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

We establish the mathematical limits of AGI safety in two forms: verifying a fixed system, and verifying that a certified safety property persists once the system self-modifies. In the static case, no algorithm can certify a highly expressive AGI's safe behaviour infallibly, completely and tractably, whether over unbounded input domains (blocked by Rice's and Godel's theorems) or over all finite hardware configurations (blocked by Trakhtenbrot's theorem, which splits into a PSPACE-hardness barrier and a co-RE-completeness barrier), forcing a Soundness-Completeness-Tractability Trilemma as a structural, not statistical, necessity. In the dynamic case, we formalise self-modification as a computable transition operator and prove that no algorithm can determine, from a system's current certified safety, whether safety survives its next self-modification step: a result that reduces to Rice's Theorem one level up, making the static and dynamic barriers two faces of one obstruction. This forces an exclusive dichotomy: persistent certification is attainable only for systems that have stopped evolving semantically, i.e. only for narrow, not general, systems. Nor can the obstruction be delegated: any supervisor adequate to audit a general AGI is itself a general AGI, so the supervisory regress never terminates. Three practical risks (finite test coverage, bounded deliberation time, restricted observation) are one phenomenon: every bounded scheme that does not reject correct evidence admits an evolution trace it certifies at every stage while the property is persistently violated. These results give formal content to the unverifiability of AI, showing it is not an engineering target deferred by current limits but a structural tension, an Expressivity Invariant governed by the same computational laws as the Halting Problem and Rice's Theorem.

An AI system's output is not the fact or world state it appears to describe, but rather an engineered representation. We propose a semantic framework to describe AI systems, to be able to examine the correctness of such representations. To do so, we distinguish what is justified by accepted domain knowledge, what reference sources say, and what the system can currently use. This allows us to give precise definitions to common failures: extrapolation, refuted or unsupported assertion, sources versus knowledge mismatch, stale or refuted source, added hypotheses, unsupported use... We hope our framework gives a useful vocabulary for specifying and checking AI systems whose outputs, citations, tool calls, and world-changing actions must be justified by reliable claims and explicit authority rather than apparent fluency.

Recent artificial intelligence (AI) systems have shown remarkable progress in mathematical reasoning. Many existing approaches, including large language models (LLMs), draw on human prior knowledge in the form of mathematical text, code, or theorem libraries. Although these approaches are highly effective in practice, it remains an open question whether an agent can autonomously discover useful theorems without such human priors. We study this question in a formal axiomatic system by developing an agent that starts from axioms and inference rules alone and gradually grows a library of useful theorems. Concretely, we propose a self-supervised theorem-discovery algorithm that alternates between proof search and useful-theorem extraction, building a theorem library whose entries are reused as lemmas for subsequent proof search. Experiments show that the agent discovers tens of thousands of theorems and finds proofs for human-written benchmark problems, suggesting that its discoveries include theorems meaningful from a human mathematical perspective. Furthermore, the discovered theorems improve LLM proof performance when provided as prompt lemmas, indicating that they can serve as external knowledge for LLM reasoning. Our results provide evidence that useful theorems can emerge from proof search without relying on human-provided theorem libraries. More broadly, they suggest a path toward self-evolving AI systems for mathematics whose discoveries remain formally verifiable.