Twin prime conjecture: arbitrarily large prime pairs separated by two

Prove that for every natural bound N there is a prime p > N such that p + 2 is also prime. Bounded-gap results do not establish the exact gap two required here. The inline primality predicate excludes 0 and 1. Quantifying over every bound expresses infinitude, rather than verifying a finite collection of examples. Research status checked 8 October 2026: no accepted general proof identified in the cited literature. This is a proposed formal statement; independent correspondence review is requested. A successful statement check establishes well-formedness, not the conjecture.

(fun prime : Nat → Prop =>
  ∀ N : Nat, ∃ p : Nat, N < p ∧ prime p ∧ prime (p + 2))
(fun p => 2 ≤ p ∧ ∀ d : Nat, d ∣ p → d = 1 ∨ d = p)

OA_statement : Prop