(U5) two points, common line: orbits of G_4 modulo <X^4, Y^4>
  [ok] X -> sX, Y -> tY sends (a, b, c) to (a s^3 t, b s^2 t^2, c s t^3); for b, c != 0 and s/t = c/b this is proportional to (ac/b^2, 1, 1)
  [ok] the invariant: for b != 0 the ratio ac/b^2 is unchanged by X -> sX, Y -> tY
  [ok] rank of (H_X, H_Y) modulo <X^3, Y^3> is two iff 9ac - 4b^2 != 0 (lambda != 4/9 on the family)
(U6) double point 2X on the line: orbits of G_4 modulo <X^4, X^3Y>
  [ok] Y -> Y + uX preserves <X^k, X^(k-1)Y> and sends (a, b, c) to (a + 3bu + 6cu^2, b + 4cu, c) modulo <X^4, X^3Y>
(A5) anchors at infinity for the single orbits
  [ok] two points, G_4 = X^2Y^2: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 + 60*Z^3, Y^6 - 180*Y*Z^2]
  [ok] two points, G_4 = X^2Y^2: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] two points, G_4 = X^2Y^2: the triple annihilates M_Q
    triple [y^6 + x*y^4 + 2*x^3*y + x*y^3 - y^2*z + 2*y*z^2, x^6 + 2*x^2*y^3 - x^3*y + x*y^3 - y^2*z - 2*z^3, -z^6 - 2*x^3*y^2 - x^3*y + 2*x*y^3 - 2*x^2*z - 2*x*z^2]: local length 40 (standard basis), 40 (stabilization)
  [ok] two points, G_4 = X^2Y^2: local length 40 <= 41 by both certificates
  [ok] two points, G_4 = X^3Y + XY^3: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 - 360*X*Z^2, Y^6 + 180*Y*Z^2 + 120*Z^3]
  [ok] two points, G_4 = X^3Y + XY^3: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] two points, G_4 = X^3Y + XY^3: the triple annihilates M_Q
    triple [y^6 - x*y^4 + x^3*y - 2*x^2*y^2 - x*y^3 + 2*x^2*z - z^3, -y^6 + 2*x*y^4 - 2*x^3*y + x^2*y^2 + 2*x*y^3 - 2*y^2*z + 2*y*z^2, x^6 + 2*x*y^4 - 2*x^3*y + x^2*y^2 + 2*x*y^3 - 2*x^2*z + x*z^2]: local length 40 (standard basis), 40 (stabilization)
  [ok] two points, G_4 = X^3Y + XY^3: local length 40 <= 41 by both certificates
  [ok] double point, G_4 = Y^4 + X^2Y^2: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 + 120*Z^3, X^5*Y]
  [ok] double point, G_4 = Y^4 + X^2Y^2: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] double point, G_4 = Y^4 + X^2Y^2: the triple annihilates M_Q
    triple [x^4*y^2 - 18*x^2*y^2 + 3*y^4 + 2*x^2*z + y*z^2, x^6 + x^2*y^3 - 12*x^2*y^2 + 2*y^4 + x^2*y*z + x^2*z - z^3, x^4*y^2 - 12*x^2*y^2 - x*y^3 + 2*y^4 + x*y*z]: local length 41 (standard basis), 41 (stabilization)
  [ok] double point, G_4 = Y^4 + X^2Y^2: local length 41 <= 41 by both certificates
  [ok] double point, G_4 = Y^4: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 + 60*Z^3, X^5*Y]
  [ok] double point, G_4 = Y^4: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] double point, G_4 = Y^4: the triple annihilates M_Q
    triple [y^6 + y^5 + 2*x^2*y^2 + x*y^3 - y^2*z + 2*y*z^2, x^6 + 2*x*y^4 - x^2*y^2 + x*y^3 - y^2*z - 2*z^3, -z^6 - 2*x^2*y^3 - x^2*y^2 + 2*x*y^3 - 2*x^2*z - 2*x*z^2]: local length 41 (standard basis), 41 (stabilization)
  [ok] double point, G_4 = Y^4: local length 41 <= 41 by both certificates
  [ok] double point, G_4 = XY^3: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 + 120*Z^3, X^5*Y]
  [ok] double point, G_4 = XY^3: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] double point, G_4 = XY^3: the triple annihilates M_Q
    triple [x^4*y^2 + 3*y^4 + 2*x^2*z + y*z^2, x^6 + x^2*y^3 + 2*y^4 + x^2*y*z + x^2*z - z^3, x^4*y^2 - x^2*y^2 + 2*y^4 + x*y*z]: local length 41 (standard basis), 41 (stabilization)
  [ok] double point, G_4 = XY^3: local length 41 <= 41 by both certificates
(U7) two points, the family G_lambda = lambda X^3Y + X^2Y^2 + XY^3: one triple for all lambda
    the triple (l = lambda): ['-x^6 + 2*x^3*y^2 + (-2)*x^3*y + (3*l - 3)*x^2*y^2 + 2*x*y^3 + 2*x^2*z + z^3', '2*y^6 + 2*x^3*y^2 + 4*x^3*y + (-6*l + 3)*x^2*y^2 + (-2)*x*y^3 + (-2)*y^2*z - x*z^2', '2*x^6 - x*y^4 + 2*x^3*y + (-3*l + 6)*x^2*y^2 + (-4)*x*y^3 + y^2*z + 2*x*z^2']
  [ok] for every lambda: no terms of degree <= 2; cubic parts kill D_3, quartic parts kill D_4(lambda) (identically in lambda), quintic parts kill D_5
  [ok] cubic and sextic parts do not depend on lambda; the cubic parts are independent
    constant cubic tails: tops of M_Q(lambda) = [X^6 + 240*Y^2*Z - 480*X*Z^2 + 120*Z^3, Y^6 + 480*Y^2*Z - 240*X*Z^2]
  [ok] (S) at lambda = 17/5, -2, 4/9, 0: M_Q(lambda) closed with H and the triple annihilates it (direct)
    dim J = 1; saturation computed in 234s; (J : m^inf) + m = BB[1]=z BB[2]=y BB[3]=x BB[4]=3888*l^3-11448*l^2-1773*l+2708
  [ok] J has dimension one, and (J : (x,y,z)^inf) + (x,y,z) = (x, y, z, p(lambda)) with p = (9l - 4)(432l^2 - 1080l - 677) up to a unit
(A7) local lengths at the three exceptional values and at the generic point
  [ok] lambda = 4/9 (D_4 = nu_4 of three points of the line): local length 41 by both certificates
  [ok] lambda = 17/5 (not exceptional): local length 40 by both certificates
  [ok] over Q(lambda): vdim = 40 (the generic local length)
    roots of 432 l^2 - 1080 l - 677 (number field): vdim 41, stabilization 41
  [ok] at the two roots of 432 l^2 - 1080 l - 677: local length 41 by both certificates (computed over the number field)
(U8) the monomial lemma for the rays of G_4 = XY^3 (and X^3Y by X <-> Y)
  [ok] {V_5 : x o V_5, z o V_5 in D_4} = <X^5, X^2Y^3, XY^4, Y^5, Y^4Z> (no X^4Y, X^4Z)
  [ok] {R_5 : y o R_5, z o R_5 in D_4} = <X^5, X^4Y, X^4Z, XY^4, Y^5> (the only monomials divisible by X^2 are X^5, X^4Y, X^4Z)
(S8) the rays on actual algebras
  [ok] (S) XY^3, sample 1: graded data <X^4, Y^4, XY^3>, and the curve of [15, Prop C.6(b)] at the point ([X], resp. [Y]) is a ray: (I : (b, c)) not in I + (b, c)
  [ok] (S) XY^3, sample 2: graded data <X^4, Y^4, XY^3>, and the curve of [15, Prop C.6(b)] at the point ([X], resp. [Y]) is a ray: (I : (b, c)) not in I + (b, c)
  [ok] (S) XY^3, sample 3: graded data <X^4, Y^4, XY^3>, and the curve of [15, Prop C.6(b)] at the point ([X], resp. [Y]) is a ray: (I : (b, c)) not in I + (b, c)
  [ok] (S) X^3Y, sample 1: graded data <X^4, Y^4, X^3Y>, and the curve of [15, Prop C.6(b)] at the point ([X], resp. [Y]) is a ray: (I : (b, c)) not in I + (b, c)
  [ok] (S) X^3Y, sample 2: graded data <X^4, Y^4, X^3Y>, and the curve of [15, Prop C.6(b)] at the point ([X], resp. [Y]) is a ray: (I : (b, c)) not in I + (b, c)
  [ok] (S) X^3Y, sample 3: graded data <X^4, Y^4, X^3Y>, and the curve of [15, Prop C.6(b)] at the point ([X], resp. [Y]) is a ray: (I : (b, c)) not in I + (b, c)
  [ok] (S) accepted ray samples: {'XY^3': 3, 'X^3Y': 3} (3 each wanted)
(S9) Lemma FQ: the free-quartic deformation on actual algebras with a free quartic generator
  [ok] (S) (R1-i), sample 1: M = M_6 + C Psi with dim M_6 = 20; M_t = M_6 + C(Psi + t W) closed with H, D_4 = <X^4, X^3L, X^3*Z + t X^2*Z^2> (t = 1, 2)
  [ok] (S) (R1-i), sample 2: M = M_6 + C Psi with dim M_6 = 20; M_t = M_6 + C(Psi + t W) closed with H, D_4 = <X^4, X^3L, X^3*Z + t X^2*Z^2> (t = 1, 2)
  [ok] (S) (R1-ii), sample 1: M = M_6 + C Psi with dim M_6 = 20; M_t = M_6 + C(Psi + t W) closed with H, D_4 = <X^4, X^3L, X^3*Y + t X^2*Z^2> (t = 1, 2)
  [ok] (S) (R1-ii), sample 2: M = M_6 + C Psi with dim M_6 = 20; M_t = M_6 + C(Psi + t W) closed with H, D_4 = <X^4, X^3L, X^3*Y + t X^2*Z^2> (t = 1, 2)
  [ok] (S) double point, X^2Y^2, sample 1: M = M_6 + C Psi with dim M_6 = 20; M_t = M_6 + C(Psi + t W) closed with H, D_4 = <X^4, X^3L, X^2*Y^2 + t Y^4> (t = 1, 2)
  [ok] (S) double point, X^2Y^2, sample 2: M = M_6 + C Psi with dim M_6 = 20; M_t = M_6 + C(Psi + t W) closed with H, D_4 = <X^4, X^3L, X^2*Y^2 + t Y^4> (t = 1, 2)
  [ok] (S) accepted free samples: {'(R1-i)': 2, '(R1-ii)': 2, 'double point, X^2Y^2': 2} (2 each wanted)
time 243.4s
48 checks, 0 failed
ALL CHECKS PASSED
