Bounded Gaps Narrow, Yet Twin Prime Conjecture Remains Unproven — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; recent advances (Maynard, Zhang) establish bounded gaps, but no full proof exists; one arXiv paper claims a constructive sieve-based proof but is not peer-validated. | MATH: Twin primes: \(p, p+2\) both prime. Zhang (2013): \(\liminf_{n\to\infty}(p_{n+1}-p_n) < 70,000,000\). Maynard (2013): \(\liminf_{n\to\infty}(p_{n+1}-p_n) \le 246\) (unconditionally, under Elliott–Halberstam: 12). No exact constant for twin primes specifically; the conjecture is \(\#\{p: p, p+2 \text{ prime}\} = \infty\). The arXiv paper (1708.07884v1) claims: for \(n\) increasing by 1, at least 3 new twin prime pairs appear below \((6n+5)^2\), using sieve of Eratosthenes density arguments — but this is not accepted proof. | CONNECTION: The modulus 6 in the sieve argument (\(6n\pm1\) forms for all primes >3) ties to base-6, not base-60, but base-60 is \(6 \times 10\) — a harmonic composite. The ratio of twin prime density to prime density asymptotically is \(2C_2/\log Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com