Constant D: A Factorial-Based Series with Applications to Game Theory, Cryptography, and Biology

We study the convergent series D = Sum_{n>=0} 1/(2·n!+1) and the constant D ≈ 0.96895437342956... to which it converges. We prove that D = e/2 − R for an explicit remainder R, and establish the bound e/4 < D < e/2. We show that D arises independently in game theory, cryptography, and biology, each time as the expected frequency of a rare event in a system with factorially growing capacity. In all three settings the condition D < 1 is the stability threshold. No closed form is currently known. The sequence is registered as OEIS A396384.