Constant d: A Factorial-Based Infinite Product, Companion to D, with Geometric Interpretations in Stochastic, Combinatorial, and Algebraic Geometry

Abstract

We study the convergent infinite product d = Product_{n>=0} (1 - 1/(2*n!+1)) ≈ 0.31991462763026131388..., the void/vacancy companion to the constant D = Sum_{n>=0} 1/(2*n!+1) ≈ 0.96895437... (OEIS A396384). We prove convergence with an explicit truncation bound, an exact power-sum identity showing D accounts for 85.02% of -ln(d), and a considerably sharper identity, d'(1)/d(1) = R, tying d's own logarithmic derivative to the exact remainder from D's founding identity. We develop d(s), the analytic continuation of d, as a meromorphic function of order 0 with explicit poles, zeros, and residues. We note that the irrationality of D and d sits precisely at an unresolved case of a live open problem, Erdős Problem #264 on factorial irrationality sequences. We present three geometric constructions — in stochastic, combinatorial, and algebraic geometry — in which d is the natural "never triggered" probability, and prove an extremal theorem and a parametric generalisation. This constant is catalogued as OEIS A395680, companion to D at OEIS A396384.