Let $S : D \to \mathbb{R}$ be the principal function for the Lagrangian $L$ with fixed initial point $(a, y_{\rm init}) \in D$, defined wherever the extremal from $(a, y_{\rm init})$ is unique. Let $u_0$ be the Jacobi field along $y_0$ with $u_0(a) = 0$, $u_0'(a) = 1$. Then: 1. $S$ is $C^2$ in a neighbourhood of $(c, y_0(c))$ if and only if $u_0(c) \ne 0$. 2. At a conjugate point $c$ where $u_0(c) = 0$, the map $y \mapsto S(x, y)$ fails to be $C^2$ near $(c, y_0(c))$: the Hessian $\partial^2 S / \partial y^2$ blows up, the field folds (different extremals of the central field coalesce), and the Hamilton–Jacobi equation has no $C^2$ solution in any neighbourhood of $(c, y_0(c))$.