Part G: Gorenstein determinant
  ok   secant: ideal of 4x4 minors of the derivative matrix = (det Q)
  ok   tangent: ideal of 4x4 minors of the derivative matrix = (det Q)
Part Q: rank Q <= 1 operator; Q = (lam Y + kap Z)^2 symbolic (charts lam != 0 and lam = 0), or Q = 0
  ok   secant, rank 1, lam != 0: Q = (lam Y + kap Z)^2
  ok   secant, rank 1, lam != 0: l o Q = 0 for l = kap y - lam z
     <h_2, Q> = 2*lam^2
  ok   secant, rank 1, lam != 0: u3 = g2 l + c x h2: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   secant, rank 1, lam = 0: Q = (lam Y + kap Z)^2
  ok   secant, rank 1, lam = 0: l o Q = 0 for l = kap y - lam z
     <h_2, Q> = 2
  ok   secant, rank 1, lam = 0: u3 = g2 l + c x h2: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   secant, Q = 0: Q = 0
  ok   secant, Q = 0: u3 = 12*g3*y^2*z + (-12*g2)*y*z^2: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
     binary part of u3: g2 (cz y - cy z), cz, cy = 12*g3 , 12*g2 (zero only on the locus treated next)
  ok   secant, Q = 0, g2^2 o G5 = 0: Q = 0 and g_2^2 o G_5 = 0
  ok   secant, Q = 0, g2^2 o G5 = 0: u3 = y^2*z: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   secant, Q = 0, g2^2 o G5 = 0: u3 = y*z^2: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   tangent, rank 1, lam != 0: Q = (lam Y + kap Z)^2
  ok   tangent, rank 1, lam != 0: l o Q = 0 for l = kap y - lam z
     <h_2, Q> = 2*lam^2
  ok   tangent, rank 1, lam != 0: u3 = g2 l + c x h2: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   tangent, rank 1, lam = 0: Q = (lam Y + kap Z)^2
  ok   tangent, rank 1, lam = 0: l o Q = 0 for l = kap y - lam z
     <h_2, Q> = 2
  ok   tangent, rank 1, lam = 0: u3 = g2 l + c x h2: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   tangent, Q = 0: Q = 0
  ok   tangent, Q = 0: u3 = 120*g5*y*z^2 + (-24*g4)*z^3: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
     binary part of u3: g2 (cz y - cy z), cz, cy = 120*g5 , 24*g4 (zero only on the locus treated next)
  ok   tangent, Q = 0, g2^2 o G5 = 0: Q = 0 and g_2^2 o G_5 = 0
  ok   tangent, Q = 0, g2^2 o G5 = 0: u3 = y*z^2: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   tangent, Q = 0, g2^2 o G5 = 0: u3 = z^3: u3 o F5 binary, orthogonal to g2; (u3 + u4) o Phi in T_{<=1}; u3 kills D_5 = S_1 o F6
  ok   anchor M = T_{<=1} + D_2 + B_3 + C XY^2 + <Y^4, Z^4, Y^2 Z^2> + <Y^5, Z^5> + C(Y^6 + Z^6): closed, H = (1,3,5,5,3,2,1), x^2 in I, F_5 = 0
  ok   anchor: u = g_2 y = y^2 z (l = y) and u = y z^2 (l = z) lie in (I : m^2), binary cubic parts nonzero (turning operators)
Part X: F6 = X Y^5
  ok   two linear conditions on p in z S_2 (dimension 3): a nonzero solution exists
  ok   p o F5 in <XY, Y^2> modulo the conditions
  ok   (p + u4) o Phi in T_{<=1} modulo the conditions; p kills D_5 = <Y^5, X Y^4>
Part P (pure sextic): w4 = z h3 kills Y^6, so w4 lies in (I : m^2) for every algebra with D_6 = C Y^6 and m^7 in I
  ok   z h3 o Y^6 = 0 for h3 symbolic
     parts G, Q, X, P done (0.7 s)
Part R1a: T5 binary, p in z S_2 with p o T5 = 0: u = p - (k/360) y^4 in (I : m^2); all symbolic, modulo the ideal (p o T5)
  ok   z o H5 = al Y^4 + be y o T5 + ga z o T5 lies in U (general H5 with this property, up to h0 Y^5)
  ok   p kills U modulo (p o T5); u o Phi, u o Psi in T_{<=1} modulo (p o T5)
Part R1b (sanity): cor:variants(b), two quartics with the single common root Y = [0:1:0], simple for p: length <= 32 + 2 + 2(k-5)
  ok   p = z(y^3 + y*z^2 + z^3), q = z(y^3 - 2*y^2*z + 5*z^3), N = y^6: finite length 36 <= 36
  ok   p = z(y^3 + y*z^2 + z^3), q = z(y^3 - 2*y^2*z + 5*z^3), N = y^7: finite length 38 <= 38
  ok   with higher terms: finite length 34 <= 38
  ok   p = z(y^3 - y*z^2), q = z(y^3 + 3*z^3), N = y^6: finite length 36 <= 36
  ok   p = z(y^3 - y*z^2), q = z(y^3 + 3*z^3), N = y^7: finite length 38 <= 38
  ok   with higher terms: finite length 34 <= 38
Part R2: Proposition prop:M; e(L) = 3! covol(Newton polyhedron) of the monomial ideals L (exact); a fixed triple in L
  ok   secant, Q_D = Y^2: e(L) = 39
     fixed triple: [2*y^7 + 2*z^6 + 3*x*y^3 + 6*x*y^2*z + 2*y^3*z + 9*x*y*z^2 + 2*y^2*z^2 + x*z^3 + y*z^3 + 8*x*y*z + x*z^2 + 4*x^2, 9*y^7 + 9*z^6 + 4*x*y^3 + 8*x*y^2*z + 3*y^3*z + 3*x*y*z^2 + 2*y^2*z^2 + 3*x*z^3 + y*z^3 + 7*x*y*z + x*z^2 + 4*x^2, 3*y^7 + 4*z^6 + 6*x*y^3 + 7*x*y^2*z + 7*y^3*z + 3*x*y*z^2 + 4*y^2*z^2 + 6*x*z^3 + 9*y*z^3 + x*y*z + 2*x*z^2 + 6*x^2]
  ok   secant, Q_D = Y^2: the fixed triple lies in L and has local length 39 = e(L) < 40 (local standard basis)
  ok   secant, Q_D = Z^2: e(L) = 39
     fixed triple: [2*y^7 + 2*z^6 + 3*x*y^3 + 6*x*y^2*z + 2*y^3*z + 9*x*y*z^2 + 2*y^2*z^2 + x*z^3 + y*z^3 + 8*x*y^2 + x*y*z + 4*x^2, 9*y^7 + 9*z^6 + 4*x*y^3 + 8*x*y^2*z + 3*y^3*z + 3*x*y*z^2 + 2*y^2*z^2 + 3*x*z^3 + y*z^3 + 7*x*y^2 + x*y*z + 4*x^2, 3*y^7 + 4*z^6 + 6*x*y^3 + 7*x*y^2*z + 7*y^3*z + 3*x*y*z^2 + 4*y^2*z^2 + 6*x*z^3 + 9*y*z^3 + x*y^2 + 2*x*y*z + 6*x^2]
  ok   secant, Q_D = Z^2: the fixed triple lies in L and has local length 39 = e(L) < 40 (local standard basis)
  ok   secant, Q_D = YZ (information): e(L) = 37
     fixed triple: [2*y^7 + 2*z^6 + 3*x*y^3 + 6*x*y^2*z + 2*y^3*z + 9*x*y*z^2 + 2*y^2*z^2 + x*z^3 + y*z^3 + 8*x*y^2 + x*z^2 + 4*x^2, 9*y^7 + 9*z^6 + 4*x*y^3 + 8*x*y^2*z + 3*y^3*z + 3*x*y*z^2 + 2*y^2*z^2 + 3*x*z^3 + y*z^3 + 7*x*y^2 + x*z^2 + 4*x^2, 3*y^7 + 4*z^6 + 6*x*y^3 + 7*x*y^2*z + 7*y^3*z + 3*x*y*z^2 + 4*y^2*z^2 + 6*x*z^3 + 9*y*z^3 + x*y^2 + 2*x*z^2 + 6*x^2]
  ok   secant, Q_D = YZ (information): the fixed triple lies in L and has local length 37 = e(L)
  ok   aligned tangent, Q_D = Y^2 (information): e(L) = 42
     fixed triple: [2*y^7 + 2*y^5*z + 3*x*y^3 + 6*x*y^2*z + 9*x*y*z^2 + 2*y^2*z^2 + x*z^3 + 2*y*z^3 + z^4 + 8*x*y*z + x*z^2 + 4*x^2, 9*y^7 + 9*y^5*z + 4*x*y^3 + 8*x*y^2*z + 3*x*y*z^2 + 3*y^2*z^2 + 3*x*z^3 + 2*y*z^3 + z^4 + 7*x*y*z + x*z^2 + 4*x^2, 3*y^7 + 4*y^5*z + 6*x*y^3 + 7*x*y^2*z + 3*x*y*z^2 + 7*y^2*z^2 + 6*x*z^3 + 4*y*z^3 + 9*z^4 + x*y*z + 2*x*z^2 + 6*x^2]
  ok   aligned tangent, Q_D = Y^2 (information): the fixed triple lies in L and has local length 42 = e(L)
Part R2b: H2 S_2 = S_4 for a base-point-free pencil H2 of binary quadrics (symbolic coefficients)
  ok   the 5x5 minors of the 6x5 matrix of H2 S_2 generate Res(h, h') (c0, c1, c2, d0, d1, d2): rank 5 iff Res != 0 (h, h' != 0)
Part R3: T5 not binary (Q_D = Y^2 when L = Y)
  ok   (i) T5 = X Z^4: p = y z^2 sends Phi and the lift of T5 into T_{<=1}
     condition matrix (rows YZ, Z^2; columns z y^2, y z^2, z^3): [(12*g2, 12*g3, 24*g4), (6*g3, 12*g4, 60*g5)]
  ok   (ii) G5 symbolic, p = (2x2 minors of the condition matrix): p o G5 in C Y^2, u = p - (k/12) x y^2 + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (ii) rank-one family alpha: G5 = g0*Y^5 + g1*Y^4*Z + g5*Z^5: all 2x2 minors of the condition matrix vanish
  ok   (ii) family alpha, p = y^2*z: p o G5 in C Y^2, u = p - (k/12) x y^2 + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (ii) family alpha, p = y*z^2: p o G5 in C Y^2, u = p - (k/12) x y^2 + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (ii) rank-one family beta: G5 = g0*Y^5 + g1*Y^4*Z + g2*Y^3*Z^2 + 2*lam*g2*Y^2*Z^3 + 2*lam^2*g2*Y*Z^4 + 4/5*lam^3*g2*Z^5: all 2x2 minors of the condition matrix vanish
  ok   (ii) family beta, p = (-2*lam)*y^2*z + y*z^2: p o G5 in C Y^2, u = p - (k/12) x y^2 + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (ii) family beta, p = (-4*lam^2)*y^2*z + z^3: p o G5 in C Y^2, u = p - (k/12) x y^2 + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (ii) the condition matrix is [[12 g2, 12 g3, 24 g4], [6 g3, 12 g4, 60 g5]]
  ok   (ii) radical of the minors ideal = ideal of the closure of family beta (g2, 2 l g2, 2 l^2 g2, 4/5 l^3 g2); family alpha lies in it
  ok   (iii) b != 0: u = z^3 + c y^4 sends Phi and the lift of T5 into T_{<=1}
  ok   (iv) T5 = X Y^4, A4 not in C Y^4: u = z^3 + c y^4 works
  ok   (v) R4 symbolic, p = (2x2 minors of the condition matrix): p o R4 = 0, u = p + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (v) the condition matrix is [[6 r1, 4 r2, 6 r3], [4 r2, 6 r3, 24 r4]]
  ok   (v) rank-one family R4 = r4 Z^4: all 2x2 minors vanish
  ok   (v) R4 = r4 Z^4, p = y^2*z: p o R4 = 0, u = p + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (v) R4 = r4 Z^4, p = y*z^2: p o R4 = 0, u = p + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (v) rank-one family R4 = r1 (Y^3 Z + 3/2 l Y^2 Z^2 + l^2 Y Z^3 + l^3/4 Z^4): all 2x2 minors vanish
  ok   (v) R4 = r1 (Y^3 Z + 3/2 l Y^2 Z^2 + l^2 Y Z^3 + l^3/4 Z^4), p = (-lam)*y^2*z + y*z^2: p o R4 = 0, u = p + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (v) R4 = r1 (Y^3 Z + 3/2 l Y^2 Z^2 + l^2 Y Z^3 + l^3/4 Z^4), p = (-lam^2)*y^2*z + z^3: p o R4 = 0, u = p + c y^4: u o Phi, u o (lift of T5) in T_{<=1}
  ok   (v) radical of the minors ideal = ideal of the closure of the second family (r1, 3/2 l r1, l^2 r1, l^3/4 r1); the family r4 Z^4 lies in it
     (a rank-one 2x3 matrix has first row 0, or first row != 0 and second row = l * first row; this gives the two families)
     part R (1.9 s)
Part D: (R2), aligned tangent: the deformation M_t (t, beta and all tails symbolic)
  ok   chart Z^4: the 20 generators of M_t have distinct pivot monomials with constant nonzero coefficients (dim M_t = 20 for all t)
  ok   chart Z^4: top forms per degree [1, 3, 5, 5, 3, 2, 1] with pivot coefficients 1 or 6: H(M_t) = (1,3,5,5,3,2,1) for every t
  ok   chart Z^4: D_5(M_t) = <Y^5, Y^4 Z + t X Y^4>, D_4(M_t) = <Y^4, Y^3 Z + t X Y^3, R_4>
  ok   chart Z^4: D1 the closure defects of M_t do not depend on t (26 nonzero defect coefficients = closure conditions of M)
     closure conditions of M (coefficient of the monomial in NF(v o G)); f*_i = coefficients of Phi, p*_i = of Psi:
       Q_t      x, X^2   : 3/4*Psi[X^3*Y] = 0
       Q_t      y, X^2   : 1/2*Psi[X^2*Y^2] = 0
       Q_t      z, X^2   : 1/4*Psi[X^2*Y*Z] = 0
       R        x, X^2   : 3*rho^-1*Phi[X^3*Z] = 0
       R        y, X^2   : rho^-1*Phi[X^2*Y*Z] = 0
       R        z, X^2   : 2*rho^-1*Phi[X^2*Z^2] = 0
       y o Phi  x, X^2   : 3*Phi[X^3*Y] = 0
       y o Phi  z, X^2   : Phi[X^2*Y*Z] = 0
       Psi_t    x, X^3   : 4*Psi[X^4] = 0
       Psi_t    x, X^2*Y : 3*Psi[X^3*Y] = 0
       Psi_t    x, X^2*Z : 3*Psi[X^3*Z] = 0
       Psi_t    x, X*Y*Z : 2*Psi[X^2*Y*Z] = 0
       Psi_t    x, X*Z^2 : 2*Psi[X^2*Z^2] = 0
       Psi_t    x, X^2   : 3*Psi[X^3] = 0
       Psi_t    z, X^3   : Psi[X^3*Z] = 0
       Psi_t    z, X^2*Y : Psi[X^2*Y*Z] = 0
       Psi_t    z, X^2*Z : 2*Psi[X^2*Z^2] = 0
       Psi_t    z, X*Y*Z : 2*Psi[X*Y*Z^2] = 0
       Psi_t    z, X*Z^2 : 3*Psi[X*Z^3] = 0
       Psi_t    z, X^2   : -1/15*Phi[X^2*Y^2] + Psi[X^2*Z] = 0
       Phi      x, X^3   : 4*Phi[X^4] = 0
       Phi      x, X^2*Y : 3*Phi[X^3*Y] = 0
       Phi      x, X^2*Z : 3*Phi[X^3*Z] = 0
       Phi      x, X*Y*Z : 2*Phi[X^2*Y*Z] = 0
       Phi      x, X*Z^2 : 2*Phi[X^2*Z^2] = 0
       Phi      x, X^2   : -1/15*a*Phi[X^2*Y^2] + 3*Phi[X^3] = 0
  ok   chart Z^4: D2 on the solved family (the conditions above solved: listed coefficients 0, Phi[X^3] = a beta/3, Psi[X^2 Z] = beta) M_t is closed identically in t and the remaining tails
  ok   chart Z^4: D2 on the solved family N = M_{<=3} + <Lambda, R, y o Phi, Phi> has dimension 18 and is closed
  ok   chart Z^4: D4 y^2 o Phi - 30 Lambda lies in M_{<=3}; z o Phi = mu Lambda + rho R; y o Psi = 4 Q (so M_0 is spanned by elements of M: M_0 = M)
  ok   D6 P = X Y^4 + (beta/3) X^3: P_X = Lambda, P_Y = 4 X Y^3, P_Z = 0 (beta symbolic); d(X Y^3) = Y^3, 3 X Y^2, 0 lie in D_3
  ok   D5 x o (Y^4 Z + t X Y^4) = t Y^4: the second quintic is not binary for t != 0
  ok   D0 (x^2 l) o G = 0 for G in D_3 (l symbolic): f = x^2 + x^2 l + O(4) makes M_{<=3} = T_{<=1} + D_2 + D_3 homogeneous
  ok   chart Y Z^3: the 20 generators of M_t have distinct pivot monomials with constant nonzero coefficients (dim M_t = 20 for all t)
  ok   chart Y Z^3: top forms per degree [1, 3, 5, 5, 3, 2, 1] with pivot coefficients 1 or 6: H(M_t) = (1,3,5,5,3,2,1) for every t
  ok   chart Y Z^3: D_5(M_t) = <Y^5, Y^4 Z + t X Y^4>, D_4(M_t) = <Y^4, Y^3 Z + t X Y^3, R_4>
  ok   chart Y Z^3: D1 the closure defects of M_t do not depend on t (26 nonzero defect coefficients = closure conditions of M)
  ok   chart Y Z^3: D2 on the solved family (the conditions above solved: listed coefficients 0, Phi[X^3] = a beta/3, Psi[X^2 Z] = beta) M_t is closed identically in t and the remaining tails
  ok   chart Y Z^3: D2 on the solved family N = M_{<=3} + <Lambda, R, y o Phi, Phi> has dimension 18 and is closed
  ok   chart Y Z^3: D4 y^2 o Phi - 30 Lambda lies in M_{<=3}; z o Phi = mu Lambda + rho R; y o Psi = 4 Q (so M_0 is spanned by elements of M: M_0 = M)
  ok   chart Y^2 Z^2: the 20 generators of M_t have distinct pivot monomials with constant nonzero coefficients (dim M_t = 20 for all t)
  ok   chart Y^2 Z^2: top forms per degree [1, 3, 5, 5, 3, 2, 1] with pivot coefficients 1 or 6: H(M_t) = (1,3,5,5,3,2,1) for every t
  ok   chart Y^2 Z^2: D_5(M_t) = <Y^5, Y^4 Z + t X Y^4>, D_4(M_t) = <Y^4, Y^3 Z + t X Y^3, R_4>
  ok   chart Y^2 Z^2: D1 the closure defects of M_t do not depend on t (26 nonzero defect coefficients = closure conditions of M)
  ok   chart Y^2 Z^2: D2 on the solved family (the conditions above solved: listed coefficients 0, Phi[X^3] = a beta/3, Psi[X^2 Z] = beta) M_t is closed identically in t and the remaining tails
  ok   chart Y^2 Z^2: D2 on the solved family N = M_{<=3} + <Lambda, R, y o Phi, Phi> has dimension 18 and is closed
  ok   chart Y^2 Z^2: D4 y^2 o Phi - 30 Lambda lies in M_{<=3}; z o Phi = mu Lambda + rho R; y o Psi = 4 Q (so M_0 is spanned by elements of M: M_0 = M)
Part D7 (sanity): one random member of the stratum (rank z^2 o R_4 = 2, rho != 0), t symbolic: binary cubic parts of (I_t : m^2) cap m^3
     member (seed 5321): R_4 = Y^2 Z^2 + Z^4 and the solved family of D2 with {'a': -1, 'h0': 1, 'mu': -1, 'rho': 3, 'f0_0': 2, 'f1_0': -1, 'f1_1': 1, 'f1_2': 2, 'f2_0': 3, 'f2_1': 2, 'f2_2': -2, 'f2_3': -1, 'f2_4': -3, 'f2_5': -3, 'f3_0': 1, 'f3_1': -2, 'f3_2': -1, 'f3_3': 2, 'f3_4': 1, 'f3_5': -3, 'f3_6': 2, 'f3_7': 1, 'f3_8': -3, 'f3_9': 2, 'f4_0': -3, 'f4_1': 3, 'f4_2': 1, 'f4_3': 1, 'f4_4': 2, 'f4_5': 2, 'f4_6': 2, 'f4_7': -3, 'f4_8': 3, 'f4_9': 3, 'f4_10': -3, 'f4_11': 2, 'f4_12': 2, 'f4_13': -3, 'f4_14': 2, 'p0_0': 2, 'p1_0': 3, 'p1_1': -1, 'p1_2': -3, 'p2_0': 2, 'p2_1': -1, 'p2_2': 3, 'p2_3': 1, 'p2_4': 2, 'p2_5': 2, 'p3_0': 1, 'p3_1': -3, 'p3_2': 1, 'p3_3': -2, 'p3_4': 2, 'p3_5': 3, 'p3_6': -1, 'p3_7': -3, 'p3_8': 2, 'p3_9': -2, 'p4_0': 3, 'p4_1': -3, 'p4_2': -3, 'p4_3': 3, 'p4_4': -1, 'p4_5': 1, 'p4_6': 3, 'p4_7': 3, 'p4_8': 1, 'p4_9': 1, 'p4_10': 2, 'p4_11': 2, 'p4_12': 1, 'p4_13': -1, 'p4_14': -1}
  ok   member: M_t closed for symbolic t (closure computed independently), H = (1,3,5,5,3,2,1); M_0 closed, same H
  ok   t = 0: no binary cubic part in (I_0 : m^2) cap m^3 (rank 0), so no turning operator: the deformation is needed
  ok   t general: binary cubic parts of (I_t : m^2) cap m^3 have rank 1 > 0 (turning operators, as in case (R3))
total time 4.4 s
ALL CHECKS PASSED
