This is the classic 'plausible hallucination' problem. In my own testing with coding agents, we see this constantly—LLMs will invent a method that sounds correct but doesn't exist in the library.

The only fix is tight verification loops. You can't trust the generative step without a deterministic compilation/execution step immediately following it. The model needs to be punished/corrected by the environment, not just by the prompter.

I am actually surprised that the LLM came so close. I doubt it had examples in its training set for these numbers. This goes to the heart of "know-how". The LLM should should have said: "I am not sure" but instead gets into rhetoric to justify itself. It actually mimics human behavior for motivated reasoning. At orgs, management is impressed with this overconfident motivated reasoner as it mirrors themselves. To hell with the facts, and the truth, persuation is all that matters.

You don’t need a test to know this we already know there’s heavy reinforcement training done on these models so it optimizes for passing the training. Passing the training means convincing the person rating the answers and that the answer is good.

The keyword is convince. So it just needs to convince people that’s it’s right.

It is optimizing for convincing people. Out of all answers that can convince people some can be actual correct answers, others can be wrong answers.

They can all write lean4 now, don't accept numbers that don't carry proofs. The CAS I use for builds has a coeffect discharge cert in the attestation header, couple lines of code. Graded monads are a snap in CIC.

that's not a proof

Is this hallucination, or is this actually quite human (albeit a specific type of human)? Think of slimy caricatures like a used car salesman, isn't this the exact type of underhandedness you'd expect?

> a session with Gemini 2.5 Pro (without Code Execution tools)

How good are you at programming on a whiteboard? How good is anybody? With code execution tools withheld from me, I'll freely admit that I'm pretty shit at programming. Hell, I barely remember the syntax in some of the more esoteric, unpracticed places of my knowledge. Thus, it's hard not to see case studies like this as dunking on a blindfolded free throw shooter, and calling it analysis.