Definition: Finite circuit certificate

A certificate for an integer $N_0$ consists of:

1. the input, output, work, and initialization conventions;

2. a finite gate list, or a finite rule that expands uniquely to that list without an oracle;

3. allowed-gate and distinct-wire checks;

4. an exact allocated-width and gate-count ledger;

5. a named resource bound, with its scope stated;

6. a proof of the support identity (eq:exact-support).

A list of observed outputs, however large, is not item 6.