=== Part U1: case (i), Lambda = V(x), D_3 = C[Y,Z]_3 + C X^3, D_3' = C[Y,Z]_4 + C X^4 ===
  [ok] (i) {P : y.P, z.P in C[Y,Z]_4 + CX^4} = C[Y,Z]_5 + X^4<Y,Z> + C X^5 (dim 9)
  [ok] (i) x^2 o P~ = 0 for every such P
  [ok] (i) {G : S_1 o G in D_3'} = C[Y,Z]_5 + C X^5 (dim 7)
  [ok] (i) every such G is free of X^4Y and X^4Z
=== Part U2: case (ii), Lambda = V(z) through [X], D_3' in W = C[X,Y]_4 + C Z^4 ===
  [ok] (ii) {P : y.P, z.P in W} = C[X,Y]_5 + C X^4Z + C Z^5 (dim 8)
  [ok] (ii) {G : S_1 o G in W} = C[X,Y]_5 + C Z^5 (dim 7)
  [ok] (ii) <(x+ay)^4 z, G> = 0 identically
  [ok] (ii) p(a) = sum_i binom(4,i) a^i <x^(4-i)y^(i+1), G_0>
  [ok] (ii) coeff of a^i in p (i=1..4) = binom(4,i)(4-i)!(i+1)! * coeff of X^(4-i)Y^(i+1)
  [ok] (ii) x'^2 o P~' has no monomial containing Z' (a binary cubic in X', Y'), all coefficients symbolic
  [ok] (ii) the dual change of coordinates (X,Y,Z) = (X', Y'+aX', Z') is the transpose of x' = x+ay (5 monomials)
=== Part U3: the exceptions of (ii) ===
  [ok] gap datum: D_4 = <X^4> + S_1 o G = <X^4, X^3Y, Z^4> has dim 3 (r = 2: sector P2)
  [ok] gap datum: D_3 = nu_3(V(z) + [Z]) = C[X,Y]_3 + C Z^3; H of the graded algebra = (1,3,6,5,3,2,1)
  [ok] gap datum: p(a) = <(x+ay)^4 y, G> = 24 for all a (no root): Prop 6.1(ii) does not apply
  [ok] gap datum: D_5 meets X^4 T_1 only in C X^5 (so it is not a tangent pencil <X^5, X^4L>)
=== Part S: actual closed algebras, rays and the colon criterion ===
  [ok] (i) P2, G = random binary in Y,Z: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (i) P1 secant <X^5,Y^5>, R in C[Y,Z]_4: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (i) P2, G = random binary in Y,Z: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (i) P1 secant <X^5,Y^5>, R in C[Y,Z]_4: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(d) P2, G = G_0 binary with root a = 3: graded datum has H; 3 actual algebras built (of 25 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(d) P1 secant <X^5,Y^5>, R binary (a = 0): graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P2, G = (Y + 2 X)^5 + 1 Z^5, root a = -2: graded datum has H; 3 actual algebras built (of 14 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P2, G = G_0 binary (e = 0), root a = 3: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P1 secant <X^5,Y^5>, R in C[X,Y]_4 + C Z^4: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P1 secant <X^5,Z^5> (second point = q): graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(d) P2, G = G_0 binary with root a = 1: graded datum has H; 3 actual algebras built (of 13 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(d) P1 secant <X^5,Y^5>, R binary (a = 0): graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P2, G = (Y + -2 X)^5 + -1 Z^5, root a = 2: graded datum has H; 3 actual algebras built (of 7 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P2, G = G_0 binary (e = 0), root a = 1: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P1 secant <X^5,Y^5>, R in C[X,Y]_4 + C Z^4: graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
  [ok] (ii)(c) P1 secant <X^5,Z^5> (second point = q): graded datum has H; 3 actual algebras built (of 3 tries); ray + colon criterion on 3 of 3
accepted samples in Part S: 48
31 checks, 0 failed
ALL CHECKS PASSED
