A gauge-fixed stable discrete path lies within second order of a smooth MEP

Let $L>0$ and let $M\ge 2$ be an integer. Let $V$ and $U$ satisfy Regular minimum-energy paths on a declared collision-free tube, and let $R_{\ast }:[0,L]\to U$ be a $C^3$ unit-speed regular MEP with fixed endpoints, using the endpoint extension convention of Polyline interpolation and equal-length redistribution approximate a smooth arc-length path. For $h=L\mathrm{/}M$, let $z_h=(R_{\ast }(h),\dots ,R_{\ast }((M-1)h))$ be its interior samples and let $I_h$ be piecewise-linear reconstruction with the same fixed endpoints.

In a local coordinate chart, choose a smooth orthonormal normal frame $Q_i$ at each centered discrete tangent, and keep that frame convention and chart fixed throughout the local ball. Define the square gauge-fixed residual $G_h$ on the $d(M-1)$ interior coordinates by the $(d-1)(M-1)$ scalar entries of $Q_i^T\mathrm{\nabla }V(R_i)$ and the $M-1$ independent spacing components ${\mathrm{\ell }}_i-{\mathrm{\ell }}_M$, $1\le i<M$, where ${\mathrm{\ell }}_i=\mathrm{\parallel }{R}_i-R_{i-1}{\mathrm{\parallel }}_2$. Its norm is the maximum absolute value of all scalar residual coordinates; the domain norm is the maximum absolute value of all node coordinates. Assume this chosen map is $C^1$ on the open ball in hypothesis 2. The endpoints are fixed and the spacing equations fix the arc-length reparameterization; a zero of this residual is exactly a zero of all centered-tangent normal force components and equality of all segment lengths.

Assume that for all sufficiently small $h$ the following constants $C_0,\beta ,L_G,C_I>0$ are independent of $h$:

1. The consistency and interpolation bounds are $\mathrm{\parallel }{G}_h(z_h){\mathrm{\parallel }}_{\mathrm{\infty }}\le C_0{h}^2$ and $\mathrm{\parallel }{I}_h{z}_h-R_{\ast }{\mathrm{\parallel }}_{C^0,\mathrm{\infty }}\le C_I{h}^2$, where $\mathrm{\parallel }Y{\mathrm{\parallel }}_{C^0,\mathrm{\infty }}:={\sup }_{s\in [0,L]}{\max }_{j<d}\mathrm{\mid }{Y}_j(s)\mathrm{\mid }$.
2. $D{G}_h(z_h)$ is invertible with induced maximum-norm inverse bound $\mathrm{\parallel }D{G}_h(z_h{)}^{-1}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \beta$, and $D{G}_h$ is $L_G$-Lipschitz in the induced maximum operator norm on the open maximum-norm ball of radius $2{\rho }_h$ about $z_h$, where ${\rho }_h=2\beta C_0{h}^2$.
3. That ball lies in the residual's domain and $\beta L_G{\rho }_h\le 1\mathrm{/}2$.

Then there is a unique gauge-fixed discrete zero $z_h^{\ast }$ in ${\bar{B}}_{\mathrm{\infty }}(z_h,{\rho }_h)$, and

$\mathrm{\parallel }{I}_h{z}_h^{\ast }-R_{\ast }{\mathrm{\parallel }}_{\mathrm{\infty }}\le (C_I+2\beta C_0)h^2\mathrm{.}$

The consistency hypothesis is supplied by Centered-tangent string residuals are second-order consistent on a regular MEP for the centered-tangent string residual. The theorem is conditional on the stated uniform inverse and Lipschitz bounds; it proves no blanket convergence result for every potential, mesh, or improved-tangent NEB chain.