# LCRF Layer 2 v1.1: Simulation Plan for Soliton Solution Search ## 1. Objective The LCRF Layer 2 formalism v1.1 [[0184_LCRF_Layer2_GA_Formalism_v1]] describes a GA multivector field `Ψ` governed by the non-linear Dirac-Hestenes equation: `ħ ∇Ψ i γ_3 - m c Ψ - λ ⟨Ψ\tilde{Ψ}⟩_S Ψ = 0` A critical requirement for this formalism to be viable (consistent with Axiom A7 - Emergence Potential) is its ability to support **stable, localized, finite-energy solutions** that can serve as analogues for fundamental particles [[0186_LCRF_Layer2_GA_Solution_Analysis]]. The existence and stability of such solutions (solitons or oscillons) in 3+1 dimensions for this specific equation are not guaranteed and require investigation. This node outlines the **simulation plan** to search for these solutions numerically, defining the methodology, parameters, initial conditions, analysis techniques, and success/failure criteria. This is a crucial validation step for Layer 2 before proceeding with quantization or further extensions. ## 2. Numerical Methodology * **Target Equation:** The classical non-linear Dirac-Hestenes equation (as above). * **Framework:** Geometric Algebra $\mathcal{G}(1,3)$. The field `Ψ` (likely even subalgebra) has 8 real components (scalar, 3 bivector components for space-space, 3 bivector components for space-time, pseudoscalar). * **Dimensionality:** 3 spatial dimensions + 1 time dimension (3+1D). *Initial tests might use lower dimensions (1+1D, 2+1D) for feasibility, but 3+1D is the target.* * **Discretization:** * **Space:** Finite-difference methods on a 3D Cartesian grid. Need sufficient resolution to capture potential soliton profiles (scale likely related to `1/m`). Appropriate boundary conditions (e.g., absorbing or sufficiently large domain). * **Time:** Stable time-stepping algorithm suitable for hyperbolic PDEs with non-linearity (e.g., Runge-Kutta, Leapfrog adapted for GA, potentially implicit methods). Ensure numerical stability (CFL). * **Implementation:** High-performance computing likely required. Need efficient GA operations (products, grade projections, norms) implemented numerically. ## 3. Simulation Parameters and Initial Conditions * **Parameters:** * `m`: Mass parameter (explore positive values). * `λ`: Non-linear coupling strength (explore positive and negative values, as stability depends on the interplay with `m`). * Set `ħ=1, c=1`. * **Initial Conditions (ICs):** Focus on localized configurations designed to potentially relax into stable solitons: * **Gaussian Packets:** Localized wave packets with varying initial widths, amplitudes, and potentially specific multivector structures (e.g., predominantly scalar+pseudoscalar or with initial bivector components). * **Perturbed Analytical Solutions (if known):** Use known solutions from related linear or non-linear equations (e.g., linear Dirac wave packet, NLKG solitons) as starting points. * **Boosted Solutions:** Apply Lorentz boosts to potentially stable static or lower-dimensional solutions. ## 4. Analysis Techniques * **Conservation Laws:** Monitor conserved quantities (Energy, Charge `Q = ∫ <Ψγ₀\tilde{Ψ}>_V d³x`, Momentum, Angular Momentum) derived from the Lagrangian [[0184]] to check numerical accuracy. * **Stability Assessment:** * Track the **localization** of the energy density derived from the energy-momentum tensor. Does an initial lump remain confined or does it disperse? * Monitor the **peak amplitude** and **spatial extent** (e.g., RMS width) of the `Ψ` field configuration. * Check if conserved quantities (especially Energy/Mass and Charge) remain constant for the localized lump. * **Solution Characterization:** If stable structures are found: * **Mass:** Calculate total energy in the rest frame. * **Charge:** Calculate total conserved charge `Q`. * **Spin:** Analyze the angular momentum density and transformation properties (bivector components) of the stable solution. * **Internal Structure:** Analyze the multivector grade distribution (`s, B, P`) and any internal oscillations (via FFT of field components within the structure). ## 5. Success and Failure Criteria (OMF Rule 5) This simulation phase tests the core viability of the Layer 2 v1.1 formalism for generating particle-like structures (Axiom A7). * **Success Criterion (Minimal):** Robust demonstration of **at least one type of stable (or extremely long-lived), localized, finite-energy, non-trivial solution** in 3+1 dimensions for physically plausible parameter ranges (`m`, `λ`). The solution should possess definite conserved charge and energy (mass). * **Stronger Success:** Finding solutions with properties suggestive of spin-1/2 (dominant bivector components, correct angular momentum). Finding a *spectrum* of different stable solutions. * **Failure Criterion (Triggering Re-Pivot/Halt):** If extensive simulations across relevant parameter space (`m`, `λ`) and diverse localized initial conditions consistently show **only dispersal of initial configurations or collapse to the trivial `Ψ=0` vacuum**, with no evidence for persistent, localized, non-trivial structures, then this specific non-linear Dirac-Hestenes equation will be deemed incapable of supporting emergent particles within LCRF Layer 2. This would falsify the v1.1 formalism and require abandoning this specific mathematical model. ## 6. Feasibility and Next Steps * **Feasibility:** Simulating non-linear GA field equations in 3+1D is computationally demanding and requires specialized numerical techniques. It is challenging but likely feasible with dedicated resources. Initial studies in 1+1D or 2+1D can provide guidance. * **Next Steps:** 1. **Develop/Adapt GA Simulation Code:** Create or adapt a numerical code capable of solving the non-linear Dirac-Hestenes equation in the desired dimensions. 2. **Perform 1+1D / 2+1D Tests:** Conduct initial, less expensive simulations in lower dimensions to identify promising parameter regimes and test numerical methods. 3. **Execute 3+1D Search:** Run simulations in 3+1D focusing on the parameter space identified in lower dimensions, using various localized initial conditions. 4. **Analyze Results:** Rigorously analyze stability, localization, and properties of any persistent structures found, applying the success/failure criteria. ## 7. Conclusion: Testing Particle Emergence This simulation plan provides a concrete strategy for testing the crucial hypothesis that the LCRF Layer 2 v1.1 formalism (non-linear Dirac-Hestenes GA field) can support the emergence of stable, particle-like solutions. The existence of such solitons is a necessary condition for the framework's viability. The results of these simulations will determine whether this specific mathematical implementation of Layer 2 is a fruitful path forward or if a different formalism or a revision of Layer 1/0 concepts is required.