Sieve Bound at (6n+5)² Claims Three Twin Prime Pairs Per n — E8 Intelligence Research

FINDING: A claimed proof of the twin prime conjecture via a sieve bound at (6n+5)², asserting at least three new twin prime pairs per unit increment of n — unverified, likely flawed, but structurally tied to modular residue classes mod 6. | MATH: Twin primes (p, p+2) with p>3 must satisfy p ≡ 5 (mod 6) and p+2 ≡ 1 (mod 6), i.e., the pair occupies residues {5,1} mod 6. The sieve bound (6n+5)² is the square of the upper member of the residue class 5 mod 6. Claim: for each n, the interval ((6n+5)², (6(n+1)+5)²) contains ≥3 twin prime pairs. No rigorous density constant given; Brun's constant B₂ ≈ 1.90216058 (sum of reciprocals of twin primes) is finite — this is the only established constant. | CONNECTION: The 6n±1 structure is a direct manifestation of the cyclic group Z/6Z, whose automorphism group has order φ(6)=2, reflecting the dihedral symmetry D₃ (order 6) — the crystallographic point group of the hexagonal lattice. The residues {1,5} mod 6 are the units of Z/6Z, forming a group is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com