# Consolidated IO Transition Rule Formalism (v2 - P_target) ## 1. Objective This node consolidates the formal developments from [[releases/archive/Information Ontology 1/0100_IO_Kappa_Refinement]], [[releases/archive/Information Ontology 1/0101_IO_Transition_Rule_v2]], [[releases/archive/Information Ontology 1/0103_IO_P_target_Dynamics]], [[releases/archive/Information Ontology 1/0097_IO_Formal_Causality]], and related nodes into a single, coherent specification for the **Information Dynamics (IO) state representation and transition rule, Version 2**. This version utilizes the Target Probability Vector (`P_target`) to represent Potentiality (κ) more richly and incorporates all core IO principles (Η, Θ, K, M, CA) into the dynamics. This serves as the baseline formalism for simulations planned in [[releases/archive/Information Ontology 1/0102_IO_Simulation_Plan_v2]]. ## 2. System Representation * **Network:** Dynamic directed graph `G = (V, E, W)` * `V`: Set of informational loci (nodes) `i`. * `E`: Set of directed edges `(j → i)` representing potential causal links. * `W`: Set of dynamic edge weights `w(j → i, S) ≥ 0` representing causal strength. * **Node State:** `State(i, S) = { ε(i, S), κΘ(i, S) }` * `ε(i, S)`: Actualized state (from a set of `N_states` discrete possibilities, e.g., `{0, 1}`). * `κΘ(i, S) = (P_target(i, S), Θ_val(i, S))`: Potentiality/Stability component. * `P_target(i, S)`: Target Probability Vector `[p_0(S), p_1(S), ..., p_{N_states-1}(S)]`, where `p_k(S)` is the intrinsic potential probability of transitioning *to* state `k`. `Σ_k p_k = 1`. Represents κ aspect. * `Θ_val(i, S)`: Accumulated stability of the current state `ε(i, S)`. Represents Θ aspect. ## 3. Parameters (Illustrative Set) * `h`: Global Entropy (Η) drive strength. * `α`: Θ resistance sensitivity parameter for `f_Θ`. * `K_min` or `γ`: Contrast (K) gating parameters for `f_K`. * `p_M`: Mimicry (M) strength parameter (bias towards causal input). * `Θ_base`, `ΔΘ_inc`, `Θ_max`: Parameters for `Θ_val` dynamics. * `Δw_inc`, `decay_rate`: Parameters for causal weight `w` dynamics (Θ on CA). * `p_target_reset_mode`, `Δp_inc`, `λ_adapt`, `p_min`: Parameters governing `P_target` dynamics (Mechanism 1, 2, 3 from [[releases/archive/Information Ontology 1/0103_IO_P_target_Dynamics]]). ## 4. Unified State Transition Algorithm (S → S+1) **For each node `i` in parallel or according to an update scheme:** **Phase 1: Calculate Influences at Step S** 1. **Local Contrast (K):** Calculate `K_local(i, S)` based on `ε(i, S)` and immediate neighbors `ε(j, S)`. (Requires metric for state difference if `N_states > 2`). 2. **Causal Input (CA):** Calculate weighted causal inputs for each potential target state `k`: `Influence_k(i, S) = Σ_{j | (j → i) ∈ E, ε(j,S)=k} w(j → i, S)` `Total_Causal_Weight(i, S) = Σ_k Influence_k(i, S)` **Phase 2: Determine Probability of State Change (Η, Θ, K)** 3. **Calculate Intrinsic Leave Potential:** `p_leave = 1 - P_target[ε(i, S)]` (Probability of *not* targeting the current state). 4. **Calculate Change Probability:** `P_change(i, S) = f_H(h, p_leave) * f_Θ(Θ_val(i, S)) * f_K(K_local(i, S))` * *Example Functions:* `f_H = min(1, h * p_leave)`, `f_Θ = 1 / (1 + α * Θ_val)`, `f_K = K_local` (or thresholded). **Phase 3: Execute State Transition (Probabilistic Update)** 5. **Determine Occurrence:** Generate `r ~ U(0, 1)`. 6. **If `r < P_change(i, S)` (Change Occurs):** a. **Determine Target State (M bias via CA on `P_target`):** i. Start with intrinsic `P_target(i, S)`. ii. Calculate modified target probabilities `P'_target = [p'_0, p'_1, ...]` incorporating Mimetic bias `p_M` based on `Influence_k` values (e.g., `p'_k ∝ p_k * (1 + p_M * Influence_k / Total_Causal_Weight)`), then normalize. iii.Sample the target state `ε_target` from the distribution `P'_target`. iv. Set `ε(i, S+1) = ε_target`. b. **Update Stability (`Θ_val`):** Reset `Θ_val(i, S+1) = Θ_base`. c. **Update Potential (`P_target`):** Apply reset rule for `P_target(i, S+1)` (e.g., Mechanism 1b or 2 from [[releases/archive/Information Ontology 1/0103_IO_P_target_Dynamics]]). Apply Η floor (Mechanism 3). d. **Flag CA Reinforcement:** Mark contributing edges `(j → i)` based on correlation between `ε(j, S)` and the transition `ε(i, S) → ε(i, S+1)`. 7. **Else (No Change Occurs):** a. **Maintain State:** `ε(i, S+1) = ε(i, S)`. b. **Update Stability (`Θ_val`):** Increment `Θ_val(i, S+1) = min(Θ_val(i, S) + ΔΘ_inc, Θ_max)`. c. **Update Potential (`P_target`):** Apply stability reinforcement rule for `P_target(i, S+1)` (e.g., Mechanism 1 or 2 from [[0103]]). Apply Η floor (Mechanism 3). d. **Flag CA Reinforcement:** Mark incoming edges `(j → i)` where `ε(j, S) == ε(i, S)`. **Phase 4: Update Causal Network (Θ Influence on CA)** 8. **Update Edge Weights `w(j → i, S+1)`:** Apply Θ reinforcement (`Δw_inc`) or decay (`decay_rate`) to edge weights based on flags set in steps 6d and 7d. **(Repeat for all nodes to complete step S+1)** ## 5. Key Features of the Formalism * **Unified Dynamics:** Integrates Η, Θ, K, M, CA into a single probabilistic rule. * **κ-ε Representation:** Models both actuality (ε) and potentiality/stability (κΘ via `P_target`, `Θ_val`). * **Context Dependence:** Transitions depend on internal state (`P_target`, `Θ_val`), global drive (`h`), local contrast (K), and causal network input (CA influencing M). * **Adaptation/Learning:** Both state stability (`Θ_val`) and causal pathways (`w`) are dynamic and reinforced by Theta (Θ). Potentiality (`P_target`) can also adapt. * **Foundation for Simulation:** Provides a concrete algorithm suitable for computational implementation and exploration [[releases/archive/Information Ontology 1/0102_IO_Simulation_Plan_v2]]. ## 6. Current Limitations and Next Steps * **Parameter Complexity:** Requires careful definition and tuning of multiple parameters and functional forms. * **`P_target` Dynamics:** The rules for updating `P_target` need further refinement and testing [[releases/archive/Information Ontology 1/0103_IO_P_target_Dynamics]]. * **Generalization:** Needs extension beyond binary states and simple network topologies. * **Validation:** Requires implementation and simulation to demonstrate emergence and compare with qualitative IO goals and eventually empirical data. ## 7. Conclusion: A Working Model for IO Simulation This consolidated v2 formalism provides a significant step towards a quantitative Information Dynamics framework. By integrating all core principles acting on a state representation that includes both actuality and a richer potentiality (via `P_target`), it offers a concrete basis for computational simulation aimed at exploring emergent complexity, self-organization, and pattern formation driven by informational dynamics. While further refinement and validation are essential, this unified rule serves as the current operational definition of IO dynamics for simulation and analysis.