Confidence is a certificate

Abstract

Most real business decisions are constrained optimization problems dressed in plain language, and a language model asked to solve one will return a schedule that reads well and cannot be checked. This paper describes Ferrox, which exposes CP-SAT, minimum-cost flow and a MIP solver as Suggestors inside the convergence loop of our substrate report, so that a proof of optimality and a fluent explanation can be produced by different participants in the same Formation. The engineering is unremarkable and the semantics are not: a heuristic answer and a proven one both enter the context, and the only thing separating them is a scalar confidence. We argue that this scalar is doing type-level work. A confidence of one means proven optimal, 0.85 means feasible with optimality not proven, and a heuristic is capped at 0.65 however good it looks — these are classes of certificate, not probabilities, and averaging them with the probability attached to a model’s output is a category error. We give the confidence rules, three benchmark instances, and one result that is worse than it looks good: on a time-windowed routing instance the solver proves that only eight of twenty customers can be served, which is a modelling finding rather than a solver failure and is reported here as such. We also report the supply-chain shape of the two solver backends we wrap, and close by proposing that a scalar is a lossy projection of a certificate lattice we have not built.