# Formalizing State Representation and Probabilistic Transitions in IO ## 1. Objective: Moving Beyond Toy Models Following the refinement strategy [[releases/archive/Information Ontology 1/0094_IO_Refinement_Strategy_v1.1]], which prioritizes formalization, we move beyond the simple cellular automata rules of previous toy models ([[releases/archive/Information Ontology 1/0086_Formalizing_Theta]], [[releases/archive/Information Ontology 1/0087_Formalizing_Mimicry]]). This node attempts a more structured formalization of: 1. The **state** of an informational locus, capturing both its actualized (ε) and potential (κ) aspects. 2. The **probabilistic transition rule (κ → ε)**, explicitly incorporating the influences of Entropy (Η) and Theta (Θ). This aims to provide a more robust foundation for quantitative modeling (Pillar 1 of the strategy). ## 2. Proposed State Representation Let the state of a fundamental informational locus `i` at sequence step `S` be represented by a structure `State(i, S)` containing at least two components: * **Actualized Component `ε(i, S)`:** Represents the definite, currently manifest state. For simplicity, let this still be binary `ε ∈ {0, 1}` initially, but it could be generalized to vectors or other structures. * **Potentiality/Stability Component `κΘ(i, S)`:** This component needs to encode both the potential for change (related to κ) and the resistance to change (related to Θ). We could represent this as a tuple or vector, for instance: * `κΘ(i, S) = (p_flip(i, S), Θ_val(i, S))` * `p_flip(i, S)`: The intrinsic potential or probability for this locus to transition *away* from its current `ε(i, S)` state in the next step, *before* considering stabilization. This captures the latent aspect of κ related to change. Higher `p_flip` means higher potential instability. * `Θ_val(i, S)`: The accumulated stability or "Theta strength" associated with the *current* `ε(i, S)` state, as developed in [[releases/archive/Information Ontology 1/0086_Formalizing_Theta]]. Higher `Θ_val` means more resistance to change. **Full State:** `State(i, S) = { ε(i, S), κΘ(i, S) } = { ε(i, S), (p_flip(i, S), Θ_val(i, S)) }` *(Note: This is a simplified representation. A full κ representation [[releases/archive/Information Ontology 1/0041_Formalizing_Kappa]] would likely need to encode probabilities for transitioning to *specific* alternative states, not just flipping, and incorporate relational potential K. This model focuses only on the Η vs. Θ dynamic for changing the current ε state).* ## 3. Formalizing the Probabilistic Transition (κ → ε influenced by Η, Θ) The transition `State(i, S) → State(i, S+1)` involves updating both `ε` and `κΘ`. **Step 1: Η Trigger & Θ Resistance (Determine if change *occurs*)** * The principle Η [[releases/archive/Information Ontology 1/0011_Define_Entropy_H]], [[releases/archive/Information Ontology 1/0071_IO_Entropy_Mechanisms]] represents the drive to explore possibilities, essentially triggering potential changes. * The principle Θ [[releases/archive/Information Ontology 1/0015_Define_Repetition_Theta]], [[releases/archive/Information Ontology 1/0069_IO_Theta_Mechanisms]] represents resistance to change based on past stability. * We combine the intrinsic potential `p_flip` with the stability `Θ_val` and a global Η influence parameter `h` (representing the overall "agitation" or exploratory drive in the system) to find the actual probability of the state `ε(i, S)` changing. * Let the probability of *actually changing* state `ε` be `P_change(i, S)`: `P_change(i, S) = f_H(h, p_flip(i, S)) * f_Θ(Θ_val(i, S))` * `f_H(h, p_flip)`: Function representing the Η-driven tendency to change, increasing with global drive `h` and intrinsic potential `p_flip`. E.g., `f_H = min(1, h * p_flip)`. * `f_Θ(Θ_val)`: Function representing Θ resistance, decreasing from 1 to 0 as `Θ_val` increases. E.g., `f_Θ = 1 / (1 + α * Θ_val)` as in [[releases/archive/Information Ontology 1/0086_Formalizing_Theta]]. * This combines the intrinsic potential (`p_flip` from κΘ) with the global drive (Η parameter `h`) and the accumulated resistance (Θ parameter `Θ_val` from κΘ). **Step 2: Update ε State** * Generate random number `r ~ U(0, 1)`. * If `r < P_change(i, S)`: * Change occurs: `ε(i, S+1) = 1 - ε(i, S)` (for binary case). * Else: * No change: `ε(i, S+1) = ε(i, S)`. **Step 3: Update κΘ Component** The potential `p_flip` and stability `Θ_val` must also evolve. * **Updating `Θ_val`:** * If change occurred (`ε` flipped): Reset stability: `Θ_val(i, S+1) = Θ_base`. * If no change occurred: Increase stability: `Θ_val(i, S+1) = min(Θ_val(i, S) + ΔΘ_inc, Θ_max)`. (As in [[0086]]). * **Updating `p_flip` (More Speculative):** How does the intrinsic potential to flip evolve? * *Option A (Static):* `p_flip` is a fixed property of the locus. * *Option B (Adaptive):* `p_flip` might decrease as `Θ_val` increases (more stable states have less potential to flip). E.g., `p_flip(i, S+1) = p_base * g(Θ_val(i, S+1))`, where `g` decreases with `Θ_val`. * *Option C (Contextual):* `p_flip` might be influenced by neighbors (related to K and M), representing relational potential. (Requires incorporating K/M). * Let's tentatively use Option B for internal consistency: `p_flip(i, S+1) = p_base / (1 + β * Θ_val(i, S+1))` (where `p_base` is baseline potential, β controls sensitivity). **Full Transition:** `State(i, S+1) = { ε(i, S+1), (p_flip(i, S+1), Θ_val(i, S+1)) }` determined by the probabilistic rules above. ## 4. Advantages of this Formalism * **Integrates κ/ε/Θ:** Represents actual state, potential for change, and stability within one structure. * **Explicit Probabilities:** Defines the probability of change based on intrinsic potential (`p_flip`), global drive (`h`), and history (`Θ_val`). * **Dynamic Stability:** Allows stability (`Θ_val`) and potential (`p_flip`) to co-evolve. * **Foundation for K/M:** Provides a state structure onto which relational potentials (K) and alignment biases (M) can be added by making `p_flip` or the *target* state of a flip depend on neighbor states. ## 5. Limitations and Next Steps * **Simplification:** Still uses binary ε, ignores relational K/M, simplifies κ representation. * **Parameter Dependence:** Introduces parameters (`h`, `α`, `β`, `Θ_base`, `ΔΘ_inc`, `Θ_max`, `p_base`) whose values need justification or fitting. * **Mechanism Choice:** Specific functions (`f_H`, `f_Θ`, `g`) are illustrative examples. * **Next Steps:** * Implement this state structure and transition rule in a simulation (e.g., extending [[0086]]/[[0087]]). * Explore parameter space to find regimes of stability and complexity. * Generalize ε to non-binary states. * Incorporate K and M by making transition probabilities/targets depend on neighbor states/contrasts. * Develop a more complete representation of κ potential [[releases/archive/Information Ontology 1/0041_Formalizing_Kappa]]. ## 6. Conclusion: Towards Quantitative IO Dynamics This node proposes a concrete step towards formalizing Information Dynamics by defining a state structure that integrates actuality (ε), potentiality-for-change (κ aspect `p_flip`), and stability (Θ aspect `Θ_val`). It outlines a probabilistic transition rule explicitly driven by Entropy (Η parameter `h`) and resisted by historical stability (Θ variable `Θ_val`). While still simplified, this formalism provides a more rigorous foundation than previous toy models for simulating IO dynamics, quantifying the interplay of core principles, and building towards more complex representations incorporating relational aspects (K, M) and richer potentiality structures (κ). It directly addresses the need for formalism identified in the refinement strategy [[releases/archive/Information Ontology 1/0094_IO_Refinement_Strategy_v1.1]].