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

We study the convergent series D = Sumn>=0 1/(2·n!+1) and the constant D ≈ 0.96895437342956027… 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 establish four structural results: D is the unique stable minimum in hierarchical roulette and D < 1 is the exact Parrondo boundary; d_n = 1 decoy per level is the unique minimum in cascading Honey Encryption satisfying E[triggers] < 1; a factorial invariance theorem showing D < 1 holds for any hierarchy depth, resolving the factorial analogue of Eigen's error-threshold paradox; and a Wilson structure theorem for the denominators. We establish a parametric identity unifying D with Erdős Problem #68 in the family Sum 1/(α·n!+β) = e/α − (β/α)·C(α,β), reducing irrationality to linear independence of e and C(2,1) over Q. The sequence is registered as OEIS A396384.