Certified pseudo-arclength traces are locally sound and conditionally terminate

The valid seed is the base of the certified list by [F1]. In an attempted step, $\tau$ is unit and the root lies within distance $r$ of its current center. Thus $\mathrm{\mid }{\tau }^T(\bar{z}-z_{\ast })\mathrm{\mid }\le r$. Writing $a={\tau }^T\widehat{t}$, the actual offset is $d={\tau }^T(z_p-z_{\ast })=ha+{\tau }^T(\bar{z}-z_{\ast })$. Hence $ha-r\le d\le ha+r$. The tests $ha>r$ and $ha+r\le \kappa r_{+}$ in [F1] put $0<d\le \kappa r_{+}$. With the whole chart angle inequality, [F2] gives a unique intersection and puts its chart parameter strictly forward of zero.

An append is permitted only after [F3] applies to that same augmented section and the whole ball is contained in the neighborhood where every zero of $F$ belongs to the supplied chart interval. Its certified augmented zero therefore lies on that chart and its section plane. By uniqueness from step 1.1 it is the selected point. [F3] supplies the recorded center error. This argument applies to exact and bounded-error candidate generators alike; their residual stops or exhausted budgets cannot replace the ball certificate. Induction over the common append block proves local soundness for the entire returned list and for each prefix on an early return.

The accepted count increases exactly once per append and starts at zero. The procedure returns finite-trace precisely when it equals $N_{\mathrm{r}\mathrm{e}\mathrm{q}}$, so that status has the stated list length. Every other continuation return preserves the already certified list. At finalization a fold field can change to certified only after its record establishes the actual fold point, its neighborhood, and every hypothesis of [F4]; applying [F4] gives precisely the local chart and curvature conclusion. Unavailable evidence leaves inconclusive, and a fold request cannot turn an unsuccessful continuation candidate into a certified section root.

At each attempt the attempted counter increases before any callback and cannot exceed $N_s$. In the exact branch [F5] allows at most $N_c$ Newton updates and a single subsequent root-ball certificate request. In the bounded-error branch [F1] allows at most $N_c$ updates and $N_c+1$ certificate requests. All other checks in an attempt are finite computations or finitely many callbacks, and by [A1] they return in finite time. Finalization likewise has finitely many terminating callback requests and no unbounded search. Hence finitely many attempts and finalization terminate, regardless of whether any certificate succeeds. The initial tests in [F1] assign the stated statuses for $N_{\mathrm{r}\mathrm{e}\mathrm{q}}=0$ and $N_s=0<N_{\mathrm{r}\mathrm{e}\mathrm{q}}$ before querying continuation callbacks; a zero corrector budget still allows the specified certificate test but no update.

The exact tangent predictor $p_{\ast }=z_{\ast }+ht$ satisfies [F6], giving $\mathrm{\parallel }F(p_{\ast }){\mathrm{\parallel }}_2\le \sqrt{n}L{h}^2\mathrm{/}2$. The actual predictor obeys $z_p-p_{\ast }=(\bar{z}-z_{\ast })+h(\widehat{t}-t)$, so $\mathrm{\parallel }{z}_p-p_{\ast }{\mathrm{\parallel }}_2\le r+he$. Apply [F7] on the admitted convex open tube and add the exact predictor bound to obtain $\mathrm{\parallel }F(z_p){\mathrm{\parallel }}_2\le D(r+he)+\sqrt{n}L{h}^2\mathrm{/}2$. The last coordinate ${\tau }^T(z_p-z_p)$ is zero irrespective of which previous unit orientation is used, so the augmented norm equals $\mathrm{\parallel }F(z_p){\mathrm{\parallel }}_2$. This proves the final claim without setting seed or tangent errors to zero.