From Relation to Phase and Relational Closure: Direction, Path Multiplicity, and Cycle Consistency in the Ω₀/A2 Programme

We organize two related reconstruction modules into a single conditional chain: persistent oriented relational dynamics can support a phase-like coordinate, and phase-bearing histories can support relative-phase-sensitive composition and cycle-closure tests. The first module derives S¹ and real rotational structure only under declared compactness, dimensionality, continuity, nonstalling, and persistence assumptions. The second establishes exact consequences of real vector combination and finite-graph cycle consistency while keeping the combination rule and physical observation map explicit assumptions. These mathematical results do not derive quantum amplitudes, the Born rule, or a physical gauge symmetry. They define a staged set of falsifiable bridges from relation to phase and from phase to candidate relational closure; physical identification remains open.