Version note
This version substantially expands the previous release into a structurally explicit academic formulation of arc-closure geometry on a smooth three-dimensional manifold with a structural 2+12+12+1 decomposition. The revised manuscript is intended to express the arc-geometric program in a form suitable for disciplined communication with mathematical physics, while preserving the distinctive internal content of the theory.
Major additions and revisions include:
- a rigorous S1S^1S1-valued phase-field formulation of phase-locking, replacing the earlier ambiguous exact-form presentation;
- the introduction of a structural closure and a closure-canonical Riemannian extension;
- explicit definitions of the horizontal Ricci scalar, temporal shear tension, and geometric action functional;
- a clarified minimal arc-closure model together with an effective elastica reduction in the thin-shell / small-curvature regime;
- a corrected and transparent derivation of the geometric eigen-action scale;
- an expanded spectral section including Cheeger-type bounds, Rayleigh-quotient motivation, a hard metric-scaling proposition, and a falsifiable strictly sublinear geometric--spectral response ansatz;
- an operational protocol for numerical or experimental scrutiny, together with representative single-arc and dual-arc schematics.
This release should be read as a substantial academic expansion of the earlier version, aimed at improving logical transparency, mathematical discipline, and communicability within the language of mathematical physics.
This record presents an expanded academic formulation of arc-closure geometry on a smooth three-dimensional manifold endowed with a structural 2+12+12+1 decomposition. The purpose of the present release is to express the internal logic of the arc-geometric program in a form suitable for disciplined communication with established mathematical physics, while preserving the distinctive structural content of the theory.
OverviewThe construction begins with a cooriented rank-222 distribution equipped with a sub-Riemannian metric and a normalized transverse field, yielding a structural closure and an associated closure-canonical Riemannian extension. This makes it possible to define, in a mathematically controlled way, a horizontal Ricci scalar, a temporal shear tension, and a purely geometric action functional.
Localized configurations are encoded by an S1S^1S1-valued phase field, so that phase-locking is formulated as a quantized period identity in terms of winding number. Nontrivial sectors arise either from topology or from defect-linking. Within a minimal arc-closure model, and under a thin-shell / small-curvature effective reduction, the framework introduces an elastica-based eigen-action scale associated with closure bookkeeping.
On the spectral side, the expanded manuscript combines Cheeger-type bounds, the Rayleigh quotient, and an exact metric-scaling proposition to motivate a geometric coupling ratio and a strictly sublinear geometric--spectral response ansatz. The paper also states an explicit falsification criterion and includes an operational protocol for numerical or experimental scrutiny.
What is new in this versionCompared with earlier versions, this release:
- reformulates phase-locking rigorously using an S1S^1S1-valued phase field;
- introduces a structural closure and a closure-canonical Riemannian extension;
- clarifies the mathematical status of the horizontal Ricci scalar and temporal shear tension;
- corrects and expands the derivation of the geometric eigen-action scale;
- adds a spectral scaling proposition and a more explicit Cheeger--Rayleigh discussion;
- distinguishes clearly between structural postulates, effective reductions, propositions, examples, and interpretive remarks;
- includes representative arc-spiral schematics and a one-page operational protocol figure.
Academic positioningThis version is not intended as a polemical rejection of established science. Its purpose is to present the arc-geometric manifold-tension program in a mathematically and academically disciplined form. In particular, the manuscript separates:
- what is structurally defined,
- what is introduced as an assumption or ansatz,
- what is established as a proposition,
- and what remains interpretive or exploratory.
The present release should therefore be read as a substantial research-program statement in mathematical physics, emphasizing internal consistency, structural clarity, and falsifiability.
Citation notePlease cite the specific version DOI corresponding to the version used. The concept DOI may be used when referring to the evolving record as a whole.
2+1张力几何体中的弧闭合本征作用和谱耦合
本记录发布的是 Arc-Closure Geometry 的扩展学术版表述：在一个带有 2+12+12+1 结构分解的光滑三维流形上，引入一种更严格、层级更清楚的弧闭合几何框架。该框架旨在以高标准数学物理语言表达弧几何计划的内部逻辑，同时保留其独特的理论结构内容。
本文从一个带次黎曼度量的余定向秩 2 分布及一个归一化横向场出发，引入结构闭包（structural closure）与相应的闭包规范 Riemann 延拓（closure-canonical Riemannian extension），从而较为可控地定义水平 Ricci 标量、时间剪切张力与纯几何作用量。
局域构型通过一个 S1S^1S1-值相位场 来编码，使得锁相被严格写成绕数所控制的周期量子化条件。非平凡扇区既可以来自流形拓扑，也可以来自缺陷链接。在最小弧闭合模型与薄壳/小曲率有效约化下，本文进一步引入基于 elastica 的几何本征作用量尺度。
在谱理论部分，本文结合 Cheeger 型界、Rayleigh 商以及严格的度量缩放命题，给出几何耦合比与一个严格次线性的几何—谱响应假设，并明确提出可证伪条件，同时加入用于数值/实验审视的操作流程图。
相较于先前版本，本版的关键改进包括：
- 以 S1S^1S1-值相位场严格化锁相写法；
- 引入结构闭包与闭包规范 Riemann 延拓；
- 更清楚地区分结构公设、有效约化、命题、例子与解释性陈述；
- 扩展谱理论部分，增强几何—谱标度的数学表达；
- 加入单弧旋与双弧旋示意图，以及一页操作协议图。
本版本并非对既有科学传统的对抗性否定，而是试图在高标准学术框架内，使弧几何的结构逻辑、数学状态与可检验性得到更清楚的表达。