Canon de Karlia — Discrete Diffusion & Homeostatic Conduction

Abstract

This repository has the exclusive objective of establishing 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, comprising a discrete diffusion operator and a localized homeostatic source term. The study focuses in particular on the spectral structure of the operator, the local stability of the coupled system, the dynamics of the injection term, and convergence toward the homogeneous state under the conditions established by the manuscript. Repository Contents: Discrete Diffusion on a Cyclic Graph — Version 3.pdf: Complete theoretical manuscript including definitions, proofs, stability criteria, spectral analysis, and numerical results. canon_karlia_sims.py: Numerical simulation script associated with the manuscript, enabling the reproduction of the presented numerical experiments and verification of the studied thresholds. canon_karlia_fig1_convergence.png: Visualization of the convergence of the 9 nodes and the global coherence index. canon_karlia_fig2_robustness.png: Comparative robustness analysis of Ginzburg–Landau and Tanh forms, as well as initial disorder absorption. canon_karlia_fig3_stability.png: Spatial stability diagram and mapping of the effective convergence rate. README.md — Canon de Karlia.md: Framing and repository description file. Reference & Contact: Author: Julien Karlia (Joules) ORCID: 0009-0004-3395-4385 INPI Reference: AxeSaros V024022026 Contact: julien.karl@gmail.com Official Website: www.julienkarlia.com Scope of the Publication: This archive constitutes the publication of the mathematical framework and its associated validation. Operational implementation elements, software architectures, execution mechanisms, and other developments not necessary for establishing the theoretical framework presented herein do not form part of this repository. Any use, extension, or publication referring to the presented theoretical framework is invited to cite the author and this publication.