Canon of Karlia — Scalar Conduction Law

Description This repository contains the complete supplementary materials, source code, and high-resolution figures for the research paper “Discrete Diffusion on a Cyclic Graph with Localized Homeostatic Source Term: Analysis of the Karlia Conduction Law”, submitted to Chaos, Solitons & Fractals (Manuscript ID: CHAOS-D-26-09350). Abstract Classical linear diffusion on networks strictly preserves spatial mean values. This work introduces the Karlia Conduction Law, breaking this conservation via a localized nonlinear feedback injection. We characterize both the stable consensus convergence regime and the post-critical transition governed by a supercritical flip (period-doubling) bifurcation. Repository Contents KarliaCanonV2.pdf: The complete manuscript containing formal theorems, proofs, and numerical validations. normal_form_v3.py: Core analytical Python script implementing cyclic Laplacian construction, critical fixed-point location, Jacobian evaluation, Floquet analysis, and cubic normal-form coefficient extraction. High-Resolution Figures: Complete set of graphical artifacts (canon_karlia_fig1_convergence.png through canon_karlia_fig4_flip.png), including convergence, spectral stability, robustness, and bifurcation-scaling results. Documentation: Comprehensive architectural scope (README_SCOPE.md) and usage guide (README.md). Key Analytical Results Critical Threshold: kc≈0.2091353228k_c \\approx 0.2091353228 at the uniform steady state. Supercritical Flip Branch: Analytical prediction of stable period-2 orbits beyond the critical point, confirmed by numerical simulations. Scaling Law Analysis: Comparison between reduced normal-form predictions and full empirical cycle measurements. Authors/Creators Iaiche Achour Karl Julien, Julien Karlia (Researcher) ORCID: 0009-0004-3395-4385 INPI Intellectual Property Reference: AxeSaros V024022026 contact : [email protected]