Twin Prime Conjecture: Unproven, With Recent Sieve Progress and Unverified Claim — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve methods and a claimed arXiv proof (1708.07884) that is not peer-validated. | MATH: Twin primes: pairs \((p, p+2)\). Maynard's theorem: \(n→∞ (pₙ₊₁-p_n) ≤ 246\) (unconditional, Zhang–Maynard–Polymath). The arXiv paper claims at least 3 new twin pairs per increment of \(n\) when setting \((6n+5)^2\) as a sieve bound — but this is not accepted mathematics. No exact constant for twin prime density; conjectured asymptotic \(~ 2C_2 x/(ln x)^2\), with \(C_2 = ∏p>2 p(p-2)/(p-1)^2 ≈ 0.66016\). | CONNECTION: The constant \(C_2\) is not a golden-ratio family number, but note \(C_2 ≈ 0.660\) — close to \(2/π ≈ 0.6366\) and \(1/φ ≈ 0.618\) but not equal. No base-60 or crystallographic symmetry appears in the valid mathematics. The sieve structure is arithmetic, not geometric. | DEPTH: 3 — The valid result (gap ≤ 246) is deep but does not Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com