==== N1: L = Y, H = X^3*Z, D_3 = [X^3, X^2*Y, X^2*Z, X*Z^2]
  [ok] N1 (U10): dim S_5 = 10, V_g in S_5 of dim 7, reps in S_5, S_5 = V_g + <r_1, r_2, r_3>
    c_3 of the four derivatives: [0, 0, 2*p3, -2/3*p1]
    rank of the B-columns: 18; reduced tau-system K*Tau = [[0, 0, 0, 8*p1, 0, 0, 72*p3, 0, 0, 0], [0, 8*p1, 0, 24*p3, 0, 0, 0, 0, 0, 0], [0, 0, 0, 8*p1, 0, 0, 72*p3, 0, 0, 0]], K*rhs = [0, 0, 0, 0, 0, 2/5*p1*p2]
==== N2: L = Z, H = X^3*Y, D_3 = [X^3, X^2*Y, X^2*Z, X*Z^2]
  [ok] N2 (U10): dim S_5 = 10, V_g in S_5 of dim 7, reps in S_5, S_5 = V_g + <r_1, r_2, r_3>
    c_3 of the four derivatives: [0, 0, 2*p3, -2/3*p2]
    rank of the B-columns: 17; reduced tau-system K*Tau = [[72*p3, 8*p2, 0, 0, 0, 0, 0, 0, 0, 0], [0, 24*p3, 0, 8*p2, 0, 0, 0, 0, 0, 0], [0, 24*p3, 0, 8*p2, 0, 0, 0, 0, 0, 0], [0, 0, 0, 24*p3, 0, 0, 24*p2, 0, 0, 0]], K*rhs = [0, 0, 0, 0, 6/5*p1*p3, 0, 0]
==== N3: L = Y, H = X^2*Y^2, D_3 = [X^3, X^2*Y, X*Y^2, Y^3]
  [ok] N3 (U10): dim S_5 = 10, V_g in S_5 of dim 7, reps in S_5, S_5 = V_g + <r_1, r_2, r_3>
    c_3 of the four derivatives: [3*p1, 0, 0, -1/2*p2 + 3*p3]
    rank of the B-columns: 18; reduced tau-system K*Tau = [[-18*p1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -6*p1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6*p2 - 36*p3, -6*p1, 0, 0, 0, 0, 0, 0, 0], [18*p2 - 108*p3, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 6*p2 - 36*p3, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 6*p2 - 36*p3, 0, 0, 0, 0, 0, 0, 0]], K*rhs = [0, 0, 0, 0, 0, 0]
  [ok] N1 (U12): c_3 = (0, 0, 2p3, -2p1/3); K*rhs = (0,0,0,0,0, 2/5 p1 p2); the tau-system has rank 2 on (p1, p3) != 0 (minors), B-rank 18
  [ok] N2 (U12): c_3 = (0, 0, 2p3, -2p2/3); K*rhs = (0,0,0,0, 6/5 p1 p3, 0,0); the tau-system has rank 3 on (p2, p3) != 0, B-rank 17
  [ok] N3 (U12): c_3 = (3p1, 0, 0, 3p3 - p2/2); K*rhs = 0; the tau-system has rank 3 exactly on (p1, p2 - 6p3) != 0, B-rank 18
  [ok] (U12) direct evaluation: consistency exactly as predicted at 13 points
(U11) the shear formula on an actual algebra (N1, p = (0,0,1)): substitute y -> y + c x^2 + c' x y and recompute M
  [ok] (U11) the inverse system of the substituted ideal has dimension 21
  [ok] (U11) its lifts of X^6, X^5Y have quintic parts A - 30 c X^4 Y and P - Y(20 c X^3 Y + 5 c' X^4) modulo D_5 (a translation in V_g)
(A12) anchors: one actual non-free member at the common point of the components, plus one more point of each component
    N1 p = (0, 0, 1): closed, H, datum, non-free: True; local length 39 (standard basis), 39 (stabilization)
      anchor member: Phi_1 = X^6 - 2*X^4 + 2*X^3*Y - 2*X^2*Y^2 - 2*X^3*Z + X^2*Y*Z - X^2*Z^2 + X^3 + 2*X^2*Y + 2*X*Y^2 - 2*Y^3 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z + 2*X*Z^2 - 2*Y*Z^2 + Z^3
      Phi_2 = X^5*Y + X^3*Z^2 + X^4 + X^3*Y - 2*X^2*Y^2 - 2/9*X*Y^3 - 2*X^3*Z - X^2*Y*Z + 1/6*X*Y^2*Z + 2*X^2*Z^2 - 2*X*Y*Z^2 + 2*X*Z^3 - 2*X^3 + 2*X^2*Y - X*Y^2 + Y^3 - X*Y*Z + 2*Y^2*Z - 2*X*Z^2 - Y*Z^2 - 2*Z^3
      triple = [-23/432*x^6 + 13/60*x^5*y - 2*x^3*y^3 + 2*x*y^5 - 2*y^6 + 2*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 + 1/180*x^5 - 2/9*x^4*y - x^3*y^2 - x^2*y^3 - x*y^4 - 2*y^5 - 2*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - x^2*y^2 + 2*x*y^3 - 2*y^4 + 23/36*x^3*z + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x*y^2 - 2*y^3 + 2*x*y*z + 2*z^3, -5/432*x^6 + 61/360*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - x*y^5 + y^6 - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 + 1/90*x^5 + 19/360*x^4*y + 2*x^2*y^3 - 2*x*y^4 + y^5 + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 + x^2*y^2 + y^4 - 1/36*x^3*z + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x*y^2 - y^3 + y^2*z, -11/720*x^6 + 53/360*x^5*y + 2*x^3*y^3 - 2*y^6 + 2*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 + 1/360*x^5 + 2/45*x^4*y + x^2*y^3 + y^5 + x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 - x^2*y^2 + 2*x*y^3 - y^4 - 1/3*x^3*z - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*y^3 - x*y*z - 2*y^2*z + y*z^2 + 2*z^3]
  [ok] N1 (A12) p = (0, 0, 1): an actual non-free member (H, graded data, M = M_6) in a CI of local length 39 <= 41 (both certificates; triple in I)
    N1 p = (1, 0, 0): closed, H, datum, non-free: True; local length 36 (standard basis), 36 (stabilization)
      anchor member: Phi_1 = X^6 + X^4*Z - 2*X^4 + 2*X^3*Y - 2*X^2*Y^2 - 2*X^3*Z + X^2*Y*Z - X^2*Z^2 + X^3 + 2*X^2*Y + 2*X*Y^2 - 2*Y^3 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z + 2*X*Z^2 - 2*Y*Z^2 + Z^3
      Phi_2 = X^5*Y + X^4 + X^3*Y - 2*X^2*Y^2 - 2*X*Y^3 - Y^4 + 2*X^3*Z - 2*X^2*Y*Z + 2*X^2*Z^2 - 2*X^3 + 2*X^2*Y - X*Y^2 + Y^3 - X*Y*Z + 2*Y^2*Z - 2*X*Z^2 - Y*Z^2 - 2*Z^3
      triple = [7/45*x^6 - 17/20*x^5*y - 2*x^3*y^3 + 2*x*y^5 - 2*y^6 + 2*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 - 3/10*x^5 - 1/15*x^4*y - x^3*y^2 - x^2*y^3 - x*y^4 - 2*y^5 - 2*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - x^2*y^2 + 2*x*y^3 - 2*y^4 + 55/6*x^3*z + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x*y^2 - 2*y^3 + 2*x*y*z + 2*z^3, 2/45*x^6 + 1/30*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - x*y^5 + y^6 - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 - 1/10*x^5 - 1/30*x^4*y + 2*x^2*y^3 - 2*x*y^4 + y^5 + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 + x^2*y^2 + y^4 + 10/3*x^3*z + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x*y^2 - y^3 + y^2*z, -29/120*x^6 + 7/5*x^5*y + 2*x^3*y^3 - 2*y^6 + 2*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 + 2/5*x^5 + 1/6*x^4*y + x^2*y^3 + y^5 + x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 - x^2*y^2 + 2*x*y^3 - y^4 - 143/12*x^3*z - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*y^3 - x*y*z - 2*y^2*z + y*z^2 + 2*z^3]
  [ok] N1 (A12) p = (1, 0, 0): an actual non-free member (H, graded data, M = M_6) in a CI of local length 36 <= 41 (both certificates; triple in I)
    N1 p = (0, 1, 1): closed, H, datum, non-free: True; local length 35 (standard basis), 35 (stabilization)
      anchor member: Phi_1 = X^6 + X^3*Y^2 - 2*X^4 + 2*X^3*Y - 2*X^2*Y^2 + 1/120*Y^4 - 2*X^3*Z + X^2*Y*Z - X^2*Z^2 + X^3 + 2*X^2*Y + 2*X*Y^2 - 2*Y^3 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z + 2*X*Z^2 - 2*Y*Z^2 + Z^3
      Phi_2 = X^5*Y + 1/6*X^2*Y^3 + X^3*Z^2 + X^4 + X^3*Y - 2*X^2*Y^2 - 2/9*X*Y^3 - 2*X^3*Z - X^2*Y*Z + 1/6*X*Y^2*Z + 2*X^2*Z^2 - 2*X*Y*Z^2 + 2*X*Z^3 - 2*X^3 + 2*X^2*Y - X*Y^2 + Y^3 - X*Y*Z + 2*Y^2*Z - 2*X*Z^2 - Y*Z^2 - 2*Z^3
      triple = [-407/10800*x^6 + 47/200*x^5*y - 2*x^3*y^3 + 2*x*y^5 - 2*y^6 + 2*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 + 1/45*x^5 - 23/90*x^4*y - x^3*y^2 - x^2*y^3 - x*y^4 - 2*y^5 - 2*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - 1/60*x^4 + 1/30*x^3*y - x^2*y^2 + 2*x*y^3 - 2*y^4 + 23/36*x^3*z + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x*y^2 - 2*y^3 + 2*x*y*z + 2*z^3, -143/10800*x^6 + 499/3600*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - x*y^5 + y^6 - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 - 1/180*x^5 + 19/360*x^4*y + 2*x^2*y^3 - 2*x*y^4 + y^5 + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 - 1/60*x^4 + 1/60*x^3*y + x^2*y^2 + y^4 - 1/36*x^3*z + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x*y^2 - y^3 + y^2*z, -13/900*x^6 + 119/900*x^5*y + 2*x^3*y^3 - 2*y^6 + 2*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 + 7/360*x^5 + 1/90*x^4*y + x^2*y^3 + y^5 + x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 - 1/30*x^3*y - x^2*y^2 + 2*x*y^3 - y^4 - 1/3*x^3*z - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*y^3 - x*y*z - 2*y^2*z + y*z^2 + 2*z^3]
  [ok] N1 (A12) p = (0, 1, 1): an actual non-free member (H, graded data, M = M_6) in a CI of local length 35 <= 41 (both certificates; triple in I)
    N2 p = (0, 1, 0): closed, H, datum, non-free: True; local length 39 (standard basis), 39 (stabilization)
      anchor member: Phi_1 = X^6 + X^4*Y - 2*X^4 + 2*X^3*Y - 2*X^2*Y^2 - 2*X^3*Z + X^2*Y*Z - X^2*Z^2 + X^3 + 2*X^2*Y + 2*X*Y^2 - 2*Y^3 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z + 2*X*Z^2 - 2*Y*Z^2 + Z^3
      Phi_2 = X^5*Z + X^4 + X^3*Y - 2*X^2*Y^2 - 2*X^3*Z - X^2*Y*Z + 2*X^2*Z^2 - 2*X*Y*Z^2 + 2*X*Z^3 - 2*X^3 + 2*X^2*Y - X*Y^2 + Y^3 - X*Y*Z + 2*Y^2*Z - 2*X*Z^2 - Y*Z^2 - 2*Z^3
      triple = [-2*x^4*y^2 + 2*x^2*y^4 - 2*x*y^5 + 2*y^6 + 17/40*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 + 1/15*x^5 - x^4*y - x^3*y^2 - x^2*y^3 - 2*x*y^4 - 2*y^5 - 1/10*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - 11/6*x^3*y - x^2*y^2 + 2*x*y^3 - 2*y^4 + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x*y^2 - 2*y^3 + 2*x*y*z + 2*z^3, -1/60*x^6 + 2*x^5*y - 2*x^4*y^2 - 2*x^3*y^3 - x^2*y^4 + x*y^5 + 1/4*x^5*z - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 + 2*x^3*y^2 - 2*x^2*y^3 + x*y^4 + 1/15*x^4*z + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 + 1/3*x^3*y + x^2*y^2 + y^4 + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x*y^2 - y^3 + y^2*z, -1/36*x^6 + 2*x^4*y^2 - 2*x*y^5 + 2*y^6 + 1/240*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 - 1/30*x^5 + x^3*y^2 + x*y^4 + y^5 - 11/60*x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 + 13/12*x^3*y - x^2*y^2 + 2*x*y^3 - y^4 - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*y^3 - x*y*z - 2*y^2*z + y*z^2 + 2*z^3]
  [ok] N2 (A12) p = (0, 1, 0): an actual non-free member (H, graded data, M = M_6) in a CI of local length 39 <= 41 (both certificates; triple in I)
    N2 p = (0, 0, 1): closed, H, datum, non-free: True; local length 36 (standard basis), 36 (stabilization)
      anchor member: Phi_1 = X^6 - 2*X^4 + 2*X^3*Y - 2*X^2*Y^2 - 2*X^3*Z + X^2*Y*Z - X^2*Z^2 + X^3 + 2*X^2*Y + 2*X*Y^2 - 2*Y^3 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z + 2*X*Z^2 - 2*Y*Z^2 + Z^3
      Phi_2 = X^5*Z + X^3*Y^2 + X^4 + X^3*Y - 2*X^2*Y^2 - 2*X*Y^3 - Y^4 + 2*X^3*Z - 2*X^2*Y*Z - 2/3*X*Y^2*Z + 2*X^2*Z^2 + 1/6*X*Y*Z^2 - 2*X^3 + 2*X^2*Y - X*Y^2 + Y^3 - X*Y*Z + 2*Y^2*Z - 2*X*Z^2 - Y*Z^2 - 2*Z^3
      triple = [-13/675*x^6 - 2*x^4*y^2 + 2*x^2*y^4 - 2*x*y^5 + 2*y^6 + 49/150*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 + 1/180*x^5 - x^4*y - x^3*y^2 - x^2*y^3 - 2*x*y^4 - 2*y^5 + 1/30*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - 25/9*x^3*y - x^2*y^2 + 2*x*y^3 - 2*y^4 - 1/10*x^3*z + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x*y^2 - 2*y^3 + 2*x*y*z + 2*z^3, 7/1800*x^6 + 2*x^5*y - 2*x^4*y^2 - 2*x^3*y^3 - x^2*y^4 + x*y^5 + 21/100*x^5*z - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 + 1/90*x^5 + 2*x^3*y^2 - 2*x^2*y^3 + x*y^4 - 11/90*x^4*z + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 - x^3*y + x^2*y^2 + y^4 - 1/10*x^3*z + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x*y^2 - y^3 + y^2*z, -161/2160*x^6 + 2*x^4*y^2 - 2*x*y^5 + 2*y^6 - 29/360*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 + 1/360*x^5 + x^3*y^2 + x*y^4 + y^5 + 29/120*x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 + 35/9*x^3*y - x^2*y^2 + 2*x*y^3 - y^4 - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*y^3 - x*y*z - 2*y^2*z + y*z^2 + 2*z^3]
  [ok] N2 (A12) p = (0, 0, 1): an actual non-free member (H, graded data, M = M_6) in a CI of local length 36 <= 41 (both certificates; triple in I)
    N2 p = (1, 1, 0): closed, H, datum, non-free: True; local length 38 (standard basis), 38 (stabilization)
      anchor member: Phi_1 = X^6 + X^4*Y + X^3*Z^2 - 2*X^4 + 2*X^3*Y - 2*X^2*Y^2 - 2*X^3*Z + X^2*Y*Z - X^2*Z^2 + 1/120*Z^4 + X^3 + 2*X^2*Y + 2*X*Y^2 - 2*Y^3 + 2*X^2*Z - X*Y*Z - 2*Y^2*Z + 2*X*Z^2 - 2*Y*Z^2 + Z^3
      Phi_2 = X^5*Z + 1/6*X^2*Z^3 + X^4 + X^3*Y - 2*X^2*Y^2 - 2*X^3*Z - X^2*Y*Z + 2*X^2*Z^2 - 2*X*Y*Z^2 + 2*X*Z^3 - 2*X^3 + 2*X^2*Y - X*Y^2 + Y^3 - X*Y*Z + 2*Y^2*Z - 2*X*Z^2 - Y*Z^2 - 2*Z^3
      triple = [-121/3600*x^6 - 2*x^4*y^2 + 2*x^2*y^4 - 2*x*y^5 + 2*y^6 + 81/200*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 + 1/20*x^5 - x^4*y - x^3*y^2 - x^2*y^3 - 2*x*y^4 - 2*y^5 - 1/12*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - 11/6*x^3*y - x^2*y^2 + 2*x*y^3 - 2*y^4 - 1/30*x^3*z + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x*y^2 - 2*y^3 + 2*x*y*z + 2*z^3, -121/3600*x^6 + 2*x^5*y - 2*x^4*y^2 - 2*x^3*y^3 - x^2*y^4 + x*y^5 + 13/60*x^5*z - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 + 2*x^3*y^2 - 2*x^2*y^3 + x*y^4 + 1/10*x^4*z + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 + 1/3*x^3*y + x^2*y^2 + y^4 + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x*y^2 - y^3 + y^2*z, -7/600*x^6 + 2*x^4*y^2 - 2*x*y^5 + 2*y^6 + 7/400*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 + x^3*y^2 + x*y^4 + y^5 - 1/5*x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 + 13/12*x^3*y - x^2*y^2 + 2*x*y^3 - y^4 - 1/30*x^3*z - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*y^3 - x*y*z - 2*y^2*z + y*z^2 + 2*z^3]
  [ok] N2 (A12) p = (1, 1, 0): an actual non-free member (H, graded data, M = M_6) in a CI of local length 38 <= 41 (both certificates; triple in I)
    N3 p = (1, 0, 0): closed, H, datum, non-free: True; local length 38 (standard basis), 38 (stabilization)
      anchor member: Phi_1 = X^6 + X^2*Y^3 + 2*X^4 - 2*X^2*Y^2 + 2*X*Y^3 - 2*Y^4 - 2*X^3*Z + X^2*Y*Z - X*Y^2*Z + 2*X^3 + 2*X^2*Y - 2*X*Y^2 + 2*Y^3 - X^2*Z - 2*X*Y*Z + 2*Y^2*Z - 2*X*Z^2 + Y*Z^2 + 2*Z^3
      Phi_2 = X^5*Y + 1/12*X*Y^4 + X^4 - 2*X^3*Y - 2*X^2*Y^2 - X*Y^3 + 2*Y^4 - 2*X^3*Z - X^2*Y*Z + 1/6*X*Y^2*Z - 1/18*Y^3*Z + 2*X^3 - X^2*Y + X*Y^2 - X^2*Z + 2*X*Y*Z - 2*Y^2*Z - X*Z^2 - 2*Y*Z^2 + Z^3
      triple = [7/90*x^6 - 46/45*x^5*y - 2*x^3*y^3 + 2*x*y^5 - 2*y^6 + 2*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 + 2/45*x^5 + 1/36*x^4*y - x^3*y^2 - x^2*y^3 - x*y^4 - 2*y^5 - 2*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - 1/2*x^2*y^2 - x*y^3 + 2*y^4 - 2*x^3*z + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x^2*z - 2*x*y*z + 2*y^2*z + 2*z^3, -23/540*x^6 + 91/360*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - x*y^5 + y^6 - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 + 1/360*x^5 + 1/12*x^4*y + 2*x^2*y^3 - 2*x*y^4 + y^5 + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 - 1/3*x^2*y^2 + x*y^3 + x^3*z + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x^2*z - x*y*z + x*z^2, 47/540*x^6 - 133/120*x^5*y + 2*x^3*y^3 - 2*y^6 + 2*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 + 1/120*x^5 + 1/360*x^4*y + x^2*y^3 + y^5 + x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 + 1/3*x^2*y^2 - x*y^3 + 2*y^4 - x^3*z - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*x*y*z - y^2*z - 2*x*z^2 + y*z^2 + 2*z^3]
  [ok] N3 (A12) p = (1, 0, 0): an actual non-free member (H, graded data, M = M_6) in a CI of local length 38 <= 41 (both certificates; triple in I)
    N3 p = (0, 1, 0): closed, H, datum, non-free: True; local length 39 (standard basis), 39 (stabilization)
      anchor member: Phi_1 = X^6 + X^3*Y^2 + 2*X^4 - 2*X^3*Y - 2*X^2*Y^2 + X*Y^3 - Y^4 + 2*X^3 + 2*X^2*Y - 2*X*Y^2 + 2*Y^3 - X^2*Z - 2*X*Y*Z + 2*Y^2*Z - 2*X*Z^2 + Y*Z^2 + 2*Z^3
      Phi_2 = X^5*Y + X^4 - 2*X^3*Y - 2*X^2*Y^2 - X*Y^3 + 2*Y^4 - 2*X^3*Z + 2*X^2*Y*Z - 2*Y^3*Z + 2*X^3 - X^2*Y + X*Y^2 - X^2*Z + 2*X*Y*Z - 2*Y^2*Z - X*Z^2 - 2*Y*Z^2 + Z^3
      triple = [-1/15*x^6 - 28/15*x^5*y - 2*x^3*y^3 + 2*x*y^5 - 2*y^6 + 2*x^5*z + 2*x^4*y*z - 2*x*y^4*z - x^4*z^2 - x^3*y*z^2 + x^2*y^2*z^2 - 2*y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 - y^3*z^3 - x^2*z^4 + 2*x*y*z^4 - y^2*z^4 - z^6 + 1/6*x^5 + 1/6*x^4*y - x^3*y^2 - x^2*y^3 - x*y^4 - 2*y^5 - 2*x^4*z - x^3*y*z - x^2*y^2*z + x*y^3*z + 2*y^4*z + 2*x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + x^2*z^3 - x*y*z^3 + x*z^4 - y*z^4 - 2*z^5 - 10*x^2*y^2 - x*y^3 + 2*y^4 - 2*x^3*z + 2*x^2*y*z + x*y^2*z - y^3*z + x^2*z^2 - 2*y^2*z^2 - x*z^3 - 2*y*z^3 - z^4 + x^2*z - 2*x*y*z + 2*y^2*z + 2*z^3, 7/360*x^6 + 11/30*x^5*y + 2*x^4*y^2 - 2*x^3*y^3 - 2*x^2*y^4 - x*y^5 + y^6 - x^4*y*z - 2*x^3*y^2*z - x^2*y^3*z - 2*x^3*y*z^2 + 2*x^2*y^2*z^2 - x^3*z^3 + 2*x*y^2*z^3 + 2*y^3*z^3 - x*y*z^4 - 2*y^2*z^4 + x*z^5 - 2*y*z^5 - 1/30*x^5 + 2/15*x^4*y + 2*x^2*y^3 - 2*x*y^4 + y^5 + 2*x^3*y*z + 2*x*y^3*z + x^3*z^2 - x^2*y*z^2 - 2*x*y^2*z^2 - y^3*z^2 + 2*x^2*z^3 - x*y*z^3 - 2*y^2*z^3 + x*z^4 - 2*y*z^4 + 2*z^5 + 2*x^2*y^2 + x*y^3 + x^3*z + 2*x^2*y*z - 2*x*y^2*z + y^3*z - 2*x*y*z^2 - 2*x*z^3 - y*z^3 + z^4 + x^2*z - x*y*z + x*z^2, 19/180*x^6 - 7/12*x^5*y + 2*x^3*y^3 - 2*y^6 + 2*x^5*z - 2*x^3*y^2*z + 2*x*y^4*z + 2*y^5*z + 2*x^4*z^2 + x^2*y^2*z^2 - 2*x*y^3*z^2 - y^4*z^2 + x^2*y*z^3 - x*y^2*z^3 + 2*y^3*z^3 + x^2*z^4 - 2*x*y*z^4 - y^2*z^4 + x*z^5 + y*z^5 - 1/10*x^5 - 1/15*x^4*y + x^2*y^3 + y^5 + x^4*z + 2*x^3*y*z - x^2*y^2*z + 2*x*y^3*z + y^4*z - x^3*z^2 + x*y^2*z^2 - 2*y^3*z^2 - x^2*z^3 + y^2*z^3 - x*z^4 - 2*y*z^4 + z^5 + 6*x^2*y^2 - x*y^3 + 2*y^4 - x^3*z - x^2*y*z + y^3*z - 2*x^2*z^2 - 2*x*y*z^2 - y^2*z^2 + x*z^3 + y*z^3 + 2*x*y*z - y^2*z - 2*x*z^2 + y*z^2 + 2*z^3]
  [ok] N3 (A12) p = (0, 1, 0): an actual non-free member (H, graded data, M = M_6) in a CI of local length 39 <= 41 (both certificates; triple in I)
(S12) random members at random points of the non-free consistent locus are closed with H and the graded data
    N1 p = (0, -3, 1) ok
    N1 p = (-2, 0, -2) ok
    N2 p = (0, 1, -1) ok
    N2 p = (1, -1, 0) ok
    N3 p = (-3, 1, -3) ok
  [ok] (S12) 5 of 5 random members accepted
time 14.0s
18 checks, 0 failed
ALL CHECKS PASSED
