Seam Geometry of Bohmian Mechanics: Geodesics, Foliations, and Nodal Singularities

This paper reframes Bohmian mechanics in seam variables built from the wavefunction modulus and phase. Writing $\psi=R\exp(iS/\hbar)$, the amplitude seam $s=\log R$ encodes the Bohm--Madelung quantum potential via $Q=-(R''/R)=-(s''+(s')^2)$ in the stationary one-dimensional (1D) convention used here. An effective seam $\tilde s:=S-\tfrac12 s$ defines a seam-induced conformal factor that serves as a prototype for higher-dimensional metric constructions.Assuming a complex coordinate $z=x+iy$ on a two-dimensional (2D) configuration plane, the quadratic differential (QD) $w=(\partial_z^2\log\psi)\,dz^2$ is introduced and its induced horizontal direction field is compared to probability-current streamlines. For the Gaussian vortex state $\psi=ze^{-|z|^2/2}$, exact alignment is proved: away from the node, the QD-horizontal direction field coincides with the probability-current direction field. More generally, a local ''vortex dominance'' proposition gives asymptotic direction-field alignment near any isolated simple node under a mild factorisation hypothesis, and explicit limitations are documented via a two-node non-example.Reproducible numerical tests for a centered vortex, a shifted vortex, and a two-node state quantify when alignment persists and when it fails.