Part 0: shapes and normalizations
  ok   Q8 in <Y^8, X Y^7, Y^7 Z>
  ok   Q7 (after Q8 = 0) in <Y^7, X Y^6, Y^6 Z>
  ok   Q6 (after Q8 = Q7 = 0), sec: in ['Y^6', 'X*Y^5', 'Y^5*Z', 'Z^6']
  ok   Q6 (after Q8 = Q7 = 0), tin: in ['Y^6', 'X*Y^5', 'Y^5*Z', 'Y^4*Z^2']
  ok   Q6 (after Q8 = Q7 = 0), tout: in ['Y^6', 'X*Y^5', 'Y^5*Z', 'X^2*Y^4']
  ok   quadratic shears: degree-8 effects 72 X Y^7, 72 Y^7 Z on Y^9
  ok   cubic shears: degree-7 effects 504 X Y^6, 504 Y^6 Z, none in degree 8
  ok   quartic shears: degree-6 effects 3024 X Y^5, 3024 Y^5 Z, none in degrees 7, 8
  ok   gauges: y o Y^9 = 9 Y^8, y^2 o Y^9 = 72 Y^7, y^3 o Y^9 = 504 Y^6 (with Q8 = 0 the gauge by y^2 o Phi changes no degree >= 6 except Y^7; with Q8 = Q7 = 0 the gauge by y^3 o Phi changes only Y^6 and degrees <= 3)
  ok   contractions: y^7 o Y^9 = 181440 Y^2, y^6 o Y^9 = 60480 Y^3, z^4 o Z^6 = 360 Z^2, y z^2 o Y^4 Z^2 = 8 Y^3, y^3 z o Y^4 Z^2 = 48 Y Z

Parts 1-2: the cases
   secant, sig != 0: Q5 = Q5_16*X*Y^4 + Q5_15*Y^5 + Q5_10*Y^4*Z + Q5_6*Y^3*Z^2 + Q5_3*Y^2*Z^3 + Q5_2*X*Z^4 + Q5_1*Y*Z^4 + Q5_0*Z^5
  ok   secant, sig != 0: high part of sig*U o Phi = 12 sig (d3 p1 + d2 p2) Y Z; U_3 kills <Y^5, Z^5>
      (any (p1, p2) != 0 with d3 p1 + d2 p2 = 0 works; the restriction y z (p1 z + p2 y) is never a cube)
   secant, sig = 0: Q5 = Q5_16*X*Y^4 + Q5_15*Y^5 + Q5_10*Y^4*Z + Q5_0*Z^5
  ok   secant, sig = 0: U = y z^2
   tangent in, lam = 0: Q5 = Q5_16*X*Y^4 + Q5_15*Y^5 + Q5_10*Y^4*Z + Q5_6*Y^3*Z^2
  ok   tangent in, lam = 0: U = y z^2 - (C0/30240) y^7, C0 = [Y^3 Z^2] Q5
   tangent in, lam != 0: Q5 = Q5_16*X*Y^4 + Q5_15*Y^5 + Q5_11*X*Y^3*Z + Q5_10*Y^4*Z + Q5_6*Y^3*Z^2 + Q5_3*Y^2*Z^3 + Q5_1*Y*Z^4 + Q5_0*Z^5
  ok   tangent in, lam != 0: high part of 48 lam U o Phi = 48 lam (60 xi p0 + 12 kap p1) Z^2; U_3 kills <Y^5, Y^4 Z>
  ok   tangent in, lam != 0: 8 lam V, V = x y^2 - B/(8 lam) y^3 z - (A/15120) y^7, V o Phi in T_{<=1}
  ok   rank-drop loci: a solution with p1 != 0 exists unless kap != 0 = xi, where every solution has p1 = 0
  ok   cube case: u = U + V has cubic part z^3 + x y^2 and d_x u = y^2 (so -(t^2/4) y^4 in G_t of [15, Thm 4.5])
  ok   y^3 o Y^9 = 504 Y^6: as Q8 = 0, the degree-6 part of f o Phi = 0 is f_3 o Y^9, so [y^3] f_3 = 0 and a cube gamma_f lies in C z^3
   tangent out: Q5 = Q5_20*X^5 + Q5_19*X^4*Y + Q5_18*X^3*Y^2 + Q5_17*X^2*Y^3 + Q5_16*X*Y^4 + Q5_15*Y^5 + Q5_11*X*Y^3*Z + Q5_10*Y^4*Z
  ok   tangent out: U = y z^2 (Q5 has Z-degree <= 1)
time 0.6 s
ALL CHECKS PASSED
