I think the solution in harder-to-verify cases is to provide AI (sub-)agents a really good set of instructions on a detailed set of guidelines of what it should do and in what ways it should think and explore and break down problems. Potentially tens of thousands of words of instructions to get the LLM to act as a competent employee in the field. Then the models need to be good enough at instruction-following to actually explore the problem in the right way and apply basic intelligence to solve it. Basically treating the LLM as a competent general knowledge worker that is unfamiliar with the specific field, and giving it detailed instructions on how to succeed in this field.

For the easy-to-verify fields like coding, you can train "domain intuitions" directly to the LLM (and some of this training should generalize to other knowledge work abilities), but for other fields you would need to supply them in the prompt as the abilities cannot be trained into the LLM directly. This will need better models but might become doable in a few generations.