Title: Arc-Closure Geometry: S^1 Phase-Locking, Eigen-Action, and Spectral Coupling in a 2+1 Tension Manifold
Abstract:
Contemporary theoretical physics frequently relies on empirical constants (e.g., G, h) and assumed flat spacetime backgrounds to describe fundamental interactions. In this foundational letter, we propose a paradigm shift by introducing Arc-Closure Geometry—a framework built upon a smooth three-dimensional manifold with a structural 2+1 decomposition.
By defining a cooriented rank-2 distribution and its canonical Riemannian extension, we formulate purely geometric tension operators: a scalar structural tension and a symmetric temporal shear tension. We demonstrate that physical quantization arises naturally as a topological constraint, governed by an S^1 phase-locking condition and its associated winding number. Furthermore, by evaluating the minimal arc-closure configuration within the thin-shell regime, we rigorously reduce these geometric tensions to an effective elastica energy, deriving a discrete, parameter-free geometric eigen-action scale.
Ultimately, utilizing Cheeger-type spectral bounds, we establish a dimensionless geometric coupling ratio and mathematically predict a falsifiable, strictly sublinear spectral scaling. This work provides a rigorous topological origin for fundamental physical couplings, completely independent of traditional phenomenological parameters, offering a purely geometric perspective on the unification of forces.