Empirical Diagnostics of Regularized Inference: A Linear-Algebraic Audit of Historical, Textual, and Psychometric Data

Quantitative workflows across economic history, computational text analysis, and behavioralpsychometrics routinely rely on bilinear projections, smoothing filters, and sparse covariance es-timators to extract latent structures from noisy observational matrices. Lacking physical groundtruth, practitioners frequently interpret smooth trajectories or sparse networks as genuine em-pirical laws. Grounded in inverse problem theory, this monograph demonstrates that structuralbreakdowns in such pipelines serve as direct signatures of unacknowledged mathematical priors.We formalize a sequential four-step linear-algebraic audit evaluating residual Hessian condition-ing (κ(H)), Effective Degrees of Freedom (DoF), singular value spectra (Σ), and eigenvector Par-ticipation Ratios (PR). Applying this protocol exclusively to real empirical datasets—postwarUnited States macroeconomic series, two centuries of presidential State of the Union addresses,and clinical psychometric cohorts—we expose four distinct failure modes: informational suf-focation under Hodrick–Prescott filtering (DoF = 12.38/203), complete orthogonal divergenceunder lexical normalization (r = −0.004), parabolic Guttman horseshoe compression of linearhistorical time (R2 = 0.4411), and artificial hub collapse under Graphical LASSO regulariza-tion (PR : 6.60 → 4.91). Together, these empirical audits establish a rigorous, domain-agnosticstandard for methodological hygiene in data-driven soft sciences.