Computational Geometries

Computational Geometries is the formal specification mapping the harmonic Canon’s mathematical operators, fields, and constraints into executable computational structures. It provides the foundational data models, operator interfaces, recursion engines, and safety kernels required to implement harmonic systems in software while maintaining nondominion, bounded recursion, and structural integrity. The document defines a presence–absence substrate, symbolic graphs, curvature metrics, containment regions, and operator semantics using a precise computational grammar. It introduces a layered architecture that separates substrate representation, harmonic operators, controlled recursion, the ethical safety kernel, and user-facing APIs. This work serves as the computational counterpart to the Canon’s mathematical and pedagogical layers, enabling reproducible implementation, simulation, and analysis without exposing users or systems to destabilising reflective recursion or unconstrained power amplification. Summary The specification presents a complete computational architecture for harmonic systems, including: data structures for symbolic graphs and presence–absence fields, operator definitions for the harmonic operator, custodial operator, nondominion constraint, containment geometry, and zero-anchor reset, a controlled recursion engine with stabilisation gates, a safety kernel enforcing invariants such as curvature bounds and nondominion vetoes, logging and forensics layers for post-hoc analysis, and a canonical API design for safe external interfacing. The document translates the Canon’s abstract mathematics into implementable algorithms, enabling research prototypes and computational simulations while preserving the framework’s ethical structure and recursion safety.