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

Release Notes: Version 8. 8 marks a fundamental paradigm shift in the presentation and formalization of the Arc Helix framework. We have transitioned from phenomenological evaluations to a strictly axiomatic geometric constraint language. Key structural updates include: Architectural Restructuring: Introduced a rigid six-layer "Support-Action Hierarchy" (from geometric admissibility to source-transport closure), mapping the topological constraints of a 1+2-dimensional manifold onto discrete observable spectra. Shift to Dimensionless Ratios: The core observables have been redefined strictly as dimensionless mass ratios (ₛ), resolving previous dimensional inconsistencies and aligning with modern renormalization-group approaches. Modular Demotion of Ad Hoc Postulates: The derivation of the absolute geometric impedance (₆₄₎₌) and the direct calculation of absolute rest masses have been systematically decoupled from the core theorems and demoted to Appendix G as an "Optional Conditional Absolute-Normalization Conjecture. " Epistemological Humility: Added explicitly formulated Limitation Statements and Falsification Protocols, clearly delineating the boundaries of the framework (e. g. , the explicit exclusion of neutrino spectra and the absence of a micro-action principle). Description: This manuscript formulates a geometry-first microformalism in which selected particle-mass hierarchies are represented as projection data on a 1+2-dimensional sub-Riemannian arc manifold. The framework is not proposed as a replacement for Quantum Field Theory (QFT) or the Standard Model; rather, it is constructed as a geometry-level constraint language for admissible mass hierarchies. Its guiding mathematical principle is that topological closure is not free: open arc configurations in an intrinsically broken manifold may be compensated only by geometrically admissible operations, and the resulting closure constraints rigorously restrict the projected discrete spectrum. The formal core of this work is a deterministic support-action hierarchy built upon six consecutive projection layers. By tracing the manifold's non-integrable distributions (contact forms) down to stationary effective numerical benchmarks, the paper demonstrates how discrete particle mass ratios can emerge as topological necessities rather than empirical inputs. Written to stringent academic standards of structure and restraint, this manuscript serves as the foundational constitutional document for the Arc Geometry framework, prioritizing mathematical consistency, modularity, and strict falsifiability.