# IO Formalism v3.0 Design: Continuous-State Network with Global Coupling ## 1. Objective and Context Following the falsification of the IO v2.x discrete-state formalism [[releases/archive/Information Ontology 1/0138_IO_Simulation_Batch1_Analysis]] and the pivot recommendation [[CEE-G-IOv2_PostMortem]], this node defines the design for the **Information Dynamics (IO) Formalism v3.0**. This version implements the conceptual model of a **continuous-state network with global field coupling** outlined in [[CEE-Sprint5-Design]], aiming to overcome the limitations of previous models by incorporating richer state representations and non-local influence. ## 2. Core Concepts * **Substrate:** A network (graph) `G=(V, E)` where nodes `i ∈ V` represent informational loci. * **State:** Each node `i` has a continuous state variable `φ(i, t) ∈ ℝ`. * **Dynamics:** Governed by a system of coupled ordinary differential equations (ODEs) or partial differential equations (PDEs if spatial derivatives are emphasized) describing `dφ/dt`. * **Influences:** The dynamics incorporate: * Intrinsic node dynamics (e.g., damping, self-interaction). * Local interactions with neighbors `j ∈ Neighbors(i)` (representing K/M/CA). * Global coupling via a field `Φ(t)` derived from the average network state (representing non-local influence). * Stochastic noise (representing Η). * Stability influence via a dynamic `Θ_val(i, t)` variable. * **Emergence:** Spacetime, particles, etc., are hypothesized to emerge as patterns and structures within the `φ` field governed by these dynamics. ## 3. Formal Specification (v3.0) ### 3.1. State Variables per Node `i` at time `t` 1. **Primary State (`ε` analogue):** `φ(i, t) ∈ ℝ` 2. **Stability State (`Θ` aspect):** `Θ_val(i, t) ∈ ℝ, Θ_val ≥ 0` *(Note: Potentiality κ is now implicitly encoded in the continuous nature of φ and the stochastic dynamics, rather than explicitly via P_target. The system always has the potential to evolve to nearby continuous states).* ### 3.2. Network Structure * **Topology:** Initially, a fixed 1D ring or 2D square lattice with nearest-neighbor connections defined by an adjacency matrix `J_{ij}` (where `J_{ij}=1` if `i, j` are neighbors, 0 otherwise). Dynamic CA weights are deferred for simplicity in v3.0. ### 3.3. Global Field * **Definition:** `Φ(t) = (1 / N) * Σ_{j ∈ V} φ(j, t)` (Mean-field coupling). ### 3.4. Dynamic Equations We propose the following coupled ODE system: 1. **State Evolution (`dφ/dt`):** ``` dφ(i, t)/dt = -μ * φ(i, t) # Intrinsic Damping (Tendency towards 0) + g * Σ_{j} J_{ij} * (φ(j, t) - φ(i, t)) # Local Interaction (Diffusion/Alignment - K/M) + λ * (Φ(t) - φ(i, t)) # Global Coupling (Tendency towards average - M analogue) - β * tanh(Θ_val(i, t)) * φ(i, t) # Stability Influence (Θ - saturating effect) + σ * η(i, t) # Stochastic Noise (Η) ``` * **Terms:** * `-μ * φ`: Damping towards `φ=0`. * `g * Σ J (φ_j - φ_i)`: Local diffusion/alignment term (Laplacian for lattice). Drives local smoothing (M) due to local differences (K). * `λ * (Φ - φ_i)`: Global coupling term, pulling node towards network average `Φ`. Strength `λ`. * `- β * tanh(Θ_val) * φ`: Stability term. High `Θ_val` increases resistance to deviation from `φ=0` (or current state if equation modified). `tanh` provides saturation. Strength `β`. * `σ * η`: Noise term representing Η. `η` is typically Gaussian white noise `N(0,1)`. Strength `σ`. 2. **Stability Evolution (`dΘ_val/dt`):** ``` dΘ_val(i, t)/dt = a # Baseline stability increase - b * |dφ(i, t)/dt| # Reduction due to change rate - c * Θ_val(i, t) # Passive decay ``` * Ensuring `Θ_val` remains non-negative might require `max(0, ...)` or adjusting the equation. * Parameters `a, b, c` control Θ dynamics. ### 3.5. Parameters * `μ`: Damping rate. * `g`: Local interaction strength. * `λ`: Global coupling strength. * `β`: Stability influence strength. * `σ`: Noise amplitude (Η strength). * `a, b, c`: Theta dynamics parameters. * `N`: System size. * Network topology (`J_{ij}`). ## 4. Justification and Connection to IO Principles * **κ/ε:** `φ` represents the continuous field where actuality manifests. Potentiality κ is implicit in the continuous state space and the stochastic/dynamic possibilities for `φ`'s evolution. The κ → ε transition is the continuous evolution `dφ/dt`. * **K/M (Local):** The `g * Σ J (φ_j - φ_i)` term models local interactions driven by difference (K) leading to alignment/smoothing (M). * **M (Global):** The `λ * (Φ - φ_i)` term models a global alignment tendency. * **Θ:** Explicitly modeled by `Θ_val` influencing `dφ/dt` and evolving based on state stability. * **Η:** Explicitly modeled by the stochastic noise term `σ * η`. * **CA:** Implicit in the network structure `J_{ij}` (fixed nearest-neighbor causality in v3.0). Dynamic CA would require evolving `J_{ij}` or adding weights `w_{ij}`. * **S:** Represented by the continuous time variable `t`. ## 5. Implementation and Simulation Plan * **Implementation:** Use numerical ODE solvers (e.g., `scipy.integrate.solve_ivp` or custom Runge-Kutta) to integrate the coupled system for `φ` and `Θ_val`. * **Initial Conditions:** Typically random `φ` values and zero `Θ_val`. * **Boundary Conditions:** Periodic for lattice simulations. * **Analysis:** Use metrics from [[0128_IO_Metrics_Implementation]] (adapted for continuous `φ`), spatial/temporal correlation functions, spectral analysis (FFT), and visualization. * **Goals:** Search parameter space for regimes exhibiting: * Pattern formation (spatial structures, waves). * Stable localized structures ("particle analogues"). * Phase transitions. * Complex dynamics ("edge of chaos"). ## 6. Conclusion: A New Formal Basis for Exploration IO Formalism v3.0 represents a significant pivot, moving to a continuous-state network model with global coupling. This design directly addresses the limitations of the v2.x discrete models by allowing for richer state representations and incorporating non-local influence via the mean field `Φ(t)`. While still requiring significant exploration and refinement (especially the specific functional forms), this provides a more promising formal foundation for investigating the emergence of complex, potentially physics-like phenomena from the core IO principles. The next step is to implement this design in code ([[releases/archive/Information Ontology 1/0140_IO_Simulation_Code_v3.0]]) and begin computational exploration.