A Thought Experiment on Double-Slit Interference from a Source with Positional Fluctuation: Push-forward of the Source-Position Distribution to the Fringe-Shift Distribution (Shape Preservation)

Thought experiment (model calculation), not a measurement. For a fixed geometry (L=10, W=5, lambda0=1) a stationary point source with positional fluctuation P(y)=cos^2(pi y/lambda) is passed through a double slit. Each trial shifts an identically shaped far-field fringe by the exact geometric path difference u(y), a nearly linear map of the source position. Over repeated trials the fringe-shift distribution is the push-forward rho(u)=P(y(u))|dy/du| of P, preserving the cos^2 shape because the map is paraxially linear; the ~3% departure is the non-paraxial nonlinearity (1/2 tan^2 theta). This is a change-of-variables (push-forward), not a derivation of a probability law. Distinguished from intensity-accumulation visibility loss (spatial coherence / van Cittert-Zernike).