Outward-rounded interval enclosures for factorable expression DAGs on boxes

Input a box $B={\prod }_j[a_j,b_j]\subseteq D_0$ with arbitrary finite ordered real endpoints, a finite real expression DAG from Real intervals, complex rectangles and ball enclosures using addition, subtraction, multiplication, and division, a precision $p\ge 2$, and a dyadic format with fixed exponent bounds. Round each input endpoint outward at precision $p$, replacing $[a_j,b_j]$ by $[{\mathrm{rd}}_p(a_j),{\mathrm{ru}}_p(b_j)]$; if an endpoint rounding overflows, return overflow. Thus every interval passed to an interval operation has dyadic endpoints and contains the original box coordinate interval. The same rule applies to a singleton input, including a midpoint point-evaluation path. The DAG must define a real-valued function at every point of $B$; for each division node its denominator range on $B$ must exclude zero. Each constant is either an exact dyadic, enclosed by $[{\mathrm{rd}}_p(c),{\mathrm{ru}}_p(c)]$, or is accompanied by a precision-indexed certified interval containing it whose width tends to zero as $p\to \mathrm{\infty }$. If any constant enclosure overflows or is unavailable, return that status with no finite bound claim. Traverse the finite DAG in topological order. Enclose each input by its outward-rounded interval, each constant by its certified interval at precision $p$, and apply the corresponding outward-rounded operation from Outward-rounded interval, rectangle and ball operations at precision $p$. Return the root enclosure $I_p(B)=[a_{-},a_{+}]$, its lower and upper endpoints, and the exact midpoint of $B$ as a candidate point. If any operation returns overflow, split-domain, domain-error, entire, or inconclusive, return that status with no finite bound claim.

For a continuous objective on an unconstrained box, the midpoint is feasible and $a_{+}$ is a verified upper bound for its objective value. On a constrained box, the midpoint is only a candidate: a separate feasibility witness is required. To refine arithmetic error, callers increase $p$ along any sequence of shrinking boxes. Fixed exponent bounds must be chosen so no exact intermediate on the certified domain overflows; otherwise the procedure returns an explicit failure status.