Twin Prime Proof Flawed: Maynard's Bound of 246 Stands — E8 Intelligence Research

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FINDING: The twin prime conjecture remains unproven; the strongest rigorous result is Maynard's theorem (infinitely many primes with bounded gaps ≤ 246), while the arXiv paper claims a flawed constructive sieve argument. | MATH: Twin primes: pairs (p, p+2). Maynard–Zhang–Polymath: liminfₙ (pₙ₊₁ − pₙ) ≤ 246. Hardy–Littlewood asymptotic: π₂(x) ~ 2C₂ ∫₂ˣ dt/(ln t)², with twin prime constant C₂ = ∏ₚ>2 [1 − 1/(p−1)²] ≈ 0.66016. The arXiv paper (1708.07884) asserts ≥3 new twin pairs per increment of n near (6n+5)² — unverified, likely erroneous. | CONNECTION: The constant C₂ = 0.66016 is not a classical harmonic ratio, but note 1/C₂ ≈ 1.5148 (near √2.29), and C₂/φ ≈ 0.408 (close to 0.382 + 0.026). The gap 246 = 2 × 3 × 41, and 6n±1 structure (all primes >3) ties to modular arithmetic mod 6 — a base-6 echo of base-60's factors (2,3,5). No crystallographic symmetry or root-system link is evident from these sources. | DEPTH: 2 — The findings are mostly popular expositions and one unverified pre Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com