(U1) excess arithmetic: 27 + 3 e_Gamma + 2 e' with e_Gamma + e' <= 5
    pairs (e_Gamma, e') with bound > 41: [(5, 0)]
  [ok] only e_Gamma = 5, e' = 0 exceeds 41 (bound 42)
  [ok] Gamma_0 with B = V(m_X^2) + q: 27 + 3*4 + 2*1 = 41
(A1) (O1): Gamma = {X, Y}, B = V(m_X^2) + [Y]
  [ok] (O1) L = Y: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 - 720*X*Y*Z, Y^6 - 720*X*Y*Z]
  [ok] (O1) L = Y: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] (O1) L = Y: the triple annihilates M_Q
    triple [x^3*z + x*y^2 + z^3, x^3*z + y^2*z + x*z^2, x^6 + y^6 + x*y*z]: local length 40 (standard basis), 40 (stabilization)
  [ok] (O1) L = Y: local length 40 <= 41 by both certificates
  [ok] (O1) L = Z: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 - 720*X*Y*Z, Y^6 - 720*X*Y*Z]
  [ok] (O1) L = Z: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] (O1) L = Z: the triple annihilates M_Q
    triple [x^3*y + x*y^2 + z^3, x^3*y + y^2*z + x*z^2, x^6 + y^6 + x*y*z]: local length 40 (standard basis), 40 (stabilization)
  [ok] (O1) L = Z: local length 40 <= 41 by both certificates
(U2) (O2): orbits of D_4
  [ok] (O2) D_3 = nu_3(V(y^2, yz, z^3)) (the cubics y^2 S_1, yz S_1, z^3 kill it; dim 4)
  [ok] Gamma_0 = V(z, y^2): Z -> Z + cX sends a X^3 Z + X^2 Z^2 to X^2 Z^2 + (a + 2c) X^3 Z + c(a + c) X^4
  [ok] Z -> Z + cX preserves D_3 = <X^3, X^2Y, X^2Z, XZ^2> and nu_k(V(z, y^2)) = <X^k, X^(k-1) Y>
  [ok] Gamma_0 = V(y, z^2): Y -> tY, Z -> uZ sends a X^3 Y + X^2 Z^2 to a t X^3 Y + u^2 X^2 Z^2 (so a != 0 is one orbit)
(A2) (O2) anchors at infinity
  [ok] (O2) Gamma_0 = V(z,y^2), D_4 = <X^4, X^3Y, X^2Z^2>: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 - 720*X*Y*Z, X^5*Y]
  [ok] (O2) Gamma_0 = V(z,y^2), D_4 = <X^4, X^3Y, X^2Z^2>: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] (O2) Gamma_0 = V(z,y^2), D_4 = <X^4, X^3Y, X^2Z^2>: the triple annihilates M_Q
    triple [x*y^2 + z^3, x^6 + x*y*z, x^3*z + y^3 + y*z^2]: local length 38 (standard basis), 38 (stabilization)
  [ok] (O2) Gamma_0 = V(z,y^2), D_4 = <X^4, X^3Y, X^2Z^2>: local length 38 <= 41 by both certificates
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^2Z^2>: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 - 720*X*Y*Z, X^5*Z]
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^2Z^2>: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^2Z^2>: the triple annihilates M_Q
    triple [x*y^2 + z^3, x^6 + x*y*z, x^3*y + y^3 + y*z^2]: local length 41 (standard basis), 41 (stabilization)
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^2Z^2>: local length 41 <= 41 by both certificates
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^3Y + X^2Z^2>: the graded data have H = (1,3,6,4,3,2,2)
    tops of M_Q: [X^6 - 720*X*Y*Z, X^5*Z]
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^3Y + X^2Z^2>: M_Q closed, dim 21, H = (1,3,6,4,3,2,2), containing D_3, D_4, D_5
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^3Y + X^2Z^2>: the triple annihilates M_Q
    triple [x*y^2 + z^3, x^6 + x*y*z, 2*x^3*y - 3*x^2*z^2 + y^3 + y*z^2]: local length 41 (standard basis), 41 (stabilization)
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = <X^4, X^3Z, X^3Y + X^2Z^2>: local length 41 <= 41 by both certificates
(S) residual data: S6.2 with random cubic tails and random triples of Ann(M_Q)
    (O2) Gamma_0 = V(z,y^2), D_4 = X^3 T_1: H = [1, 3, 6, 4, 3, 2, 2], local lengths [42, 42]
  [ok] (O2) Gamma_0 = V(z,y^2), D_4 = X^3 T_1: H correct; S6.2 with random tails gives > 41 (these data are settled otherwise: l21_1364322_r1.sage, l21_1364322_r2.sage)
    (O2) Gamma_0 = V(y,z^2), D_4 = X^3 T_1: H = [1, 3, 6, 4, 3, 2, 2], local lengths [42, 42]
  [ok] (O2) Gamma_0 = V(y,z^2), D_4 = X^3 T_1: H correct; S6.2 with random tails gives > 41 (these data are settled otherwise: l21_1364322_r1.sage, l21_1364322_r2.sage)
    (O3) Gamma = {X,Y}, G_4 = X^3 Y: H = [1, 3, 6, 4, 3, 2, 2], local lengths [42, 42]
  [ok] (O3) Gamma = {X,Y}, G_4 = X^3 Y: H correct; S6.2 with random tails gives > 41 (these data are settled otherwise: l21_1364322_r1.sage, l21_1364322_r2.sage)
    (O3) Gamma = 2X on the line, G_4 = X^2 Y^2: H = [1, 3, 6, 4, 3, 2, 2], local lengths [42, 42]
  [ok] (O3) Gamma = 2X on the line, G_4 = X^2 Y^2: H correct; S6.2 with random tails gives > 41 (these data are settled otherwise: l21_1364322_r1.sage, l21_1364322_r2.sage)
30 checks, 0 failed
ALL CHECKS PASSED
