Finite-budget dimer rotation and saddle translation

Let $U,V$ be as in Regular minimum-energy paths on a declared collision-free tube. Inputs: $x_0\in U$, a unit vector $v_0$, finite-difference distance $ϵ>0$, force-evaluation error bound $\eta \ge 0$, rotation step $\alpha >0$, rotation tolerance ${\tau }_{\mathrm{r}\mathrm{o}\mathrm{t}}\ge 0$, force tolerance ${\tau }_F\ge 0$, rotation cap $J\in {\mathbb{N}}_0$, translation step $\sigma >0$, and translation cap $K\in {\mathbb{N}}_0$. Return the last point, direction, reported residuals, a termination status from force-tolerance, rotation-cap, translation-cap, geometry-failure, rotation-failure, or nonfinite-evaluation, and mode_status=mode-unverified.

Initialize the translation-update counter $k=0$, the current point $x=x_0$, and the current unit direction $v=v_0$. At each translation iteration, start with the current $v$, set the rotation-update counter $j=0$, and repeat the following residual test and possible update:

1. If $x\pm ϵv\mathrm{\notin }U$, return geometry-failure. Evaluate approximate gradients ${\widehat{g}}_{\pm }$ at $x\pm ϵv$ with the declared error bound, and form $D_{ϵ}=({\widehat{g}}_{+}-{\widehat{g}}_{-})\mathrm{/}(2ϵ)$. Nonfinite data return nonfinite-evaluation.
2. Set $r_{\mathrm{r}\mathrm{o}\mathrm{t}}=\mathrm{\parallel }(I-v{v}^T)D_{ϵ}{\mathrm{\parallel }}_2$. If $r_{\mathrm{r}\mathrm{o}\mathrm{t}}\le {\tau }_{\mathrm{r}\mathrm{o}\mathrm{t}}$, end this rotation loop. If the test fails and $j=J$, return rotation-cap with the last direction. Otherwise set $w=v-\alpha (I-v{v}^T)D_{ϵ}$. If $w=0$, return rotation-failure; else replace $v$ by $w\mathrm{/}\mathrm{\parallel }w{\mathrm{\parallel }}_2$, set $j\leftarrow j+1$, and repeat the rotation test.

Thus $J$ counts direction updates, and the current direction is tested before the cap is returned, including when $J=0$. A small rotational residual by itself does not certify that the direction is the lowest eigenmode. This procedure does not take or validate a spectral-gap or mode-tracking certificate, so it always returns mode_status=mode-unverified.

After a rotation residual passes, evaluate $\widehat{g}(x)$ with error at most $\eta$. If it is nonfinite, return nonfinite-evaluation. If $\mathrm{\parallel }\widehat{g}(x){\mathrm{\parallel }}_2\le {\tau }_F$, return force-tolerance together with the mode status. If $k=K$, return translation-cap together with the mode status; perform an update only when $k<K$.

$x^{+}=x+\sigma [-\widehat{g}(x)+2(\widehat{g}(x{)}^{T}v)v]\mathrm{.}$

If $x^{+}\mathrm{\notin }U$, return geometry-failure; otherwise set $x\leftarrow x^{+}$, set $k\leftarrow k+1$, and begin the next translation iteration. Thus $K$ counts translation updates; the force-tolerance test is made before the cap test, including when $K=0$. mode_status=mode-unverified is metadata about the direction, not a termination status. This finite procedure makes no arbitrary-start, termination, saddle-convergence, or global-transition claim.