Circle Mathematics Framework: A Generative Model of Physical Reality from Circle Topology

The Circle Mathematics Framework (CMF) proposes a paradigm shift in fundamental physics from a descriptive to a generative model. It posits that the universe is not composed of primary particles but emerges from a computational process based on the geometry and topology of the circle, $S^1$. In this framework, a quantum state is a superposition of integer winding configurations on the circle. A single integer, the winding number $n$, serves a dual function: its magnitude determines a particle’s mass via a precise formula, while its prime factorization encodes the particle’s identity and interaction properties—its fundamental “source code.” A topological resonance condition, based on the golden ratio $\phi$, selects stable particles from the infinite set of integers, explaining the discrete particle spectrum. The CMF resolves long-standing puzzles by re-contextualizing the graviton as a quantum of the projection geometry, dark matter as high-$n$ topological resonances with neutral prime signatures, and the photon as a symmetric combination of patterns with zero net winding. The framework derives fundamental constants, such as the fine-structure constant and mass ratios, with remarkable precision and proposes concrete experimental verification protocols through gravitational wave analysis and computational dark matter searches.