The Triangle Registry- Primitive Recursion Units and Interaction Rules in Octomorphic Geometry

The Triangle Registry: Primitive Recursion Units and Interaction Rules in Octomorphic Geometry This paper defines the Triangle Registry, a finite, explicitly enumerated set of primitive recursion units forming the discrete substrate of the Octomorphic Framework. The registry consists of 84 lawful, returnable triangular seats, each defined by integer coordinates subject to fixed coherence and lawfulness constraints. These primitives are not energetic objects; they are structural units from which higher-order physical, chemical, and computational structures are assembled. The paper provides: A formal definition of primitive triangles and their derived parameters The complete lawfulness constraints governing registry inclusion Explicit rules for interaction, adjacency, and global closure A finite construction of the registry without probabilistic assumptions A clear separation between structure (registry) and energy (emergent from constraint and collapse) The Triangle Registry functions as the structural backbone for subsequent work in the Octomorphic Framework, including spectral topology, non-associative merge systems, and the dual-channel energy decomposition developed in the Reaction Atlas v2.0. This document is intentionally foundational. It does not assign energies, forces, or dynamics to registry elements. Instead, it establishes the discrete grammar required for any physically meaningful transition, interaction, or assembly within the framework. The complete registry table is provided as a companion dataset (CSV) and constitutes the authoritative reference for all registry-based calculations. Relation to Other Works Serves as the structural substrate for Reaction Atlas v2.0 Supplies the registry and adjacency rules used in octal resonator analysis Provides the formal basis for lawful, sublawful, and gradient edge classifications Discrete geometry, registry systems, recursion units, graph grammar, coherence geometry, octomorphic framework, structural foundation.