# LCRF Layer 2 v1.3B: Simulation Plan for Coupled Ψ-Θ GA Field ## 1. Objective This node outlines the simulation plan to test the LCRF Layer 2 formalism v1.3B defined in [[0203_LCRF_Layer2_GA_Option1B_Formalism]]. The primary goal is to determine if explicitly coupling the GA multivector field `Ψ` to a dynamic stability field `Θ(x, t)` leads to the emergence of **stable, localized, finite-energy solutions (solitons)** in 3+1 dimensions, potentially succeeding where the v1.1 formalism failed [[0197_LCRF_Layer2_GA_Sim_3D_Search]]. ## 2. Numerical Methodology * **Target Equations:** The coupled system for `Ψ` and `Θ`: 1. `ħ ∇Ψ i γ_3 - m c Ψ - λ ⟨Ψ\tilde{Ψ}⟩_S Ψ - β' Θ Ψ = 0` 2. `∂_t Θ = a - b * ||∂_t Ψ|| - c * Θ` (ensuring `Θ ≥ 0`) * **Framework:** GA $\mathcal{G}(1,3)$ for `Ψ`, scalar field for `Θ`. * **Dimensionality:** Target 3+1D, with initial tests in 1+1D / 2+1D. * **Discretization:** Finite-difference methods for spatial derivatives, stable time-stepping (e.g., Runge-Kutta) suitable for the coupled system. Need appropriate GA norm `||...||` for `||∂_t Ψ||`. * **Implementation:** Requires solving a coupled system involving GA multivectors and scalars. High-performance computing likely needed for 3+1D. ## 3. Simulation Parameters and Initial Conditions * **Parameters:** `m`, `λ` (from v1.1), plus new parameters `β'`, `a`, `b`, `c` governing `Θ` dynamics and coupling. Explore ranges for these new parameters, particularly the coupling `β'` and the relative rates `a, b, c`. * **Initial Conditions:** * `Ψ(x, 0)`: Localized configurations (e.g., Gaussian packets) as in [[0195_LCRF_Layer2_GA_Soliton_Search_Plan]]. * `Θ(x, 0)`: Typically start with low initial stability, e.g., `Θ(x, 0) = 0` or small uniform value. ## 4. Analysis Techniques * **Conservation Laws:** Monitor Energy, Charge `Q`, etc. Note that energy conservation might be more complex due to the postulated, potentially non-Lagrangian nature of the `Θ` equation. Check consistency. * **Stability Assessment:** Track localization of energy density (`Ψ` part), peak amplitude/width of `Ψ`, and evolution of `Θ` field. Stable solutions should show persistent localization of `Ψ` accompanied by a corresponding stable, localized high-value region in the `Θ` field. * **Solution Characterization:** Analyze mass, charge, spin characteristics of `Ψ` component, and the profile/dynamics of the associated `Θ` field for any stable structures found. ## 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`-`Θ` solutions in 3+1D. The `Ψ` component should resemble a particle analogue, stabilized by a corresponding localized high-`Θ` region. * **Failure Criterion (Triggering STOP/Re-Pivot for Option 1B):** If simulations consistently show only dispersal/collapse of `Ψ` (similar to v1.1), or if the `Θ` field dynamics do not lead to effective stabilization, or if numerical instabilities prevent reliable results despite reasonable effort. ## 6. Feasibility Assessment * **Moderate-High Risk:** Introduces a new field and coupling, increasing complexity. The non-Lagrangian nature of the `Θ` equation might raise consistency issues (e.g., with energy conservation) that need careful handling. Numerical solution of the coupled system is challenging. However, it directly implements an IO principle (Θ) intended to provide stability. ## 7. Conclusion This plan outlines the simulation strategy for testing Option 1B. By explicitly modeling the stability principle Θ as a dynamic field coupled to `Ψ`, this approach directly tests if IO's conceptual stabilization mechanism can operate within the GA formalism to produce particle analogues. The outcome will determine the viability of this specific path.