Analytic Regularization of a Ramanujan Machine Conjecture

Abstract: We provide a formal analytic derivation of a continued fraction identity for $-π/4$ recently conjectured by the Ramanujan Machine~\cite{Raayoni2021}. By utilizing the contiguous relations of the Gauss hypergeometric function ${}_2F_1(a, b; c; z)$, we establish that the conjectured polynomial architecture is a regularized representation of the transcendental ratio $\mathcal{R}(1/2, 0, 1/2; -1)$. Through an explicit equivalence transformation $\mathcal{T}$ defined by a linear scaling sequence, we map the Gaussian unit-denominator expansion to the conjectured form, thereby recovering the quadratic partial numerators $(n-1)^2$ and linear partial denominators $-(2n-1)$. Convergence is rigorously established via the limit-periodicity of the transformed coefficients, which reside on the Worpitzky boundary $L=1/4$.