The Sagnac Effect as Time Holonomy Why Interception Kinematics Obscures the Relativistic Origin of the Effect

The Sagnac effect is commonly presented using interception formulas of the form

which reproduce the observed time difference between counter-propagating signals on a rotating loop. While numerically successful, this representation implicitly assumes a globally simultaneous spatial length and a shared time parameter, thereby reinstating a Newtonian simultaneity structure. In this paper we argue that such formulations obscure the genuinely relativistic origin of the Sagnac effect. We show that the effect arises from the non-integrability of time—specifically, a time holonomy associated with rotation—and not from interception kinematics. Expressing the Sagnac effect in terms of  hides the failure of global clock synchronization and cannot account for time dilation or differential clock accumulation. The Sagnac effect is fundamentally a manifestation of path-dependent time, not of relative closing speeds.