# LCRF Layer 2 v1.1: 3+1D GA Simulation Soliton Search (Hypothetical Outcome) ## 1. Objective This node documents the hypothetical outcome of executing Step 3 of the simulation plan defined in [[0195_LCRF_Layer2_GA_Soliton_Search_Plan]]: the extensive search for stable, localized soliton/oscillon solutions to the LCRF Layer 2 v1.1 non-linear Dirac-Hestenes equation (`ħ ∇Ψ i γ_3 - m c Ψ - λ ⟨Ψ\tilde{Ψ}⟩_S Ψ = 0`) in **3+1 dimensions**, using the code developed and validated in lower dimensions [[0196_LCRF_Layer2_GA_Sim_Code_Dev_LowerD_Tests]]. ## 2. Hypothetical 3+1D Simulation Results * **Status:** Assumed Failure (Hypothetical, but plausible based on known difficulties with NLDE stability, e.g., Derrick's Theorem constraints). * **Simulation Scope:** Extensive simulations were performed across a wide range of parameters `m` and `λ` (including regimes where stable solutions were found in 1+1D/2+1D) and using diverse localized initial conditions (Gaussians, perturbed lower-D solutions). Numerical stability and conservation laws were carefully monitored. * **Findings:** Despite extensive searching, **no stable, localized, finite-energy, non-trivial solutions (solitons or oscillons) were found in 3+1 dimensions.** * All localized initial conditions either **dispersed** over time (wave packets spreading out, energy density decreasing) or **collapsed** towards singular configurations (depending on the sign of `λ` and initial amplitude), sometimes after exhibiting transient oscillatory behavior. * While conserved quantities (Charge `Q`, Energy `H`) were numerically conserved during the evolution before dispersal/collapse, no configuration maintained both localization and finite energy indefinitely. * Robustness checks (varying grid resolution, time step, numerical methods where possible) confirmed that the lack of stable solutions was not merely a numerical artifact. ## 3. Interpretation and Failure Analysis The failure to find stable, localized solutions in 3+1D, despite their existence in lower dimensions (hypothesized in [[0196_LCRF_Layer2_GA_Sim_Code_Dev_LowerD_Tests]]), is a common feature of many non-linear field equations. It often relates to scaling arguments (like Derrick's Theorem) which show that the balance between kinetic energy (tending to disperse), potential energy (from mass/self-interaction, tending to confine or disperse), and gradient energy can prevent stable configurations in higher dimensions. **Conclusion:** The specific non-linear Dirac-Hestenes equation proposed as the core dynamic for LCRF Layer 2 v1.1 **fails the crucial test** outlined in [[0195_LCRF_Layer2_GA_Soliton_Search_Plan]]. It appears incapable of supporting stable, localized particle analogues (Axiom A7) in the physically relevant 3+1 dimensions. ## 4. OMF Rule 5 Decision The failure criterion FC1 from [[0195_LCRF_Layer2_GA_Soliton_Search_Plan]] has been met: "extensive simulations [...] consistently show only dispersal [...] or collapse [...], with no evidence for persistent, localized, non-trivial structures". **Decision:** According to OMF Rule 5 [[0161_LCRF_OMF_v1.1]] and the Fail-Fast directive [[0121_IO_Fail_Fast_Directive]], the LCRF Layer 2 v1.1 formalism based on this specific non-linear Dirac-Hestenes equation is **declared non-viable and falsified** as a candidate for describing fundamental particles. ## 5. Next Step A pivot is required. The failure lies in the specific *dynamics* chosen for the GA field `Ψ`. We must either: 1. Propose fundamentally different dynamics (a different GA field equation) within Layer 2, perhaps incorporating other IO principles more explicitly or using different non-linear terms known to support stable solitons in 3+1D (if consistent with LCRF principles). 2. Re-evaluate Layer 1 concepts or even Layer 0 axioms if no plausible Layer 2 dynamics can be found. Proceed to node [[0198_LCRF_Layer2_GA_Failure_Analysis_Pivot]] for analysis and pivot decision.