ORM or PRM for a Multi-Step Geometry Proof Assistant?

You're building a system that generates step-by-step geometry proofs and want to add search-against-a-verifier at inference time (best-of-N or beam search) to improve accuracy. You have automatic final-answer checking (the proof's stated conclusion either matches the target theorem or doesn't) but no human-annotated step-level correctness labels, and collecting them would be slow and expensive given how much geometric/logical expertise each label requires.

1. Would an ORM or a PRM support beam search over partial proofs? Explain why the other one specifically cannot.
2. Given the stated constraint (no step-level human labels, but automatic final-answer checking exists), propose a concrete way to get PRM-quality supervision anyway, and explain the mechanism.
3. Name one way the resulting auto-labeled signal could still be systematically wrong for this specific domain, distinct from a generic "the labels are noisy" complaint.