Part A: sextic tails
  ok   L6 = [Xq^6, Xq^5*Yq, Xq^5*Zq, Xq^4*Yq^2]
Part B (sanity): the criterion (L5)-(L3) agrees with direct closedness (random data, both branches; random.seed(7) here)
   trial 0 (delta = 0): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
   trial 1 (delta = 0): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
   trial 2 (delta = 0): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
   trial 3 (delta = 0): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
   trial 4 (delta = 1): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
   trial 5 (delta = 1): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
   trial 6 (delta = -3): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
   trial 7 (delta = 2): conditions True, closed True, H = [1, 3, 6, 5, 2, 2, 1, 1]; perturbed: conditions False, closed False
  ok   (S) criterion agrees with direct closedness, H = (1,3,6,5,2,2,1,1) on 8 solved members
Part C: delta = 0
  ok   (L4) holds identically for Phi5 = Phi5^0(beta,gamma) + <X^5,X^4Y,X^4Z,X^3Y^2>; d = -10/21*be^2 + 2*ep, d' = 0
  ok   homogeneous (L4) (delta = 0): solution space of dimension 4 (= 4)
   (f) W general: <(Y^2, Z^3), (Z^2, Y^3)> [sigma 10]: dim A_D = 6, r_raw = 12; sigma + r_raw = 22, sigma + 2 dim A_D = 22
   (f) W = <(Y^2,0),(YZ,0)> [sigma 8]: dim A_D = 8, r_raw = 15; sigma + r_raw = 23, sigma + 2 dim A_D = 24
   (f) W = grad of (Y^3 + Z^3, 0) = <(Y^2,0),(Z^2,0)> [sigma 8]: dim A_D = 8, r_raw = 15; sigma + r_raw = 23, sigma + 2 dim A_D = 24
   (f) W = <(Y^2,0),(0,Y^3)> [sigma 8]: dim A_D = 8, r_raw = 15; sigma + r_raw = 23, sigma + 2 dim A_D = 24
   (f) W = <(0,Y^3),(0,Z^3)> [sigma 8]: dim A_D = 8, r_raw = 14; sigma + r_raw = 22, sigma + 2 dim A_D = 24
   (f) W = <(0,Y^2 Z),(Y^2, Y^3)> [sigma 8]: dim A_D = 7, r_raw = 13; sigma + r_raw = 21, sigma + 2 dim A_D = 22
   (c4) nu_3(C4), z = y^2 + y^3, + Z^3 [sigma 7]: dim A_D = 7, r_raw = 14; sigma + r_raw = 21, sigma + 2 dim A_D = 21
   (c4) nu_3(C4), z = y^3, + random G [sigma 7]: dim A_D = 6, r_raw = 12; sigma + r_raw = 19, sigma + 2 dim A_D = 19
   (c4) nu_3(C4) on the line, + X Z^2 [sigma 7]: dim A_D = 7, r_raw = 13; sigma + r_raw = 20, sigma + 2 dim A_D = 21
   (l) C[X,Y]_3 + Z^3 [sigma 5]: dim A_D = 7, r_raw = 14; sigma + r_raw = 19, sigma + 2 dim A_D = 19
   (l) C[X,Y]_3 + X Z^2 [sigma 5]: dim A_D = 7, r_raw = 13; sigma + r_raw = 18, sigma + 2 dim A_D = 19
  ok   (S) bad-flag representatives: r_raw <= 2 dim A_D, sigma + r_raw <= 24, sigma + 2 dim A_D <= 25
  ok   (U) nu_3(C4) = [1/6*XX^3, 1/2*XX^2*YY, 1/2*XX*YY^2 + 1/2*aa*XX^2*ZZ, 1/6*YY^3 + 1/2*bb*XX^2*ZZ + aa*XX*YY*ZZ]: the 3x3 minors of (z o, y o) generate the unit ideal of QQ[a,b]
   count: Q has dimension 15 + 3 + 4 + 30 + 20 = 72, section rank 4*5 = 20, components >= 52; bad part <= max(24 + 7 + 20, 25 + 6 + 20) = 51
  ok   (U) 51 < 52
  ok   (S) dim A_D <= 7 on random flags
Part D: delta != 0
  ok   (L4z) holds identically for Phi5 = Phi5^0 + phi, phi in C[X,Y]_5 + <X^4Z, X^3YZ>
  ok   (L3Psi z): z o Psi4 lies in E0 identically: (1/(2*de)*q2 + (-5*be*ga)/(21*de))*XD^3 + (2*ga)/7*XD^2*YD
   wbar = class of (L3Psi y) modulo E0 = (6/de*p4 + (-4*de)/35)*XD*YD^2 + 10/de*p5*YD^3
  ok   wbar lies in <XY^2, Y^3> and equals ((12 p4 - 8 de^2/35) X Y^2 + 20 p5 Y^3)/(2 de)
Part E: members of the two delta != 0 families with certified complete intersections
   first family (wbar = 0): D = <X^3, X^2Y, XZ^2, YZ^2, XYZ>, Phi = X^7 + X^4Y^2 + X^3Y^2 + (2/105) XY^4: closed True, H = [1, 3, 6, 5, 2, 2, 1, 1], delta = 1, wbar = 0
      cubic parts [y^3, x^2*z, x*y^2 + y^2*z + z^3] pair to zero with XYZ: True; local standard basis: local length 32
      complete intersection (1/420*x^4*y - 1/105*x^3*y + y^3, x^2*z, -1/420*x^5 - 1/105*x^4 + x*y^2 + y^2*z + z^3): local length 32 (dim S/(C + m^N) = 32 for N = 10 and N = 11)
  ok   (A) first family (wbar = 0): D = <X^3, X^2Y, XZ^2, YZ^2, XYZ>, Phi = X^7 + X^4Y^2 + X^3Y^2 + (2/105) XY^4: member of the family, all three generators in Ann(M), local length 32 = 32 < 42
   second family (wbar != 0, wbar in D/E0): D = <X^3, X^2Y, XY^2, XZ^2, XYZ>, Phi = X^7 + X^4Y^2: closed True, H = [1, 3, 6, 5, 2, 2, 1, 1], delta = 1, wbar = -4/35*Xq*Yq^2
      cubic parts [y^3, x^2*z, y^2*z + z^3] pair to zero with XYZ: True; local standard basis: local length 33
      complete intersection (y^3, x^2*z, -1/105*x^5 + x^2*y^2 + y^2*z + z^3): local length 33 (dim S/(C + m^N) = 33 for N = 10 and N = 11)
  ok   (A) second family (wbar != 0, wbar in D/E0): D = <X^3, X^2Y, XY^2, XZ^2, XYZ>, Phi = X^7 + X^4Y^2: member of the family, all three generators in Ann(M), local length 33 = 33 < 42
total time 12.6 s
ALL CHECKS PASSED
