Geometric Constraints on Discrete Particle-Mass Hierarchies from a 1+2-Dimensional Arc Manifold - 弧论坛

Positioning note
This manuscript is written as a piece of the highest-level academic literature, organized to top-journal standards of structure, restraint, and falsifiability, but it is not written with top-journal publication as its purpose. Its aim is conceptual,
mathematical, and programmatic clarity.

Abstract
We formulate a geometry-first microformalism in which selected particle-mass hierarchies are represented as projection data on a 1+2-dimensional arc manifold. The framework is not proposed as a replacement for quantum field theory; rather, it is a geometry-level constraint language for admissible mass hierarchies. Its guiding principle is that closure is not free: open arc configurations may be compensated only by geometrically admissible operations, and the resulting closure constraints restrict the projected spectrum.

The formal core of the paper is a support-action hierarchy built from six benchmark layers: geometric admissibility, structural compatibility, structural curvature, holonomy-transport structural field, connection-holonomy balance, and derived balance-current / source-transport closure data. Each downstream layer is obtained as a projection or coarse image of the next upstream one. Under stated compactness, coercivity, and regularity hypotheses, the associated support
actions admit stationary realizations, the resulting normalized supports are continuous on the benchmark class, and descriptor-level determinacy and perturbative stability propagate down the hierarchy.

Within this framework, we prove a discrete leading-order law for dimensionless mass ratios, show that residual-only sector selection becomes asymptotically degenerate at large ladder depth, construct a mixed residual-plus-geometric selector, and compress the action hierarchy into a master theorem linking closure-level support actions to their downstream descriptor, centroid, and penalty-matrix limits. The formalism is illustrated on a restricted benchmark set of charged leptons, heavy-quark fragments, and electroweak bosons.

The resulting framework is not an ab initio derivation of all Standard Model masses and does not replace the Higgs mechanism or quantum-field-theoretic dynamics. Its claim is narrower and more disciplined: closure, compensation, transport, compatibility, curvature, balance, and closure-law support actions on an arc manifold can impose nontrivial, mathematically explicit, and falsifiable constraints on discrete mass hierarchies.

Data & Code Availability: The theoretical framework and matrix data described in this manuscript are archived at Zenodo.
DOI: 10.5281/zenodo.19391221