Function verify_threshold 

pub fn verify_threshold(proof: &PredicateProof, threshold: u64) -> bool

Verify a PredicateProof produced by prove_threshold against a public threshold. Returns true iff the proof is valid for that exact threshold — a proof made for one threshold does not verify against a different one.

The proof carries its own verifying key (per-statement setup), so no external VK is needed. threshold is supplied here as the public input.