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

==== Abstract ====

This paper introduces the Mixed-Base Invariance Theorem, a novel mathematical framework that unifies prime number theory and phase-locking phenomena within a discrete computational structure.
By defining cross-base groups between distinct prime bases aaa and bbb, we demonstrate that the digit-difference vector between consecutive elements exhibits invariance or mirror symmetry under the exchange of bases.
This invariant relationship constitutes a discrete analogue of physical phase locking, enabling integer-based modeling of resonance, synchronization, and self-organizing systems.
The theorem bridges the gap between abstract number theory and physical synchronization, providing both a new mathematical language and a computational tool for engineering applications.