On the Rigidity of Prime-Preserving Bijections

Abstract

We prove that any bijection f : N → N preserving primality must

fix all but finitely many natural numbers. Our proof refines previous

heuristic arguments using explicit counting, analytic number theory, and

density estimates. We demonstrate that an infinite prime permutation

forces an injectivity violation, resolving a conjecture on the rigidity of

prime-preserving permutations.