Energy return does not by itself certify industrial closure

Energy-return metrics are widely used to compare energy technologies, but a favourable value does not show whether the same industrial plan restores every function required for another productive cycle. This preprint formalizes common-plan functional restitution through the Physical Energy Restitution Ratio (PERR). Under a fixed closed catalogue, PERR≥1 is equivalent to one-generation coverage, and a scalar can certify that decision only when equal scalar values cannot correspond to opposite outcomes. A controlled pair has identical horizon, EROI=2 and aggregate electrical exergy but opposite PERR/COVER decisions. A published photovoltaic representation provides an external non-identifiability test under TOPCon metallization refinement. An analytic energy-coordinate bound excludes a frozen four-route power-to-liquid/direct-air-capture class, and deterministic certificates exclude the complete declared Almería and Pelham 16-plan families and the declared Salt Creek family under favourable accounting. One-generation coverage is distinguished from indefinite recurrence. This record is a preprint and has not undergone peer review. It contains the scientific manuscript and public supplementary material. No source code or general-purpose implementation is included in this public release.