=== Part A: exact rational arithmetic ===
e < 2.7183 (rational upper bound for e): True
1.134^8 > 2.7183 > e, so e^(1/8) < 1.134: True
(1/2) * 1.134 = 0.567 and 1 - 0.567 = 0.433 > 0.43: True
(1/2)(1 + 1/(8m))^(m-1) < 0.567 for m = 1..3000 (exact): True
1/4 + (8/9)(1/8) = 13/36: True
(9/8)/(2 * (13/36)) = 81/52, and 81/(52 pi) < 1/2 because pi > 3.1415 > 81/26: True
=== Part B: the profile of the note, a = 1/(4m), b = 1/(2m) ===
m=  1 (n=  3): F=0.374182345504  m*F=0.374182346  h(b)=1.125000000000  max m f h^(m-1)=0.374182 <= 0.500000  min coeff=0.6258  Sc in (9.94e-168, 29.1192]  rel|(3)-(5)|=4.7e-16  ok=True
  m=1: all checks of Part B: True
m=  2 (n=  4): F=0.187088345621  m*F=0.374176691  h(b)=1.062500000000  max m f h^(m-1)=0.397563 <= 0.531250  min coeff=0.6026  Sc in (3.81e-167, 117.966]  rel|(3)-(5)|=5.4e-16  ok=True
  m=2: all checks of Part B: True
m=  3 (n=  5): F=0.124724924471  m*F=0.374174773  h(b)=1.041666666667  max m f h^(m-1)=0.406006 <= 0.542535  min coeff=0.5942  Sc in (8.45e-167, 266.565]  rel|(3)-(5)|=5.6e-16  ok=True
  m=3: all checks of Part B: True
m=  4 (n=  6): F=0.093543452033  m*F=0.374173808  h(b)=1.031250000000  max m f h^(m-1)=0.410360 <= 0.548355  min coeff=0.5898  Sc in (1.49e-166, 474.915]  rel|(3)-(5)|=6.7e-16  ok=True
  m=4: all checks of Part B: True
m=  5 (n=  7): F=0.074834645380  m*F=0.374173227  h(b)=1.025000000000  max m f h^(m-1)=0.413017 <= 0.551906  min coeff=0.5872  Sc in (2.32e-166, 743.013]  rel|(3)-(5)|=6.4e-16  ok=True
  m=5: all checks of Part B: True
m=  6 (n=  8): F=0.062362139759  m*F=0.374172839  h(b)=1.020833333333  max m f h^(m-1)=0.414807 <= 0.554299  min coeff=0.5854  Sc in (3.33e-166, 1070.86]  rel|(3)-(5)|=7.2e-16  ok=True
  m=6: all checks of Part B: True
m=  7 (n=  9): F=0.053453222962  m*F=0.374172561  h(b)=1.017857142857  max m f h^(m-1)=0.416095 <= 0.556021  min coeff=0.5841  Sc in (4.52e-166, 1458.46]  rel|(3)-(5)|=6.8e-16  ok=True
  m=7: all checks of Part B: True
m=  8 (n= 10): F=0.046771544017  m*F=0.374172352  h(b)=1.015625000000  max m f h^(m-1)=0.417067 <= 0.557319  min coeff=0.5832  Sc in (5.89e-166, 1905.82]  rel|(3)-(5)|=6.0e-16  ok=True
  m=8: all checks of Part B: True
m=  9 (n= 11): F=0.041574687751  m*F=0.374172190  h(b)=1.013888888889  max m f h^(m-1)=0.417825 <= 0.558333  min coeff=0.5824  Sc in (7.44e-166, 2412.93]  rel|(3)-(5)|=6.0e-16  ok=True
  m=9: all checks of Part B: True
m= 10 (n= 12): F=0.037417205977  m*F=0.374172060  h(b)=1.012500000000  max m f h^(m-1)=0.418434 <= 0.559146  min coeff=0.5818  Sc in (9.18e-166, 2979.79]  rel|(3)-(5)|=5.8e-16  ok=True
  m=10: all checks of Part B: True
m= 11 (n= 13): F=0.034015632123  m*F=0.374171953  h(b)=1.011363636364  max m f h^(m-1)=0.418933 <= 0.559813  min coeff=0.5813  Sc in (1.11e-165, 3606.4]  rel|(3)-(5)|=5.4e-16  ok=True
  m=11: all checks of Part B: True
m= 12 (n= 14): F=0.031180988720  m*F=0.374171865  h(b)=1.010416666667  max m f h^(m-1)=0.419350 <= 0.560371  min coeff=0.5809  Sc in (1.32e-165, 4292.76]  rel|(3)-(5)|=6.1e-16  ok=True
  m=12: all checks of Part B: True
m= 25 (n= 27): F=0.014966854259  m*F=0.374171356  h(b)=1.005000000000  max m f h^(m-1)=0.421751 <= 0.563580  min coeff=0.5785  Sc in (5.7e-165, 18652.8]  rel|(3)-(5)|=6.4e-16  ok=True
  m=25: all checks of Part B: True
m=100 (n=102): F=0.003741710040  m*F=0.374171004  h(b)=1.001250000000  max m f h^(m-1)=0.423429 <= 0.565823  min coeff=0.5768  Sc in (9.09e-164, 298679]  rel|(3)-(5)|=6.0e-16  ok=True
  m=100: all checks of Part B: True
=== Part C: other admissible parameters (Remark 3.3) ===
40 random admissible (m, a, b): Sc > 0 at all interior grid points: True
=== Part D: negative controls (parameters violating m b (1+(b-a)/2)^(m-1) < 1) ===
  m=1, a=1.2, b=2.0: m b (1+(b-a)/2)^(m-1) = 2.000 >= 1;  min Sc on the shell = -1.666
  m=1, a=1.2, b=2.0: Sc < 0 somewhere (as expected): True
  m=2, a=0.9, b=1.5: m b (1+(b-a)/2)^(m-1) = 3.900 >= 1;  min Sc on the shell = -6.301
  m=2, a=0.9, b=1.5: Sc < 0 somewhere (as expected): True
  m=3, a=0.5, b=1.0: m b (1+(b-a)/2)^(m-1) = 4.688 >= 1;  min Sc on the shell = -10.14
  m=3, a=0.5, b=1.0: Sc < 0 somewhere (as expected): True
=== Part E: Phi pulls the flat metric back to dr^2 + F^2 dtau^2 + (r-c)^2 g_{S^m} ===
m=1: Phi^*(d sigma^2 + |dy|^2) == dr^2 + F^2 dtau^2 + (r-c)^2 g_S: True
m=2: Phi^*(d sigma^2 + |dy|^2) == dr^2 + F^2 dtau^2 + (r-c)^2 g_S: True
m=3: Phi^*(d sigma^2 + |dy|^2) == dr^2 + F^2 dtau^2 + (r-c)^2 g_S: True
=== Part F: the lattice decomposition of Proposition 5.1 ===
200 random rational lattices: w_i = t_i w_1 + u_i, u_i orthogonal to w_1 and linearly independent: True
=== Part G: the constants of Proposition 5.2 (m = 1) ===
m=1: F = 0.374182346 >= 13/36 = 0.361111111: True
m=1: rho_1/|w_1| = rho_0/(2 pi F) = 0.478508 < 1/2: True
m=1: h <= 9/8 on [a, b]: True
Z^(n-1), n = 3..8: distance from w_(n-1) to span(w_1..w_(n-2)) >= |w_1|: True
ALL LEAD CHECKS PASSED
