# Incorporating Contrast (K) and Mimicry (M) into the Formal State Transition ## 1. Objective: Adding Relational Dynamics Node [[releases/archive/Information Ontology 1/0095_IO_State_Formalism]] established a formal state representation `State(i, S) = { ε(i, S), (p_flip(i, S), Θ_val(i, S)) }` and a transition rule driven by internal potential (`p_flip`), global entropy (`h`), and internal stability (`Θ_val`). However, this model lacks the crucial relational aspects of Information Dynamics (IO): **Contrast (K)** [[releases/archive/Information Ontology 1/0003_Define_Contrast_K]] enabling interaction, and **Mimicry (M)** [[releases/archive/Information Ontology 1/0007_Define_Mimicry_M]] biasing transitions towards alignment. This node extends the formalism by incorporating K and M into the probabilistic transition rule, making state changes dependent on the local neighborhood context. ## 2. Incorporating Contrast (K): Modulating Interaction Probability Contrast (K) measures the potential difference between states, enabling interaction [[releases/archive/Information Ontology 1/0073_IO_Contrast_Mechanisms]]. We can model this by making the probability of an Η-triggered change *also* dependent on the local contrast. High contrast facilitates change; low contrast inhibits it (relative to the Η drive). * **Local Contrast Measure:** Define a local contrast measure `K_local(i, S)` for node `i` based on its difference from neighbors. For binary states `ε ∈ {0, 1}` and neighbors `j ∈ Neighbors(i)`: `K_local(i, S) = (1 / |Neighbors(i)|) * Σ_{j ∈ Neighbors(i)} |ε(i, S) - ε(j, S)|` This gives a value between 0 (all neighbors same as `i`) and 1 (all neighbors different from `i`). * **Modified Change Probability:** Modify the probability `P_change` from [[releases/archive/Information Ontology 1/0095_IO_State_Formalism]] to include `K_local`. `P_change(i, S) = f_H(h, p_flip(i, S)) * f_Θ(Θ_val(i, S)) * f_K(K_local(i, S))` * `f_K(K_local)`: Function representing K's influence, increasing from 0 towards 1 as `K_local` increases. E.g., `f_K = K_local` (linear dependence) or `f_K = (1 + γ * K_local)` normalized appropriately, or perhaps a threshold function `f_K = 1 if K_local > K_min else 0`. * **Interpretation:** Change is most likely when driven by Η (`f_H` high), resisted weakly by Θ (`f_Θ` high), AND enabled by sufficient local difference (`f_K` high). Low contrast suppresses the likelihood of change, even if Η and Θ allow it. ## 3. Incorporating Mimicry (M): Biasing the Target State Mimicry (M) represents the tendency towards alignment or pattern replication [[releases/archive/Information Ontology 1/0070_IO_Mimicry_Mechanisms]]. We model this by biasing the *target state* of a change towards the state of the neighbors, similar to [[releases/archive/Information Ontology 1/0087_Formalizing_Mimicry]]. * **Neighborhood Influence:** When a state change *does* occur (i.e., `r < P_change(i, S)` in Step 2 of [[0095]]), the new state `ε(i, S+1)` is chosen probabilistically based on neighbor states. * **Target State Probability:** Let `N_0` be the number of neighbors in state 0, and `N_1` be the number of neighbors in state 1. Let `N_total = N_0 + N_1`. * Probability of transitioning *to* state 0: `P(target=0 | change occurs) = (N_0 / N_total) * p_M + (1 - p_M) * (1 - ε(i, S))` * Probability of transitioning *to* state 1: `P(target=1 | change occurs) = (N_1 / N_total) * p_M + (1 - p_M) * ε(i, S)` *(Self-correction: This combines M bias with a residual probability (1-p_M) of simply flipping, allowing non-mimetic change. If neighbors are balanced N0=N1, the M term gives 0.5 probability for either target state.)* * `p_M` (Mimicry strength, `0 ≤ p_M ≤ 1`): Controls how strongly the target state is biased by neighbors. `p_M=1` means the target state is purely determined by the majority neighbor state (or random if balanced). `p_M=0` means the target state is always the opposite of the current state (simple flip, no mimicry). * **Updated State Change Rule (Step 2a):** * If change occurs (`r < P_change(i, S)`): 1. Calculate `N_0`, `N_1`. 2. Calculate `P_target_0 = (N_0 / N_total) * p_M + (1 - p_M) * (1 - ε(i, S))` (handle `N_total=0` case if needed). 3. Generate `r3 ~ U(0, 1)`. 4. If `r3 < P_target_0`: `ε(i, S+1) = 0`. 5. Else: `ε(i, S+1) = 1`. ## 4. The Full Transition Rule (Η, Θ, K, M) Combining [[releases/archive/Information Ontology 1/0095_IO_State_Formalism]] and the modifications above: 1. **For each node `i` at step `S`:** 2. **Calculate Inputs:** Determine `ε(i, S)`, `p_flip(i, S)`, `Θ_val(i, S)`. Calculate `K_local(i, S)` based on neighbors `ε(j, S)`. 3. **Calculate Change Probability:** `P_change(i, S) = f_H(h, p_flip(i, S)) * f_Θ(Θ_val(i, S)) * f_K(K_local(i, S))`. 4. **Determine Occurrence:** Generate `r ~ U(0, 1)`. 5. **If `r < P_change(i, S)` (Change Occurs):** a. **Determine Target State (M bias):** Calculate `N_0`, `N_1` from neighbors. Calculate `P_target_0 = ...` using `p_M`. Generate `r3 ~ U(0, 1)`. Set `ε(i, S+1)` to `0` if `r3 < P_target_0`, else `1`. b. **Update Stability/Potential:** Reset/decay `Θ_val(i, S+1)`. Update `p_flip(i, S+1)` based on new `Θ_val`. 6. **Else (No Change Occurs):** a. **Maintain State:** `ε(i, S+1) = ε(i, S)`. b. **Update Stability/Potential:** Increment `Θ_val(i, S+1)`. Update `p_flip(i, S+1)` based on new `Θ_val`. *(Note: Causality CA is implicitly represented by the dependence on the state at step S to determine the state at S+1, propagating through the network structure).* ## 5. Implications and Expected Behavior * **Interaction Gating (K):** Interactions (state changes) are less likely in homogeneous regions (low `K_local`), promoting stability beyond just Θ. Change is favored at boundaries or in diverse regions. * **Alignment/Smoothing (M):** When changes do occur, they tend to align with the local consensus (`p_M > 0`), leading to domain growth, smoothing of boundaries, and pattern replication. * **Rich Dynamics:** The interplay of Η (random drive), Θ (inertia/memory), K (interaction gating), and M (alignment bias) should allow for complex emergent dynamics, including pattern formation, competition, and potentially computation-like behavior, depending on parameter tuning (`h`, `α`, `β`, `K_min`/`γ`, `p_M`). ## 6. Conclusion: Towards a Complete Local Dynamic This node extends the formal IO state transition model to incorporate the relational principles of Contrast (K) and Mimicry (M). K modulates the *probability* of interaction based on local difference, while M biases the *outcome* of interactions towards local similarity. This provides a more complete picture of the local dynamics governed by the interplay of all five core principles (Η, Θ, K, M, implicit CA). While still simplified (binary states, specific functional forms), this formalism offers a concrete basis for simulation and analysis, allowing exploration of how complex, self-organizing patterns might emerge from these fundamental informational rules, paving the way for more sophisticated models.