This work proposes a two-sheeted phase model for the classical problem of parallels. Its central claim is that Lobachevskian nonsecance can be reconstructed without treating Euclid’s original fifth postulate as intrinsically false in the transported geometry. In the model, two homologous affine transverse lines co-rotate in the desynchronized regime. When the comparison is made in the transported configuration, a transversal joining both lines preserves the Euclidean same-side angular sum of two right angles. The Lobachevskian profile appears only under a fixed-frame, non-transported metric reading: one transported line is compared with a reference line kept in the neutral frame. In that projection, the apparent angular defect, the nonsecant window, the two limiting directions, and the classical angle-of-parallelism profile can be recovered. The article therefore separates three levels that are often conflated: Euclidean angularity in the transported configuration, Lobachevskian nonsecance in the fixed-frame reading, and later hyperbolic metric codification. It argues that Lobachevsky’s imaginary law may be understood as the formal trace of a phase-transported two-sheeted geometry read through a non-transported metric frame, rather than as a direct failure of Euclid’s angular postulate in the underlying configuration.