=== Part 0 (U): the trichotomy rank three / secant / tangent
  ok   (U) rad(3x3 minors of [X^4; T_X; T_Y]) = ideal of the cone over the rational normal quartic (prime)
  ok   (U) (Y+mX)^5 - m^5X^5 has coefficients binom(5,i) m^(5-i); at s = 0 the point is (t1,0,0,0,0), i.e. T = X^4Y
=== Part 1 (U): symbolic identities
  ok   (U) Q_6 = int T dY has (Q_6)_Y = T and no X^6, X^5Y terms
  ok   (U) z o F_6 = b X^5 and y o F_6 = a X^5 + dl T for F_6 = cX^6 + aX^5Y + bX^5Z + dl Q_6
  ok   (U) a sextic F_6 with y o F_6, z o F_6 free of Z lies in C[X,Y]_6 + C X^5 Z (kernel of dimension 8 = 7 + 1)
  ok   (U) exp(k Y d_X^2): degree 7 unchanged, degree 6 gains 42 k X^5 Y
  ok   (U) exp(k Y d_X^3): degrees 7, 6 unchanged, degree 5 gains 210 k X^4 Y
  ok   (U) exp(k Z d_X^2): degree 7 unchanged, degree 6 gains 42 k X^5 Z
  ok   (U) exp(k Z d_X^3): degrees 7, 6 unchanged, degree 5 gains 210 k X^4 Z
  ok   (U, spot check) adjointness <f(x,y,z+3q), G> = <f, exp(3 Z q(d)) G> for q = x^2, x^3 (60 monomials f, 2 forms G)
  ok   (U) p_4 o F' = (element of C[X]) + dl<p_4,T_X> XY + dl<p_4,T_Y>/2 Y^2 + <p_4,(F_5)_Y> Y + <p_4,(F_5)_Z> Z
  ok   (U) p_4 o (T + G_4 + G_3) = (element of C[X]) + <p_4,T_Y> Y
  ok   (U) <p_4, X^4> = 24 p_0
=== Part 2 (S): actual members of the rank-three and secant sectors; the normalization of the proof and the colon criterion
--- rank3 T=XY^4: 128 conditions, 55 eliminated, 5 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- rank3 T=X^2Y^3: 128 conditions, 55 eliminated, 5 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- rank3 T=X^3Y^2: 128 conditions, 55 eliminated, 5 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- rank3 T=Y^5+X^2Y^3: 128 conditions, 55 eliminated, 5 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- rank3 T=Y^5+X^3Y^2+X^4Y: 128 conditions, 55 eliminated, 5 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- secant T=Y^5, R=X^3Y: 134 conditions, 58 eliminated, 3 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- secant T=Y^5, R=X^2Y^2: 134 conditions, 58 eliminated, 3 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- secant T=Y^5, R=XY^3: 134 conditions, 58 eliminated, 3 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: no point found (skipped)
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
--- secant T=Y^5, R=X^3Y+XY^3: 134 conditions, 58 eliminated, 3 residual equations
    dense 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    dense 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 0: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 1: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    active 2: ray (y,z) after the shears (active: True; steps (True, True, True, True, True))
    inactive 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    inactive 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    cubic 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 0: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
    Zsq 1: ray (y,z) after the shears (active: False; steps (True, True, True, True, True))
  ok   (S) 107 actual members tested (30 active), the normalization and the colon criterion hold for 107
time 50.7 s
ALL CHECKS PASSED
