(D) the graded datum, I* and L_def (U)
  [ok] the graded inverse system T_<=2 + D_3 + ... + D_7 is closed, of dimension 21, with H = (1,3,6,3,2,2,2,2)
  [ok] dim I*_k = 7, 13, 19, 26, 34 for k = 3..7
  [ok] I*_3 = (y,z)^2 S_1 (the cubics without x^3, x^2y, x^2z)
  [ok] (L_def)_k = I*_k for k = 3..6 and (L_def)_7 = S_7 (the D_k^perp form an ideal: M_g is closed)
  [ok] L_def contains m^7 and nothing of degree < 3
  [ok] the cubic base scheme: the saturation of (I*_3) is (y,z)^2 = m_X^2, of length 3 (Hilbert polynomial 3), supported at [X]
(K) K_v(L_def) at v = [X] in the chart x = rho, y = rho eta, z = rho zeta (U)
  [ok] K_v(L_def) = ((eta,zeta)^2, rho (zeta, eta^2), rho^4) (equality of ideals of QQ[rho,eta,zeta], hence of O_v)
  [ok] K_v(L_def) = ((eta,zeta)^2, rho zeta, rho^4) as well (rho eta^2 lies in (eta,zeta)^2): five minimal generators
  [ok] b_3 = (eta,zeta)^2 and b_4 = b_5 = b_6 = (zeta, eta^2) (local ideals of the base schemes of (L_def)_k at v)
  [ok] degrees k >= 8 add nothing: K + (rho^5 c(1,eta,zeta) : c in S_8) = K
(J) the parameter ideal J = (eta^2, zeta (rho + eta), zeta^2 + rho^4) (A)
  [ok] J lies in K_v(L_def)
    colength of J in O_v: 12 (local standard basis = stabilization at N = 6)
  [ok] the colength of J is 12 = 2*4 + 4
  [ok] C = V(eta^2, zeta^2 + rho^4) is a curve through v (dimension one; a complete intersection, hence Cohen-Macaulay)
    on C: l(O/(C, zeta)) = 8 (= l(O/(eta^2, rho^4))), l(O/(C, rho + eta)) = 4 (= l(O/(eta^2, zeta^2)))
  [ok] zeta and rho + eta are parameters on C, cutting lengths 8 and 4, and 8 + 4 = 12 = colength of J
(E) J is a reduction of K = K_v(L_def): e(K) = 12 (U)
  [ok] J K is contained in K^2 and J K^2 in K^3 (as J is contained in K)
    colengths in O_v: J K^2 99, K^3 99; J K 39, K^2 38
  [ok] K^3 = J K^2 in O_v (equal colengths of nested ideals), so K is integral over J: J is a reduction of K
  [ok] K^2 != J K in O_v (the reduction number is exactly 2)
    hence e(K_v(L_def)) = e(J) = 12 (J a parameter ideal of the regular local ring O_v: e(J) = colength)
    (S) Hilbert-Samuel lengths l(O/K^n), n = 1..6: [9, 38, 99, 204, 365, 594]; third differences [12, 12, 12]
  [ok] (S) the third differences of l(O/K^n) are constant 12 (leading term e(K) n^3/6 with e(K) = 12)
(W) Proposition prop:WM: e(L_def) = 27 + sum_v e(K_v(L_def)) (A)
  [ok] B is the single point [X], so e(L_def) = 27 + 12 = 39 < 42 = 2d; Proposition prop:WM(c) applies
(T) an explicit triple of L_def with independent cubic parts (A)
  [ok] f1, f2, f3 lie in L_def
  [ok] their cubic parts w_i lie in I*_3 and are linearly independent
  [ok] the cubic parts meet in P^2 exactly in the base scheme: the saturation of (w_1, w_2, w_3) is (y,z)^2
    local length of (f1, f2, f3) at the origin: 39 (local standard basis = stabilization at N = 13)
  [ok] the triple generates an ideal of local length 39 <= 41 < 42 = 2d in S_m (Corollary cor:cubic applies)
  [ok] the transforms factor as rho^3 times the proper transforms
    proper transforms at v: local length 12; 27 + 12 = 39
  [ok] cross-check of Proposition prop:WM(a) for this triple: the proper transforms meet at v in length 12, and 27 + 12 = 39
(Q) the inverse system M_Q of eq. (wcq) for this triple (A)
  [ok] C_1, C_2, C_3 in T_3 with <w_i, C_j> = delta_ij
  [ok] M_Q = T_<=2 + D_3 + ... + D_6 + {F + Q(F) : F in D_7} (21 independent generators) is closed (its closure has dimension 21), with H = (1,3,6,3,2,2,2,2)
  [ok] Q(X^7) = -<x^7, X^7> C_3 = -5040 C_3 != 0 and Q(X^6 Y) = 0: the cubic tail moves
  [ok] f1, f2, f3 annihilate M_Q
(R) three random elements of L_def (S)
    local lengths of four random triples of L_def: [39, 39, 39, 39]
  [ok] (S) random triples of L_def have local length 39 = e(L_def)
30 checks, 0 failed
ALL CHECKS PASSED
