Proper Time as a Coupling Integral in Special Relativity: A Geometric Interpretation via Thales Partitioning

This paper presents a geometric reinterpretation of proper time in special relativity using constraint geometry and Thales partitioning. By normalizing the relativistic energy–momentum relation, the standard Lorentz factor is shown to arise naturally as a mass-channel amplitude on a Pythagorean constraint surface. Proper time is then identified exactly as a coupling integral, representing accumulated participation in the mass channel along a worldline. Within this framework, time dilation corresponds to partial decoupling from the mass channel, velocity emerges as a ratio of cross-channel to single-channel coupling, and causal structure follows directly from channel-ordering constraints. A discrete postal correspondence model is introduced to illustrate the mechanism in an accessible way, reproducing relational time dilation and topology-dependent convergence without invoking spacetime geometry. Connections are drawn to harmonic mean dynamics and per-orbit contraction in gravitationally bound systems, clarifying how coupling strength and relational density regulate convergence across scales. No modification of special or general relativity is proposed; all results follow from standard relations. The contribution is interpretive and geometric, providing a constraint-based coordinate system in which proper time, velocity, and causality can be read directly as coupling diagnostics.