弧闭合几何：2+1 张力流形中的 S^1 相锁定、特征作用量与谱耦合

Arcman

弧闭合几何：2+1 张力流形中的 S^1 相锁定、特征作用量与谱耦合

https://doi.org/10.5281/zenodo.19246505

核心提要：

现代物理学长期依赖经验常数（如万有引力常数 G、普朗克常数 h）以及平直时空背景的假设。本篇快报（Letter）旨在打破这一形式代数的壁垒，正式提出“弧闭合几何（Arc-Closure Geometry）”的基础数学框架。

本研究在具备 2+1 结构分解的光滑三维流形上，构建了共向秩-2分布及典范黎曼扩张。借此，我们将物理学中的“力”还原为纯粹的几何张力算子：标量结构张力与对称时间剪切张力。研究严密论证了：

• 量子化的拓扑起源： 局域构型受 $S^1$ 相位场约束，通过环绕数（Winding Number）自然实现了相锁定（Phase-locking）与周期的拓扑量子化。
• 特征作用量的演生： 在薄壳机制下求解最小弧闭合模型，几何张力被严格降维至等效的 Elastica 弯曲弹性能，从而纯几何地推导出一个离散的“特征作用量尺度”。
• 宏观引力耦合的终极预言： 结合 Cheeger 谱界限分析，本文推导出一个无量纲的几何耦合比，并给出了一个可被实验证伪的、严格的亚线性谱缩放假设（Sublinear Spectral Scaling）。
  
  

这是一次向物理学本体论的回归。本框架证明：宇宙的基本耦合与尺度，并非源于造物主随机设定的经验参数，而是“弧流形”在寻求时空拓扑闭合时的必然几何代价。

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.