 Numerical upper-bound pruning without a solution witness

Suppose $0\le h(n)\le h^{\ast }(n)\le u(n)$ in a search problem, with $h^{\ast }=\infty$ at dead ends. Every finite $g(n)+u(n)$ bounds some reachable solution cost from above, so their running minimum $B$ is at least the global optimum $C^{\ast }$. Every node of an optimal path has $g+h\le C^{\ast }\le B$, and therefore survives the displayed strict pruning rule. Ordinary A-star search can continue to a selected goal without receiving a path witnessing $u$.

Equality pruning is unsafe without a stored incumbent solution: when $B=C^{\ast }$, it could delete every optimal continuation. A numerical upper bound does not by itself provide a plan to return. This distinguishes an upper-bound certificate from a constructed solution in branch and bound.