== (1) N = 4, symbolic rhombus
  ok   rhombus: vertices on the ellipse
  ok   rhombus: every side tangent to the caustic
  ok   rhombus: reflection law at every vertex
    A      = 2*a*b
    Adag   = 2/(a*b)
    Apol   = 4*a**3/b**5
    Adual  = 2*b**5/a**3
    Apt    = b**5/a**3
    Aped   = 4*a**3*b**3/(a**2 + b**2)**2
    Aant   = 4*a*b
    Aip    = 4*b**5/(a**3*(a**2 + b**2)**2)
  ok   A = 2ab
  ok   A_dag = 2/(ab)
  ok   A_pol = 4a^3/b^5
  ok   A_dual = 2b^5/a^3
  ok   A_pt = A_dual/2
  ok   A_ped = 4a^3b^3/(a^2+b^2)^2
  ok   A_ant = 4ab
  ok   A_pol/A_dag = 2a^4/b^4
  ok   A_pol/A_dual = 2a^6/b^10
  ok   A_ped/A_dual = 2a^6/(b^2(a^2+b^2)^2)
  ok   A*A_dag = 4
  ok   rhombus: polar sides touch the circle (O_p, a/b^2)
    A_ip at the rhombus = 4*b**5/(a**3*(a**2 + b**2)**2)

== (2) N = 6, symbolic axial hexagon
  ok   hexagon: vertices on the ellipse
  ok   hexagon: every side tangent to the caustic
  ok   hexagon: reflection law at every vertex
    A      = 2*a*b*s*(2*a + b)/(a + b)**2
    A_dag  = s*(2*a - b)*(2*a + b)/(2*a**2*b**3)
    A_ant  = -2*a*s*(a - 2*b)/b
    R(alpha,beta,lambda_6) = -(a - 2*b)*(a + b)**2/(b**2*(2*a + b))
  ok   R(lambda_6) = (2b-a)(a+b)^2/(b^2(2a+b))
  ok   A_ant/A = R at the axial hexagon (FerAnt Lemma 4, independent check)
    A/A_dag = 4*a**3*b**4/((a + b)**2*(2*a - b))
  ok   A/A_dag = 4a^3b^4/((2a-b)(a+b)^2) (formula quoted in 10.5281/zenodo.23094118)
    A_dag/A_ant (row k818, N = 6) = -(2*a - b)*(2*a + b)/(4*a**3*b**2*(a - 2*b))
  ok   A_dag/A_ant = (4a^2-b^2)/(4a^3 b^2 (2b-a))
    value at (a,b) = (1, 4/5): 35/16 = 2.18750000000
    roots of R(lambda_6) = 0 in b: [a/2]
  ok   R(lambda_6) = 0 iff b = a/2 (positive b)
  ok   A_pol/A_dag at hexagon, a=1,b=1/2: 64/15

== (3) the derivation in the proof of Theorem 5(c)
    solutions of (a-x1)^2 = beta x1^2/alpha: [a**2/(a - b), a**2/(a + b)]
  ok   solutions are a^2/(a+b) and a^2/(a-b)
  ok   a^2/(a-b) > a (excluded), a^2/(a+b) < a
  ok   lambda = (a - x1)^2 = a^2b^2/(a+b)^2
  ok   R = 0 needs b/a < 1/sqrt3: lambda(a-b)^2 = ab(a-3b) with alpha=a^2.. sign check

ALL SYMBOLIC CHECKS PASSED
