Collatz v4

This paper addresses the non-existence of non-trivial cycles in the Collatz dynamical system using a 2-adic and Diophantine approach. We study the accelerated Collatz map T*(n) = (3n+1)/2^{v₂(3n+1)} and prove that it admits no periodic orbit of positive odd integers other than {1}.
The proof is structured in three layers: (1) an algebraic identity showing the uniform valuation distribution v_i = 2 yields uniquely n = 1; (2) a non-invariance theorem demonstrating that no 2-adic neighborhood of −1/3 is forward-invariant under T*; and (3) a no-cycle condition for non-uniform distributions, proved analytically for L = 2 and verified computationally for L up to 12 via a new Recursive Divisibility Lemma.
The general case is reduced to a single Primitive Prime Factor Conjecture, supported by Zsygmondy's theorem and Baker's effective bounds on linear forms in logarithms.
Keywords: Collatz conjecture, cycle equation, 2-adic integers, primitive prime divisors, Baker's theorem, Zsygmondy's theorem, recursive divisibility.

Note:"Version 3.0 — Layers 1 and 2 are unconditionally proved. The general case of Layer 3 depends on Conjecture 5.1 (Primitive Prime Factor Conjecture)."