Finite-budget single shooting for a nonlinear two-point BVP

Let $U\subseteq {\mathbb{R}}^d$ be the open trial-initial-state set. Let $D_B\subseteq {\mathbb{R}}^d\times {\mathbb{R}}^d$ be the open domain of the $C^1$ boundary map $B$, where $C^1$ means total differentiability with continuous derivative entries. Inputs are a certified evaluator from Certified differentiable IVP approximation interface for shooting, an initial trial $s_0\in U$, residual tolerance $\tau \ge 0$, and a correction budget $M\in {\mathbb{N}}_0$. The evaluator may return failure on a queried trial. Every successful response must supply the approximate trajectory, the endpoint $\tilde{\mathrm{\Phi }}(s)$ and the exact derivative $D\tilde{\mathrm{\Phi }}(s)$ of the declared $C^1$ numerical endpoint map, with the interface's stated uniform trajectory-error bound. Success on every queried trial is not an input assumption.

For $k=0,1,\dots ,M$, in this order:

1. Query the evaluator at $s_k$. If it fails, return the last accepted trial with status evaluator-failure and no residual certificate. Before evaluating $B$, check $(s_k,\tilde{\mathrm{\Phi }}(s_k))\in D_B$; if this fails or cannot be established, return the last accepted trial with status invalid-endpoint-pair and no residual certificate. On success form $\tilde{R}(s_k)=B(s_k,\tilde{\mathrm{\Phi }}(s_k))$ and $D\tilde{R}(s_k)=D_{1}B(s_k,\tilde{\mathrm{\Phi }}(s_k))+D_{2}B(s_k,\tilde{\mathrm{\Phi }}(s_k))D\tilde{\mathrm{\Phi }}(s_k)$ using Derivative of the nonlinear shooting residual. A failure to evaluate these quantities returns evaluator-failure without a residual certificate.
2. If $\mathrm{\parallel }\tilde{R}(s_k){\mathrm{\parallel }}_2\le \tau$, return $s_k$, its approximate trajectory, and status root-tolerance-met. This status asserts only the displayed residual tolerance.
3. If $k=M$, return the last trial and status iteration-budget-exhausted.
4. Call Newton correction for a square vector residual with $A=D\tilde{R}(s_k)$ and $r=\tilde{R}(s_k)$. If it reports singular-jacobian, stop with that status. Otherwise let $u_k$ be its returned correction.
5. If the proposed trial $s_k+u_k$ is outside $U$ or its membership cannot be established, return the last accepted trial with status invalid-trial; otherwise accept $s_{k+1}=s_k+u_k$. Initially the last accepted trial is $s_0$; accepting a trial supplies no boundary residual certificate until step 1 succeeds there.

On a sequence of successful evaluations with endpoint pairs in $D_B$, this has the same residual, singularity, trial-domain and budget checks as Finite-budget Newton for a square vector residual. It makes no global-start, exact-root, or discretization-error claim. The numerical residual is $C^1$ on the open subset of $U$ where $(s,\tilde{\mathrm{\Phi }}(s))\in D_B$; no boundary value is evaluated outside that subset.