Finite-Sample Representation Framework: Property-Specific Preservation and Estimator-Dependent Downstream Consequences

This preprint presents the Finite-Sample Representation Framework (FSRT), an empirical framework for studying how finite samples preserve different properties of their source populations and how realized sample–population discrepancies relate to downstream statistical and machine-learning estimators. The central premise is that finite-sample representativeness should not be treated as a single universal quantity. Instead, different population properties may be preserved at different rates and with different levels of reliability, and the coordinates that matter for downstream behavior may depend on the estimator or learning algorithm being used. The study combines controlled synthetic experiments with real-data empirical-population analyses. It examines property-specific preservation, fitted sample-size scaling behavior, and downstream consequences for ordinary least squares, regularized logistic regression, and Random Forests. Key findings include: finite-sample preservation is property- and population-dependent; a synchronized replicate-level bootstrap supports substantial differences between some fitted preservation trajectories, including a strong mean–kurtosis contrast under a Student-t5 population; for smooth estimators, classical perturbation, influence, and Newton quantities often explain parameter displacement as well as or better than FSRT-motivated representation coordinates; for Random Forests, task-aligned leaf-response representation shows incremental predictive value beyond tested feature-only and label-aware generic balance controls in a subset of experimental regimes; after Holm correction, 11 of 16 comparisons in the primary label-aware Random-Forest analysis are statistically supported, while unsupported and boundary cases are retained and reported explicitly. The manuscript therefore does not claim a universal representation metric or a replacement for classical estimator theory. Instead, FSRT is proposed as an organizing framework for distinguishing generic sample–population discrepancy, estimator-relevant representation, estimator response, and downstream consequence. This revised version incorporates confirmatory paired-bootstrap analyses, multiplicity-adjusted Random-Forest comparisons, stronger classical estimator baselines, explicit negative and boundary results, and expanded reproducibility documentation. The accompanying reproducibility materials include analysis code, protocol-history documentation, replicate-level outputs used for the principal confirmatory comparisons, multiplicity-adjustment results, and synchronized bootstrap materials. Author: Abdulrazzag Al AlewyORCID: 0009-0007-4017-9219Email: [email protected]: https://github.com/marsman1976/puzzle-framework-paper1 Status: Revised preprint. This version is intended for journal submission and has not yet been published as a peer-reviewed journal article.