C.29.1:10.3 - Why the method allows inequalities and returns

Exact preservation is one strong route to result reuse. Sound inclusion and bounds are another. If an account covers more possibilities than the source, a claim holding for all those possibilities also holds for the covered source possibilities. An apparent bad case in the larger account may require refinement before it becomes a source counterexample.

The method uses that asymmetry to retain useful consequences. A relaxation bound can settle a cost threshold even when the relaxed optimizer cannot be returned. A temperature interval can settle a sampled threshold without identifying both temperatures.

Returns therefore follow the failed dependency. The practitioner can refine a state, restrict a domain, recover a premise or ask a weaker question. The unaffected mathematical work remains available.