Part A. Lemma 3.1 (disk model) against a direct simulation
  2000 random (p, beta, t<=60): max |direct - model| = 1.06e-12
  ok    model (Lemma 3.1) agrees with the simulation
Part B. Lemmas 3.2 and 3.3
  g_(1,0)(0) = 0.250000000000 (exactly 1/4: arccos 0 = pi/2)
  g_(1,0)(1-1e-12) = 0.159154943, 1/(2 pi) = 0.159154943
  ok    limit k1/(2 pi) at rho -> 1
  ok    1/4 differs from 1/(2 pi) (pi > 2)
  ok    |g_k| large near rho = 1 when k2 != 0 (min over |k1|<=5, |k2|<=3: 35354.5)
  ok    EXACT: |h_rho(f)|^2 = rho^2 + (1-2f)^2 (1-rho^2) on a 39 x 41 rational grid
  ok    radial law of Lemma 3.3 (Monte Carlo, 2e5 samples, max dev 1.7e-03)
Part C. Corollary 3.5 (ball in R^3 via the plane through the axis)
  1000 random photons in the unit 3-ball: max |gamma_v(t) - E_w(g_beta(t))| = 1.69e-12
  ok    planar reduction of Corollary 3.5
Part D. Ellipse: kappa, the reflection, the frequencies (Section 4)
  kappa of (4) before and after 3000 random reflections: max difference 6.2e-15
  ok    kappa is a first integral of the billiard
  a=2.0, b=1.7321, kappa=-0.0900 (from (4): -0.0900): bounces/(T w) = 0.99979, x-sign changes/(2 T w_nu) = 0.99987
  ok    frequencies (9) reproduce the simulated rates
  a=2.0, b=1.7321, kappa=-0.8100 (from (4): -0.8100): bounces/(T w) = 0.99985, x-sign changes/(2 T w_nu) = 0.99985
  ok    frequencies (9) reproduce the simulated rates
  a=2.0, b=1.7321, kappa=-2.2500 (from (4): -2.2500): bounces/(T w) = 0.99991, x-sign changes/(2 T w_nu) = 0.99997
  ok    frequencies (9) reproduce the simulated rates
  a=2.0, b=1.7321, kappa=+0.9900 (from (4): +0.9900): bounces/(T w) = 0.99997, x-sign changes/(2 T w_nu) = 1.00009
  ok    frequencies (9) reproduce the simulated rates
  a=2.0, b=1.7321, kappa=+0.7500 (from (4): +0.7500): bounces/(T w) = 0.99997, x-sign changes/(2 T w_nu) = 1.00025
  ok    frequencies (9) reproduce the simulated rates
  a=2.0, b=1.7321, kappa=+0.1900 (from (4): +0.1900): bounces/(T w) = 0.99999, x-sign changes/(2 T w_nu) = 0.99980
  ok    frequencies (9) reproduce the simulated rates
  a=1.0, b=0.6000, kappa=-0.1600 (from (4): -0.1600): bounces/(T w) = 1.00004, x-sign changes/(2 T w_nu) = 0.99985
  ok    frequencies (9) reproduce the simulated rates
  a=1.0, b=0.6000, kappa=+0.5500 (from (4): +0.5500): bounces/(T w) = 1.00009, x-sign changes/(2 T w_nu) = 1.00011
  ok    frequencies (9) reproduce the simulated rates
Part E. Limits of the frequencies (Lemmas 4.3, 4.4)
  ellipse a=2.0, b=1.7321: perimeter 11.7396977 > 4a = 8.0
  ok    perimeter > 4a
    rotational, eps = B(1-1e-7): omega = (6.422e+02, 0.0851811); 1/perimeter = 0.0851811
  ok    whispering end: omega_mu -> inf, omega_nu -> 1/perimeter
    rotational, eps = 1e-04: omega = (0.25665, 0.12167); limit (0.25000, 0.12500)
    rotational, eps = 1e-08: omega = (0.25352, 0.12324); limit (0.25000, 0.12500)
    rotational, eps = 1e-12: omega = (0.25239, 0.12381); limit (0.25000, 0.12500)
    librational, eps = 1e-6: P_mu - P_nu = -2.197225, D_0 = -2.197225; T_mu = 4.000000 (4(a-c) = 4.000000), T_nu = 4.000000 (4c = 4.000000)
  ok    P_mu - P_nu -> D_0 < 0
    librational, eps = 1e-6: omega = (0.127300, 0.122700); (11) to first order: (0.127485, 0.122515)
  ok    expansion (11), first order
    librational, eps = 1-1e-10: omega = (0.1443376, 0.0962250); limit (0.1443376, 0.0962250)
  ok    minor-axis end of the librational regime
  ellipse a=1.0, b=0.6000: perimeter 5.1053998 > 4a = 4.0
  ok    perimeter > 4a
    rotational, eps = B(1-1e-7): omega = (1.748e+03, 0.1958711); 1/perimeter = 0.1958710
  ok    whispering end: omega_mu -> inf, omega_nu -> 1/perimeter
    rotational, eps = 1e-04: omega = (0.54522, 0.24435); limit (0.50000, 0.25000)
    rotational, eps = 1e-08: omega = (0.52322, 0.24710); limit (0.50000, 0.25000)
    rotational, eps = 1e-12: omega = (0.51562, 0.24805); limit (0.50000, 0.25000)
    librational, eps = 1e-6: P_mu - P_nu = -5.493061, D_0 = -5.493061; T_mu = 0.800000 (4(a-c) = 0.800000), T_nu = 3.200000 (4c = 3.200000)
  ok    P_mu - P_nu -> D_0 < 0
    librational, eps = 1e-6: omega = (0.265341, 0.246165); (11) to first order: (0.265904, 0.246024)
  ok    expansion (11), first order
    librational, eps = 1-1e-10: omega = (0.4166667, 0.1706940); limit (0.4166667, 0.1706940)
  ok    minor-axis end of the librational regime
  ok    tan x - 4x/pi < 0 on a grid of (0, pi/4) (max -2.15e-06)
  ok    cos(phi) - 1 + (2 phi/pi) sin(phi) > 0 on a grid of (0, pi/2) (min 3.37e-11)
  ok    b + c - a - (2c/pi) arctan(b/c) > 0 for a = 1, 0 < b < 1 (min 2.73e-05)
Part F. Proposition 4.5, EXACT (a, b, c) = (5, 4, 3), rational directions
  ok    EXACT: 1428 rational directions from F2 reflect into the segment towards F1, total length |F2X| + |XF1| = 2a
Part G. Section 5: the numbers of the construction (EXACT) and the focal property
  ok    EXACT: I_0 = (-7/8, 7/8), flat top [7/8, 9/8]
  ok    EXACT: the focus (1, 0) lies on the flat top
  ok    EXACT: max of x^2/4 + y^2/3 on [1.5,1.7]x[-0.6,-0.4] = 337/400 < 1
  ok    EXACT: lower corners at y = -sqrt(165)/8 = -1.6057, -sqrt(117)/8 = -1.3521
  ok    EXACT: V lies to the right of the right tooth
  ok    psi is convex on (0, 1] (sign of psi'')
  ok    focal property: 20000 random reflections, separation of the foci preserved

ALL CHECKS PASSED
