The Finite Interpreter and the Explanatory Halting Theorem

For any subject that must organize, sustain, and deliver explanations within finite time—a finite interpreter—what structural constraints on explanation are impossible to evade? This paper establishes three connected conclusions. First, whenever explanatory activity forms a nonempty finite modular structure, the explanatory graph must contain frontier nodes with no incoming edges. Frontiers cannot be eliminated; they can only be displaced. Second, for every linguistic interpretive practice still maintained by living agents, the frontier is necessarily open. Language has no meaning outside the communities that maintain it. So long as successor maintenance remains physically possible, no finite historical prefix can reliably clear the future space of unfolding-claims. To be alive is to be open; not to be open is to be dead; there is no third possibility. Third, no "absolutely terminal" certification can be legitimately completed in finitely many steps. At the weak level, no closure-certification procedure grounded in a finite historical prefix is reliable across all physically continuable futures. At the strong level, whenever the maintaining community can stably encode finite processes and embed frontier-problems into the halting problem, EXPAND is Σ₁-complete and UNEXPAND is Π₁-complete. Crucially, denying that a node satisfies the openness condition is itself a Π₁-complete claim—any attempt to evade the weak-level theorem falls under the strong-level constraint. Frontiers may be occupied, frozen, replaced, or improved in quality; they cannot be crowned.