This revised theoretical preprint defines the closure conditions under which a metric-admissible Lorentzian regime becomes a realized metric regime. It follows the public sequence established by Pre-Metric Information Geometry and Coherence-Conditioned Emergence of Spacetime, Coherence Field Dynamics and the Emergence of Metric Structure, and Emergent Metric Structure: From Pre-Metric Informational Organization to Metric-Admissible Spacetime. The paper begins only after the predicate MetricAdmissibleU (g) has been assigned and introduces MetricClosedU (g) as the predicate governing metric closure on a nontrivial domain U. Metric closure is defined as the condition under which an admissible Lorentzian tensor becomes differentially closed, causally coherent, volume-stable, and robust as a domain-level geometric regime under coherence constraints. The paper distinguishes metric closure from information-induced tensoriality, reduced local coherence support, metric admissibility, gravitational field equations, empirical validation, external realization, prediction, and control. Its contribution is formal and classificatory: it specifies when the metric problem is closed in the restricted geometric sense without deriving downstream gravitational dynamics.
Vien Nguyen Son (Sun,) studied this question.