Version 2. Revised and expanded axiomatic presentation of the framework.Clarified geometric setting and structure.No physical interpretation or empirical claims added. This work presents an axiomatic and fully dimensionless framework for the normalization of periodic geometric systems. The construction is based on internal geometric relations rather than empirical measurement or physical calibration. A central result is the emergence of a universal normalization identity, Ψ = 1, which remains invariant across circular and spherical extensions of the system. The framework begins with two-dimensional circular normalization and is extended algebraically to three-dimensional scaling through a regulator ratio that bridges periodic angular structure with decimal scaling. Recursive normalization leads to the natural emergence of large-scale numerical magnitudes while preserving structural consistency through a defined reset gap at the periodic boundary. All quantities introduced are dimensionless and symbolic, and no direct identification with known physical constants is asserted. The contribution of this work is conceptual and axiomatic, providing a coherent geometric normalization scheme that may serve as a mathematical foundation for further investigations into periodicity, scaling, and abstract geometric symmetry-breaking.
Georgios Bouras (Sun,) studied this question.