From Prepared Electron Flux to Localized Detection Events: Schrödinger Flow, Boundary Transport, and the Structure of Born Statistics

The Born rule provides the empirically successful statistical connection between the Schrödinger wave function and the outcomes of quantum experiments. Yet two logically distinct questions are often compressed into this rule: what the wave function describes during propagation, and why repeated localized particle events exhibit the statistics associated with its squared amplitude. In an earlier analysis, we proposed a deliberately restricted interpretation of the single-particle Schrödinger wave function as a complex-amplitude representation of the spatial--directional structure of quantum propagation. The present paper advances that proposal by one dynamical step. We begin not from an abstract probability distribution, but from the operational structure of electron-scattering experiments. Irrespective of the detailed electron-generation mechanism, a completed preparation procedure produces an actual electron flux crossing a source-exit surface. We therefore introduce a preparation boundary Σ₀, and define N₀ as the actual number of electrons crossing Σ₀ during a specified time interval. A second surface, ΣD, represents an idealized detection boundary. For a Schrödinger state Ψ( r,t), the exact continuity structure defines\[ρ_Ψ=|Ψ|^2, j_Ψ=/mIm(Ψ^*∇Ψ),\]and, wherever ρ_Ψ≠0,\[ v_Ψ= j_Ψ/ρ_Ψ.\] We investigate the trajectory hypothesis that a localized electron configuration crossing Σ₀ at ( X₀,t₀) evolves according to\[d X/dt= v_Ψ( X,t).\]This generates a boundary-to-boundary flow map\[Φ_Ψ:( X_0,t_0)( X_D,t_D).\] An ensemble of actual source crossings therefore defines a physical source measure μ₀, while the corresponding distribution of detector crossings is the push-forward\[μ_D=(Φ_Ψ)_*μ_0.\] The Born problem can then be reformulated sharply: under what physical conditions does the source-generated measure μ₀, transported by the Schrödinger flow, produce the standard Born distribution at ΣD? The trajectory law and the equivariance of |Ψ|² are closely related to established de Broglie--Bohm theory, and the connection between quantum flux and asymptotic surface crossings is rigorously developed in the flux-across-surfaces literature. We do not claim these mathematical structures as new. The contribution of the present work is instead to reconstruct the quantum-statistical problem from the complete experimental chain\[{aligned}preparation& crossingödinger flow\& crossing,{aligned}\]and thereby isolate a distinct source-measure problem. If Born statistics are to be dynamically accounted for rather than imposed at the outset, the remaining question is not only how |Ψ|² is transported, but why the microscopic preparation process produces the particular source ensemble whose boundary transport yields Born statistics. The present paper does not claim to solve that final problem. It formulates it as a concrete boundary-transport problem that can be addressed analytically, numerically, and ultimately through explicit models of electron preparation.