Twin Prime Conjecture Unproven; Maynard–Tao Bounds Gaps at 246, Not 2 — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; best partial result is bounded prime gaps (Maynard–Tao: infinitely many gaps ≤ 246), not infinitely many gaps = 2. MATH: - Twin primes: pairs (p, p+2), both prime. Conjecture: infinitely many such pairs. - Hardy–Littlewood (1923) heuristic: π₂(x) ~ 2C₂ · x / (ln x)², with twin-prime constant C₂ = ∏p>2 [1 − 1/(p−1)²] ≈ 0.6601618... - Brun's theorem (1919): Σp, p+2 prime (1/p + 1/(p+2)) converges to Brun's constant B₂ ≈ 1.90216058... (finite sum ⇒ twin primes are sparse, but does not preclude infinitude). - Maynard (2013) / Polymath8b: lim infn→∞ (pₙ₊₁ − p_n) ≤ 246 unconditionally. - Under Elliott–Halberstam / GEH: gap bound reduces to 6 (Polymath8b), and 2 under stronger hypotheses. - Sieve of Eratosthenes structure: twin primes are exactly n such that 6n±1 both prime (for n ≥ 1), i.e., candidates lie on the arithmetic progression 6n±1. - The arXiv:1708.07884 claim uses (6n+5)² as a bound and asserts ≥3 new twin pairs per inc Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com