Arc-Closure Eigen-Action and Spectral Coupling in a 2 + 1 Tension Geometry

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+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 S¹-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+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. Overview The construction begins with a cooriented rank-2 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 S¹-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 version Compared with earlier versions, this release: reformulates phase-locking rigorously using an S¹-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 positioning This 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 note Please 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.