Positive determinant of $\mathrm{THess}$ on the energy component implies positive definiteness

Throughout, $\mathrm{Phase}=\mathbb R^4$ with coordinates $s=(z_1,z_2,w_1,w_2)$ of the Levi-Civita regularization $q+\mu=2z^2$, $p=w/\bar z$ of the planar circular restricted three-body problem at the primary of mass $1-\mu$ located at $(-\mu,0)$. The regularized Hamiltonian at the Jacobi level $H=-c$ is $K_{\mu,c}=P-\mu Z/r$, where $Z=z_1^2+z_2^2$, $r=\lvert 2z^2-1\rvert=\sqrt{D}$ with $D=(2(z_1^2-z_2^2)-1)^2+(4z_1z_2)^2$, and $P$ is the polynomial part of eq. (2.2). The set $U=\{D>0\}$ excludes the (unregularized) second collision.

Let $0\le\mu\le\tfrac12$ and $c\ge\tfrac{21}{10}$, and let $\Sigma_{\mu,c}$ be the left energy component. Suppose that the symmetrized tangential Hessian has positive determinant everywhere on it:

Then $\mathrm{THess}_{\mu,c}(s)$ is positive definite for every $s\in\Sigma_{\mu,c}$.

This isolates the purely numerical content of the positive-tangential-Hessian clause of Proposition 4.4. Once the determinant is known to be positive along the component, definiteness propagates from the collision point by connectedness.