Structural Results Toward the Twin Prime Conjecture via Mod-6 Obstruction Theory

We develop structural infrastructure for the twin prime conjecture using the Platonic proof kernel. The key insight is that mod-6 arithmetic provides an obstruction mechanism: for any prime p > 3 with p ≡ 1 (mod 6), its potential twin partner p + 2 ≡ 3 (mod 6) is divisible by 3, hence composite. This forces all twin primes p > 3 to satisfy p ≡ 5 (mod 6). We formalize number theory predicates (NT.IsPrime, NT.IsTwinPrime, NT.Dvd), the twin prime sequence NT.TwinPrimeSeq, and Brun's partial sum NT.BrunPartial with 85 verified theorems across five layers (L1–L4). The proof architecture includes 153 base hypotheses encoding numerical data, structural facts about prime divisibility, and the monotone bounded convergence principle. We verify bounded monotonicity for the first 49 twin prime pairs and establish sieve-theoretic connections to Type II damping transfer from the Riemann hypothesis approach. The formalization identifies three infrastructure gaps blocking a full proof: ε-δ analysis for Brun convergence, Nat.mod/Nat.dvd predicates in the kernel, and inclusion-exclusion sieve theory.

Keywords: twin prime conjecture, mod-6 arithmetic, Brun's constant, sieve methods, formal verification

MSC 2020: 11N05 (Distribution of primes), 11A41 (Primes), 11Y11 (Primality)