Why AI Hallucination Is Mathematically Inevitable
Four independent research groups. Four different proof techniques. The same conclusion: every future model will hallucinate.
Not “current models hallucinate.” Every model. The ones that exist today. The ones that ship next year. The ones we haven't imagined yet.
The four proofs
Xu et al. (2024)
Hallucination is Inevitable
Any computable learner with finite training data will hallucinate on some inputs. This is a formal impossibility result, not an empirical observation.
No amount of training data eliminates hallucination.
Banerjee & Monsalve (2024)
Information-Theoretic Bound
Generative models that minimize cross-entropy loss must hallucinate when the training distribution does not cover the full output space. The bound is tight.
Better loss functions do not fix this.
Karpowicz (2024)
Godelian Limit
Self-verification of generative outputs is equivalent to the halting problem for sufficiently expressive models. Models cannot reliably verify their own outputs.
Models cannot verify themselves.
LeCun (2023)
Autoregressive Trap
Token-by-token generation compounds small errors across sequence length. Each token conditions on potentially incorrect previous tokens. Error accumulation is structural.
Architecture improvements reduce but cannot eliminate this.
What this means
The industry has two strategies for hallucination. One is to train it away. The other is to detect and correct it externally. The proofs above show the first strategy has a ceiling. A mathematical one.
RLHF, DPO, constitutional AI, chain-of-thought — all useful. None sufficient. They reduce hallucination rates. They do not eliminate them. They cannot.
Verification is permanent infrastructure
If hallucination is inevitable, then external verification is not a stopgap. It is permanent infrastructure. Every generation of models needs it. Every domain that uses AI needs it.
This is why we built Assay as Layer 2 — a verification substrate that sits below the model layer. Models generate. Layer 2 verifies. The same architecture that makes TCP/IP reliable over unreliable networks.
Full paper and proofs:
DOI 10.5281/zenodo.18522644The question is not whether your AI will hallucinate. It will. The question is whether you catch it before your users do.
Drop a repo link. I'll run it for free.
— Ty