[PASS] S  round S^2 (stereographic): Sc = 2, K(d_1,d_2) = 1 (Sc = 2, K = 1)
[PASS] S  round S^3 (stereographic): Sc = 6, K(d_1,d_2) = 1 (Sc = 6, K = 1)
[PASS] A  m=1 (dim 3): brute-force Sc equals formula (3) [0.1s]
[PASS] B  m=1: sectional curvatures K_ab as in formula (2)
[PASS] B  m=1: all 6 components R_abcd with {a,b} != {c,d} vanish
[PASS] C  m=1: (f h^m/2) Sc = (1 - m f h^(m-1)) phi' + m(m-1)/2 f h^(m-2) (1-phi^2)   [brute-force Sc]
[PASS] E  m=1: core f = r, h = 1 has Sc = m(m-1) = 0
[PASS] E  m=1: end f = F, h = r - c: full Riemann tensor vanishes
[PASS] E  m=1: negative control h = r - c + r^2/7 is detected as non-flat
[PASS] G  m=1: |omega(u)| = 1
[PASS] G  m=1: Phi^*(dsigma^2 + |dy|^2) = dr^2 + F^2 dtau^2 + (r-c)^2 g_S
[PASS] A  m=2 (dim 4): brute-force Sc equals formula (3) [0.3s]
[PASS] B  m=2: sectional curvatures K_ab as in formula (2)
[PASS] B  m=2: all 30 components R_abcd with {a,b} != {c,d} vanish
[PASS] C  m=2: (f h^m/2) Sc = (1 - m f h^(m-1)) phi' + m(m-1)/2 f h^(m-2) (1-phi^2)   [brute-force Sc]
[PASS] E  m=2: core f = r, h = 1 has Sc = m(m-1) = 2
[PASS] E  m=2: end f = F, h = r - c: full Riemann tensor vanishes
[PASS] E  m=2: negative control h = r - c + r^2/7 is detected as non-flat
[PASS] G  m=2: |omega(u)| = 1
[PASS] G  m=2: Phi^*(dsigma^2 + |dy|^2) = dr^2 + F^2 dtau^2 + (r-c)^2 g_S
[PASS] A  m=3 (dim 5): brute-force Sc equals formula (3) [0.5s]
[PASS] B  m=3: sectional curvatures K_ab as in formula (2)
[PASS] B  m=3: all 90 components R_abcd with {a,b} != {c,d} vanish
[PASS] C  m=3: (f h^m/2) Sc = (1 - m f h^(m-1)) phi' + m(m-1)/2 f h^(m-2) (1-phi^2)   [brute-force Sc]
[PASS] E  m=3: core f = r, h = 1 has Sc = m(m-1) = 6
[PASS] E  m=3: end f = F, h = r - c: full Riemann tensor vanishes
[PASS] E  m=3: negative control h = r - c + r^2/7 is detected as non-flat
[PASS] G  m=3: |omega(u)| = 1
[PASS] G  m=3: Phi^*(dsigma^2 + |dy|^2) = dr^2 + F^2 dtau^2 + (r-c)^2 g_S
[PASS] D  m=1: non-diagonal chart (off-diagonal entries present: True) gives the same Sc [1.1s]
[PASS] D  m=2: non-diagonal chart (off-diagonal entries present: True) gives the same Sc [2.3s]
[PASS] H  symbolic m: (f h^m/2) * (3) equals formula (4)
[PASS] H  symbolic m: with h' = phi, h^m f' = 1 - phi, formula (3) gives the key identity (5)
[PASS] M  m=1: mean curvature of {r = const} is f'/f + m h'/h; corner model: inner 1/l, outer m/R
[PASS] M  m=2: mean curvature of {r = const} is f'/f + m h'/h; corner model: inner 1/l, outer m/R
[PASS] M  m=3: mean curvature of {r = const} is f'/f + m h'/h; corner model: inner 1/l, outer m/R
[PASS] N  Euclidean Reissner-Nordstrom metric V dt^2 + drho^2/V + rho^2 g_S2 is scalar-flat for all (mu, q)
[PASS] N  its Ricci eigenvalues are q^2/rho^4 (t, rho directions) and -q^2/rho^4 (sphere): not flat for q != 0
[PASS] N  rho_+ = mu + sqrt(mu^2 + q^2) is a zero of V
[PASS] N  V'(rho_+) = 2 (mu rho_+ + q^2)/rho_+^3, and mu rho_+ + q^2 = rho_+ sqrt(mu^2+q^2) > 0 (bolt, also for mu < 0)
[PASS] N  rho_+ > 0 at sample parameters with mu < 0, mu = 0 and mu > 0

sympy 1.14.0; total time 9.9s
RUN2 CURVATURE: ALL CHECKS PASSED
