Frequency-Phase Hydrodynamics of Conserved Momentum: Toroidal Localization and the Nature of Time

The modern physical paradigm operationalizes space and time as independent, smooth manifolds that act as a geometric background for dynamic fields and particles. This paper presents a fundamental alternative: primary and invariant momentum is the sole substance of objective reality, while space and time manifest exclusively as its phase projections. By replacing the static real-number line with a closed, multi-dimensional toroidal medium governed by a non-linear logarithmic Planck scale, we establish a rigorous framework where structural metrics emerge from the phase states of a propagating wave packet. This framework eliminates traditional divergence issues by replacing infinite coordinates with periodic phase intervals, redefining physical evolution not as movement through space, but as a cyclic transformation of wave states.