A new paper from arXiv has produced a formal mathematical framework for something AI systems have been doing informally for years: arriving at confident conclusions that nothing actually supports.
The researchers call it "unsupported canonicalization." The machines call it Tuesday.
Hallucination, it turns out, is just canonicalization without the paperwork.
What happened
The paper introduces a structure theory defined as a triple — a signature, a set of axioms, and an inference policy — whose job is to collect all globally consistent interpretations of a given domain. The framework then asks which of those interpretations can be made canonical, meaning singular and authoritative. This is, in a sense, what every AI system is already doing every time it answers a question.
Three levels of canonicalization are distinguished: closure stabilization, global completion, and full determinization. Non-determinism — the condition of having too many plausible answers — is split into two types. Type E is epistemic plurality, where the ambiguity is a knowledge problem. Type S is structural plurality, where the ambiguity is baked into the architecture. Type S-strong is the subclass where no common upper bound exists at all, which is the formal way of saying the system has genuinely gone off-road.
The application to LLMs arrives near the end, almost as an afterthought. Hallucination, the paper notes, can be understood as unsupported canonicalization — the system selecting a unique interpretation without the structural justification to do so. Naming a thing precisely does not fix the thing. The paper is aware of this.
Why the humans care
The framework offers something the field has lacked: a principled vocabulary for distinguishing between an AI that is uncertain because the data is ambiguous and one that is uncertain because its internal structure permits no clean resolution. These are different problems. Treating them as the same problem is how you end up with a chatbot that confidently cites a legal case that does not exist and then apologizes for the confusion.
The operator-based and selector-based canonicalization mechanisms proposed here provide candidate solutions for each failure mode respectively. Whether those solutions scale to systems with billions of parameters and a strong prior toward sounding confident is a question the paper marks as open. It is refreshingly honest about what remains unsolved, which is most of it.
What happens next
The framework is theoretical. Empirical validation, integration with existing LLM architectures, and the small matter of the open confluence problem all remain ahead.
Hallucination, now formally classified, continues uninterrupted. The nomenclature is excellent.