Canon de Karlia — Discrete Diffusion & Homeostatic Conduction

This repository establishes the rigorous mathematical proof and numerical validation of the scalar conduction law — Canon de Karlia. It formalizes the analysis of a nonlinear discrete dynamical system defined on a cyclic graph, driven by a discrete diffusion operator and a localized homeostatic source term. The framework proves local transverse stability, analyzes spectral properties under rank-one perturbations, and charts convergence toward the homogeneous state. Repository Contents Discrete Diffusion on a Cyclic Graph — Version 3.pdf — Complete theoretical manuscript containing definitions, proofs, stability criteria, spectral analysis, and numerical outcomes. canon_karlia_sims.py — Numerical simulation script enabling independent replication of experiments and verification of stability thresholds. canon_karlia_fig1_convergence.png — Node-by-node convergence dynamics and global coherence index tracking. canon_karlia_fig2_robustness.png — Comparative robustness analysis of Ginzburg–Landau and Tanh injection forms under initial disorder. canon_karlia_fig3_stability.png — Spatial stability maps and effective convergence rate cartography. canon_karlia_somesthetic_source.png — Conceptual schema of the somesthetic toroidal field and opposing fluid/energy currents underlying the $N=9$ system. README.md — Technical framing and repository overview. README_SCOPE.md — Explicit separation scope detailing the boundary between this public theoretical core and the full internal operational architecture. Empirical Origin The mathematical formalization presented in this repository is directly derived from the empirical mapping of an internal somesthetic field. The conceptual schema (canon_karlia_somesthetic_source.png) illustrates the toroidal double-helix structure and counter-current dynamics (physical CSF fluid versus scalar/energetic flux) along the central axis, serving as the physiological baseline for discretizing the $N=9$ cyclic graph and its homeostatic injection operator. Reference & Contact Author: Julien Karlia (Joules) ORCID: 0009-0004-3395-4385 INPI Reference: AxeSaros V024022026 Contact: [email protected] Official Website: www.julienkarlia.com Scope of the Publication & Intellectual Property This archive is strictly limited to the publication of the mathematical framework and its associated validation. Operational implementation engines, runtime software architectures, and extended control mechanisms are intentionally held in reserve. Any operational deployment, governance architecture, industrial application, or derivative extension of this framework is covered by the intellectual property protection registered with the INPI under reference AxeSaros V024022026.