This is a fascinating problem space in LLMs! A broadly trained foundational model is gonna be dogshit at mathematical reasoning, as its main corpus of knowledge is mainly general knowledge and writings. So it sounds right that it would have a lay-person's understanding of those subjects.
There are techniques called "fine tuning" where you take a base foundational model (like GPT4 or Claude 4.0 or whatever LLM is your fancy) and then do a differential training run on top and introduce in your own corpus of documents. It's SUPER expensive compared to just doing Retrieval Augmented Generation, however it results in a LLM that has a better "feel" for the "music" of the theory than a generalist LLM.
All that to say though - they're still pretty dogshit at correctness. To meaningfully improve upon correctness it would take a true Generative Adversarial Network where you're training both a generative model and a "judge" model (academically, it's called a "discriminator" model) that decides if the generated output is dogshit or not. It's can result in very finely tuned and believable generative models, however they have an extremely narrow domain of utility and still take a SHITTON of electricity to produce.
And to even start building a GAN architecture, it takes a mountain of data to train the generator and disciminator. Data that doesn't really exist as of right now, and as you correctly point out, is probably going to be generated with AI and result in a Hapsburg AI model