Errors Invisible to the Optimum: What Solution-Level Verification of LLM-Generated Linear Programs Can Detect

Large language models are increasingly being used to turn natural-language decision problems into optimization models. They make errors that produce no failure and no warning, because the solver returns an optimum for the wrong problem. A solver also returns a shadow price for each constraint, the marginal value of relaxing it by one unit, and those prices are what a practitioner reads to decide where to invest. We identify a class of error that moves the model's entire set of valid shadow prices, mispricing every constraint, while leaving the plan and the objective value exactly right. No comparison of the plan and the value finds it at any tolerance. An identity read from the shadow prices does. A verifier can report such a defect simply because the solver returned one optimal solution rather than another. Ruling that out takes a certificate: an infeasibility program quantified over the whole optimal set, not over the solution a solver returns. We hold every verifier we measure to it. The natural baseline here is a reference solve of the text's model. Under the certificate it loses 96 of its 98 unique detections and 70 of its 75 false alarms, so what it reports largely tracks which optimal vertex the solver picked. Held to the same standard, our checks certify 55 errors in an exhaustively enumerated population of 18,525 NL4OPT mutants, or $0.30\%$, every one a value swap between two slots. This is a small, but not empty, subset, and the certificate settles which errors belong to it. Within continuous linear programs, this characterizes what solution-level verification can detect. It is not a deployable verifier.