Paraboloid f=x²−y²: grad(0)=0; Hessian diag(2,−2); det −4; Morse index 1

Sympy-exact Morse-critical-point witness for the canonical saddle f(x, y) = x² − y² on R². Step 1 — gradient: ∇f(x, y) = (2x, −2y), vanishing at the origin: ∇f(0, 0) = (0, 0)^T. Step 2 — Hessian: H = diag(2, −2), a constant symmetric…