Hardy–Littlewood Series, Sieve Admissibility, and Maynard–Tao Bounded Prime Gaps — E8 Intelligence Research

FINDING: Hardy–Littlewood singular series governs the density of admissible k-tuple prime constellations; sieve theory (Eratosthenes, Selberg, interval sieve) realizes exactly the admissible residue classes, and Maynard–Tao gives unconditional bounded gaps. MATH: - Singular series: 𝔖(𝒽) = ∏_p (1 − ν_p(𝒽)/p)(1 − 1/p)^{−k}, where ν_p(𝒽) = # distinct residues of 𝒽 = {h₁,…,h_k} mod p. - Admissibility: ν_p(𝒽) < p for all primes p (equivalently, no p divides ∏_{i2}(1 − 1/(p−1)²) ≈ 1.3203236… - Twin-prime constant: C₂ = ∏_{p>2} p(p−2)/(p−1)² ≈ 1.32032. - Maynard–Tao: lim inf (p_{n+m} − p_n) ≤ C(m), bounded for all m; m=1 gives ≤ 246 (Polymath8b). - Second HL conjecture: π(x+y) ≤ π(x) + π(y) — known false for large x (incompatible with first HL). CONNECTION: Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com