(U1) bounds
  [ok] Gamma reduced: 27 + 5 + 3 + 2 = 37 < 42
  [ok] Gamma = double point at X + point: 27 + 5*2 + 3 = 40 < 42
  [ok] Gamma = X + double point elsewhere: 27 + 5 + 3*2 = 38 < 42
  [ok] Gamma curvilinear triple at X: 27 + 15 = 42 (fails)
(U2) normal form of an active D_5
  [ok] collinear: V_0 + c X^4Y -> V_0 + (c + 2 mu) X^4Y modulo X^5
  [ok] noncollinear: nu_5(Gamma) = <X^5, X^4Y, 2X^3Y^2 + X^4Z>
  [ok] noncollinear: V_0 + c X^4Y -> V_0 + (c + 9 mu) X^4Y modulo X^5
  [ok] noncollinear: the substitution preserves nu_3(Gamma) for every mu
  [ok] noncollinear: the substitution preserves nu_4(Gamma) for every mu
  [ok] collinear: Y -> Y + mu X preserves nu_3(Gamma) for every mu
  [ok] collinear: Y -> Y + mu X preserves nu_4(Gamma) for every mu
  [ok] collinear: Y -> Y + mu X preserves nu_5(Gamma) for every mu
(A) the two active graded data: certified triples in I*
  [ok] collinear: graded data have H = (1,3,6,3,3,2,1,1,1)
  [ok] collinear: dim S_1 o D_5 = 3 (active)
  [ok] collinear: triple lies in I*
    collinear: local length 39 (standard basis), 39 (stabilization)
  [ok] collinear: local length <= 41 by both certificates
  [ok] noncollinear: graded data have H = (1,3,6,3,3,2,1,1,1)
  [ok] noncollinear: dim S_1 o D_5 = 3 (active)
  [ok] noncollinear: triple lies in I*
    noncollinear: local length 39 (standard basis), 39 (stabilization)
  [ok] noncollinear: local length <= 41 by both certificates
(W) conic curvilinear Gamma, tangent D_5: Corollary S6.2 (anchor at infinity)
  [ok] M_Q = T_<=2 + nu_3(Gamma) + nu_4(Gamma) + <X^5, X^4 Y> + X^6 + X^7 + C(X^8 - 6720 Y^3): closed, dim 21, H = (1,3,6,3,3,2,1,1,1)
  [ok] its graded data are those of the stratum (D_5 = <X^5, X^4 Y>, D_3 = nu_3(Gamma))
  [ok] x^2 z - x y^2, y^3 + x^8, z^3 + x^5 y annihilate M_Q
    local length 41 (standard basis), 41 (stabilization)
  [ok] local length 41 <= 41 by both certificates
(F) fat Gamma, D_5 = <X^5, X^4 Y>, delta != 0: closedness with all tails symbolic over Q(delta)
  [ok] frame: the 8 lifts, T_<=2 and X^2 T_1 have distinct pivots with coefficient 1 (21 independent vectors, a basis of M)
  [ok] C_5 general (18 coefficients): the remainders of degree >= 4 generate the ideal of the 12 coefficients of z o C_5 outside X^3 T_1
    C_5 in <X^3YZ, X^3Z^2, X^3Y^2, X^2Y^3, XY^4, Y^5>; nonzero remainders: [('Phi', 'z'), ('y Phi/(2 delta)', 'y')]
  [ok] exactly two nonzero remainders: (Phi, z) and (y Phi/(2 delta), y)
  [ok] remainder of y o (y Phi/(2 delta)) = rho_1 = (6 c_{XY^4} X Y^2 + 10 c_{Y^5} Y^3)/delta
  [ok] remainder of z o Phi = rho_2 = z o H_4 - c_{X^3YZ} (xy o C_5)/(8 delta) - 2 c_{X^3Z^2} K_3 modulo X^2 T_1
  [ok] J_3 occurs in no remainder
  [ok] rho_2 = 0 is 7 equations, affine in the 7 coefficients h_{XY^2Z}, h_{Y^3Z}, h_{XYZ^2}, h_{Y^2Z^2}, h_{XZ^3}, h_{YZ^3}, h_{Z^4} with an invertible constant matrix; rho_1 involves none of them
    => closed iff c_{XY^4} = c_{Y^5} = 0 and the 7 coefficients are given polynomials in delta^{+-1}, c_{X^3YZ}, c_{X^3Z^2},
       c_{X^3Y^2}, c_{X^2Y^3}, the other 8 coefficients of H_4 and K_3; J_3 free: a graph over C^* x C^4 x C^8 x C^10 x C^10, irreducible
(A1) anchor of the fat family: Phi = X^8 + X^4 Y^2, W = X^3 Z (delta = 1, all other tails 0)
  [ok] M_0 closed, dim 21, H = (1,3,6,3,3,2,1,1,1)
  [ok] graded data of M_0: D_3 = X^2 T_1 (fat Gamma), D_4 = X^3 T_1, D_5 = <X^5, X^4 Y>
  [ok] M_0 is the span of the frame at delta = 1, C_5 = H_4 = J_3 = K_3 = 0 (a member of the normalized family; rho_1 = rho_2 = 0 there)
  [ok] y^3, x z^2, z^3 + x y^2 - x^5/840 annihilate M_0
    local length 39 (standard basis), 39 (stabilization); by hand: weights (3,6,5), degrees 18, 13, 15, 18*13*15/90 = 39
  [ok] local length 39 by both certificates
(S) sanity: e(I*) on the two strata (random triples of I*)
    conic, tangent D_5: [42, 42]
  [ok] conic, tangent D_5: e(I*) > 41
    fat: [47, 47]
  [ok] fat: e(I*) > 41
38 checks, 0 failed
ALL CHECKS PASSED
