Certified lower, upper, and emptiness bounds on a box

Fix an instance $(D_0,C,f)$ from Compact-box global minimization problems and a closed subbox $B\subseteq D_0$. A certified box report may contain any combination of the following fields:

• A feasible witness $(x_B,U_B)$, with $x_B\in B\cap C$ and a verified upper bound $f(x_B)\le U_B$.
• A lower bound ${\mathrm{\ell }}_B\in \mathbb{R}$ satisfying ${\mathrm{\ell }}_B\le f(x)$ for every $x\in B\cap C$. This pointwise condition is vacuous when $B\cap C=\mathrm{\varnothing }$; when $B\cap C\mathrm{\ne }\mathrm{\varnothing }$, it is equivalent to ${\mathrm{\ell }}_B\le {\inf }_{x\in B\cap C}f(x)$.
• An empty-box proof establishing $B\cap C=\mathrm{\varnothing }$.
• A non-finite or unavailable status, which makes no feasibility, lower-bound, or optimality claim.

A box is closed for a requested tolerance $\epsilon >0$ only when it has a lower bound and ${\mathrm{\ell }}_B\ge U-\epsilon$, where $U$ is the best currently verified feasible upper bound. Closing a box stops further subdivision but retains its lower bound in the global ledger. A report lacking a required field is not repaired by a numerical guess.