This paper resolves the interface problem between viscous and inviscid closure mechanisms by establishing a precise notion of closure continuity in the vanishing viscosity limit for the three-dimensional incompressible Navier–Stokes equations. Building on the viscous high-frequency packet contraction mechanism developed in Paper I and the inviscid defect-driven closure framework of Paper II, this work places both regimes on a common analytic carrier: the high-frequency tail energy above a dyadic cutoff. On this carrier, the paper derives exact packetized drift identities for Leray–Hopf Navier–Stokes solutions and for weak Euler limits admitting a Duchon–Robert local energy balance. The central result shows that viscous persistence reorganizes, in the limit of vanishing viscosity, into inviscid anomalous dissipation in a precise, scale-localized sense. At fixed cutoff, the packetized viscous dissipation terms converge to packetized defect contributions, while nonlinear transfer terms and cutoff remainders converge appropriately. This yields stability of the packet drift identity under the inviscid limit and establishes continuity of the closure architecture across regimes. Under a uniform strict high-frequency margin hypothesis, the paper proves that one-sided packet contraction transfers from the viscous family to the inviscid limit. As a consequence, any Euler limit obtained under such a margin inherits deterministic tail decay and conditional regularization at the cutoff scale. The argument is entirely implication-based: no new compactness theory is proposed, and no claims are made regarding generic regularity or convergence beyond explicitly stated hypotheses. The paper also identifies all obstruction channels to closure continuity, including loss of cutoff-level compactness, persistence leakage, and defect delocalization. These obstructions exhaust the ways in which viscous irreversibility can fail to pass to the inviscid limit on the chosen carrier. As the third paper in the series Closure and Regularity in Partial Differential Equations, this work supplies the missing logical connection between viscous and inviscid closure mechanisms, showing that they are not disjoint arguments but two endpoints of a single high-frequency packet architecture. All results are formulated entirely in standard PDE and harmonic analysis language. Related papers in the series Closure and Regularity in Partial Differential Equations: Part I — From High-Frequency Surplus to Regularity in 3D Navier–Stokeshttps://doi.org/10.5281/zenodo.18371918 Part II — Inviscid Closure and Anomalous Dissipation in the Euler Equationshttps://doi.org/10.5281/zenodo.18371954 Part IV — Shock Formation, Dissipation, and Closure in Compressible Flowhttps://doi.org/10.5281/zenodo.18372079 Part V — A Universality Test of the Strict Margin Closure Mechanismhttps://doi.org/10.5281/zenodo.18372181 Part VI — Failure Modes and Obstructions to Analytic Closurehttps://doi.org/10.5281/zenodo.18372250