# LCRF Layer 2 v1.2: Simulation Plan for Gauged GA Field Theory ## 1. Objective This node outlines the simulation plan to test the LCRF Layer 2 formalism v1.2 defined in [[0206_LCRF_Layer2_GA_Option1C_Formalism]]. The primary goal is to determine if coupling the non-linear GA multivector field `Ψ` to a U(1) gauge field `A` (electromagnetism analogue) leads to the emergence of **stable, localized, finite-energy solutions (solitons)** in 3+1 dimensions, succeeding where the un-gauged v1.1 formalism failed [[0197_LCRF_Layer2_GA_Sim_3D_Search]]. ## 2. Numerical Methodology * **Target Equations:** The coupled system for `Ψ` and `A_μ` derived from `L_{total}` [[0206_LCRF_Layer2_GA_Option1C_Formalism]]: 1. `ħ γ^μ (∂_μ - i(q/ħc)A_μ) Ψ i γ_3 - m c Ψ - λ ⟨Ψ\tilde{Ψ}⟩_S Ψ = 0` (Gauged NL Dirac-Hestenes) 2. `∂_ν F^{νμ} = (q/c) J^μ` (Maxwell's equations with source `J^μ = c ⟨ Ψ γ^μ i γ_3 \tilde{Ψ} ⟩_S` - using appropriate operator ordering/definition for quantum current `hat{J}^μ`). * **Framework:** Classical field theory simulation initially (treating `Ψ` and `A` as classical fields) to search for stable soliton configurations. Quantization effects considered later. GA $\mathcal{G}(1,3)$ for `Ψ`. Vector field `A_μ`. * **Dimensionality:** Target 3+1D. Initial tests in 1+1D / 2+1D recommended. * **Discretization:** Requires stable numerical schemes for coupled non-linear GA PDEs and Maxwell's equations (e.g., finite-difference time-domain (FDTD) adapted for GA, constrained transport for Maxwell). Gauge fixing (e.g., Lorenz gauge `∂_μ A^μ = 0`) will be necessary. * **Implementation:** Highly complex. Requires robust handling of GA algebra, coupled PDEs, and gauge constraints. HPC essential for 3+1D. ## 3. Simulation Parameters and Initial Conditions * **Parameters:** `m`, `λ` (from v1.1), plus the gauge coupling constant `q` (related to fine-structure constant `α_em`). Explore different values of `q` and `λ`. * **Initial Conditions:** * `Ψ(x, 0)`: Localized configurations (e.g., Gaussian packets for `Ψ`). * `A_μ(x, 0)`: Typically start with zero gauge field, or potentially a configuration corresponding to the field generated by the initial `Ψ` charge distribution (solving the constraint part of Maxwell's equations). ## 4. Analysis Techniques * **Conservation Laws:** Monitor Energy (total energy of `Ψ` and `A` fields), Charge `Q`, Momentum, Angular Momentum. Check gauge constraint satisfaction (e.g., `∂_μ J^μ ≈ 0`). * **Stability Assessment:** Track localization of energy density (both `Ψ` and `A` contributions), peak amplitude/width of `Ψ`. Look for convergence to stable, non-dispersing coupled field configurations. * **Solution Characterization:** If stable structures found, analyze total mass/energy, total charge `Q`, spin characteristics (from `Ψ`), field profiles (`Ψ` and `A`), binding energy (compared to free `Ψ` energy). ## 5. Success and Failure Criteria (OMF Rule 5) * **Success Criterion (Minimal):** Robust demonstration of stable (or long-lived), localized, finite-energy, non-trivial coupled `Ψ`-`A` solutions in 3+1D. The solution should represent a confined `Ψ` configuration bound by its own gauge field `A`. * **Failure Criterion (Triggering STOP/Re-Pivot for Option 1C):** If simulations consistently show only dispersal/collapse of `Ψ` (potentially modified by `A` but still unstable), or charge screening effects prevent localization, or numerical instabilities related to the coupled system prove insurmountable despite reasonable effort. ## 6. Feasibility Assessment * **Very High Risk/Complexity:** Simulating coupled non-linear GA field equations with gauge fields in 3+1D is at the forefront of computational physics. Significant challenges exist in algorithm development, stability, and computational cost. Success is far from guaranteed. However, the approach is physically well-motivated. ## 7. Conclusion This plan outlines the simulation strategy for testing Option 1C (LCRF v1.2). Investigating the gauged theory is the most promising direction following the failure of the un-gauged model. The key question is whether the electromagnetic interaction introduced via the U(1) gauge coupling is sufficient to stabilize localized configurations of the non-linear `Ψ` field. The complexity is high, but a positive result would be a major validation for the LCRF Layer 2 formalism.