# Defining Adjacency and Emergent Locality in Information Dynamics ## 1. The Challenge: Generating Space from Information Standard physical theories often treat space either as a passive background container (Newtonian physics, SR, QM) or as a dynamic geometric field intertwined with matter and energy (GR). The Information Dynamics (IO) framework, aiming for a deeper foundation, cannot presuppose space. Just as time is proposed to emerge from the fundamental Sequence (S) of State Changes (Δi) [[releases/archive/Information Ontology 1/0004_Define_StateChange_Sequence]], **space must also emerge** from the underlying informational network and its dynamics. The challenge is to explain how the seemingly continuous, typically three-dimensional spatial manifold we experience arises from discrete or potentially non-geometric informational interactions. ## 2. Hypothesis: Adjacency from Interaction Potential and Causality The core idea is to define spatial relationships not geometrically, but **informationally**, based on the potential and history of interactions within the network: * **Fundamental Entities:** The network consists of informational nodes or regions existing in Potentiality (κ) and undergoing transitions to Actuality (ε) [[releases/archive/Information Ontology 1/0012_Alternative_Kappa_Epsilon_Ontology]]. * **Defining Adjacency:** Two informational entities (nodes or κ-regions) are considered **adjacent** if they have a **high probability of direct interaction**. This probability is determined by: * **Sufficient Contrast (K):** They must possess potential differences (K, derived from their κ states [[0012]]) that enable interaction [[releases/archive/Information Ontology 1/0003_Define_Contrast_K]]. * **Strong Causal Links (CA):** Past interactions may have established strong causal dependencies ([[releases/archive/Information Ontology 1/0008_Define_Causality_CA]]), making future interactions between them more likely. Pathways reinforced by Theta (Θ) [[releases/archive/Information Ontology 1/0015_Define_Repetition_Theta]] would contribute significantly to adjacency. * **Minimal Intermediaries:** Direct interaction implies that the influence doesn't primarily rely on propagating through a chain of other intermediate nodes. Adjacency is therefore a dynamic, relational property reflecting the strength and directness of potential informational coupling. ## 3. Locality as an Interaction Gradient **Locality** emerges as a consequence of this definition of adjacency. Interactions are fundamentally local in the informational sense: * **Primary Interactions:** Direct interactions (κ → ε events involving two or more entities) occur predominantly between adjacent entities. * **Influence Attenuation:** The probability or strength of influence between two non-adjacent entities decreases rapidly with the "informational distance" – the number of intermediate interaction steps (Δi events along S) required to connect them via causal pathways (CA). What we perceive as spatial locality is hypothesized to be a macroscopic manifestation of this underlying informational locality. Actions primarily affect their immediate informational neighbors. ## 4. Network Structure and Emergent Dimensionality The structure of the perceived spatial dimensions arises from the **connectivity patterns** of the underlying information network: * **Connectivity:** How many other nodes is a typical node "adjacent" to? What is the average "path length" (number of steps) between arbitrary nodes? * **Dimensionality:** The effective dimensionality of the emergent space reflects the scaling relationship between the number of nodes within an "informational radius" and the radius itself. A network structure where the number of nodes grows roughly as R^3 with informational distance R would appear three-dimensional at a macroscopic scale. * **Dynamic Structure:** The network structure is not necessarily static. The creation of new information states (ε), the strengthening or weakening of causal links (CA) via Θ and Η, could lead to an evolving network topology, potentially corresponding to the expansion of space or changes in local curvature. ## 5. Relation to Physical Space and Spacetime * **Emergent Manifold:** The smooth spacetime manifold of physics is viewed as a coarse-grained, statistical approximation of the underlying, potentially discrete or fractal, information network structure and its dynamics (Sequence S). * **Physical Distance:** Measured distances in physical space correspond to the average informational distance (e.g., minimum number of interaction steps, or inverse probability of direct influence) between corresponding regions of the network. * **Speed of Light (c):** The maximum speed for propagating influence (c) reflects the fundamental rate at which State Changes (Δi) can propagate sequentially (S) across adjacent nodes in the network. It's the speed limit inherent in the informational processing itself. * **Gravity as Network Geometry:** The curvature of spacetime in GR might correspond to variations in the density and connectivity patterns of the information network, influencing the pathways of causal influence (CA) – analogous to how mass/energy warps the network structure. ## 6. Challenges and Open Questions * **Formalism:** Developing a precise mathematical or computational formalism for "informational distance," network connectivity, and their relation to emergent dimensionality is a major challenge. Graph theory, network science, and potentially quantum information geometry might provide tools. * **Explaining Observed Properties:** How does this model specifically explain the observed 3+1 dimensions, the local Euclidean nature of space, quantum entanglement's apparent non-locality (perhaps reflecting deeper network connections outside the emergent spatial metric?), and the specific dynamics described by GR? * **Discreteness vs. Continuum:** Is the underlying network fundamentally discrete, or does the κ-potential allow for a form of continuum? How does the smooth manifold of GR emerge? ## 7. Conclusion: Space as Network Connectivity Defining adjacency and locality based on interaction potential (K, CA, Θ) within the IO framework offers a plausible route to understanding space not as a pre-existing container, but as an **emergent property of the information network's dynamic connectivity**. It shifts the ontological foundation from geometry to relational information processing. While significant formal challenges remain, this perspective provides a conceptual basis for integrating the emergence of space alongside time (S) within a unified informational ontology.