The Principle of Optimal Applicability of Probability: A Unified Framework for Epistemic Boundaries of Probabilistic Reasoning

This work proposes the Principle of Optimal Applicability of Probability which distinguishes between the universal formal applicability of probability theory and the conditions under which its application is epistemically optimal. The principle states that probability is optimally applicable where (1) a law constrains outcomes to an effectively bounded possibility space and (2) expectations are formulated at the class level rather than for individual realizations. Rather than claiming wholly original discovery this work synthesizes and formalizes convergent insights from frequentist propensity logical and Bayesian traditions including von Mises collective theory Knights risk-uncertainty distinction Reichenbachs reference class problem and recent work on ergodicity economics into a unified methodological criterion. It provides formal definitions a structured justification with corollaries explicit connections to ergodic theory and non-ergodicity and detailed practical applications across medicine finance engineering artificial intelligence and policy-making including diagnostic tools for practitioners. The framework addresses persistent controversies over unique events extreme risks and individual outcomes where probabilistic reasoning is often accused of irrelevance or deception. It clarifies that many alleged failures of probability stem from expectation misalignment rather than formal defects offering a methodological layer between pure probability theory and empirical modeling practice. Keywords: probability theory epistemic optimality reference class problem Knightian uncertainty ergodicity non-ergodic processes Bayesian epistemology decision theory methodological foundations The document includes philosophical foundations a formal criterion with proof sketch corollaries practical guidelines a diagnostic checklist and a discussion of limitations. It is intended as a contribution to the philosophy of probability and methodology of science suitable for researchers in statistics epistemology risk analysis and interdisciplinary applications.