Detecting Physical and Procedural Bias in Lottery Draws: A Number-Theoretic and Statistical Study

Abstract

Physical lottery systems are designed to approximate uniform sampling without replacement, yet practical implementations involve latent mechanical and procedural factors that can induce weak, time-varying departures from ideal randomness. This paper develops a hybrid inferential and predictive framework that integrates regime diagnostics, dependence-aware multiplicity control, bounded-confounding identification, staged transfer evaluation, and reliability-constrained integrated scoring. The objective is explicitly non-deterministic: we test reproducible structure and uncertainty bounds rather than deterministic prediction of winning combinations. We formalize five optimization/identification programs with explicit decision variables, feasible sets, and optimality criteria, and we provide complete theorem and lemma proofs for the key guarantees used by the pipeline. On a long-horizon historical draw corpus, evidence is asymmetric: confounding-robust directional interpretation and strict false-discovery control are strong, while segmentation stability and integrated score dominance remain below pre-registered gates. The resulting contribution is methodological and practical: robust bias-screening claims can be made with high transparency under severe observability limits, while integrated-superiority claims should remain conditional until targeted reruns resolve the remaining gates.