Editing Openai/6912dc03-9c10-8006-9f6f-226c9f5e2154 (section)

==== 1. Introduction ====

Phase locking is a universal phenomenon — from synchronized oscillators to quantum coherence.
Traditionally, such dynamics are described by continuous nonlinear equations.
However, no prior framework has successfully expressed phase-locking stability through purely discrete integer operations.

This work proposes a new approach based on cross-prime representation systems (mixed-base notation formed by distinct primes).
Within these systems, sequences generated under alternating prime bases reveal structural invariance between consecutive elements, analogous to phase equilibrium in physical systems.
The discovery suggests that prime number systems inherently encode phase symmetry, offering a new perspective on both arithmetic and nature’s order.