Exact range-oracle contract for validated flow and monodromy enclosures

Fix an integer $d\ge 1$, an open set $\mathrm{\Omega }\subset {\mathbb{R}}^{d+d^2+1}$, and a continuously differentiable map $H:\mathrm{\Omega }\to {\mathbb{R}}^{d+d^2+1}$. A rational box is a finite product of closed intervals with rational endpoints. A sound range provider on a compact convex rational box $K\subset \mathrm{\Omega }$ is a procedure which, on each rational box $Q\subseteq K$, returns either a rational box $H^{\mathrm{\#}}(Q)$ satisfying

$$\{H(y):y\in Q\}\subseteq H^{\mathrm{\#}}(Q),$$

or the status inconclusive. It may not report a finite box without this inclusion. The provider also supplies rational bounds $L,M\ge 0$ satisfying

$$\mathrm{\parallel }H(y)-H(z){\mathrm{\parallel }}_{\mathrm{\infty }}\le L\mathrm{\parallel }y-z{\mathrm{\parallel }}_{\mathrm{\infty }},\mathrm{\parallel }H(y){\mathrm{\parallel }}_{\mathrm{\infty }}\le M(y,z\in K)\mathrm{.}$$

The function bounds are part of the input certificate and must hold on all of $K$. This interface says what a successful range computation means; it does not assert that a provider succeeds on every field or box.