Finite-budget index-one eigenvector-following Newton step

Let $U,V$ be as in Regular minimum-energy paths on a declared collision-free tube. Inputs: an initial point $x_0\in U$, a unit initial mode reference $v_{\mathrm{r}\mathrm{e}\mathrm{f}}\in {\mathbb{R}}^d$, a local-ball center $x_{\mathrm{r}\mathrm{e}\mathrm{f}}\in U$, radius $r_{\mathrm{l}\mathrm{o}\mathrm{c}}>0$, gradient tolerance ${\tau }_g\ge 0$, eigenvalue margin $\gamma >0$, eigenpair residual tolerance ${\tau }_{\mathrm{e}\mathrm{i}\mathrm{g}}\ge 0$, mode-overlap threshold $0\le {\tau }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}}<1$, update budget $N\in {\mathbb{N}}_0$, and a supplied spectral-data routine $E$. On input $(x,H(x))$, $E$ must return in finite time either failure or an orthogonal matrix $Q$, ordered approximate eigenvalues ${\widehat{\lambda }}_j$, and validated ordered intervals $I_j$ with their checked certificates as specified below. This input routine supplies spectral data; the enclosure definition does not compute them. Return the last point, the most recently accepted mode reference, and a termination status from geometry-failure, outside-local-basin, gradient-residual-tolerance, budget-exhausted, eigensolve-tolerance-failure, index-one-branch-failure, mode-tracking-failure, or singular-jacobian. The mode reference starts at $v_{\mathrm{r}\mathrm{e}\mathrm{f}}$ and changes only after a successful eigenpair and overlap check, storing the signed $q_1$ from that check. A positive residual tolerance does not in general certify that $q_1$ is an exact lowest eigenvector, and a reference accepted at an earlier iterate carries no spectral certification at the returned point. When called at $x_k$ in step 2, the input routine $E$ supplies an orthogonal matrix $Q=[q_1,\dots ,q_d]$, ordered approximate eigenvalues ${\widehat{\lambda }}_1\le \cdots \le {\widehat{\lambda }}_d$, and validated intervals $I_j=[{\mathrm{\ell }}_j,u_j]$ in the sense of Validated ordered eigenvalue enclosures for an index-one Hessian. Require every residual $\mathrm{\parallel }H(x_k)q_j-{\widehat{\lambda }}_j{q}_j{\mathrm{\parallel }}_2\le {\tau }_{\mathrm{e}\mathrm{i}\mathrm{g}}$, $\sqrt{d}{\tau }_{\mathrm{e}\mathrm{i}\mathrm{g}}\le \gamma \mathrm{/}2$, and ${\widehat{\lambda }}_j\in I_j$. The Hessian $H={\mathrm{\nabla }}^{2}V$ is symmetric by The Hessian of a $C^2$ scalar field is symmetric.

For $k=0,\dots ,N$, in order:

1. If $x_k\mathrm{\notin }U$, return geometry-failure. If $x_k\mathrm{\notin }{B}_2(x_{\mathrm{r}\mathrm{e}\mathrm{f}},r_{\mathrm{l}\mathrm{o}\mathrm{c}})$, return outside-local-basin. Compute $g_k=\mathrm{\nabla }V(x_k)$. If $\mathrm{\parallel }{g}_k{\mathrm{\parallel }}_2\le {\tau }_g$, return gradient-residual-tolerance; if $k=N$, return budget-exhausted.
2. Evaluate $H(x_k)$ and call $E(x_k,H(x_k))$. If it returns failure, return eigensolve-tolerance-failure; otherwise use its returned $Q$, ${\widehat{\lambda }}_j$, and $I_j$. If an eigenpair residual exceeds ${\tau }_{\mathrm{e}\mathrm{i}\mathrm{g}}$, $\sqrt{d}{\tau }_{\mathrm{e}\mathrm{i}\mathrm{g}}>\gamma \mathrm{/}2$, or some ${\widehat{\lambda }}_j\mathrm{\notin }{I}_j$, return eigensolve-tolerance-failure. If the validated intervals do not satisfy $u_1\le -\gamma$ and ${\mathrm{\ell }}_j\ge \gamma$ for all $j\ge 2$ (with the dimension-one convention in the definition), return index-one-branch-failure. If $\mathrm{\mid }{q}_1^T{v}_{\mathrm{r}\mathrm{e}\mathrm{f}}\mathrm{\mid }\le {\tau }_{\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}}$, return mode-tracking-failure; otherwise retain $q_1$ when $q_1^T{v}_{\mathrm{r}\mathrm{e}\mathrm{f}}>0$ and replace it by $-q_1$ when $q_1^T{v}_{\mathrm{r}\mathrm{e}\mathrm{f}}<0$. Use the resulting direction as the next mode reference. This overlap check screens changes of the computed reference direction; it does not certify exact mode identity or alter the Newton correction.
3. Call Newton correction for a square vector residual with $A=H(x_k)$ and $r=g_k$. If it returns singular-jacobian, return that status. Its correction $p_k$ is the exact Newton solution of $H(x_k)p_k=-g_k$ in the mathematical model; the eigenpair data certifies the index-one branch but does not replace this linear solve.
4. If $x_k+p_k\mathrm{\notin }U$ or lies outside $B_2(x_{\mathrm{r}\mathrm{e}\mathrm{f}},r_{\mathrm{l}\mathrm{o}\mathrm{c}})$, return outside-local-basin. Otherwise set $x_{k+1}=x_k+p_k$ and continue.

An approximate eigensolve has a finite residual-tolerance branch and an explicit failure status. The definition makes no global-convergence claim. Local quadratic behavior is stated separately and requires the hypotheses of Updated-Jacobian Newton has local quadratic convergence at a simple root.