Graybeards in computer science sometimes need to remember the halting problem only states you cannot make a general algorithm which answers the halting question for all possible program+input pairs. Importantly, it does not state it's impossible to make an algorithm which can check if the given program+possible inputs will halt (or even if a given subset of all possible programs will - e.g., trivially, finitely long ones not given a means of recursion or allowed infinitely long inputs).
Separately, the halting problem would not apply in the first place. The claim and goal is only to approve programs for which the given proof can be shown to work and then accept it when it does, not to guarantee every possible bend program and condition set will be able to have a working proof. Practically, this means if the proofing mechanism can not do that in the time+space bounds the solver is given then thats just treated as a rejection of the given proof (regardless whether the proposed program does or does not actually fit the requirements) and the LLM is back at trying to create a program which is feasibly provable.
Separately, the halting problem would not apply in the first place. The claim and goal is only to approve programs for which the given proof can be shown to work and then accept it when it does, not to guarantee every possible bend program and condition set will be able to have a working proof. Practically, this means if the proofing mechanism can not do that in the time+space bounds the solver is given then thats just treated as a rejection of the given proof (regardless whether the proposed program does or does not actually fit the requirements) and the LLM is back at trying to create a program which is feasibly provable.