Projected Euler with certified inexact projection solves

Input a nonempty compact, closed, and convex set $C\subseteq {\mathbb{R}}^d$, $d\ge 1$, an exact force evaluator $f:[0,T]\times C\to {\mathbb{R}}^d$, $T>0$, integer step count $N\ge 1$, initial state $x_0\in C$, tolerances ${\epsilon }_k\ge 0$, finite budgets $B_k\in \mathbb{N}$ for $k=0,\dots ,N-1$, and a projection-certificate provider $S$. The explicit closedness condition matches Projected contact dynamics and explicit projection Euler step. An admissible provider, queried with $(C,y,\epsilon ,B)$, must return after at most $B$ candidate checks, in either case: unresolved or certified(p,\Pi) with a finite exact certificate $\mathrm{\Pi }$ proving $p\in C$ and

${\sup }_{z\in C}⟨y-p,z-p⟩\le {\epsilon }^2\mathrm{.}$

Set $h=T\mathrm{/}N$, $t_k=kh$, and start with the given $x_0$. For $k=0,\dots ,N-1$, form $y_k=x_k+hf(t_k,x_k)$ and call $S(C,y_k,{\epsilon }_k,B_k)$. If the response is unresolved or is not a sound certified(p,\Pi) response of the stated form, stop and return unresolved with the valid prefix $x_0,\dots ,x_k$. Otherwise set $x_{k+1}=p$ and continue. After all $N$ calls succeed, return complete, the nodes, their affine interpolant, and the per-step tolerances. The procedure assumes exact force evaluations and sound exact certificates; arithmetic and force-evaluation errors are not included. For exact projection-Euler requests, set every ${\epsilon }_k=0$ and require the same residual certificate of A certified projection variational residual bounds the projection error.

Allow at most $N$ provider calls, and enforce each call's stated finite candidate-check budget. Failure to certify a projection is reported; no convergence or accuracy is inferred from an unverified stopping tolerance. The provider is an explicit input: this definition does not construct a solver for arbitrary compact convex sets.