=== Part C (U): closedness conditions = J0 for every U containing G (rho symbolic)
  ok   c_U = <q4, A_Y> = <y q4, A> (adjointness)
  ok   U = q4^perp contains G = <X^4, X^3Y> (q4 = y^2 rho)
  ok   the closedness conditions of P, P_X, Q (derivatives) span exactly J0 (7 independent polynomials): A_Y in U and phi(A_Y) = Z b_Y; all other derivatives give no condition
   J0 = [12*r0*a3 + 24*r1*a4 + 120*r2*a5, 3*a3*f00 + 4*a4*f10 + 5*a5*f20 - b1, 3*a3*f01 + 4*a4*f11 + 5*a5*f21 - 2*b2, 3*a3*f02 + 4*a4*f12 + 5*a5*f22 - 3*b3, 3*a3*f03 + 4*a4*f13 + 5*a5*f23, 3*a3*f04 + 4*a4*f14 + 5*a5*f24, 3*a3*f05 + 4*a4*f15 + 5*a5*f25]
  ok   B2: every generator of J0 has degree 2 (c_U is bilinear in (rho, A); the E-equations are bilinear in (phi~, A) plus linear in b)
=== Part R (U): the corrected cubics over rho = (x + s y)(x + t y), all tails symbolic
  ok   U = <X^4, X^3Y, v1, v2> with v1 = 3(s+t)X^2Y^2 - 2XY^3, v2 = 6st X^2Y^2 - Y^4 (polynomial basis)
  ok   the four quartic tops are independent for all (s,t) (a 4x4 minor is the constant 2): H = (1,3,6,4,4,2,1)
  ok   21 spanning elements
   Groebner basis of J0: 76 elements (0.0 s)
  ok   closedness: all 7 conditions from the derivatives of all 21 spanning elements lie in J0
  ok   and they span J0 (so the closedness ideal is exactly J0 on this chart)
  ok   l1 = x + s y: Y-coefficient of (5) in J0 True; (l u) o m = 0 mod J0 for all 21 m True; (z u) o m = (y^2 u) o m = 0 for m != P True; (z u) o P = k, (y^2 u) o P = h True; ((z - al y^2) u) o P = k - al h True  (ALL TAILS SYMBOLIC)
      k = <y^2 l', b> = 6*t*b3 + 2*b2,   h = <y^4 l', A> = 120*t*a5 + 24*a4
  ok   l2 = x + t y: Y-coefficient of (5) in J0 True; (l u) o m = 0 mod J0 for all 21 m True; (z u) o m = (y^2 u) o m = 0 for m != P True; (z u) o P = k, (y^2 u) o P = h True; ((z - al y^2) u) o P = k - al h True  (ALL TAILS SYMBOLIC)
      k = <y^2 l', b> = 6*s*b3 + 2*b2,   h = <y^4 l', A> = 120*s*a5 + 24*a4
=== Part V (U): the vanishing argument and the detector
  ok   c_U, h_1, h_2 are linear forms in (a3, a4, a5) only; det = 34560 (s - t) = 34560*s - 34560*t: for s != t, c_U = h_1 = h_2 = 0 iff a3 = a4 = a5 = 0, i.e. A in <X^5, X^4Y, X^3Y^2>, i.e. A_Y in G (w = 0)
  ok   on w = 0, J0 forces b_Y = 0 (b1 = b2 = b3 = 0), hence k_1 = k_2 = 0
  ok   conversely, w = 0 gives h_1 = h_2 = 0 (so on the stratum w = 0 both factors have (k, h) = (0, 0))
  ok   l1: L = (-s)*X + Y, Psi = L^3 + 6 al L Z: l o Psi = (z - al y^2) o Psi = 0 for every al; <u, Psi> = -6*s + 6*t, nonzero iff s != t
  ok   l2: L = (-t)*X + Y, Psi = L^3 + 6 al L Z: l o Psi = (z - al y^2) o Psi = 0 for every al; <u, Psi> = 6*s - 6*t, nonzero iff s != t
=== Part I (U): the strata and their irreducibility
  ok   A -> A_Y: B_5 -> B_4 is onto with kernel C X^5; preimage of G is <X^5, X^4Y, X^3Y^2>; <X^2Y^3, XY^4, Y^5> maps isomorphically onto <X^2Y^2, XY^3, Y^4>
  ok   the pairing y^2 R_2 x (B_4/G) -> C is perfect (det 576); the incidence form <y^2 rho, w> = 4*p0*w0 + 6*p1*w1 + 24*p2*w2 is a quadric of rank 6, hence absolutely irreducible (the incidence {(rho, w) : <y^2 rho, w> = 0} is irreducible, of dimension 5)
  ok   the map (phi~, b) -> phi~(w) - Z b_Y (6 x 22): b does not enter the E_high rows; the E_high block has the 3x3 minors w0^3, w1^3, w2^3; the b-block (Z d_Y : B_3 -> Z B_2) has rank 3; so the rank is 6 for w != 0 and 3 for w = 0
   dimensions of auxiliary parameter spaces (rho in C^3, not P^2, and phi~ defined on C[X,Y]_4/G, three-dimensional, rather than on U/G): stratum w = 0: rho 3 + A 3 (a0,a1,a2) + phi~ 18 + b 1 (b0) = 25; stratum w != 0: incidence 5 + A 3 + (phi~, b) 22 - 6 = 16 -> 24; plus 22 free coefficients (B, g, C, chi) in both, so 47 and 46.  They exceed the dimensions of the strata (one scaling coordinate of rho and six coordinates of phi~ are redundant); only the irreducibility of these spaces, hence of their images, is used
  ok   fixed U (s, t) = (1, -2): J0 has exactly 2 minimal primes, of dimensions [17, 18], one of them (a3,a4,a5,b1,b2,b3) (stratum w = 0); so the closedness locus is NOT an affine space (B2)
  ok   fixed U (s, t) = (1/2, 3): J0 has exactly 2 minimal primes, of dimensions [17, 18], one of them (a3,a4,a5,b1,b2,b3) (stratum w = 0); so the closedness locus is NOT an affine space (B2)
ZERO
  ok   s, t symbolic: J0 has exactly 2 minimal primes, of dimensions [19, 20] (= 2 + the dimensions at fixed U), one of them (a3,a4,a5,b1,b2,b3)
  ok   rho = r0 x^2 + r1 xy + r2 y^2 symbolic (every U containing G): J0 has exactly 2 minimal primes, of dimensions [20, 21], one of them (a3,a4,a5,b1,b2,b3): the closedness locus is the union of the two irreducible strata closures
=== Part S (S): actual algebras
  ok   independent CP-T member 0 (l1 = x - y, l2 = x - 3*y) (expected (k,h) [(250, 180), (106, 28)]): closed, H ok; normal form after the two shears True; factors ['x - 3*y', 'x - y']; (k,h) per factor [(106, 28), (250, 180)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 3*y, z - (53/14) y^2): detector True, EXACT colon True
  ok   independent CP-T member 1 (l1 = x - 2*y, l2 = x - y) (expected (k,h) [(110, -30), (202, -84)]): closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x - y']; (k,h) per factor [(110, -30), (202, -84)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 2*y, z - (-11/3) y^2): detector True, EXACT colon True
  ok   independent CP-T member 2 (l1 = x - 2*y, l2 = x - y) (expected (k,h) [(-42, 0), (-50, 24)]): closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x - y']; (k,h) per factor [(-42, 0), (-50, 24)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - y, z - (-25/12) y^2): detector True, EXACT colon True; factor with h = 0 != k: [('x - 2*y', -42)], there y^2 u in I and z u not in I (no ray from its u): True
  ok   independent CP-T member 3 (l1 = x + 3*y, l2 = x - 2*y) (expected (k,h) [(-108, -48), (132, -108)]): closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x + 3*y']; (k,h) per factor [(132, -108), (-108, -48)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 2*y, z - (-11/9) y^2): detector True, EXACT colon True
  ok   independent CP-T member 4 (l1 = x + 2*y, l2 = x + 3*y) (expected (k,h) [(-27, 78), (-25, 36)]): closed, H ok; normal form after the two shears True; factors ['x + 2*y', 'x + 3*y']; (k,h) per factor [(-27, 78), (-25, 36)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x + 2*y, z - (-9/26) y^2): detector True, EXACT colon True
  ok   independent CP-T member 5 (l1 = x + 2*y, l2 = x - 2*y) (expected (k,h) [(29, -54), (-27, -42)]): closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x + 2*y']; (k,h) per factor [(-27, -42), (29, -54)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 2*y, z - (9/14) y^2): detector True, EXACT colon True
  ok   independent CP-T member 6 (l1 = x + 3*y, l2 = x - 3*y) (expected (k,h) [(267, -60), (-237, -84)]): closed, H ok; normal form after the two shears True; factors ['x - 3*y', 'x + 3*y']; (k,h) per factor [(-237, -84), (267, -60)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 3*y, z - (79/28) y^2): detector True, EXACT colon True
  ok   the member with (k, h) = (-42, 0) for one factor occurs among them, and the other factor gives the ray
  ok   own member rho = (x + 1 y)(x + -2 y), mode gen, scrambled True: closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x + y']; (k,h) per factor [(-1620, 480), (4980, -960)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 2*y, z - (-27/8) y^2): detector True, EXACT colon True
  ok   own member rho = (x + 1/2 y)(x + 3 y), mode gen, scrambled True: closed, H ok; normal form after the two shears True; factors ['x + 3*y', 'x + 1/2*y']; (k,h) per factor [(128, 12), (-17, -288)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x + 3*y, z - (32/3) y^2): detector True, EXACT colon True
  ok   own member rho = (x + -1 y)(x + 3 y), mode gen, scrambled False: closed, H ok; normal form after the two shears True; factors ['x - y', 'x + 3*y']; (k,h) per factor [(-8628, -1392), (4396, 528)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - y, z - (719/116) y^2): detector True, EXACT colon True
  ok   own member rho = (x + 2 y)(x + -3 y), mode gen, scrambled True: closed, H ok; normal form after the two shears True; factors ['x - 3*y', 'x + 2*y']; (k,h) per factor [(-13335, -1080), (11715, 1920)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 3*y, z - (889/72) y^2): detector True, EXACT colon True
  ok   own member rho = (x + 1 y)(x + -2 y), mode w0, scrambled True: closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x + y']; (k,h) per factor [(0, 0), (0, 0)] (formulas = actual u: True); w = 0: True; dichotomy True; ray (x - 2*y, z - (0) y^2): detector True, EXACT colon True
  ok      stratum w = 0: both factors (0, 0), straight ray
  ok   own member rho = (x + -3 y)(x + 2 y), mode w0, scrambled True: closed, H ok; normal form after the two shears True; factors ['x - 3*y', 'x + 2*y']; (k,h) per factor [(0, 0), (0, 0)] (formulas = actual u: True); w = 0: True; dichotomy True; ray (x - 3*y, z - (0) y^2): detector True, EXACT colon True
  ok      stratum w = 0: both factors (0, 0), straight ray
  ok   own member rho = (x + 1 y)(x + -2 y), mode h1, scrambled True: closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x + y']; (k,h) per factor [(-2380, 1440), (1100, 0)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x - 2*y, z - (-119/72) y^2): detector True, EXACT colon True; factor with h = 0 != k: [('x + y', 1100)], there y^2 u in I and z u not in I (no ray from its u): True
  ok      stratum w != 0 with h_1 = 0: the other factor has h != 0
  ok   own member rho = (x + -2 y)(x + 1 y), mode h1, scrambled True: closed, H ok; normal form after the two shears True; factors ['x - 2*y', 'x + y']; (k,h) per factor [(110, 0), (-1030, -360)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x + y, z - (103/36) y^2): detector True, EXACT colon True; factor with h = 0 != k: [('x - 2*y', 110)], there y^2 u in I and z u not in I (no ray from its u): True
  ok      stratum w != 0 with h_1 = 0: the other factor has h != 0
  ok   own member rho = (x + 3 y)(x + 1 y), mode h1, scrambled False: closed, H ok; normal form after the two shears True; factors ['x + y', 'x + 3*y']; (k,h) per factor [(1525, -1200), (125, 0)] (formulas = actual u: True); w = 0: False; dichotomy True; ray (x + y, z - (-61/48) y^2): detector True, EXACT colon True; factor with h = 0 != k: [('x + 3*y', 125)], there y^2 u in I and z u not in I (no ray from its u): True
  ok      stratum w != 0 with h_1 = 0: the other factor has h != 0
  ok   actual members accepted: 16 of 16
time 5.3 s
ALL CHECKS PASSED
