# exact_cert.py (verifier A, AMR-050-0011)

== E4  (N=4, tau=1): N=4, a^2=16/9, b=1, lam=16/25
  T: all 4 chords tangent to the caustic (exact identity in R): True   [0.0s]
  A*A_M independent of u, all M (exact): True   [1.4s] (sizes n=161, d=161)
     value at u=0: A*A_M = 2048/625
  A_M/A independent of u, all M (exact): False   [2.2s]
     value at u=0: A_M/A = 288/625
  A'_M/A' independent of u, all M (exact): True   [2.6s]
     value at u=0: A'_M/A' = 1/2
  (control) A'*A'_M independent of u, all M: False   [2.9s]
# exact_cert.py (verifier A, AMR-050-0011)

== E8  (N=8, tau=1): N=8, a^2=8/3, b=1, lam=8/35
  T: all 8 chords tangent to the caustic (exact identity in R): True   [0.0s]
  A*A_M independent of u, all M (exact): True   [67.0s] (sizes n=1347, d=449)
     value at u=0: A*A_M = 256*(8*X**2 + 3*Y**2 + 13)/175
  A_M/A independent of u, all M (exact): False   [113.2s]
     value at u=0: A_M/A = 24*(8*X**2 + 3*Y**2 + 13)/343
  A'_M/A' independent of u, all M (exact): True   [133.4s]
     value at u=0: A'_M/A' = 3*(8*X**2 + 3*Y**2 + 15)/50
  (control) A'*A'_M independent of u, all M: False   [150.0s]

== E8s (N=8, tau=3): N=8, a^2=32/27, b=1, lam=32/35
  T: all 8 chords tangent to the caustic (exact identity in R): True   [0.0s]
  A*A_M independent of u, all M (exact): True   [66.0s] (sizes n=1347, d=449)
     value at u=0: A*A_M = -512*(32*X**2 + 27*Y**2 - 13)/4725
  A_M/A independent of u, all M (exact): False   [110.6s]
     value at u=0: A_M/A = -4*(32*X**2 + 27*Y**2 - 13)/343
  A'_M/A' independent of u, all M (exact): True   [131.1s]
     value at u=0: A'_M/A' = -(32*X**2 + 27*Y**2 - 85)/600
  (control) A'*A'_M independent of u, all M: False   [148.5s]

== E6  (N=6, tau=1; generator dd = sqrt2 dn u): N=6, a^2=4, b=1, lam=4/9
  T: all 6 chords tangent to the caustic (exact identity in R): True   [0.0s]
  A*A_M independent of u, all M (exact): False   [5.1s] (sizes n=870, d=289)
  A_M/A independent of u, all M (exact): True   [8.0s]
     value at u=0: A_M/A = (4*X**2 + Y**2 + 7)/9
  A'_M/A' independent of u, all M (exact): False   [9.6s]
     value at u=0: A'_M/A' = 4*(4*X**2 + Y**2 + 9)/45
  (control) A'*A'_M independent of u, all M: True   [10.9s]
