Bounded Prime Gaps at 246, but Twin Prime Conjecture Remains Unproven — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve methods shows infinitely many primes with bounded gaps (≤246), but not specifically gap-2 pairs. The arXiv paper claims a constructive proof via sieve of Eratosthenes and Mersenne numbers, but this is not peer-validated. MATH: - Twin primes: pairs (p, p+2) both prime. - Maynard–Tao theorem: lim inf (pₙ₊₁ − p_n) ≤ 246 (unconditional, 2013–2014). Under Elliott–Halberstam, gap ≤ 6; under generalized EH, gap = 2 (i.e., twin primes) would follow — but EH is unproven. - The arXiv paper (1708.07884v1) claims: setting n → n+1 increases twin prime pairs by at least 3, using (6n+5)² as a sieve boundary. This is a specific, checkable claim — but no rigorous proof of infinitude is established in mainstream literature. - No new constants or ratios appear in the mainstream results; the gap bound 246 is a combinatorial sieve artifact, not a harmonic ratio. CONNECTION: - The sieve of Eratosthenes operates on re Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com