# LCRF Layer 2 v1.3A: Simulation Plan for Higher-Order GA Field ## 1. Objective This node outlines the simulation plan to test the LCRF Layer 2 formalism v1.3A defined in [[0200_LCRF_Layer2_GA_Option1A_Formalism]]. The primary goal is to determine if the addition of the proposed higher-order (Skyrme-like) terms to the non-linear Dirac-Hestenes equation leads to the emergence of **stable, localized, finite-energy solutions (solitons)** in 3+1 dimensions, addressing the failure of the v1.1 formalism [[0197_LCRF_Layer2_GA_Sim_3D_Search]]. ## 2. Numerical Methodology * **Target Equation:** The highly non-linear partial differential equation derived from `L_{v1.3A}` [[0200_LCRF_Layer2_GA_Option1A_Formalism]], involving up to third or fourth derivatives of `Ψ`. * **Framework:** GA $\mathcal{G}(1,3)$, `Ψ` likely even subalgebra. * **Dimensionality:** Target 3+1D. Initial tests in 1+1D / 2+1D crucial due to complexity. * **Discretization:** Requires stable numerical schemes capable of handling higher-order spatial and temporal derivatives accurately (e.g., higher-order finite differences, spectral methods might be considered). Numerical stability will be a major challenge. * **Implementation:** Significant challenge. Requires advanced numerical PDE techniques implemented for GA multivectors. High-performance computing essential. ## 3. Simulation Parameters and Initial Conditions * **Parameters:** `m`, `λ` (from v1.1), and the new higher-order coupling `κ`. Explore different relative strengths and signs. * **Initial Conditions:** Similar to [[0195_LCRF_Layer2_GA_Soliton_Search_Plan]]: localized Gaussian packets, potentially configurations with non-trivial topology (e.g., twists in the bivector field) designed to seed topological solitons. ## 4. Analysis Techniques * **Conservation Laws:** Monitor energy, charge `Q`, momentum, angular momentum. Check for conservation of any potential *topological* charge associated with the solutions. * **Stability Assessment:** Track localization of energy density, peak amplitude, spatial extent. Look for convergence to a stable, non-dispersing profile. * **Solution Characterization:** If stable solutions found, analyze mass, charge, spin characteristics, internal structure, and topological properties. ## 5. Success and Failure Criteria (OMF Rule 5) * **Success Criterion (Minimal):** Robust demonstration of stable, localized, finite-energy, non-trivial solutions in 3+1D, possessing conserved energy and charge, emerging from generic localized initial conditions for some parameter range. Identification of a conserved topological charge would be strong supporting evidence. * **Failure Criterion (Triggering STOP/Re-Pivot for Option 1A):** If simulations consistently show only dispersal, collapse, or numerical instability despite significant effort in developing stable numerical schemes and exploring parameters, OR if stable solutions found are clearly numerical artifacts or lack physical relevance (e.g., infinite energy). ## 6. Feasibility Assessment * **High Risk:** Implementing stable numerical solvers for higher-order non-linear GA equations is extremely challenging. Theoretical justification for the specific higher-order terms from LCRF principles is weak. High risk of numerical instability or finding only artifacts. ## 7. Conclusion This plan outlines the necessary simulation effort to test Option 1A. Given the significant implementation challenges and weaker theoretical motivation compared to incorporating known physical interactions (like gauge fields), this path carries substantial risk. The simulation should only proceed if simpler options (like Option 1C) definitively fail.