# Formal Definition of Ontological Closure (OC) via Tension Minimization for AGE v0.1 **AFKB Artifact ID:** AUTX_A1_AGEv0.1_OC_TensionMin_V1 **Date:** [Actual Date of Generation by Autologos] **Project:** AUTX-A.1: Formal Mathematical and Computational Modeling of the Autaxic Generative Engine **SOW Reference:** AUTX-SOW-P6.2.2-TM2-OC **Based On:** * `_25161103619.md` (Toy Model Proposal - Phase 1, Step 3 OC definitions) * `AUTX-A0-CCD-TM-001.md` (Conceptual Clarification Document, section on OC and Tension) * `AUTX_A1_FormalNotation_AGEv0.1_V1.md` (Formal Notation) * `AUTX_A1_AGEv0.1_Primitives_V1.md` (Formal Primitives) * `AUTX_A1_AGEv0.1_CoreRules_V1.md` (Formal Core Rules) ## 1. Introduction This document provides the formal mathematical definition of "Relational Tension" for the Autaxic Generative Engine (AGE) v0.1, and defines Ontological Closure (OC) in terms of a subgraph being at a local minimum of this tension function with respect to the core algorithm rules. This definition complements the "Fixed Point" definition of OC. ## 2. Relational Tension Function for AGE v0.1 Let `S` be a subgraph of the total graph `G_t = (V_t, E_t)`. For AGE v0.1, we consider the tension of the entire graph `G` at a given step. The Relational Tension `Tension(G)` is defined as a scalar value based on the proto-properties of its Distinctions (D-nodes). ### 2.1. Components of Tension The tension function for AGE v0.1 considers two main sources: 1. **Unsatisfied ProtoValence:** Distinctions desire to form a certain number of relations. Unfulfilled valence contributes to tension. 2. **Frustrated ProtoPolarity:** Non-neutral Distinctions (`PP != 0`) that still have unsatisfied `ProtoValence` represent an unbalanced potential for interaction, contributing to tension. ### 2.2. Formal Definition of `Tension(G)` Let `V(G)` be the set of all Distinction nodes in graph `G`. For each Distinction `D_i ∈ V(G)`: * Let `D_i.PV` be its `ProtoValence`. * Let `D_i.PP` be its `ProtoPolarity`. * Let `num_relations(D_i)` be the current number of relations incident to `D_i`. * The **Unsatisfied Valence** for `D_i` is `UV_i = max(0, D_i.PV - num_relations(D_i))`. The total Relational Tension `Tension(G)` is defined as: **`Tension(G) = w_v * Σ_{D_i ∈ V(G)} (UV_i) + w_p * Σ_{D_i ∈ V(G) where UV_i > 0 AND D_i.PP ≠ 0} (|D_i.PP|)`** Where: * `w_v`: A non-negative weight parameter for the contribution of unsatisfied valence to tension. * `w_p`: A non-negative weight parameter for the contribution of frustrated polarity (on nodes with unsatisfied valence) to tension. * `|D_i.PP|`: Absolute value of `ProtoPolarity`. This definition aligns with the proposal in `AUTX-A0-CCD-TM-001.md` (Sec 3.3). ## 3. Ontological Closure (OC) via Tension Minimization A graph `G` achieves Ontological Closure via Tension Minimization, denoted `is_OC_TM(G)`, if it represents a local minimum of the `Tension(G)` function with respect to the core "Cosmic Algorithm" rules defined for AGE v0.1. ### 3.1. Core Algorithm Rules for AGE v0.1 The relevant core rules from `AUTX_A1_AGEv0.1_CoreRules_V1.md` are: 1. `GenesisRule(p_new_D)` 2. `FormationRule(D_a, D_b)` 3. `AnnihilationRule(R_x)` (which implies sub-rules for relation annihilation or node+relation annihilation based on `inconsistent_pair`) * For AGE v0.1, based on `AUTX-A0-CCD-TM-001`, `AnnihilationRule` can be considered as two sub-types for clarity in the TM check: * `RelationAnnihilationRule(R_x)`: If `R_x(D_a -> D_b)` exists and `inconsistent_polarities(D_a.PP, D_b.PP)` (e.g., same non-zero polarity), then `R_x` is removed. * `NodeAnnihilationRule(D_i)`: If `D_i` has `D_i.PV = 0` and `num_relations(D_i) = 0`, then `D_i` is removed. * *(Note: The original `AnnihilationRule(R)` from `_25161103619.md` implied removal of D nodes too. The CCD suggested this for opposite polarities. If we simplify for v0.1 that the AnnihilationRule (when checking for tension decrease) only acts on relations if nodes have *same* polarity, and a separate NodeAnnihilationRule handles isolated inert nodes, this might be cleaner for Tension Minimization check. Or, the `AnnihilationRule(R)` could be defined to remove D1, D2, R if D1.PP = +1 and D2.PP = -1 as per `AUTX-A0-CCD-TM-001`)* * **Decision for TM Check:** For Tension Minimization check, we consider structural change rules: `FormationRule` and `AnnihilationRule`. `GenesisRule` always *adds* new elements, often increasing tension initially, so it's typically not considered a "tension minimizing" step for an *existing* subgraph, but rather a perturbation. ### 3.2. Formal Definition of `is_OC_TM(G)` A graph `G_t` at simulation step `t` is considered to achieve Ontological Closure via Tension Minimization (`is_OC_TM(G_t) = True`) if and only if: For every possible single application instance `op` of a structural rule (`FormationRule` or `AnnihilationRule`) that is applicable to `G_t`: Let `G'_t = op(G_t)` be the graph state resulting from applying that single rule instance `op` to `G_t`. Then, `Tension(G'_t) ≥ Tension(G_t)`. In other words, no single applicable structural rule can be applied to `G_t` that would result in a graph state with strictly lower tension. ### 3.3. Notes on Definition * This definition focuses on "local" minima with respect to single rule applications. * The set of applicable rule instances must be determined based on the current state `G_t` and the conditions specified in `AUTX_A1_AGEv0.1_CoreRules_V1.md`. * The `GenesisRule` is generally not considered in this check for an existing graph's stability, as its role is to introduce new potential (and often new tension) from S0. ## 4. Relationship to Fixed Point OC * If a graph `G` achieves Fixed Point OC (i.e., `is_OC_FP(G) = True`), meaning no structural rules are applicable, then it trivially satisfies `is_OC_TM(G) = True` because the set of applicable operations `op` that could reduce tension is empty. * However, a graph `G` can satisfy `is_OC_TM(G) = True` without being a Fixed Point if applicable rules exist but their application does not decrease (or might even increase) tension. ## 5. Scope and Future Evolution This definition of Relational Tension and Tension Minimization OC is for AGE v0.1. Future expansions (as per `AUTX-A0-Conceptual-AGE-Expansion-Speculations-V1.md`) may include: * More complex tension functions incorporating other proto-properties (e.g., `TemporalInertia`, `InformationContent`), relation properties, or even topological features of subgraphs. * Differentiated tension types (`ValenceTension`, `PolarityTension`, `InformationTension`, `VacuumTension`). * Localized tension measures for subgraphs or individual entities. * OC definitions based on dynamic attractors or limit cycles within the tension landscape. * Rules that are dynamically selected or prioritized based on tension gradients or types. This initial definition provides a necessary component for implementing and testing OC within the AGE v0.1 simulation.