A complete pointwise theory of when the induced metric can convert a non-convex loss into a geodesically convex one. The Riemannian Hessian is (Hess_g L)_ij = (H_ij - C_ij)/(1 + ξ‖∇L‖²_γ), where C_ij = Γ^k_ij(γ)·∇_k L is a curvature correction depending on ∇γ and ∇L but not on H. Three-level hierarchy: (1) constant γ gives C ∝ H — eigenvalue signs are preserved, saddles cannot be fixed; (2) scalar γ = e^s I gives trace-constrained C (tr(C) = 0 in 2D), which can fix asymmetric saddles but not symmetric ones; (3) diagonal γ = diag(e^{s_i}) gives unrestricted C via exponential anisotropy ratios e^{s_i - s_j}, enabling sign-flipping for all N. The diagonal-N theorem is constructively proved via a Gershgorin argument and formally verified in Lean 4 (zero sorry). A universal critical-point obstruction (C ∝ ∇L vanishes at saddles) precludes global geodesic convexity for any function with stationary points, but the geodesic convexity basin around any minimum can be dramatically enlarged beyond the Euclidean convexity basin (demonstrated on the quartic well, four disconnected basins merge into one). The optimal correction has anti-correlation structure (∇s_i with sign opposite to H_ii) which motivates the curvature-aware diagonal variant.