The critical exponent Alpha_ℙ of the Alpha family over the primes

For k > 0 let S(k) = Σ_p 1/(p (log p)^k). We prove that S is real-analytic and strictly convex on (0, ∞) and has a unique minimizer k_ℙ, with three equivalent characterizations: variational, as an exact balance between the negative atom at p = 2 and the primes p ≥ 3, and as an exponential-tilt centering of log log p. A prime-zeta Mellin representation gives k_ℙ = 1.60499334177498705010… to 109 significant digits, verified by three independent high-precision runs. We prove monotone convergence of truncated minimizers, embed k_ℙ in a strictly decreasing critical curve k_ℙ(s) with an explicit two-prime asymptotic, give a criterion for filtered prime families, and compare with the all-integers minimizer k_ℕ = 2.48810101048…

The constants k_ℙ and k_ℕ were denoted k₀ and k₀^E in earlier work of the author.

Files: the paper (PDF and DOCX); kP_high_precision.py, the Python/mpmath script that computes k_ℙ and k_ℕ and performs the Diophantine tests; and kP_log_110.json, the log with the 109 verified digits and all run parameters.