Part 0/1: parametrization and rays (S)
  [ok] r = 0, sample 0: H, X^4 in M, X^4 not in <y,z> o M, colon criterion
  [ok] r = 0, sample 1: H, X^4 in M, X^4 not in <y,z> o M, colon criterion
  [ok] r = 2, rho = 5, sample 0: D_3 = V, H, functional u, colon criterion
  [ok] r = 2, rho = 5, sample 1: D_3 = V, H, functional u, colon criterion
  [ok] r = 2, rho = 5, sample 2: D_3 = V, H, functional u, colon criterion
  [ok] r = 1, d_0 = 3, q = 3, sample 0: H, functional u, colon criterion
  [ok] r = 1, d_0 = 3, q = 3, sample 1: H, functional u, colon criterion
  [ok] r = 1, d_0 = 3, q = 3, sample 2: H, functional u, colon criterion
  [ok] secant A = Y^5, q = 3, sample 0: H, functional u, colon criterion
  [ok] secant A = Y^5, q = 3, sample 1: H, functional u, colon criterion
  [ok] secant A = Y^5, q = 3, sample 2: H, functional u, colon criterion
  [ok] tangent A = X^3Y^2: X^3 lies in S_0 = <A_XY, A_YY, R_Y, R_Z> (the functional of the rays is not available)
    accepted samples: {'r0': 2, 'r2rho5': 3, 'd03q3': 3, 'secq3': 3}
Part 2: the d_0 = 2 normal forms (U)
  [ok] classes of A_XY, A_YY mod X^3 in the basis X^2Y, XY^2, Y^3: rows (6al, 6be, 4ga), (6be, 12ga, 20de)
  [ok] al != 0, be != 0, d_0 = 2: A = c0[(X+tY)^5 - X^5 - 5tX^4Y] with t = be/al, c0 = al/(10t^2)
  [ok] (13) gives rank one
  [ok] secant change: A(X, (Y - X)/t) = c0 (Y^5 + 4X^5 - 5X^4Y)
Part 3: dimensions of the strata
    tangent A = X^3Y^2: d_0 = 2, affine dims of {R : q <= 2}, {R : q <= 1} = 9, 5
  [ok] tangent A = X^3Y^2 (U): d_0 = 2, dim{q <= 2} <= 9, dim{q <= 1} <= 5; incidence strata 9 - 3 + 2 + 8 + ... : q = 2 part <= 9 + 8 = 17 < 19, q <= 1 part <= 5 + 12 = 17 < 19
    secant A = Y^5: d_0 = 2, affine dims of {R : q <= 2}, {R : q <= 1} = 9, 5
  [ok] secant A = Y^5 (U): d_0 = 2, dim{q <= 2} <= 9, dim{q <= 1} <= 5; incidence strata 9 - 3 + 2 + 8 + ... : q = 2 part <= 9 + 8 = 17 < 19, q <= 1 part <= 5 + 12 = 17 < 19
    random binary A (d_0 = 3): dims 10, 6
  [ok] d_0 = 3 sample 0 (S): dim{q <= 2} <= 10 (10 + 4 = 14 < 15), dim{q <= 1} <= 6 (6 + 8 = 14 < 15)
    random binary A (d_0 = 3): dims 10, 6
  [ok] d_0 = 3 sample 1 (S): dim{q <= 2} <= 10 (10 + 4 = 14 < 15), dim{q <= 1} <= 6 (6 + 8 = 14 < 15)
    random binary A (d_0 = 3): dims 10, 6
  [ok] d_0 = 3 sample 2 (S): dim{q <= 2} <= 10 (10 + 4 = 14 < 15), dim{q <= 1} <= 6 (6 + 8 = 14 < 15)
    accepted d_0 = 3 samples: 3
  [ok] tangent: R = Z^4 + X^2Z^2 + Y^2Z^2 has q = 3
  [ok] (S) single h in L: dim hM_3 = 9 (A-space 9; [h] in P^4: 13)
  [ok] (S) coprime pencil in L: dim NM_3 = 15 (A-space 3; Gr(2,5): 9)
  [ok] (S) N = l<y,z>, l not in (y,z): dim NM_3 = 12 (A-space 6; 2 + 6 = 8)
  [ok] (S) N = l<y,z>, l in (y,z): dim NM_3 = 12 (A-space 6; 1 + 6 = 7)
  [ok] (S) N = l<p1,p2>, l in (y,z), <p1,p2> != <y,z>: dim NM_3 = 13 (A-space 5; 3 + 5 = 8)
  [ok] (S) normalized quintics: (y,z)^2 S_3 has dim 18 and meets <x^5, x^4y, x^4z> trivially
Part 4: excess components (S)
  [ok] three points X, X+Y, X+Z: length 3 and contains [X]
    three points X, X+Y, X+Z: r = 2, rho = 2, solutions 2, base scheme = Gamma: True, excess bound 27 + 3e(Gamma) = 36
  [ok] rho = 2, three points X, X+Y, X+Z: actual algebra, B_3 = Gamma, excess <= 36 < 42
  [ok] X and a double point at Y (along Z): length 3 and contains [X]
    X and a double point at Y (along Z): r = 2, rho = 2, solutions 2, base scheme = Gamma: True, excess bound 27 + 3e(Gamma) = 36
  [ok] rho = 2, X and a double point at Y (along Z): actual algebra, B_3 = Gamma, excess <= 36 < 42
  [ok] double point at X (along Y) and the point Z: length 3 and contains [X]
    double point at X (along Y) and the point Z: r = 2, rho = 2, solutions 2, base scheme = Gamma: True, excess bound 27 + 3e(Gamma) = 36
  [ok] rho = 2, double point at X (along Y) and the point Z: actual algebra, B_3 = Gamma, excess <= 36 < 42
  [ok] double point at X (along Y) and the point X+Y: length 3 and contains [X]
    double point at X (along Y) and the point X+Y: no r = 2 quintic (dim of solutions 2) -- type does not occur with r = 2
  [ok] curvilinear triple at X (conic type): length 3 and contains [X]
    curvilinear triple at X (conic type): r = 2, rho = 2, solutions 2, base scheme = Gamma: True, excess bound 27 + 3e(Gamma) = 36
  [ok] rho = 2, curvilinear triple at X (conic type): actual algebra, B_3 = Gamma, excess <= 36 < 42
  [ok] collinear triple at X: length 3 and contains [X]
    collinear triple at X: no r = 2 quintic (dim of solutions 2) -- type does not occur with r = 2
  [ok] fat point at X: length 3 and contains [X]
    fat point at X: r = 2, rho = 2, solutions 3, base scheme = Gamma: True, excess bound 27 + 3e(Gamma) = 39
  [ok] rho = 2, fat point at X: actual algebra, B_3 = Gamma, excess <= 39 < 42
    accepted rho = 2 samples: 5
    q = 0, X, Y, X+Y on z = 0: d_0 = 3, q = 0, base scheme = Gamma: True (excess bound 27 + 3*3 = 36)
  [ok] r = 1, d_0 = 3, q = 0, X, Y, X+Y on z = 0: actual algebra, B_3 = Gamma, excess <= 36
    q = 0, double point at X along Y and the point X+Y: d_0 = 3, q = 0, base scheme = Gamma: True (excess bound 27 + 3*3 = 36)
  [ok] r = 1, d_0 = 3, q = 0, double point at X along Y and the point X+Y: actual algebra, B_3 = Gamma, excess <= 36
    q = 0, triple at X on z = 0: d_0 = 3, q = 0, base scheme = Gamma: True (excess bound 27 + 3*3 = 36)
  [ok] r = 1, d_0 = 3, q = 0, triple at X on z = 0: actual algebra, B_3 = Gamma, excess <= 36
    accepted q = 0 samples: 3
Part 5: the tangent anchor (A)
  [ok] anchor: R not in U_0 = <X^4, X^3Y>, A_0 + dR in D_3 (dim 6), B = 0
    anchor lies on the stratum q = 2
  [ok] anchor: H = (1,3,6,6,3,1,1)
  [ok] anchor: the three generators lie in I
    local length: 30 (standard basis), 30 (stabilization)
  [ok] anchor: local length 30 <= 41 by both certificates
Part 6: Corollary S6.2 on the graded datum of the anchor (S, cross-check)
    M_Q with Q(X^6) = Y^3: H = [1, 3, 6, 6, 3, 1, 1]; a random triple of Ann(M_Q) has local length 33 (standard basis)
  [ok] S6.2 cross-check: H(M_Q) = H and local length <= 41
Part 3b: the strata rho = 3, 4 of Step (2d), uniformly (U; second review, 2026-10-12)
  [ok] (U) x^3-coefficient of p q equals p_x q_xx (p linear, q quadric; symbolic)
  [ok] (U) for h in L and any linear l, h l lies in M_3 (no x^3 term): so h_1 h_2 S_1 lies in h_1 M_3 cap h_2 M_3
  [ok] (U) dim{rho <= 3} <= max(dim stratum + i) = 9, dim{rho <= 4} <= 4 + 9 = 13
  [ok] (U) fibres Gr(1, T_3/V) (rho = 4, dim 4) and Gr(2, T_3/V) (rho = 3, dim 8): strata of dim <= 17 < 18 = 38 - 20
  [ok] (A) dim N M_3 = 18 - i(N) at one pencil of each type: <xy, z^2> (coprime) 15, y<y,z> 12, y<x, y+z> 13, (x+y)<y,z> 12
time 8.8s
53 checks, 0 failed
ALL CHECKS PASSED
