--- 1. sanity: round spheres (graph chart) ---
S^2: Sc = 2 True
S^3: Sc = 6 True
--- 2. Sc of dr^2 + f^2 dtau^2 + h^2 g_{S^m} vs formula (3) ---
m=1 symbolic: Sc == formula (3)  [0.3s] True
m=2 symbolic: Sc == formula (3)  [1.0s] True
m=3 symbolic: Sc == formula (3)  [4.3s] True
m=4 at chart point (-1/10, 3/10, -21/100, -1/20): Sc == formula (3)  [400.0s] True
--- 3. identity (F3), symbolic m ---
(f h^m/2) Sc == (1 - m f h^(m-1)) phi' + m(m-1)/2 f h^(m-2)(1-phi^2)  [symbolic m] True
   m=1 True
   m=2 True
   m=3 True
   m=4 True
   m=5 True
   m=6 True
   m=7 True
   m=8 True
   f'' formula along ODE, m=1 True
   f'' formula along ODE, m=2 True
   f'' formula along ODE, m=3 True
   f'' formula along ODE, m=7 True
--- 4. end metric dr^2 + F^2 dtau^2 + (r-c)^2 g_{S^m}: full Riemann tensor = 0 ---
m=1: all R^a_bcd vanish (non-diagonal chart) True
m=1: negative control detected non-flat True
m=2: all R^a_bcd vanish (non-diagonal chart) True
m=2: negative control detected non-flat True
m=3: all R^a_bcd vanish (non-diagonal chart) True
m=3: negative control detected non-flat True
ALL INDEPENDENT SYMBOLIC CHECKS PASSED
