Admissibility, Irreversibility, and Finite Resolution in Navier–Stokes Dynamics

Paper V: Thermodynamic closure: irreversibility and entropy arise necessarily from admissibility enforcement under finite resolution. Constraint Geometry Series track. This paper provides a thermodynamic closure for the Constraint Program: it argues that irreversibility and entropy-like asymmetry arise as structural consequences of admissibility under finite resolution. Rather than introducing new primitives or proposing a new thermodynamic theory, the paper identifies a general implication: when a system’s evolution is constrained by admissibility (in the sense developed across the series), the space of permissible continuations is not symmetric under reversal. In this framing, “constraint cost” names the structural fact that excluding inadmissible continuations changes what transitions remain reachable. The analysis treats irreversibility as neither assumed nor optimized, and it does not rely on statistical inference as a primary premise. The emphasis is on interpretation: entropy is presented as a downstream signature of constrained evolution rather than as an axiom or an informational proxy. This document is observer-only and non-procedural. It provides no diagnostic method, calibration, thresholds, or implementation guidance. Structural claims and results are presented at the level of formal consequence and conceptual closure within the series. CRL-0 · Stage-0 · observer-only · non-authoritative · no methods, thresholds, procedures, or operational guidance. Revisionv1.2 This revision aligns the document to CRL-0 licensing posture(observer-only, non-authoritative, non-procedural)and strengthens non-operational boundaries. Structural claims and results unchanged. Published: Jan 1, 2026 Revised: Jan 25, 2026 (v1.2) Revision v1.3 (June 2026): Title-page affiliation and series-position language updated for consistency with the Constraint Geometry Series. Structural claims, abstract, and body content unchanged.