Analytically Consistent Reconstruction of Finite Data Using Pad\'e Sequences
Reconstructing the analytic structure of a function from finite datasets is a fundamental problem across theoretical, numerical, and experimental physics. While Pad\'e approximants provide a natural framework, finite-information effects, as well as statistical and systematic uncertainties, may obscure the underlying analytic structure and limit reconstruction reliability. In this work, we reinterpret the appearance of Froissart doublets not merely as numerical artifacts but as \textit{diagnostic objects} carrying information about the analytic consistency of the input data. Accordingly, we develop a general Pad\'e-based algorithm that exploits the dynamics of Froissart doublets along Pad\'e sequences to identify localized inconsistencies and iteratively reconstruct the analytic structure most compatible with the data. The method requires no model for the origin of the inconsistencies and distinguishes genuine analytic features from spurious structures induced by finite-information effects. We validate it using Stieltjes functions, realistic pseudo-experimental datasets with statistical and systematic uncertainties, and general holomorphic functions. The complete algorithm is provided as a supplementary Mathematica notebook in an open GitLab repository.