A Bidirectional Translation Between Tier-0 Closure and Probabilistic Inference

This paper provides a rigorous, bidirectional translation between the Tier-0 closure criterion and standard probabilistic inference. It serves as a stress test of the Tier-0 framework in the inferential domain, following earlier work that established a corresponding translation between Tier-0 and orthodox partial differential equation (PDE) closure methods. The Tier-0 framework formulates admissibility of mathematical and physical laws as a fixed-point condition under representation, persistence, and completion. While earlier papers demonstrated this structure within nonlinear PDE analysis, the present work examines whether the same law-level criterion survives contact with probabilistic inference, a domain traditionally regarded as conceptually distinct from PDE theory. The main result is a definition-level equivalence theorem: a probabilistic update rule is lawfully admissible if and only if it satisfies a concrete inference-side closure package consisting of representation invariance, persistence under refinement, stability under limits, and minimal completion. This package is expressed entirely in standard probabilistic and information-theoretic language, without requiring acceptance of Tier-0 terminology. Classical inference procedures, including Bayesian conditioning, KL-projection, and maximum-entropy updating, are recovered as fixed points of this closure process. Common inference paradoxes are classified as precise failure modes of closure, rather than as foundational inconsistencies. When a strict persistence margin is present, uniqueness of the admissible update rule follows. Together with the earlier PDE translation paper, this work demonstrates that the Tier-0 admissibility criterion is not tied to any specific mathematical sector. Instead, it acts as a sector-independent law-level organizer, capable of being instantiated faithfully in both analytic PDE theory and probabilistic inference. The paper can be read independently, but it completes a two-domain validation of Tier-0 as a universal closure principle.