In standard treatments of the Michelson—Morley experiment, the round-trip time is calculated along a central path that identifies reflection with the geometric center of the mirror. This choice is adopted without proof: the reflector is treated as point-like, or the central ray is selected on symmetry grounds, without verifying that the reduction preserves the property being calculated.This article provides that verification. By parameterizing the reflecting surface with a continuous coordinate, the total round-trip time is calculated as a function of the reflection point. In the geometric time-of-flight model with point-like source and receiver, the physically relevant path is the one that minimizes the total time. In general, the corresponding minimum point coincides neither with the geometric center of the reflector nor with the first point reached by the incident wavefront.The main result is that, although the minimum-time reflection point is displaced from the geometric center by an amount of order O(β^2 )L, the round-trip time evaluated along the central path differs from the minimum time only at order O(β^4 ), with β=vu2044c, whereas the anisotropy predicted by Galilean kinematics already appears at order O(β^2 ).As a corollary, not all geometrically plausible reductions have the same status. The first-contact construction—which identifies the relevant reflection point with the first point reached by the incident wavefront during the outward propagation alone—produces an error already at order O(β^2 ) and cancels the entire angular dependence. Two geometrically analogous reductions of the same system therefore have opposite methodological status, and the difference can be diagnosed from the perturbative structure of the error.The immediate motivation for this work comes from an acoustic analogue proposed by Norbert Feist. In that context, the author had previously adopted the first-contact construction as an explanation of an experimentally reported isotropic result. The present article shows that this construction was incorrect and identifies the error: minimizing the outward propagation time alone is not equivalent to minimizing the total round-trip time.