Height on S²: perfect Morse, c=(1,0,1), Σ(−1)^i c_i = χ(S²) = 2

Sympy-exact arithmetic witness of the Morse inequalities equality case on S². The height function f(x, y, z) = z restricted to the round sphere S² ⊂ R³ has exactly two non-degenerate critical points: a minimum at the south pole S = (0, 0,…