A Continuous Drift-Diffusion Transport Framework for Logarithmic Trajectory Scaling in Pseudo-Random Collatz Dynamics

The asymptotic behavior of piecewise arithmetic maps often resistspurely deterministic methods due to high local parity oscillations. Inthis work, we formulate an ensemble transport framework that bridgesthe discrete parity dynamics of the Collatz map with a continuousadvection-diffusion partial differential equation (PDE). By evaluatingresidue invariants modulo 3 alongside a binary stochastic assumption,we derive the continuous drift (µ =12ln(3/4) ≈ −0.1438) and macroscopic diffusion parameters via a Kramers-Moyal continuum expansionin logarithmic space z = ln(x). We demonstrate that the ensembleprobability density ρ(x, t) over linear coordinates undergoes advectivefunneling toward the singular absorbing state x = 1. Furthermore,treating the boundary z ≤ 0 as an absorbing barrier yields analyticalestimates for first-passage and total stopping time distributions characterized by an Inverse Gaussian (Wald) profile, alongside a powerlaw scaling exponent for maximum trajectory excursions (∼ Γ−0.892).Empirical Monte Carlo orbit samplings show strong structural concordance with the theoretical first-passage density, providing a tractablecontinuous hydrodynamic analog for understanding the global contractive tendencies of arithmetic parity dynamical systems.