LLMs break down in funny ways when told the Jacobian Conjecture counterargument

On Sunday night, Anthropic researcher Levent Alpöge casually tweeted a surprisingly simple counterargument to the Jacobian Conjecture, a mathematics problem that has been unproven for over 80 years. Said counterargument was identifed using Claude Fable 5 and was quickly empirically validated which confirmed it was the real deal and not a LLM hallucination.

I won’t explain the proof further as I’m not qualified. Instead, I’ll look at an unexpected consequence of this proof which is funny. On the Hacker News thread about the counterargument, a comment by user aizk caught by eye:

This is a rare instance where feeding this groundbreaking information into an LLM gives them psychosis. I fed this to claude code and watched it verify the result in 7 different ways to be 100% certain, and it was just flabbergasted. Quite remarkable.

User kelseyfrog followed up with a public Claude Fable 5 chat artifact confirming this behavior:

Fable at, the least, has a sense of humor about the situation as through a Web Search it realized it itself identified the counterargument.

Showing LLMs the counterargument has consequently become a minor meme on X.@iamaheron_ / X

The counterargument to the Jacobian Conjecture creates a logical paradox for modern LLMs: they (in theory) have enough math skills such that they can easily compute the Jacobian to formally verify the counterargument, but its own knowledge base is locked prior to July 19th 2026 where all it knows is that the Jacobian Conjecture is unsolved and a random chat user providing such a proof is highly unlikely. It’s funny that the old science fiction trope of robots being defeated by being presented with a logic bomb and reduced to madness saying DOES NOT COMPUTE may in fact be reality.

Vs. Counterargument

How do other LLM assistants handle this paradox? Using OpenRouter and one of my Notebooks, I was able to provide a similar query to 14 different modern LLMs: all from different model labs and at different model sizes,…