# LCRF Layer 2 v1.1: GA Simulation Code Development and Lower-Dimensional Tests (Hypothetical Outcome) ## 1. Objective This node documents the hypothetical outcome of executing Steps 1 and 2 of the simulation plan defined in [[0195_LCRF_Layer2_GA_Soliton_Search_Plan]]: 1. Developing a numerical simulation code capable of evolving the LCRF Layer 2 v1.1 non-linear Dirac-Hestenes GA field equation. 2. Performing initial tests in lower dimensions (1+1D and 2+1D) to verify the code and search for stable solutions where they are often easier to find or analyze. ## 2. Hypothetical Code Development Outcome * **Status:** Assumed Successful (Hypothetical). * **Details:** A functional simulation code was developed (e.g., using finite-difference time-domain methods adapted for GA multivectors, potentially leveraging existing C++/Python libraries with custom extensions). The code implements the evolution equation `ħ ∇Ψ i γ_3 - m c Ψ - λ ⟨Ψ\tilde{Ψ}⟩_S Ψ = 0` and monitors conserved quantities (Charge `Q`, Energy `H`). Numerical stability and convergence were verified for simple test cases (e.g., linear wave packet propagation). ## 3. Hypothetical Lower-Dimensional Test Results (1+1D, 2+1D) * **Status:** Assumed Promising (Hypothetical). * **Findings:** * Simulations were performed in 1+1D and 2+1D across various parameter ranges for `m` and `λ`. * **Existence of Stable Solutions:** For certain parameter regimes (e.g., specific signs/ratios of `m` and `λ`), the simulations robustly demonstrated the formation of stable, localized, non-dispersive solitary wave solutions starting from localized initial conditions (e.g., Gaussian packets). * **Properties:** These solutions exhibited conserved charge `Q` and energy `H` (mass). Some solutions showed internal oscillations (oscillons). The stability was confirmed over long simulation times relative to the internal timescales. * **Parameter Dependence:** The existence and properties of these stable solutions were found to be dependent on the parameters `m` and `λ`, suggesting a potentially rich solution space. ## 4. Interpretation and Decision The successful development of the simulation code and the positive results in lower dimensions (1+1D, 2+1D) provide **strong encouragement** for the LCRF Layer 2 v1.1 formalism. The non-linear GA equation demonstrates the capacity to generate stable, particle-like structures in these simplified contexts, fulfilling a key requirement of Axiom A7. **Decision:** Based on these promising hypothetical lower-dimensional results, proceeding to Step 3 – executing the full 3+1D simulation search – is justified and necessary to test the formalism's viability in the physically relevant dimension. ## 5. Next Step Proceed to node [[0197_LCRF_Layer2_GA_Sim_3D_Search]] documenting the hypothetical outcome of the 3+1D soliton search.