Relational Phase, Stationary Action and Quantized Closure

Action principles, phase accumulation and quantized closure appear across classical mechanics, quantum theory, semiclassical dynamics, superfluidity and geometric phase phenomena. Their coexistence is familiar; their structural relationship is usually taken as part of the established formalism. Here we reconstruct that relationship in the opposite direction. Starting from persistent relational dynamics, we show how a real two-component rotation supplies a phase coordinate without assuming a primitive complex number, how a relational canonical one-form can connect phase increments to action-like increments, and how stationary phase and global phase consistency impose two distinct selection conditions. The local condition yields stationary action, δA_R = 0. The global condition yields integral symplectic periods. Under the explicit convention F_R = −iΩ_R/κ_A for a U(1) connection curvature, global compatibility requires [Ω_R/(2πκ_A)] ∈ H²(M_R,Z), equivalently ∫_Σ Ω_R = 2πnκ_A on closed two-cycles. This separates the origin of discreteness from the origin of the dimensionful conversion scale. Topology fixes integer classes; it does not determine κ_A. We therefore identify a precise remaining problem: what selects a unique, universal and non-zero κ⋆ across coherently composable sectors? The framework makes this question falsifiable by requiring cross-scale closure, causal ancestry, intervention consistency and independence from observer compression. It also distinguishes a universal phase–action conversion scale from scale-dependent effective couplings. The result is a sharpened bridge from relational phase to stationary dynamics and quantized closure, with a mathematically explicit boundary between what is derived, what is inherited from established geometry, and what remains to be selected dynamically. The revision formalizes an Action-Scale Factorization Theorem together with Exact-Action and Curvature-Compatibility no-go results, thereby separating the existence of quantized closed-path sectors from the still-open physical selection of a universal non-zero scale such as ħ.