A Non-Formulable Theorem: A Fundamental Limit of Finite Syntactic Systems and Its Consequences for Security and AI

The paper addresses a foundational question in the theory of finite syntactic systems: whether such a system, when coherent and sufficiently expressive, can autonomously generate all of its own theorems. Buono (2026) answers this negatively by proving a metatheorem — a theorem about theorems — that asserts the existence of at least one theorem that any such system S cannot produce on its own.

The scope of the claim is deliberately broad. The author states that the result applies to *every* finite syntactic system, explicitly enumerating security mechanisms, AI systems, formal verifiers, legal systems, economic models, and — reflexively — the formal system in which the metatheorem is itself proved. This universality is the central contribution: rather than demonstrating an isolated incompleteness phenomenon within a particular formalism, the paper claims a structural limit that binds any finite syntactic system meeting the stated conditions of coherence and sufficient expressiveness.

The motivation spans multiple domains. In cybersecurity, the limit bears on whether a finite mechanism can autonomously enumerate every attack or vulnerability it is meant to detect. In AI, it bears o

The principal result is the metatheorem itself: for every coherent and sufficiently expressive finite syntactic system $S$, there exists at least one theorem $T$ such that $T\in \mathrm{Thm}(S)$ but $S$ cannot produce $T$ autonomously. Equivalently, the autonomously producible fragment of any such system is strictly smaller than its full theorem set.

The result is universal in two respects. First, it quantifies over all finite syntactic systems meeting the coherence and expressiveness conditions, rather than over a specific formalism. Second, it is self-applicable: the formal system in which the metatheorem is proved is itself one of the systems to which the metatheorem applies. This reflexivity means the theorem does not exempt its own host system from the limit it establishes.

The paper enumerates the domains to which the result extends. Security mechanisms are finite syntactic systems, so the metatheorem implies that at least one of their theorems cannot be autonomously produced by them. The same holds for AI systems, formal verifiers, legal systems, and economic models. In each case, the consequence is structural: no finite syntactic system in the specified class is autonomo