# IO Phase v1.0: Numerical Simulation Specification for Eq. IO-2' ## 1. Introduction and Context Phases v0.1-v0.4 of the Information Ontology (IO) exploration established a conceptual framework based on a continuous Geometric Algebra (GA) multivector field $\mathbf{\Psi} \in \mathcal{G}(1,3)$ governed by intrinsic dynamics. Analytical investigation identified the first-order, non-linear equation **Eq. IO-2': $\nabla \mathbf{\Psi} + (m_0 + \lambda \langle \tilde{\mathbf{\Psi}}\mathbf{\Psi} \rangle_0) \mathbf{\Psi} = 0$** as the most promising candidate, possessing theoretical stability (via Hamiltonian analysis [[v0.4_Report_Refined]]) and the necessary structure (GA, $U(1)_I$ symmetry) to potentially support emergent particles with mass, spin (S=0, S=1/2), and charge. However, analytical methods proved insufficient to determine if Eq. IO-2' actually admits the required stable, localized, non-perturbative solutions (oscillons/solitons) [[v0.4_Report_Refined]]. The existence of such solutions is the **critical, unverified hypothesis** upon which the viability of this IO implementation rests. Therefore, Phase v1.0 is dedicated to specifying the **numerical simulation required to definitively test this hypothesis**. This specification outlines the objectives, methodology, parameters, analysis, and success criteria for this crucial computational investigation. ## 2. Simulation Objectives The primary objectives of the numerical simulation are: 1. **Existence Verification:** Determine if Eq. IO-2' admits stable or long-lived, localized, non-dispersive solutions in 3+1 dimensions for physically reasonable parameter ranges ($m_0 < 0, \lambda > 0$). 2. **Solution Characterization:** If such solutions exist, characterize their fundamental properties: * **Mass (M):** Calculated from the integrated Hamiltonian density $\mathcal{H}$ [[v0.4_Report_Refined]]. * **Spin (S):** Determined by analyzing the solution's transformation under rotations and the structure of its angular momentum tensor, specifically looking for S=0 (scalar/pseudoscalar dominant) and S=1/2 (bivector dominant) characteristics. * **Charge (Q):** Calculated from the conserved Noether charge associated with the $U(1)_I$ symmetry. * **Stability:** Assessed by long-term evolution, monitoring energy conservation and spatial localization. * **Internal Structure:** Analyze the multivector composition (scalar, vector, bivector, pseudoscalar parts) and any internal oscillatory frequencies. 3. **Spectrum Mapping:** Explore how solution properties (existence, stability, M, S, Q) depend on the input parameters $m_0$ and $\lambda$. Identify if a discrete spectrum of stable states emerges. ## 3. Numerical Methodology * **Target Equation:** Eq. IO-2', expanded using the GA basis: $\gamma^\mu \partial_\mu \mathbf{\Psi} + (m_0 + \lambda \langle \tilde{\mathbf{\Psi}}\mathbf{\Psi} \rangle_0) \mathbf{\Psi} = 0$. * **Framework:** Geometric Algebra $\mathcal{G}(1,3)$. The field $\mathbf{\Psi}$ has 16 real components. * **Dimensionality:** 3 spatial dimensions + 1 time dimension (3+1D). * **Discretization:** * **Space:** Finite-difference methods (e.g., centered differences of appropriate order) on a 3D Cartesian grid with sufficient resolution to capture expected solution scales (related to $1/|m_0|$). Boundary conditions: Periodic or absorbing, depending on the test case. * **Time:** A stable, explicit or implicit time-stepping algorithm suitable for first-order hyperbolic PDEs (e.g., Runge-Kutta methods like RK4, potentially Leapfrog or Crank-Nicolson if adapted for GA structure and non-linearity). Ensure numerical stability (CFL condition for explicit methods). * **Implementation:** Requires high-performance computing resources due to the 16 components and 3D grid. Code development likely needed in C++, Fortran, or potentially GPU-accelerated Python (e.g., using CuPy/JAX with custom GA kernels). Libraries for GA operations might need development or adaptation. ## 4. Simulation Parameters and Initial Conditions * **Parameters:** Systematically vary the fundamental parameters $m_0$ (negative values) and $\lambda$ (positive values) across orders of magnitude relevant to potential particle scales. Set $c_0=1$ initially. * **Initial Conditions (ICs):** Explore a diverse range of localized ICs designed to potentially seed stable solutions: * Gaussian profiles with varying widths and amplitudes. * ICs with specific multivector structures (e.g., pure scalar+pseudoscalar, pure even subalgebra, configurations with initial bivector "spin"). * Boosted versions of potentially stable 1D analytical solutions (if any can be found). * Perturbed versions of known analytical solutions to related NLDEs. * Random localized fluctuations. ## 5. Analysis Techniques * **Conservation Laws:** Monitor numerically conserved quantities (Energy H, Momentum P, Charge Q, Angular Momentum J) to verify code accuracy and stability. * **Stability Assessment:** Track the peak amplitude, spatial extent (e.g., RMS width), and integrated energy (Mass M) of localized structures over long simulation times. Stable solutions should maintain localization and conserved quantities. * **Property Extraction:** * **Mass (M):** Integrate the Hamiltonian density $\mathcal{H}$ over the simulation volume. * **Spin (S):** Calculate the angular momentum density and integrate. Analyze the transformation properties of the solution under numerical rotations or decompose into irreducible representations of the rotation group. Identify dominant multivector components (scalar/pseudoscalar vs. bivector). * **Charge (Q):** Integrate the conserved $U(1)_I$ charge density. * **Internal Frequency:** Perform Fourier analysis (FFT) of field components at specific points within the localized structure to identify dominant internal oscillation frequencies ($\omega$). ## 6. Success and Failure Criteria (OMF Rule 5 Applied) This simulation phase constitutes a critical test under OMF Rule 5. * **Success Criterion (Compelling Validation):** The simulation must robustly demonstrate the existence of **at least one stable (or extremely long-lived), localized, non-perturbative solution with non-zero mass (M>0) AND properties consistent with either S=0 or S=1/2**. Finding *both* S=0 and S=1/2 type solutions would be a major success. The existence must be verified across a reasonable parameter range and for different ICs, demonstrating robustness. * **Failure Criterion (Falsification):** If, after extensive simulation across relevant parameter space and diverse ICs, **no stable, localized, non-trivial solutions are found** (all initial configurations disperse or collapse), OR if the only stable solutions found have properties fundamentally incompatible with basic particle physics (e.g., all solutions are massless, or only S=0 states exist despite seeding S=1/2 structures), then **Eq. IO-2' and the IO v1.0 framework built upon it will be considered falsified.** ## 7. Resource Considerations Executing these simulations represents a significant computational undertaking, likely requiring access to high-performance computing clusters and specialized code development expertise in numerical relativity, computational physics, and potentially GA libraries. This specification serves as the blueprint for such an effort, which likely requires external resources beyond the current conceptual exploration phase. ## 8. Conclusion: The Decisive Computational Test Phase v1.0 transitions the IO framework from analytical/conceptual exploration to computational validation. The numerical simulation specified here is designed to directly test the central, unverified hypothesis: the existence of stable, particle-like solutions emerging from the preferred dynamics (Eq. IO-2') within the GA framework. The outcome of this simulation, whether success, partial success, or failure according to the defined criteria, will be decisive for the future of this specific implementation of Information Ontology. It addresses the primary weakness identified in previous phases and adheres to the rigorous validation requirements of the OMF.