VERIFICATION IS NOT PROOF Independence Results for Lehmer's Conjecture, the Twin Prime Conjecture, and the Structural Limits of the Jensen–P´olya Program

We establish three results concerning the provability of classical number-theoreticconjectures within ZFC set theory. First, we prove that the Jensen–P´olya criterion,while providing a valid reformulation of the Riemann Hypothesis, is structurallyincapable of resolving RH: no finite collection of Jensen polynomial verificationsconstitutes a proof, regardless of how large the verified region becomes.Second, we prove that Lehmer’s Conjecture (the non-vanishing of Ramanujan’stau function) is independent of ZFC—no finite certificate can establish universalnon-vanishing across an infinite domain. Third, we prove that the Twin PrimeConjecture is independent of ZFC—no finite mechanism forces the infinite recurrenceof prime gaps equal to 2.All three results follow from a unified principle: statements about infinite collectionsthat lack finite certificates are independent of any finitely-axiomatized theory.We call this the Finite Data Principle. The results demonstrate that verificationbasedapproaches to these conjectures—however computationally impressive—arecategorically distinct from proof and can never bridge the gap to resolution.