Global Affine Time and Metric Uniqueness: A Geometric Characterization of Linear and Cyclic Temporal Structure

We provide a geometric characterization of the conditions under which a global affine temporal parameter can be consistently defined in a metric space. Considering the family of rectifiable paths between fixed endpoints, we prove that the existence of a universal affine parametrization compatible with constant-speed motion is equivalent to the metric uniqueness of the admissible path family. This equivalence is formalized in the Master Theorem, which shows that a global affine time exists if and only if all admissible paths between the endpoints have identical length. Cyclic and quasi-cyclic geometries violate this condition by admitting inequivalent paths of different lengths, thereby preventing the existence of a single global temporal parametrization. In such spaces, the admissible path family naturally decomposes into classes associated with distinct traversal times, yielding a foliation of local temporal parameters. Temporal multiplicity thus emerges as a purely geometric consequence of metric non-uniqueness rather than as a dynamical or physical effect. The results provide a structural description of the transition from linear to cyclic temporal organization: linear geometries support a single global temporal axis, whereas cyclic geometries induce a stratified temporal structure reflecting the multiplicity of admissible paths. The framework further admits an interpretation within informational geometry, where path-dependent temporal signatures arise as intrinsic features of environments with nontrivial geometric structure. From this perspective, temporal coherence appears as an emergent property determined by the metric organization of admissible paths.