The improved NEB monotone branch is generally only first-order consistent

At $n=0$, [A1] gives ${\mathrm{\partial }}_{s}X=T(s)$ and ${\mathrm{\partial }}_{n}X=N(s)$, an orthonormal coordinate basis. The potential has coordinate derivatives ${\mathrm{\partial }}_{s}V=W^{\mathrm{\prime }}(s)$ and ${\mathrm{\partial }}_{n}V=n=0$, so $\mathrm{\nabla }V(R(s))=W^{\mathrm{\prime }}(s)T(s)$. Therefore its component normal to $T(s)$ vanishes and $R$ is a regular MEP by [F2].

We have $W^{\mathrm{\prime }}(s)=4s(s^2-a^2)>0$ for $-a<s<0$ and $W^{\mathrm{\prime }}(s)<0$ for $0<s<a$, so the energy is strictly increasing then decreasing. Since $V=W(s)+n^2\mathrm{/}2\ge 0$ with equality exactly at $(s,n)=(\pm a,0)$, those two endpoints are global minima on the sector.

On an equal-arc mesh of width $h$ in the increasing subarc $(-a,0)$, [F1] selects the forward chord. For a circle, $R(s+h)-R(s)$ points in direction $T(s+h\mathrm{/}2)$, so the angle between its normalized direction and $T(s)$ is exactly $h\mathrm{/}(2R_0)$. The true gradient magnitude there is $\mathrm{\mid }{W}^{\mathrm{\prime }}(s)\mathrm{\mid }$, and its projection onto the normal complement of $T(s+h\mathrm{/}2)$ has norm $\mathrm{\mid }{W}^{\mathrm{\prime }}(s)\mathrm{\mid }\sin (h\mathrm{/}(2R_0))$.

Fix $s$ strictly inside the subarc, so $\mathrm{\mid }{W}^{\mathrm{\prime }}(s)\mathrm{\mid }>0$. By [F3], the ratio of this force error to $h$ tends to $\mathrm{\mid }{W}^{\mathrm{\prime }}(s)\mathrm{\mid }\mathrm{/}(2R_0)>0$ as $h\to 0$. The chord length is $2R_0\sin (h\mathrm{/}(2R_0))>0$ for small $h$, so there is no tangent degeneracy. A quantity whose ratio to $h$ tends to a positive number cannot be $O(h^2)$; this proves the claim and, by [F4], distinguishes this one-sided estimate from the centered-string estimate.