Bounded Gaps Proven, Twin Prime Conjecture Still Unresolved — 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 no proof of infinitely many pairs with gap exactly 2. The arXiv paper claims a flawed constructive proof. | MATH: Maynard–Tao theorem: \(\liminf_{n\to\infty} (p_{n+1}-p_n) \leq 246\). Twin prime pairs: \((p, p+2)\). The arXiv paper (1708.07884v1) asserts ≥3 new twin pairs per increment of \(n\) near \((6n+5)^2\) — this is unverified and likely erroneous. No new constants or exact ratios emerge. | CONNECTION: None direct. However, the sieve of Eratosthenes operates on lattice-like residue classes mod 6 (all primes >3 are \(6k\pm1\)), which aligns with base-6 (not base-60) periodicity. The gap 2 corresponds to the smallest non-trivial difference in the residue lattice \(\{6k-1, 6k+1\}\) — a symmetry of order 2. No golden ratio, 0.618, or crystallographic symmetry appears in the known mathematics. | DEPTH: 3/10 — The bounded-gap result is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com