Exact Functional Solution Space: Auditing Neural Analyses Across Exact-Equivalent Parameterizations

Neural-network analyses are typically computed from a particular parameterization but often interpreted as properties of the learned solution. This interpretation is problematic because different parameterizations can realize exactly the same input-output function while changing internal quantities and the outcomes of downstream procedures. We address this problem using Exact Functional Solution Space (EFSS), which identifies a neural solution with its realized function rather than with a particular parameter vector. This yields a simple criterion: a solution-level claim should remain consistent across exact-equivalent parameterizations under the appropriate semantics of its output. We turn this criterion into an output-aware audit that distinguishes direct invariance, correspondence under relabeling or transport, and consistency of model-producing procedures. Across Transformer interventions, pruning, and model editing, we find that similar task performance can conceal representation-dependent scientific judgments, while unstable internal measurements can leave reported decisions unchanged. These findings show that consistency must be assessed at the level of the claim being made. EFSS makes exact functional equivalence a practical basis for tracing such dependence through an analysis and guiding targeted repairs under the tested transformations. By separating the consistency of scientific conclusions from task utility, the framework helps researchers assess what their analyses establish about a learned function.