In program-based harness generation, candidate harnesses are evaluated within nested formal spaces:

$\mathcal{G} \supseteq \mathcal{H}^{\text{syn}} \supseteq \mathcal{H}_\Pi \supseteq \mathcal{H}_\Pi^{\text{exec}}$

1. Raw Generation Space ($\mathcal{G}$): The unrestricted output space of programs emitted by a generative model.
2. Syntactically Valid Harnesses ($\mathcal{H}^{\text{syn}}$): The subset of programs that parse cleanly without syntax errors.
3. Protocol-Compliant Harnesses ($\mathcal{H}_\Pi$): Programs adhering strictly to the shared protocol $\Pi$, which specifies modular schemas, lifecycle interfaces, validation rules, and execution semantics for (M, P, A, F).
4. Executable Harness Subset ($\mathcal{H}_\Pi^{\text{exec}}$): The set of protocol-compliant harnesses that run successfully without runtime exceptions, tool-call failures, or interface crashes under an environment kernel.