Emergent Metric Structure: From Pre-Metric Informational Organization to Metric-Admissible Spacetime

This revised theoretical preprint defines metric admissibility as the intermediate regime condition between pre-metric informational organization and realized Lorentzian metricity. The paper is situated after the pre-metric information-geometric construction developed in Pre-Metric Information Geometry and Coherence-Conditioned Emergence of Spacetime and the reduced local coherence-dynamical layer developed in Coherence Field Dynamics and the Emergence of Metric Structure. The analysis begins from a smooth base manifold without primitive metric, affine connection, causal structure, temporal order, or gravitational dynamics. A statistical state section induces an information-geometric tensor by Fisher–Rao pullback, but this tensor is not treated as a spacetime metric. The paper then defines the conditions under which a coherence-conditioned candidate tensor becomes admissible as a non-degenerate, temporally oriented Lorentzian metric on a nontrivial domain. The central predicate introduced is MetricAdmissibleU (g). It requires information-geometric regularity, reduced local coherence support, candidate-tensor definition, non-degeneracy, Lorentzian signature, coherent temporal orientation, stable global coherence range, and domain-level persistence. On this basis, metric structure is classified as a bounded regime property rather than as a primitive object, an automatic consequence of informational geometry, or a direct result of local coherence support. The paper also distinguishes metric admissibility from information-induced tensoriality, local coherence support, causal admissibility, realized metricity, metric closure, gravitational field equations, empirical prediction, external realization, and operational control. It identifies the principal failure modes of the transition, including subcritical coherence, threshold degeneracy, upper-bound instability, local coherence without metric admissibility, Lorentzian signature without coherent temporal orientation, transient Lorentzianization, and domain fragmentation.