MATH.20:2 - Problem

A plausible estimate can be close in familiar cases while lying on the wrong side of an unknown elsewhere. A bound on an intermediate quantity can also be mistaken for a bound on the requested result. For example, a small equation residual need not imply a small error in its solution.

Bounds themselves can be uninformative. A wide interval may leave the receiving choice unchanged, and separately sharp coordinate bounds may never be attained together. The work needs both a justified relation and a way to tell whether improving that relation would help.