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The paper studies whether an agent can expand what it computes without relying on rewards, a judge model, or an answer key, while remaining exactly correct on structured questions. A single frozen model operates two nested loops over a small exact domain-specific language. In the inner loop, a new answer is accepted only when two independently written expressions agree on every record. In the outer loop, the agent identifies a missing capability and proposes an operator, which is admitted only when an independent second implementation agrees and the result satisfies a conservation law. This mechanism roughly triples the answerable-question set, while a control without induction answers none of the newly enabled questions. On an independent reference, the admission rule rejects all 138 of 138 buggy operator variants, including a variant written by another model family; simpler signals such as whether the code runs, self-consistency, or library compression admit most of them. Without modification, the system expands its answerable sets by more than tenfold on a business-ledger domain and nearly twentyfold on a clinical-encounter domain, both with unrelated schemas. Further runs with other proposers preserve the soundness of the gate and produce comparable self-extension, although the number of induced operators follows each proposer’s DSL fluency.