Scale-Invariant Quantum Spin and Topological Phase Inversion in the Logarithmic Throat: A Formal Proof in Lean 4

This paper presents a rigorous integration of the author's apparatus of Frequency Mathematics with a dynamic model of quantum spin under continuous scale transformations (N). Using the interactive theorem prover Lean 4, we have carried out a full formal verification that the fundamental 4π-periodicity (geometric invariant of spin) remains absolutely unchanged when crossing the critical scale boundary N = 1. Based on the axiomatics of Frequency Mathematics and every wave packet, we have strictly derived the topological inversion of the phase process: in the macroscopic regime, the phase increment follows standard metrics, whereas within the logarithmic throat (N < 1), the direction of winding is mirror-inverted. The formal proof confirms that the discrete topological configuration (spin sign) returns безошибочно to its initial state exactly after four elementary steps in both scale regions, guaranteeing the dynamic indestructibility of the quantum.