Two-Field Integer Propagation Modeland a Riemann–P Realization of Collatz Dynamics

This paper proposes a geometric propagation model on the plane formed by two alternating integer fields placed on parallel layers y=n. Odd layers carry an expansion field and even layers carry a collapse field. Local directions are specified through explicit gradient (tangent-slope) laws for the fields Ψ, yielding parallel corridors, trajectory merging, and event-driven switching at inter-layer boundaries. This gradient field is connected to the Riemann–P (three-singularity Fuchsian) differential equation: choosing a half-integer local exponent produces square root scaling, so dilations of the independent variable generate multiplicative amplitude updates. When integers are embedded as a half-integer leaf u=n+1/2 and a first-return-to-leaf rule selects dyadic contraction depth, the induced return map is exactly the Collatz map. We provide vector-field examples, switching rules, a formal equivalence, and a numerical propagation example illustrating why this reformulation is useful in the digamma form. This model naturally leads to the Ψ-function and a new Collatz constant, β=0.93982.