*y^5 - 5*y^6 - 6*x^5*z - 2*x^4*y*z + 6*x^3*y^2*z + x^2*y^3*z - 4*y^5*z - 5*x^4*z^2 - 6*x^3*y*z^2 + 2*x^2*y^2*z^2 + 6*x*y^3*z^2 - 6*y^4*z^2 + 2*x^3*z^3 + 5*x^2*y*z^3 - 2*x*y^2*z^3 + 4*y^3*z^3 + x^2*z^4 - 4*x*y*z^4 + 6*y^2*z^4 - 4*x*z^5 + 6*z^6 + 6*x^3*y^2 + 6*x^2*y^3 - 6*x*y^4 - 4*x^4*z - 2*x^3*y*z + 4*x^2*y^2*z + 4*x*y^3*z + 7*y^4*z + 4*x^3*z^2 + 2*x^2*y*z^2 + 6*x*y^2*z^2 + y^3*z^2 + y^2*z^3 - 2*x*z^4 + 5*y*z^4 + 4*z^5 - 2*x^2*y^2 + 3*y^4 - x^3*z + 2*x^2*y*z + 6*x*y^2*z - 6*y^3*z - 42*x^2*z^2 - 7*x*y*z^2 - 4*y^2*z^2 + x*z^3 + 3*y*z^3 + 7*z^4 - 469/22*x*y^2 + 3*x*y*z + y^2*z + 3*y*z^2, 2*x^6 + 7*x^5*y - 3*x^4*y^2 + 5*x^3*y^3 - x^2*y^4 - 6*x*y^5 - y^6 + 2*x^5*z + x^4*y*z - 3*x^3*y^2*z + 5*x^2*y^3*z + x*y^4*z + 4*y^5*z + 6*x^4*z^2 + 2*x^3*y*z^2 - x^2*y^2*z^2 + 4*x*y^3*z^2 + 3*y^4*z^2 - 7*x^3*z^3 - 7*x^2*y*z^3 + 5*x*y^2*z^3 + 2*y^3*z^3 - 3*x^2*z^4 - x*y*z^4 + 7*x*z^5 + y*z^5 + 7*z^6 + 7*x^3*y^2 - 5*x^2*y^3 - 4*x*y^4 + 7*y^5 - 4*x^4*z - 3*x^3*y*z + x^2*y^2*z - 6*x*y^3*z + 5*y^4*z - 7*x^3*z^2 + 5*x^2*y*z^2 - 5*x*y^2*z^2 + 7*y^3*z^2 + 3*x^2*z^3 - 7*x*y*z^3 + 4*y^2*z^3 - 6*x*z^4 + 3*y*z^4 - 7*z^5 + x^2*y^2 - 4*x*y^3 - y^4 - 4*x^3*z + 2*x^2*y*z + x*y^2*z - 6*y^3*z - 30*x^2*z^2 + 2*x*y*z^2 - 7*y^2*z^2 + 7*y*z^3 + 5*z^4 + 11533/66*x*y^2 + 3*y^3 + 30*x^2*z - x*y*z - 6*y^2*z - 7*y*z^2 - 5*z^3]
  [ok] (A, class S) m = 0, piece beta': actual member closed with H, non-free (e = -3, dim M' = 21), D_3 = N_3 of dim 5 computed from S_1 o M, rank(h3, h4 mod P) = 0; S6.2 triple (cubic parts independent, in L_def) of local length ds = 35, stabilization = 35, in [21, 41]
   member (m = -2, piece alpha): Phi = X^6 + A + B + C, Psi = X^4Y + P_4 + P_3 with
      A = 3*X^5 + 4*X^4*Y + X^3*Y^2 - 3*X^4*Z + 2*X^3*Z^2 - 2/3*X^2*Z^3 - 2/5*Z^5
      B = 17*X^3*Y + 1193/180*X^2*Y^2 - 44/75*X*Y^3 - 67/1500*Y^4 - X^3*Z + 671/90*X^2*Y*Z + 356/45*X*Y^2*Z - 53/50*Y^3*Z + X^2*Z^2 + 112/5*X*Y*Z^2 - 159/20*Y^2*Z^2 + X*Z^3 - 1313/30*Y*Z^3 + 4*Z^4
      C = 4*X^3 + 5*X^2*Y - X*Y^2 - Y^3 + X^2*Z - 2*X*Y*Z + Y^2*Z + 3*X*Z^2 + 2*Y*Z^2 - Z^3
      P4 = -X^4 - 5*X^3*Y - 4*X^2*Y^2 + 2/75*X*Y^3 + 1/250*Y^4 + X^3*Z + 64/45*X^2*Y*Z - 3/5*X*Y^2*Z + 2/25*Y^3*Z - 3*X^2*Z^2 - 3*X*Y*Z^2 + 3/5*Y^2*Z^2 - 3*X*Z^3 + 2*Y*Z^3 + 5*Z^4
      P3 = 4*X^3 + X^2*Y + 4*X*Y^2 + Y^3 - 2*X*Y*Z - 4*X*Z^2 - 2*Y*Z^2
      D_3 = N_3 = [1/2*X^2*Z + Z^3, -1/10*X*Y^2 + 1/75*Y^3 - 43/6*X^2*Z - X*Y*Z + 1/5*Y^2*Z + Y*Z^2, -3*X^2*Z + X*Z^2, X^2*Y, X^3]
      S6.2 triple: [4*x^6 + 4*x^5*y + 3*x^4*y^2 + 5*x^3*y^3 + x^2*y^4 - 2*x*y^5 + y^6 + 5*x^5*z + 4*x^4*y*z + 7*x^3*y^2*z + 6*x^2*y^3*z + 4*x*y^4*z + 3*y^5*z + 4*x^4*z^2 - 3*x^3*y*z^2 - 3*x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 - 4*x^3*z^3 - 5*x^2*y*z^3 - 4*y^3*z^3 - 2*x^2*z^4 - 7*x*y*z^4 + 7*y^2*z^4 + 7*x*z^5 - 3*y*z^5 + z^6 - 4*x^3*y^2 + 3*x^2*y^3 + 3*x*y^4 - 2*y^5 - x^3*y*z - 7*x^2*y^2*z - 6*x*y^3*z - 5*y^4*z - 2*x^3*z^2 - 3*x^2*y*z^2 + 5*x*y^2*z^2 - 6*y^3*z^2 + 4*x^2*z^3 - x*y*z^3 - 3*y^2*z^3 + 7*x*z^4 + y*z^4 - 6*z^5 + 5*x^2*y^2 + x*y^3 + 6*y^4 - 13*x^3*z + 2*x^2*y*z + 4*x*y^2*z + 3*y^3*z - 3*x^2*z^2 - x*y*z^2 + 7*y^2*z^2 - 5*x*z^3 - 6*y*z^3 - 6*z^4 + 3992/5*x*y^2 + 6*y^3 - 12*x^2*z + 3*y^2*z - 36*x*z^2 - 7*y*z^2 + 2*z^3, -6*x^6 + 2*x^5*y + 7*x^4*y^2 - x^3*y^3 + 7*x^2*y^4 - 7*x*y^5 - 7*y^6 - 7*x^5*z - 7*x^3*y^2*z + 7*x^2*y^3*z - 4*x*y^4*z - 7*y^5*z + 3*x^4*z^2 + 4*x^3*y*z^2 - 2*x^2*y^2*z^2 + 4*x*y^3*z^2 + 6*y^4*z^2 - 6*x^3*z^3 + 3*x^2*y*z^3 - 2*x*y^2*z^3 + 7*y^3*z^3 - 4*x^2*z^4 - x*y*z^4 - 2*y^2*z^4 + 5*x*z^5 + y*z^5 + 2*x^3*y^2 + 6*x^2*y^3 - 4*x*y^4 - y^5 + 7*x^4*z - 4*x^3*y*z + 2*x*y^3*z - 7*x^3*z^2 - 3*x^2*y*z^2 - 2*x*y^2*z^2 + y^3*z^2 - 3*x^2*z^3 + 5*y^2*z^3 - 3*x*z^4 + 4*y*z^4 + 4*z^5 + 4*x^2*y^2 - 6*x*y^3 + y^4 + 13*x^3*z + 6*x^2*y*z - 7*x*y^2*z + 4*y^3*z + 3*x^2*z^2 - 7*x*y*z^2 - 6*y^2*z^2 - y*z^3 + 6*z^4 - 2130*x*y^2 + 30*x^2*z - 2*x*y*z + 90*x*z^2 + y*z^2 - 5*z^3, -5*x^5*y + 6*x^4*y^2 + 4*x^3*y^3 - 4*x^2*y^4 - 7*x*y^5 - 2*y^6 - 5*x^5*z + 5*x^4*y*z + 2*x^3*y^2*z + 2*x^2*y^3*z + x*y^4*z + 2*y^5*z - 2*x^3*y*z^2 - x^2*y^2*z^2 + 6*x*y^3*z^2 - 6*y^4*z^2 - x^3*z^3 - 3*x^2*y*z^3 + 3*x*y^2*z^3 - y^3*z^3 + 5*x^2*z^4 + 2*x*y*z^4 + 4*y^2*z^4 + 5*y*z^5 - 5*z^6 + 6*x^3*y^2 + 4*x^2*y^3 + 6*x*y^4 + 4*y^5 - 7*x^4*z - 3*x^3*y*z + 2*x^2*y^2*z + 7*x*y^3*z + y^4*z + 4*x^3*z^2 + 3*x^2*y*z^2 - 5*x*y^2*z^2 - y^3*z^2 + 7*x^2*z^3 - x*y*z^3 - 4*y^2*z^3 + 5*x*z^4 - 3*y*z^4 + 2*z^5 - 5*x^2*y^2 + 3*x*y^3 - 6*y^4 - 13*x^3*z + 2*x^2*y*z + 7*x*y^2*z - 6*y^3*z - 3*x^2*z^2 + 4*x*y*z^2 - 4*y^2*z^2 + 2*x*z^3 + 3*y*z^3 - 6*z^4 + 15024/5*x*y^2 - 3*y^3 - 42*x^2*z - 7*y^2*z - 126*x*z^2 + y*z^2 + 7*z^3]
  [ok] (A, class S) m = -2, piece alpha': actual member closed with H, non-free (e = -2, dim M' = 21), D_3 = N_3 of dim 5 computed from S_1 o M, rank(h3, h4 mod P) = 1; S6.2 triple (cubic parts independent, in L_def) of local length ds = 35, stabilization = 35, in [21, 41]
   member (m = -2, piece beta): Phi = X^6 + A + B + C, Psi = X^4Y + P_4 + P_3 with
      A = 4*X^5 - 2*X^3*Y^2 - 4*X^4*Z + 4*X^3*Z^2 - 4/3*X^2*Z^3 - 4/5*Z^5
      B = 4*X^4 + 4*X^3*Y + 4*X^2*Y^2 - X*Y^3 + 2*Y^4 - 3*X^3*Z + 2*X^2*Y*Z + 2*X*Y^2*Z + 5*X^2*Z^2 - X*Y*Z^2 - 3*X*Z^3 - 4*Y*Z^3 - 3*Z^4
      C = 3*X^3 - 5*X^2*Y + 2*X*Y^2 - 4*Y^3 - 5*X^2*Z - 4*X*Y*Z + Y^2*Z - 3*X*Z^2 + 4*Y*Z^2 + 4*Z^3
      P4 = -X^4 + X^2*Y^2 - 2/15*X*Y^3 + 3*X^3*Z + 38/45*X^2*Y*Z - 19/2*X^2*Z^2 + 3*X*Z^3 + 4*Z^4
      P3 = 3*X^3 + 4*X^2*Y - 5*X*Y^2 - 3*Y^3 + 5*X^2*Z + X*Y*Z - 5*Y^2*Z - 3*X*Z^2 + 3*Y*Z^2 + 5*Z^3
      D_3 = N_3 = [1/2*X^2*Z + Z^3, -3*X^2*Z + X*Z^2, -45/256*X*Y^2 + 177/128*Y^3 - 4365/512*X^2*Z + X*Y*Z, X^2*Y, X^3]
      S6.2 triple: [-2*x^6 + 5*x^5*y - 5*x^4*y^2 - 7*x^3*y^3 - 7*x^2*y^4 + 5*x*y^5 - 6*y^6 - 7*x^5*z + x^4*y*z + 2*x^3*y^2*z + 4*x^2*y^3*z + 6*x*y^4*z - y^5*z - 3*x^4*z^2 - 5*x^3*y*z^2 - 2*x^2*y^2*z^2 + 7*x*y^3*z^2 + 6*y^4*z^2 - 4*x^3*z^3 - 2*x^2*y*z^3 + 7*x*y^2*z^3 - 5*y^3*z^3 - x^2*z^4 - 2*x*y*z^4 - 4*y^2*z^4 + 2*x*z^5 - 5*y*z^5 - 5*z^6 + 7*x^3*y^2 - 7*x^2*y^3 + x*y^4 - 5*y^5 + 5*x^4*z - 4*x^3*y*z + 2*x^2*y^2*z + 6*x*y^3*z + 7*y^4*z + 5*x^3*z^2 + 3*x^2*y*z^2 - 4*x*y^2*z^2 + 2*y^3*z^2 + 7*x^2*z^3 - 5*x*y*z^3 - 6*y^2*z^3 - 7*x*z^4 - y*z^4 + 4*z^5 - 3*x^2*y^2 + 2*x*y^3 - 2*y^4 + 13*x^3*z - 5*x^2*y*z - 2*x*y^2*z + y^3*z - 3*x^2*z^2 - 6*x*y*z^2 + 2*y^2*z^2 - 7*x*z^3 - 5*y*z^3 + 7*z^4 + 11777/45*x*y^2 - y^3 - 6*x^2*z - 2*x*y*z - 3*y^2*z - 18*x*z^2 + y*z^2 + z^3, -x^6 + 7*x^5*y - 4*x^4*y^2 - 6*x^3*y^3 + 3*x^2*y^4 - 2*x*y^5 - 2*y^6 + 4*x^4*y*z - 3*x^3*y^2*z + 6*x^2*y^3*z - x*y^4*z + 3*y^5*z + 3*x^4*z^2 - 3*x^3*y*z^2 + 3*x^2*y^2*z^2 - 2*x*y^3*z^2 - 7*y^4*z^2 + x^3*z^3 + x^2*y*z^3 + 2*x*y^2*z^3 + y^3*z^3 + 3*x^2*z^4 + 5*x*y*z^4 + 2*y^2*z^4 + 7*x*z^5 + 2*y*z^5 - 5*z^6 + 6*x^3*y^2 + 7*x^2*y^3 + 2*x*y^4 + 6*y^5 - 2*x^4*z - x^3*y*z - 4*x^2*y^2*z - 2*x*y^3*z - 6*y^4*z - 3*x^3*z^2 - 6*x^2*y*z^2 - 2*x*y^2*z^2 - 4*y^3*z^2 - 6*x^2*z^3 + 5*x*y*z^3 + 5*y^2*z^3 - 2*x*z^4 - y*z^4 + 6*x^2*y^2 + 4*x*y^3 - 4*y^4 - 13*x^3*z - 6*x^2*y*z + x*y^2*z + y^3*z + 3*x^2*z^2 + 2*x*y*z^2 - 7*y^2*z^2 + 4*y*z^3 - 7*z^4 - 48388/45*x*y^2 + 4*y^3 + 24*x^2*z - 2*x*y*z + 4*y^2*z + 72*x*z^2 + 6*y*z^2 - 4*z^3, 4*x^6 - 5*x^5*y + x^4*y^2 + 5*x^3*y^3 - 3*x^2*y^4 + 7*x*y^5 - 7*y^6 - 2*x^5*z + 7*x^4*y*z + 7*x^3*y^2*z - 6*x^2*y^3*z + 5*x*y^4*z - 3*y^5*z - 4*x^4*z^2 - 6*x^3*y*z^2 - 2*x^2*y^2*z^2 - 7*x*y^3*z^2 + 4*y^4*z^2 + 4*x^3*z^3 - 7*x^2*y*z^3 + 4*x*y^2*z^3 + 2*y^3*z^3 - x^2*z^4 + 6*x*y*z^4 + 6*y^2*z^4 + 2*y*z^5 - z^6 + 5*x^3*y^2 + 6*x^2*y^3 + x*y^4 - 7*y^5 - 7*x^4*z + 4*x^3*y*z + 6*x^2*y^2*z - 3*x*y^3*z + 7*y^4*z + 4*x^3*z^2 - 6*x^2*y*z^2 - 3*x*y^2*z^2 + 3*y^3*z^2 - 7*x^2*z^3 - 5*x*y*z^3 + 4*y^2*z^3 - 6*x*z^4 + y*z^4 + z^5 - 6*x^2*y^2 - 4*x*y^3 - 4*y^4 - 16/3*x^3*z + 3*x^2*y*z - 4*x*y^2*z + 4*y^3*z + 2*x^2*z^2 + 6*x*y*z^2 + 7*y^2*z^2 + x*z^3 - 7*y*z^3 - 3*z^4 - 3657/5*x*y^2 + 6*y^3 + 18*x^2*z - 7*y^2*z + 54*x*z^2 + 2*y*z^2 - 3*z^3]
  [ok] (A, class S) m = -2, piece beta': actual member closed with H, non-free (e = -4, dim M' = 21), D_3 = N_3 of dim 5 computed from S_1 o M, rank(h3, h4 mod P) = 0; S6.2 triple (cubic parts independent, in L_def) of local length ds = 35, stabilization = 35, in [21, 41]
   [time 10.7 s]
====================================================================================================
PART B (free algebras, every m): Lemma FQ with W = X^3Z, and one Lemma-QT triple identically in m
  [ok] (B1) S_1 o W in D_3 for W = X^3Z (x o W = 3X^2Z, y o W = 0, z o W = X^3)
  [ok] (B1) R_m + t X^3Z = R_{m+t} identically in (m, t): M_t = M' + C(Xi + tW) has the (F8) graded datum at m + t
  [ok] (B2) Lemma QT data: <u^6, X^6> = 720 for u = x + y; N = <y, z>; R' = X^2YZ in K_N (y o R' = X^2Z, z o R' = X^2Y in D_3); G = C(X^5 - X^4Y)
   M_inf(m) = T<=2 + D_3 + <X^4, X^3Y, R_m> + C(X^5 - X^4Y) + C(6*X^5 + X^2*Z + 2*X*Y*Z) + C(X^6 + X^2*Y*Z)
  [ok] (B2) M_inf(m) is closed for every m (all derivatives in the span with coefficients in Q[m]), its 21 tops are independent for every m (a 21x21 minor of the top matrix is -6), H = [1, 3, 6, 5, 3, 2, 1]
   triple:
      f_1 = -1/360*x^5 - 1/72*x^4*y - 3*x^2*z^2*m + 2*x^3*z - 6*x^2*z^2 + z^4 + y^3 + x*y*z
      f_2 = -6*x^2*z^2 + z^4 + x*y^2 - y*z^2
      f_3 = -3*x^2*z^2*m + 2*x^3*z + y^2*z
  [ok] (B3) f_1, f_2, f_3 lie in Ann(M_inf(m)) identically in m (pairing with the 21 generators vanishes in Q[m])
  [ok] (B3) shape: f_i = w_i + q4_i(m) + q5_i + q6_i, w_i in D_3^perp independent and independent of m, q4_i in D_4(m)^perp identically, degree of f_i in m = [1, 0, 1]
  [ok] (B4) at m_0 = 17/5: M_inf(m_0) (r1lib M_QT) is closed with H, the specialized triple lies in its annihilator, and its local length is ds = 40, stabilization = 40, in [21, 41]
  [ok] (B5) cross-check (S): over Q(m) (Singular ring (0,m),(x,y,z),ds) the local length of (f_1, f_2, f_3) is 40
   [info, S] local lengths (ds, stabilization) at further m: [(0, 40, 40), (1, 40, 40), (-2, 40, 40), (1/2, 40, 40), (-3/8, 40, 40)]
   [time 12.9 s]
  [ok] (B6, S) m = 17/5: 2 actual free algebras (dim M' = 20) in 5 tries; M_t = M' + C(Xi + tX^3Z) closed with H, D_3 unchanged and D_4(t) = <X^4, X^3Y, R_{m+t}> for t = 1, 2 on 2 of 2
  [ok] (B6, S) m = 0: 2 actual free algebras (dim M' = 20) in 19 tries; M_t = M' + C(Xi + tX^3Z) closed with H, D_3 unchanged and D_4(t) = <X^4, X^3Y, R_{m+t}> for t = 1, 2 on 2 of 2
  [ok] (B6, S) m = -2: 2 actual free algebras (dim M' = 20) in 11 tries; M_t = M' + C(Xi + tX^3Z) closed with H, D_3 unchanged and D_4(t) = <X^4, X^3Y, R_{m+t}> for t = 1, 2 on 2 of 2
   accepted free samples: 6
certified values: {(17/5, 'alpha'): 35, (17/5, 'beta'): 35, (0, 'alpha'): 35, (0, 'beta'): 35, (-2, 'alpha'): 35, (-2, 'beta'): 35, 'B': 40}
   [time 23.6 s]
30 checks, 0 failed
ALL CHECKS PASSED
