=== Part A: brute force Sc vs closed formula (F1) ===
m=1 (dim 3) brute-force Sc == F1   [0.1s]: True
m=2 (dim 4) brute-force Sc == F1   [0.2s]: True
m=3 (dim 5) brute-force Sc == F1   [0.4s]: True
m=4 (dim 6) brute-force Sc == F1   [0.9s]: True
m=5 (dim 7) brute-force Sc == F1   [2.6s]: True
=== Part B: sanity checks ===
m=1: round S^3 has Sc = 6: True
m=1: F1 on round S^3: True
m=1: core R^2 x S^m(1) has Sc = m(m-1): True
m=1: end S^1(F) x (R^(m+1)\B) has Sc = 0 (brute force): True
m=2: round S^4 has Sc = 12: True
m=2: F1 on round S^4: True
m=2: core R^2 x S^m(1) has Sc = m(m-1): True
m=2: end S^1(F) x (R^(m+1)\B) has Sc = 0 (brute force): True
m=3: round S^5 has Sc = 20: True
m=3: F1 on round S^5: True
m=3: core R^2 x S^m(1) has Sc = m(m-1): True
m=3: end S^1(F) x (R^(m+1)\B) has Sc = 0 (brute force): True
m=4: round S^6 has Sc = 30: True
m=4: F1 on round S^6: True
m=4: core R^2 x S^m(1) has Sc = m(m-1): True
m=4: end S^1(F) x (R^(m+1)\B) has Sc = 0 (brute force): True
=== Part C: identity (F2) for symbolic m ===
(f h^m/2) Sc == -(h^m f')' + f h^(m-2)[m(m-1)/2 (1-h'^2) - m h h''] (symbolic m): True
=== Part D: Sc for the profile h' = phi, h^m f' = 1 - phi (symbolic m) ===
Sc == 2[(1 - m f h^(m-1)) phi' + m(m-1)/2 f h^(m-2)(1-phi^2)]/(f h^m)  (symbolic m): True
d/dr[(1-phi)/h^m] == -phi'/h^m - m(1-phi)phi/h^(m+1) when h' = phi: True
  integer m=1: (F3) holds: True
  integer m=2: (F3) holds: True
  integer m=3: (F3) holds: True
  integer m=4: (F3) holds: True
  integer m=5: (F3) holds: True
=== Part E: product with a flat circle ===
m=1: Sc(g + dz^2) == F1: True
m=2: Sc(g + dz^2) == F1: True
=== Part F: the end metric dr^2 + F^2 dtau^2 + (r-c)^2 g_{S^m} has vanishing full Riemann tensor ===
m=1: end metric is flat (all R^a_bcd = 0): True
m=1: core metric dr^2 + r^2 dtau^2 + dth^2 is flat (all R^a_bcd = 0): True
m=2: end metric is flat (all R^a_bcd = 0): True
m=2: negative control, core R^2 x S^2(1) is detected as NOT flat: True
m=3: end metric is flat (all R^a_bcd = 0): True
m=3: negative control, core R^2 x S^3(1) is detected as NOT flat: True
ALL CHECKS PASSED
