Finite-image projected-gradient string with arc-length redistribution

Let $U,V,a,b$ satisfy Regular minimum-energy paths on a declared collision-free tube. Inputs: an integer $M\ge 2$, images $R_0=a,R_1,\dots ,R_{M-1},R_M=b$ in $U$, a step size $\eta >0$, force and spacing tolerances ${\tau }_F,{\tau }_s\ge 0$, and a finite update budget $N\in {\mathbb{N}}_0$. Output: the last image chain and one of tolerances-met, geometry-failure, nonfinite-evaluation, or budget-exhausted. A stop status is an algorithmic report, not a continuum-error certificate.

At iteration $k=0,\dots ,N$, perform the following in order. After every arithmetic operation or norm evaluation below, return nonfinite-evaluation if its scalar or vector result is nonfinite, before using that result in a comparison, normalization, segment lookup, or subsequent operation. In particular, this check applies to chords, lengths, cumulative lengths, target lengths, and interpolated images.

1. For each interior $i=1,\dots ,M-1$, form the centered chord $c_i=R_{i+1}-R_{i-1}$. If $c_i=0$, return geometry-failure. Otherwise put $t_i=c_i\mathrm{/}\mathrm{\parallel }{c}_i{\mathrm{\parallel }}_2$ and $F_i^{\perp }=-(I-t_i{t}_i^T)\mathrm{\nabla }V(R_i)$. If any force is nonfinite, return nonfinite-evaluation.
2. For the current chain set ${\mathrm{\ell }}_i=\mathrm{\parallel }{R}_i-R_{i-1}{\mathrm{\parallel }}_2$ for $1\le i\le M$, $\bar{\mathrm{\ell }}=M^{-1}{\sum }_{i=1}^M{\mathrm{\ell }}_i$, $r_F={\max }_{1\le i<M}\mathrm{\parallel }{F}_i^{\perp }{\mathrm{\parallel }}_2$, and $r_s={\max }_{1\le i\le M}\mathrm{\mid }{\mathrm{\ell }}_i-\bar{\mathrm{\ell }}\mathrm{\mid }$. If $r_F\le {\tau }_F$ and $r_s\le {\tau }_s$, return tolerances-met.
3. If $k=N$, return budget-exhausted. Otherwise set $Y_0=a$, $Y_M=b$, and $Y_i=R_i+\eta F_i^{\perp }$ for interior images. Reject any nonfinite $Y_i$ with nonfinite-evaluation; if any $Y_i\mathrm{\notin }U$, return geometry-failure.
4. Compute cumulative lengths $s_0=0$, $s_j={\sum }_{i=1}^j\mathrm{\parallel }{Y}_i-Y_{i-1}{\mathrm{\parallel }}_2$, and total length $S=s_M$. If $S=0$ or the computed cumulative lengths are not strictly increasing, return geometry-failure. For $j=1,\dots ,M-1$, place the new image at the unique point of the piecewise-linear polygon $Y_0,\dots ,Y_M$ whose cumulative length is $jS\mathrm{/}M$; if that target is a vertex, use that vertex, and otherwise use linear interpolation on its unique containing segment. If any redistributed image lies outside $U$, return geometry-failure. Set $R_0^{k+1}=a,R_M^{k+1}=b$ and use the checked redistributed points as the new interior images.

This is the specified projected-force string variant. No convergence from arbitrary initial images, no automatic endpoint minimization, and no error estimate follow from the procedure alone.