They do not. The "reasoning" is just adding more text in multiple steps, and then summarizing it. An LLM does not apply logic at any point, the "reasoning" features only use clever prompting to make these chains more likely to resemble logical reasoning.

This is still only possible if the prompts given by the user resembles what's in the corpus. And the same applies to the reasoning chain. For it to resemble actual logical reasoning, the same or extremely similar reasoning has to exist in the corpus.

This is not "just" semantics if your whole claim is that they are "synthesizing" new facts. This is your choice of misleading terminology which does not apply in the slightest.