Arc-Closure Geometry: S¹ 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 S¹ 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.