Finite-budget phase-fixed periodic-orbit correction and validation

For a state dimension $d\ge 1$, the vector-field input is a $C^2$ map $f:U\to {\mathbb{R}}^d$ in fixed-parameter mode or a $C^2$ one-parameter family $f:U\times V\to {\mathbb{R}}^d$ with scalar parameter $\mu \in V\subset \mathbb{R}$ in section mode, where $U\subset {\mathbb{R}}^d$ and $V\subset \mathbb{R}$ are open. Also supply an affine phase with nonzero rational covector and rational reference point, a compact convex validation tube, and the sound range provider of Exact range-oracle contract for validated flow and monodromy enclosures. Supply positive finite step counts for each requested validated jet run, finite arithmetic and candidate-update budgets, a rational initial candidate, a dyadic point preconditioner $C$, a rational inverse witness $W$, a positive rational minimum box width $\eta$, and a finite Krawczyk box budget. In fixed-parameter mode the rational candidate box is $Z=X\times \mathrm{\Theta }\subset U\times (0,\mathrm{\infty })$ with the parameter held fixed, so both the variable $z=(x,\theta )$ and residual $F=G$ have dimension $d+1$. In section mode require exactly one scalar continuation parameter $\mu \in \mathbb{R}$ (the $p=1$ case of Certified residual, Jacobian, trajectory, and monodromy enclosures for periodic shooting), take $Z=X\times \mathrm{\Theta }\times \mathrm{\Lambda }$ with rational $\mathrm{\Lambda }\subset V\subset \mathbb{R}$, and set $z=(x,\theta ,\mu )$. Then $G$ has dimension $d+1$, the affine section adds one scalar equation, and both $z$ and $F$ have dimension $d+2$. Independent validation retains every admitted rational endpoint of $Z$ and uses the exact affine dyadic unit-box chart below; non-dyadic physical endpoints are not a failure condition. Supply an open convex shooting domain containing $Z$ on which the exact flow exists through every admitted positive period, so $F$ is $C^1$ there. All rational point, endpoint, interval and matrix operations below are exact or return an explicit arithmetic-failure status within the supplied cap.

Write $F=G$ in fixed-parameter mode. In section mode, used for continuation, require one scalar continuation parameter $\mu \in \mathbb{R}$ and write $z=(x,\theta ,\mu )$ and $F(z)=(G(x,\theta ,\mu ),{\tau }^{\mathrm{\top }}(z-z_p))$, where $G$ is the parameterized map in Certified residual, Jacobian, trajectory, and monodromy enclosures for periodic shooting, $\tau$ is a supplied unit normal, and $z_p$ is a supplied predictor. Supply certified rational enclosures for the coefficients and constant term of this affine row at every requested accuracy; if they are rational use their exact values, and otherwise use outward interval arithmetic. If a finite row enclosure cannot be obtained within the arithmetic cap, return inconclusive. For the one-parameter field use the static-parameter augmentation $g(x,\mu )=(f(x,\mu ),0)$ in that definition and supply its sound augmented range provider and tube. For every section-mode jet query, first reorder its input $(x,\theta ,\mu )$ to $(q,\theta )=(x,\mu ,\theta )$. Reorder the returned Jacobian columns from $(x,\mu ,\theta )$ to $(x,\theta ,\mu )$, leaving its residual rows unchanged. Then append a sound outward enclosure of the affine section row to every residual and Jacobian enclosure (the exact row when its data are rational). Apply this rule both to candidate updates and to every independent-validation callback. In fixed-parameter mode, $z=(x,\theta )$ and $F=G$ both have dimension $d+1$; in section mode, $z=(x,\theta ,\mu )$ and $F=(G,{\tau }^{\mathrm{\top }}(z-z_p))$ both have dimension $d+2$.

The procedure retains the following candidate modes; candidate generation alone never validates a root.

1. Enclosure-guided single shooting: this mode needs only the supplied range provider, not a pointwise differentiable numerical IVP evaluator. Start at the supplied rational $z_0\in Z$. At each of at most the supplied number of updates, run the validated shooting-jet procedure of Certified residual, Jacobian, trajectory, and monodromy enclosures for periodic shooting on the singleton $\{z_k\}$, using its validated Euler method and the supplied finite caps. If a tube, range, domain or arithmetic check fails, return its inconclusive status and the last candidate. Otherwise let $r_k$ and $A_k$ be the coordinatewise rational midpoints of the returned boxes for $F(z_k)$ and $DF(z_k)$. Record $z_k$ as a candidate and attempt the independent validation below. If it does not validate and updates remain, call Newton correction for a square vector residual on $(A_k,r_k)$. A singular matrix returns singular-jacobian; otherwise its rational correction $u_k$ proposes $z_{k+1}=z_k+u_k$. Reject a proposal outside $Z$ with invalid-trial. The midpoint matrix is only a proposal matrix: it is not claimed to be the exact derivative of a numerical residual map. Exhaustion returns the last candidate and candidate-budget-exhausted.
2. Differentiable single shooting: supply, as an additional input for this mode, a certified differentiable evaluator satisfying Certified differentiable IVP approximation interface for shooting for the augmented BVP, including its approximate trajectories, $C^1$ endpoint map, exact numerical-map derivative, and uniform trajectory-error bound. Run Finite-budget single shooting for a nonlinear two-point BVP on the phase-bordered BVP. In section mode augment the state by the constant parameter and append the affine section condition to its boundary residual. An interval jet alone is not this additional evaluator; if it is absent, return evaluator-unavailable without making the call.
3. Discretized candidate: either run Finite-budget multiple shooting for a nonlinear two-point BVP on the augmented segment BVP with separately supplied certified differentiable segment evaluators satisfying the same interface, or run Finite-budget implicit-midpoint collocation for a nonlinear BVP using the pointwise field/boundary values and their exact derivatives required by its residual. Use the residuals of Multiple-shooting and midpoint-collocation residuals for a phase-fixed periodic orbit; in section mode include the constant parameter and affine section row. Missing pointwise/segment evaluation data returns evaluator-unavailable. Use the last finite tuple's first physical state and positive period (and parameter in section mode) as a shooting candidate. Reject a candidate outside $Z$ or a nonrational candidate without a supplied rational proposal; rationalization is only a new candidate, not a trajectory or residual certificate.

All modes propagate evaluator-failure, invalid-endpoint-pair, invalid-trial, singular-jacobian, inconclusive and exhausted-budget statuses. No candidate mode is assumed to succeed from arbitrary data. In particular, the first mode composes the available interval-enclosure provider directly; it does not infer a $C^1$ point evaluator from interval boxes.

Independent validation: retain the exact rational candidate box $Z={\prod }_{j=1}^N[a_j,b_j]$ in the declared physical variable order, where $N=d+1$ or $d+2$. If any side is degenerate, return inconclusive for validation, retaining the candidate: a box with empty ambient interior cannot pass the strict-interior inclusion required for a unique-root return. Degenerate boxes remain admitted inputs, and candidate-generation queries may still use their rational singleton or lower-dimensional boxes. Otherwise put $D=\mathrm{diag}(b_j-a_j)$, $T(u)=a+Du$, $B=[0,1{]}^N$ and $\widehat{F}(u)=F(T(u))$. Use the supplied open convex shooting domain containing $Z$, where the exact residual is $C^1$; failure to certify that domain returns inconclusive. The chart has an open convex preimage domain, and $D\widehat{F}(u)=DF(T(u))D$, by Rational boxes admit dyadic enclosure boundaries and exact affine root charts. Run Finite-budget multidimensional Krawczyk root certification on $\widehat{F}$ and the finite dyadic box $B$, using the supplied dyadic $C$, rational $W$, box budget and normalized minimum width $\widehat{\eta }=\eta \mathrm{/}{\max }_j(b_j-a_j)>0$. Here $C$ is used as a preconditioner for this normalized residual; no approximate-inverse quality was assumed of it. The witness check remains the exact check $WC=I$. A normalized terminal box of maximum width at most $\widehat{\eta }$ has physical image of maximum width at most $\eta$; the budget still bounds all processed boxes. For every processed dyadic subbox $A\subseteq B$, let $m_A$ be its dyadic midpoint and set $Y=T(A)$ and $m_Y=T(m_A)$, the exact rational physical midpoint. The residual callback runs the validated shooting jet on $\{m_Y\}$ and appends the section-row enclosure when present. The Jacobian callback runs it on the rational box $Y$, appends the physical affine section row when present, and multiplies column $j$ of this physical Jacobian enclosure by $D_{jj}$. Exact rational endpoint operations followed by outward dyadic hull conversion give finite dyadic enclosures $F_0\supseteq \widehat{F}(m_A)$ and $\widehat{J}(A)\supseteq D\widehat{F}(A)$. A failed conversion or overflow marks $A$ unresolved; it is never supplied as a finite interval. This uses the actual midpoint of the current normalized box, whose image is the actual midpoint of $Y$, even when it differs from the candidate center. A failed query or tube inclusion marks that box unresolved. Normalized Krawczyk interval operations are outward under Finite real interval operations enclose their exact values. Retain the complete normalized Krawczyk output partition, including every terminal box and its status, and its exact physical image partition under $T$. Status labels on physical images mean the corresponding root conclusions transferred by the affine bijection, not a Krawczyk computation with an unscaled physical Jacobian. Continue only if the overall Krawczyk status is complete, exactly one normalized terminal box $A_Y$ has status unique-root, and every other terminal box has status root-free; if there is no such unique box, more than one, or any existence-only/unresolved box, return inconclusive. Set $Y=T(A_Y)$. Let $X_Y$ and ${\mathrm{\Theta }}_Y$ be the physical-state and period projections of this rational unique-root image box, and include ${\mathrm{\Lambda }}_Y$ in section mode. Form $Q_Y=(X_Y\times {\mathrm{\Lambda }}_Y)\times \{I_{d+p}\}\times {\mathrm{\Theta }}_Y$ in section mode, or $Q_Y=X_Y\times \{I_d\}\times {\mathrm{\Theta }}_Y$ in fixed-parameter mode. Check $Q_Y$ is contained in the supplied augmented tube and query the same sound range provider for its augmented field enclosure. Divide the first $d$ coordinates by the positive interval ${\mathrm{\Theta }}_Y$ with exact rational endpoint arithmetic to obtain an interval box containing $f(X_Y,{\mathrm{\Lambda }}_Y)$ (or $f(X_Y)$ in fixed-parameter mode). Require at least one component interval to exclude zero; if inclusion, query or division fails, or no component excludes zero, return inconclusive. Finally require a successful full-box jet run on $Y$ to enclose the entire trajectory and monodromy; a failed query or tube check returns inconclusive.

Return validated-root only for such a box $Y$, and report it as the output box together with the complete terminal partition, the affine chart $(a,D)$, unique normalized box $A_Y$, the actual physical midpoint residual enclosure, physical Jacobian enclosure, normalized Jacobian enclosure, inverse witness, normalized strict inclusion and contraction inequalities, field component excluding zero, trajectory tube and monodromy enclosure. The bijection and complete mapped partition prove that $Y$ contains the only zero in the original rational box $Z$. If subdivision was used the output box is $Y$, not the original larger $Z$. Otherwise return inconclusive or the precise finite candidate failure status. Candidate midpoints and correction residuals never replace the independently validated callback enclosures.