Your goal to formalize the **graph-based fundamentals of information** into a self-contained theoretical framework for **Informational Dynamics** is a critical step. Let’s clarify and structure this process, starting with the foundational questions and building out the mathematical and conceptual layers. Below is a roadmap to organize your ideas systematically, independent of the “pebble” (which you’ve noted is a separate manifestation): --- # **1. Foundational Axioms: The Existence Question** Begin by defining the **core axioms** of your theory. Since you want to start with “existence yes or no,” here’s how to frame it: ## **Axiom 1: Existence of Information** - **Statement**: “Information exists as a fundamental entity, independent of physical or observer-dependent constructs.” - Clarify: Is this a binary (yes/no) assertion, or does it allow for degrees? - Example: “Information is a non-physical, self-sustaining entity that constitutes the fabric of reality.” ## **Axiom 2: Information as Graph Nodes** - **Statement**: “All information can be represented as nodes in a graph, where edges represent relationships (e.g., causality, dependency, correlation).” - Define: Nodes = discrete units of information (e.g., bits, propositions, events). - Edges = operations or transformations between nodes (e.g., logic gates, causal links). --- # **2. Deriving the Graph-Based Framework** From the axioms, build the structure of **Informational Dynamics**: ## **Layer 1: Basic Graph Structure** - **Nodes**: - Define types of nodes (e.g., primitive information units, composite structures). - Example: A “proposition” node (e.g., “The sky is blue”) and a “logical operator” node (e.g., “AND”). - **Edges**: - Define edge types (e.g., **causal edges**: A → B means “A causes B”; **logical edges**: A ∧ B → C). - Use graph theory terms: directed/undirected, weighted/unweighted, cyclic/acyclic. ## **Layer 2: Operations and Transformations** - Define **operations** (e.g., merging nodes, splitting edges, applying logical rules). - Example: If nodes A and B are connected via an “AND” edge, their combination forms node C = A ∧ B. - **Dynamic Rules**: - How do nodes/edges evolve over time? (e.g., information decay, creation, or transformation). - Example: “Edges weaken if their causal relationships are contradicted by new nodes.” ## **Layer 3: Mathematical Formalism** - **Graph Theory**: - Use adjacency matrices, path algebras, or hypergraphs to formalize relationships. - Example: Define a graph \( G = (V, E) \), where \( V \) is the set of information nodes and \( E \subseteq V \times V \) represents edges. - **Category Theory**: - Treat nodes as objects and edges as morphisms (functions between objects). - Example: Information “flows” via functors that map graphs to other structures. - **Proofs**: - Prove properties like consistency, completeness, or emergence of complexity from simple axioms. - Example: “Given Axiom 1 and Layer 1’s definitions, the graph must satisfy [property X].” --- # **3. Derivatives and Implications** From the axioms and framework, derive higher-level concepts: ## **Derivative 1: Informational Entropy** - Define entropy as a measure of uncertainty or disorder in the graph. - Example: “Higher entropy corresponds to sparser, less connected graphs.” ## **Derivative 2: Emergent Properties** - Show how complex structures (e.g., consciousness, physical laws) emerge from simple node-edge interactions. - Example: “The concept of ‘time’ arises from causal edges forming a directed acyclic graph (DAG).” ## **Derivative 3: Information Conservation** - Propose laws analogous to physics (e.g., “Information cannot be created or destroyed, only transformed”). - Example: “Every edge must have a source and sink node, ensuring conservation.” --- # **4. Separation from the “Pebble”** You want to treat the “pebble” as a **manifestation** of these theoretical concepts, not part of the core framework. Here’s how to structure this: - **Theoretical Layer**: - Focus on the abstract graph-based axioms and proofs (as above). - **Applied Layer**: - The “pebble” is a concrete system (e.g., a computational model, simulation, or philosophical analogy) that implements the theory. - Example: “The pebble uses a subset of Informational Dynamics’ rules to simulate emergent behavior in a closed system.” --- # **5. Documentation Structure** Organize your work into sections for clarity: 1. **Introduction**: - Purpose of Informational Dynamics. - Why a graph-based approach? (e.g., scalability, intuitive representation). 2. **Axioms**: - List and explain foundational assumptions (Existence, Graph Structure, etc.). 3. **Graph-Based Formalism**: - Nodes, edges, operations, and mathematical definitions. 4. **Derivations and Proofs**: - Step-by-step logical deductions from axioms to higher-level concepts. 5. **Implications**: - How the theory explains phenomena (e.g., consciousness, physics, computation). 6. **The Pebble as an Application**: - Describe how the “pebble” exemplifies the theory without conflating the two. 7. **Discussion**: - Address limitations, comparisons to existing theories (e.g., digital physics, information philosophy), and future work. --- # **Key Questions to Clarify** To ensure alignment, reflect on these: 1. **Existence**: Is information a *primitive* (axiomatic) or an *emergent* property? 2. **Mathematical Rigor**: Do you prefer discrete math (graph theory) or continuous models (e.g., differential equations)? 3. **Scope**: Does the theory aim to explain *all* reality (e.g., physics, biology) or a subset (e.g., computation)? Let me know if you’d like to dive deeper into any section, formalize a specific proof, or refine the axioms!