Continuous Transport Modeling and Density Compression Dynamics in Modularly Restricted Collatz Trajectories

This paper presents a formal dynamical systems framework for analyzing the distribution and convergence behavior of trajectories under the Collatz map. By partitioning the integer domain into modular residue classes modulo 3, we formulate the iterative dynamics asa discrete-time Markov process over invariant residue spaces. We evaluate the average logarithmic contraction rate across parity-transition cycles and establish the continuum limit transition from discrete parity maps to a macroscopic advection-diffusion Transport Partial Differential Equation (PDE). Driven by a constant macroscopic drift coefficient (DI = −0.25), derived from the expected geometric step decay, the continuous probability density function ρ(x, t) undergoes spatial concentration towards a Dirac delta distribution centered at the trivial absorbing cycle {1, 2, 4}. Furthermore, empirical orbit verificationacross an extended numerical domain up to N = 108 confirms the absence of non-trivial absorbing cycles within the evaluated bounds, providing a unified macro-micro mechanical model for heuristic trajectory convergence.