### **The Formal Hypothesis of Autaxys (v2.0)** **Preamble:** The Autaxys framework introduces a fundamental reorientation of ontological perspective, moving decisively away from the traditional view (termed "ANWOS" - Assumed Nature of What Is). ANWOS typically posits a universe comprising elementary, independently existing entities (particles, fields, strings, etc.) embedded within a pre-existing spacetime arena, governed by immutable, externally imposed laws derived from observation. In stark contrast, Autaxys proposes a universe fundamentally understood as a dynamic, self-organizing, self-consistent network of attributed relations undergoing continuous computation. This framework seeks to provide a generative, first-principles explanation for the emergence of observed physical reality in its entirety, encompassing its intricate structure across all scales, its fundamental dynamics, the properties of its constituents, the nature of its forces, and even the fabric of spacetime itself. This emergence is driven by a process of iterative relational processing and emergent self-consistency, guided by a fundamental principle of maximizing "existential fitness" or "relational aesthetics" within a continuously evolving attributed graph structure. The Autaxys hypothesis posits that the universe's observed regularities, its fundamental particles and forces, the emergent properties of spacetime, and the very "laws" that govern their behavior are not axiomatic irreducible primitives or externally imposed constraints. Rather, they arise intrinsically and dynamically from the iterative transformations of this fundamental relational substrate, governed by an underlying optimization principle. Specifically, the central hypothesis is that a sufficiently simple, minimal, and precisely defined set of initial principles governing this substrate and its dynamics – collectively termed an "Autaxys Configuration," acting as a cosmic "seed" or "algorithm" – can computationally generate the complexity, specificity, and fine-tuning characteristic of the physical universe. This generative approach stands in sharp contrast to traditional descriptive or inferential methodologies that model reality by postulating fundamental entities and laws based on observed patterns. Autaxys aims instead to *derive* the observed patterns, the fundamental constituents, and the effective laws of physics *from* the generative principles, thereby illuminating *why* reality takes its specific form, rather than merely describing *what* it is or *how* it behaves according to pre-defined, observationally inferred rules. It represents a form of computational natural philosophy dedicated to identifying the minimal generative principles – the simplest "source code" – capable of producing the observed universe as a preferred, stable, and highly fit outcome of a fundamental computational process, akin to discovering the most parsimonious algorithm that compiles into the complex and specific universe we perceive. The transition from a high-level conceptual framework (v1.9) to a formal, rigorous, and computationally grounded theory (v2.0) is a critical step towards rendering this hypothesis scientifically testable, amenable to large-scale computational simulation, and ultimately subject to empirical validation and potential falsification. This document formally defines the core postulates, derived concepts, and central hypothesis of the v2.0 framework, serving as the theoretical foundation and operational blueprint for all subsequent modeling and empirical validation efforts. It systematically outlines the fundamental ontology (P1) based on an attributed relational graph, the fundamental dynamics (P2) as a graph rewriting system, the fundamental state selection principle (P3) driven by Autaxic Lagrangian optimization, and the nature of emergent phenomena (DC1-DC4), including stable patterns, their quantifiable properties (AQNs), self-maintenance (Ontological Closure), and the intrinsic drive towards complexity and diversity (Exploration Drive). These foundational elements culminate in the central hypothesis (H1), which links these principles to the generation and properties of observed physical reality (H1.1-H1.5). This framework represents a unified attempt to derive the entire spectrum of physical reality, from fundamental particles and forces to spacetime and cosmology, from a minimal set of generative axioms rooted in relational processing and emergent self-consistency guided by a principle of existential fitness. The philosophical implications of this generative, relational ontology are profound, challenging traditional notions of substance, locality, and the nature of physical law, suggesting that reality's essence lies not in static things or external rules, but in dynamic, attributed relationships and the computational process of their evolution towards preferred states of coherence and fitness. **Core Postulates (Axioms of the System):** 1. **P1 (Relational Substrate):** The fundamental substrate of reality at any discrete state $t$ is exhaustively and solely represented by a finite, dynamic **Attributed Relational Graph (ARG)**, denoted $G_t = (D_t, R_t, f_{D,t}, f_{R,t})$. The entire history and structure of the universe are constituted by the sequence of states $G_0, G_1, G_2, ..., G_t, G_{t+1}, ...$, where the evolution from $G_t$ to $G_{t+1}$ occurs over discrete, sequential time steps. The universe *is* this evolving graph; it is not embedded within or dependent upon any external, pre-existing space or time, nor does it subsist upon a fundamental, pre-defined manifold. Instead, spatial relationships, temporal durations, causal connections, and geometric properties (such as dimensionality and curvature) are not fundamental primitives but are *created*, *defined*, or *emerge* from the intrinsic structure and dynamics of the graph itself (H1.5). This relational ontology is fundamental, positing that existence, identity, and properties are defined solely by relationships and intrinsic potentials encoded as attributes, not by independent substance, inherent location, or pre-existing presence in an external arena. The graph serves as the sole container, descriptor, and constituent of reality at the most fundamental level. The specific choice of the ARG formalism (directed, multiset, attributed) represents a deliberate hypothesis regarding the minimal necessary characteristics of this fundamental substrate required for the emergence of observed reality in v2.0. While acknowledging that reality might ultimately be better described by richer or alternative formalisms (e.g., Attributed Hypergraphs, Attributed Simplicial Complexes, Category-Theoretic Structures, Process Calculi, Formal Languages/Grammars, Abstract Algebraic Structures - see 2.1.6 in D-P6.2-2), the ARG is selected for v2.0 due to its balance of expressive power, intuitive representation, and computational tractability for initial modeling and simulation efforts. The ARG provides a flexible, non-manifold, non-metric structure at the fundamental level, from which spacetime and geometry are expected to emerge. The initial state $G_0$ from which the universe evolves is also a parameter of the configuration, representing the fundamental initial conditions of the substrate. The specific properties of $G_0$ (size, density, proto-property distribution, initial structure) are part of the configuration hypothesis and can influence the specific history generated, though robust configurations should ideally converge to similar emergent statistical properties, suggesting convergence to a dominant attractor basin or ensemble distribution. * **$D_t$**: A finite set of unique, transient identifiers for **Distinctions** (nodes) existing at time $t$. $D_t \subset \mathbb{N}$, where natural numbers serve merely as transient labels for distinct entities or potential 'points' of relation at a given time step. Distinctions represent the fundamental 'units' of individuation or difference within the substrate; they acquire identity solely through their participation in relations and their assigned proto-properties. They possess no inherent spatial location, temporal duration, or intrinsic substance independent of the graph state. They are dimensionless points of differentiation or localized computational processes within the relational network, analogous to abstract points or events in a causal set, but crucially endowed with rich intrinsic properties and capable of forming complex, attributed relations. Their transience implies that distinctions can be created or destroyed by the dynamics (P2), reflecting potential changes in the fundamental granularity or degrees of freedom of the substrate, potentially relating to particle creation/annihilation events in emergent physics or phase transitions in the fundamental structure of reality. The size of the set $|D_t|$ can grow or shrink over time, representing changes in the fundamental degrees of freedom of the system, potentially driven by the $L_A$ optimization (P3). The uniqueness of identifiers is necessary for tracking and computation at a given step; their fundamental, persistent identity is relationally defined by their stable involvement in recurrent emergent patterns ($P_{ID}$s, DC1) and their characteristic, statistically derived AQNs (DC2), analogous to how particles are identified by their properties and interactions. * **$R_t$**: A finite multiset of directed **Relations** (edges) between distinctions at time $t$. $R_t \subset D_t \times D_t \times \mathbb{N}$, where $(u, v, k)$ signifies the $k$-th distinct relation instance of a potentially specified type from distinction $u$ to distinction $v$. $u, v \in D_t$. Relations define the instantaneous structure, connections, dependencies, and influence pathways between distinctions. They establish a network topology that is not embedded in a pre-existing space but *constitutes* the substrate's relational structure. Relations are the 'bonds', 'interactions', 'information channels', 'dependencies', 'correlations', or 'causal links' between distinctions. They are the dynamic connections that weave the distinctions into a coherent structure. The use of a multiset allows for multiple distinct relation instances of the same source, target, and even type or properties between the same pair of distinctions, potentially representing different interaction channels, parallel connections, different aspects of a composite interaction, or even different emergent forces acting between the same entities simultaneously. This multiset capability adds significant expressive power, allowing for richer and more nuanced interactions and states than simple graphs, potentially modeling phenomena like multiple force carriers between particles, different types of quantum correlations, layered dependencies in complex systems, or different interaction channels (e.g., electromagnetic vs. strong interaction between two particles). The directed nature is fundamental for modeling causality, information flow, asymmetric dependencies, directed processes, or gradients, potentially giving rise to an emergent causal structure (H1.5) and an arrow of time (H1.1, H1.5). Relations themselves are first-class entities in the ontology, not just abstract links; they possess properties, can be created, destroyed, and transformed by the dynamics (P2), and play an active role in shaping the system's evolution by enabling or constraining rule applications and influencing $L_A$. The multiset nature could also potentially model emergent field excitations, where a set of relation instances between regions represents a distributed field state or flux. Relations are the carriers of connection and influence in the system. * **$f_{D,t}$**: An attribute function assigning a set of intrinsic **Proto-properties** to each distinction in $D_t$. $f_{D,t}: D_t \to \mathcal{P}(\Pi_D)$. $\Pi_D$ is a universal, finite set of all possible *types* of proto-properties for distinctions. $\Pi_D = \{\pi_{D,1}, \pi_{D,2}, ..., \pi_{D,m}\}$. Proto-properties are not passive labels but represent active potentials, constraints, internal states, computational flags, inherent capabilities, or affordances that dictate how a distinction can participate in relations and transformations. They constitute the fundamental qualitative 'alphabet' or 'feature set' of the substrate, serving as analogues to fundamental charges, internal quantum numbers, or computational states in physics, albeit at a more primitive, pre-geometric, pre-spacetime level. The specific set of proto-properties assigned to a distinction instance ($f_{D,t}(d) \subseteq \Pi_D$) defines its instantaneous qualitative nature, its role in the network, and its potential for interaction. Examples of proto-property types include abstract 'polarities', 'valence limits', 'interaction types', 'identity markers', 'state indicators', 'computational flags', 'binding potentials', 'topological potentials', 'information capacity', 'color charges', 'weak isospin states', 'flavor types', 'quantum state potential', and 'relativistic potential'. The specific universal set $\Pi_D$ and the rigorous semantics of its proto-properties (i.e., how their presence or absence constrains rule applicability, influences $L_A$, determines transformation outcomes, and potentially participates in algebraic structures or state machines) are fixed for a given Autaxys Configuration. The structure of $\Pi_D$ itself (e.g., whether properties are orthogonal, hierarchical, form a lattice, a group representation space, an algebraic structure like a field or ring, a category, a specific computational state machine with defined transitions and states, or elements of a non-commutative algebra like a Clifford algebra or Jordan algebra) is a critical part of the configuration definition; for instance, proto-properties could form representations of abstract algebraic structures, and rule applications might require compatibility with these structures, potentially encoding conservation laws. The semantics of proto-properties are critical; they define the "meaning" or "behavioral potential" of distinctions in terms of their dynamic potential and interaction capabilities, essentially providing a primitive "type system," "state space," or "algebraic foundation" for the fundamental units, enabling differentiation and selective interaction. Proto-properties can also be viewed as defining the internal degrees of freedom of distinctions, analogous to internal quantum states. The definition of $\Pi_D$ and its semantics is a key component of the Autaxys configuration hypothesis, encoding the fundamental qualities and their rules of combination and interaction. The set of proto-properties defines the "alphabet" of existence. * **$f_{R,t}$**: An attribute function assigning a set of intrinsic **Proto-properties** to each relation in $R_t$. $f_{R,t}: R_t \to \mathcal{P}(\Pi_R)$. $\Pi_R$ is a universal, finite set of all possible *types* of proto-properties for relations. $\Pi_R = \{\pi_{R,1}, \pi_{R,2}, ..., \pi_{R,p}\}$. These govern the 'type', 'strength', 'directionality constraints', 'mediating potential', 'information capacity', 'causal nature', 'temporal ordering potential', 'spatial analogue properties', or 'contextual role' of a relation instance, influencing which rules can apply to it, the effects of rule application involving it, and how it affects the relational context of connected distinctions. Examples include 'binding strength levels', 'information flow potential', 'relation categories' (representing different interaction channels), 'temporal ordering potential', 'spatial analogue properties' (representing relational 'nearness' or 'farness'), 'interaction channel identifiers', 'causal type', 'mediator flags' (indicating the relation can mediate a specific interaction type, potentially linking to emergent force carriers), 'topological significance', 'coherence potential', 'information content', 'quantum correlation indicator', and 'metric contribution'. Relations are not merely structural links but are also attributed entities capable of carrying information, state, or mediating specific types of interactions. They can act as channels for influence or as proxies for fields or forces. The specific universal set $\Pi_R$ and the rigorous semantics of its proto-properties (including potential algebraic composition rules, ordering, or compatibility constraints, and their influence on rule application and $L_A$) are also fixed for a given Autaxys Configuration. Similar to $\Pi_D$, the structure and semantics of $\Pi_R$ are part of the configuration; for instance, relation properties could define an ordering or partial order on relations, or have composition rules under rule application that determine resulting relation properties, potentially forming algebraic structures themselves. The definition of $\Pi_R$ and its semantics is another key component of the Autaxys configuration hypothesis, encoding the fundamental types of connections and their inherent qualities and dynamics. These properties define the "alphabet" of connections. Proto-properties can also be viewed as internal states or degrees of freedom of relations. * The initial graph state $G_0$ (which could be minimal, e.g., an empty graph, a few distinctions with random properties, or could represent a highly ordered initial state, or even a state generated by a separate "pre-cosmic" process), the universal sets $\Pi_D, \Pi_R$ with their defined algebraic structures and semantic constraints, the finite set of attributed graph rewrite rules $\mathcal{R}$ (including any context dependencies or rule-specific parameters), and the computable function $L_A$ with its parameters, functional form, and any associated stochastic/exploration mechanisms (DC4) collectively define a specific **Autaxys Configuration** or "Universe Seed." Discovering the minimal, most parsimonious configuration that robustly generates patterns quantitatively and qualitatively matching observed reality is a core scientific goal of the Autaxys program (H1, 5.2 in AUTX-M1). The initial state $G_0$ is also a parameter, potentially influencing the specific history generated by a configuration, though robust configurations should ideally lead to similar emergent patterns and statistical properties regardless of simple $G_0$ variations, suggesting convergence to a dominant attractor basin or ensemble distribution. The space of all possible Autaxys Configurations is the space of all possible fundamental generative principles, and finding the one that matches our universe is the Grand Challenge, analogous to finding the correct set of fundamental laws and constants in traditional physics, but here, they are the rules and fitness function of a generative process operating on a relational substrate. The configuration itself can be seen as the "program" being executed to generate reality (4.3 in AUTX-P1.0-ANWOS). The goal is to find the simplest such program that compiles into the observed universe (5.2 in AUTX-M1). The size of the configuration space is vast, emphasizing the need for efficient search strategies (see AUTX-M1). The initial distribution of proto-properties in $G_0$ is also part of the configuration, representing the initial conditions of the fundamental substrate. The definition of this configuration space is a key component of the theoretical framework. 2. **P2 (Dynamic Evolution via Rule Application):** The state of the graph evolves over discrete, sequential time steps $t \to t+1$ via the application of precisely one instance of a rule selected from a finite, fixed set, the **Cosmic Algorithm ($\mathcal{R} = \{r_i\}$)**. The Cosmic Algorithm is the fundamental generative grammar of the universe, defining all possible local and potentially global, property-constrained transformations of the relational substrate. It is a set of attributed graph rewrite rules. The discrete nature of time steps is a fundamental postulate, implying that change occurs in quantized steps, which may relate to fundamental temporal discreteness or the operational definition of a "moment" as a unit of transformation, potentially linking to Planck time in emergent spacetime (H1.5). This discreteness also simplifies computational simulation and provides a natural framework for a step-by-step generative process. This postulate defines the fundamental granularity of change in the system, the elementary "event" of the universe's evolution. Each step corresponds to the application of a single selected rule instance, chosen from the set of all possible applications according to the Autaxic Action Principle (P3). This process defines the flow of time as a sequence of discrete transformations. * Each rule $r_i \in \mathcal{R}$ is a directed **Attributed Graph Rewrite Rule** of the form $L_i \Rightarrow R_i$. $L_i$ (the left-hand side) and $R_i$ (the right-hand side) are small, connected, attributed relational graphs, typically with designated 'interface' nodes/edges that specify how the rule connects to the surrounding context of the larger graph $G_t$ during application. The rule explicitly defines a mapping between elements (nodes and edges) of $L_i$ and $R_i$, specifying which elements are preserved, deleted, created, and how their proto-properties are transformed upon application. Rule application adheres to algebraic graph rewriting semantics, typically following the Double Pushout (DPO) or Single Pushout (SPO) approach, extended to accommodate attributed graphs and multisets. DPO semantics requires a "context" graph $K$ (the part of $G_t$ not matched by $L_i$) and ensures that elements in $L_i$ that are connected to the context $K$ are preserved in $R_i$ via the interface mapping, ensuring coherent integration of the rewritten part into the global graph. SPO is simpler, directly replacing $L_i$ with $R_i$ based on the mapping and potentially requiring explicit rules for connecting $R_i$ to the context. The specific choice of semantics (DPO vs. SPO vs. other variants like sesqui-pushout or hyperedge replacement, or even more complex rule types like nested graph rewriting or rules operating on specific algebraic structures of properties) impacts how rules interact with their context and is part of the configuration definition. Rules can be categorized by their primary effect on graph structure and attributes, reflecting the fundamental types of processes possible in the substrate: Creation, Deletion, Transformation, Structural, and Composite rules. Rules can also be context-dependent, requiring the presence or absence of certain structures, proto-properties, or even global graph properties in the vicinity of the match or elsewhere in the graph for applicability (negative application conditions - NACs, positive application conditions - PACs, global application conditions - GACs). Context conditions can range from simple checks on neighboring nodes/edges/properties to complex requirements on the local or global graph structure or even the historical trajectory. The structure of $L_i$ and $R_i$, the mapping between them, and any context conditions define the specific local causality and transformation potential of the rule instance. Rules represent the "micro-actions" possible within the substrate, the pre-geometry, pre-physics operations from which observed dynamics emerge. They are the fundamental interactions at the most primitive level, defining the allowed state transitions of the fundamental substrate. The set $\mathcal{R}$ itself, its size, and the complexity of its rules are parameters of the Autaxys Configuration. The design of $\mathcal{R}$ is a critical part of the configuration hypothesis, encoding the fundamental "grammar" of change and the potential complexity of interactions. The rules encode the "laws" of how the substrate transforms itself. They define the "transition probabilities" or "allowed transitions" between different microstates, constrained by proto-properties and context. The rule set can be interpreted as the "instruction set" of the Cosmic Algorithm, defining the fundamental computational steps. The formal properties of the GRS defined by $\mathcal{R}$ are inherent theoretical properties of the configuration that may relate to fundamental limits or properties of the universe. Rules can also include algebraic operations on proto-properties as part of the transformation, ensuring compatibility with the defined algebraic structure of $\Pi_D, \Pi_R$, potentially encoding conservation laws. * **Applicability**: A rule $r_i$ is applicable to the current graph $G_t$ if there exists at least one valid occurrence (a valid subgraph match) of $L_i$ within $G_t$. A match is an injective mapping $\mu: L_i \to G_t$ that preserves the graph structure (nodes, edges, directedness, multiset nature of edges) and satisfies all specified proto-property constraints defined for the rule $r_i$. Proto-property matching can involve complex boolean logic, value comparisons, set operations, algebraic compatibility checks, or even pattern matching on sets of properties, defined rigorously for each rule. The rule may also specify constraints on properties *not* present in $L_i$ but required in the context of $\mu(L_i)$ in $G_t$ (negative application conditions - NACs), or require specific global conditions on $G_t$ (e.g., total number of nodes, average degree, presence of specific global patterns, or global $L_A$ thresholds, or conditions related to emergent spacetime properties like local curvature or time rate, or conditions related to the system's history like total elapsed steps or total $L_A$ accumulated). Let $\mathcal{M}(G_t, r_i)$ be the set of all valid structural and property-constrained matches of $L_i$ in $G_t$. Let $\mathcal{M}(G_t) = \{ (r_i, \mu) \mid r_i \in \mathcal{R}, \mu \in \mathcal{M}(G_t, r_i) \}$ be the finite set of all possible rule application instances (rule-match pairs) available in $G_t$. If $\mathcal{M}(G_t)$ is empty, the system halts in a terminal state where no further evolution is possible under the given rule set. The computational challenge of finding $\mathcal{M}(G_t)$ involves potentially solving multiple instances of the NP-complete subgraph isomorphism problem, complicated by attribute matching, complex context constraints (local and global), and the potentially large and dynamic size of $G_t$. Efficient algorithms, indexing structures, heuristic pruning, and potentially machine learning approaches are critical for practical simulation (see AUTX-M1). The number of possible matches can be vast, representing a branching tree of potential futures at each step. Rules can also be prioritized or weighted based on their properties, context, or recent application history, influencing the selection process (P3). * **Transformation**: Applying rule $r_i$ at a specific match $\mu \in \mathcal{M}(G_t, r_i)$ transforms $G_t$ into a potential immediate successor state $G'_{t+1}$. This transformation is performed according to the rule's mapping, typically following a double pushout (DPO) or single pushout (SPO) semantics from algebraic graph rewriting theory, extended to attributed graphs and handling the multiset nature of edges. The subgraph matched by $\mu(L_i)$ is conceptually removed (or elements are identified for preservation/transformation), and the graph $R_i$ is inserted, with connections to the remaining context of $G_t$ established as specified by the rule's interface mapping. This process can add/remove nodes/edges and change proto-property assignments of elements within the matched $\mu(L_i)$ that map to $R_i$, or even elements in the context based on rule specifications. The transformation must preserve global invariants or satisfy post-conditions specified by the rule or the configuration (e.g., conservation of specific proto-property sums if defined by the underlying algebraic structure of $\Pi_D, \Pi_R$ and the rules, or conservation of topological invariants like cycle counts or Betti numbers, or conservation of total $L_A$ contribution from specific pattern types if the rule is designed to model a specific interaction). Let $\mathcal{A}(G_t, r_i, \mu)$ denote the unique graph state resulting from applying rule $r_i$ via match $\mu$ in $G_t$. Rules can also specify the creation of new, non-local relations or correlations between distant parts of the graph, which is important for modeling entanglement or field propagation. Rules can also specify the creation of new proto-properties on existing or new elements, or the removal of properties. The transformation of proto-properties is a critical part of the rule definition, specifying how the qualitative nature of entities changes upon interaction, potentially analogous to particle transformations or state changes. The transformation process can also involve complex algebraic operations on proto-properties, determined by the rules and the structure of $\Pi_D, \Pi_R$. * Let $\mathcal{P}(G_t) = \{ \mathcal{A}(G_t, r_i, \mu) \mid (r_i, \mu) \in \mathcal{M}(G_t) \}$ be the finite set of all possible immediate successor graph states reachable from $G_t$ by applying exactly one rule instance. The size of $\mathcal{P}(G_t)$ can be very large, equal to $|\mathcal{M}(G_t)|$. The selection of which one of these potential futures is actualized is governed by P3. While the potential for multiple simultaneous rule applications across spatially or relationally separated regions could be modeled as parallel processes, v2.0 postulates a single rule application per discrete step for simplicity and computational tractability, defining the most granular unit of cosmic change and providing a clear temporal ordering. Future versions might explore parallel rule application semantics, multi-step lookahead in the optimization, or modeling synchronous vs. asynchronous rule application. The computational process of finding $\mathcal{P}(G_t)$ involves generating these potential states efficiently. 3. **P3 (The Autaxic Action Principle & State Selection):** The selection of the unique state transition from $G_t$ to $G_{t+1}$ at each discrete time step is governed by a single, fundamental guiding principle. This constitutes a variational principle analogous to the Principle of Least Action, but framed as one of maximization, or alternatively, a probabilistic sampling of the fitness landscape. This principle provides the "telos" or directedness to the cosmic evolution, acting as a fundamental selection pressure on the possible transformations defined by the Cosmic Algorithm. It embodies the fundamental drive towards increasing structure, organization, and complexity, counteracting simple entropic decay. * **The Autaxic Lagrangian ($L_A$):** There exists a universal, computable function $L_A: \text{AttributedGraphs} \to \mathbb{R}$ that maps any valid attributed relational graph state $G$ to a single scalar value (P3 in `D-P6.2-4`). This value quantifies the state's "Relational Aesthetics," "existential fitness," intrinsic "coherence," "organization," "potential for sustained, complex existence," or "thermodynamic viability" within the Autaxys framework. It embodies the *Economy of Existence*, favoring states that exhibit robust, organized structure, efficient relational processing, high information content relative to their underlying resource cost, and potentially the capacity for further complex evolution or adaptability. $L_A(G)$ is a flexible, *computable* function that can include diverse terms reflecting various desirable properties of a graph state hypothesized to contribute to its "fitness." The specific functional form and parameters of $L_A$ are part of the Autaxys Configuration. Defining an $L_A$ that generates physical reality is a central scientific challenge, requiring a balance between favoring stable structure (exploitation) and favoring conditions that lead to discovering more complex structures (exploration). The structure of $L_A(G)$ can be hierarchical, aggregating contributions from local patterns, regional properties, and global metrics, and can include non-linear interactions between terms or threshold effects. $L_A$ calculation requires real-time or near-real-time pattern identification (DC1), AQN computation (DC2), and global metric calculation within the potential successor state $G'$. The computational efficiency of $L_A$ evaluation is a major challenge, potentially involving NP-hard problems. Efficient algorithms, indexing structures, heuristic pruning, and incremental $L_A$ updates based on localized rule effects are critical for practical simulation (see AUTX-M1). The design of $L_A$ is a hypothesis about the fundamental values or preferences inherent in reality. The specific terms included and their functional form are critical components of the Autaxys Configuration. > $ L_A(G) = F \left( \Lambda_{patterns}(G), \Lambda_{global}(G), \Lambda_{exploration}(G), \Lambda_{constraints}(G), \Lambda_{spacetime}(G), \Lambda_{quantum}(G), \Lambda_{algebraic}(G), \Lambda_{thermodynamic}(G), \Lambda_{computational}(G) \right) $ where these components are functions aggregating diverse metrics: * $\Lambda_{patterns}(G)$: Aggregates contributions from identified instances of distinct $P_{ID}$ types currently present in $G$, favoring states with high total "fitness contribution" from stable, complex patterns. * $\Lambda_{global}(G)$: Aggregates contributions from computable metrics describing the overall state, structure, and resource utilization of the graph, potentially linking to cosmological principles or fundamental constants. * $\Lambda_{exploration}(G)$: Specifically designed to counteract premature convergence and promote discovery of complex, diverse, and globally favorable states (DC4), introducing a drive towards novelty or potential for future growth. * $\Lambda_{constraints}(G)$: Terms that penalize states violating specific structural or property constraints disfavored by the system, guiding evolution towards desired emergent properties without hardcoding them in rules. * $\Lambda_{spacetime}(G)$: Terms related to the emergence of spacetime-like structures, rewarding configurations exhibiting properties hypothesized to correspond to observed spacetime features, geometry, and gravity. * $\Lambda_{quantum}(G)$: Terms related to the emergence of quantum-like phenomena, rewarding configurations exhibiting properties analogous to superposition, entanglement, quantization, and probabilistic outcomes. * $\Lambda_{algebraic}(G)$: Terms related to rewarding algebraic structures in proto-properties or relations hypothesized to underlie emergent conservation laws and symmetries. * $\Lambda_{thermodynamic}(G)$: Terms related to emergent thermodynamic properties, penalizing states with excessive free energy analogue or rewarding states exhibiting desirable entropy characteristics or phase transitions. * $\Lambda_{computational}(G)$: Terms related to the intrinsic computational cost or efficiency of the graph state, viewing the universe as a computation, penalizing expensive or non-terminating configurations and rewarding information processing capacity. * The function $F$ combines these components into a single scalar value. $F$ could be a simple weighted sum, a non-linear function, or a more complex aggregation. The specific functional form $L_A(G)$, its parameters, and the weighting factors constitute a critical part of the Autaxys Configuration. The choice of $L_A$ encodes the fundamental "aesthetic," "goal," or "selection pressure" that shapes the emergent reality. Different $L_A$ functions will lead to universes with different emergent properties and fundamental "laws." Defining and refining $L_A$ is a major scientific challenge within the framework (see AUTX-M1). The $L_A$ function might itself be learned or evolved over cosmic time in more advanced theoretical extensions, but for v2.0 it is fixed for a given configuration. * **The Selection Principle:** At each discrete step $t$, from the finite set of all possible immediate successor graph states $\mathcal{P}(G_t)$, the system selects the unique state $G_{t+1}$. This selection is governed by a principle that favors states with higher values of the Autaxic Lagrangian. The primary mechanisms are either deterministic maximization or probabilistic sampling based on $L_A$. This implements a non-greedy selection from $\mathcal{P}(G_t)$ when probabilistic methods are used, which is crucial for incorporating the Exploration Drive (DC4) and potentially modeling quantum probability. > $ G_{t+1} = \underset{G' \in \mathcal{P}(G_t)}{\text{argmax}} (L_A(G')) \quad \text{(Standard Greedy Selection)} $ > $ P(G_{t+1} = G') = \frac{f(L_A(G'), \text{ExplorationParams}, G_t, \mathcal{R})}{\sum_{G'' \in \mathcal{P}(G_t)} f(L_A(G''), \text{ExplorationParams}, G_t, \mathcal{R})} \quad \text{(Probabilistic Selection, e.g., Boltzmann, with DC4 influence and context dependence)} $ * In the deterministic greedy model, ties where multiple successor states yield the exact same maximum $L_A$ must be resolved by a fixed, deterministic tie-breaking rule (e.g., based on a canonical ordering of rule indices, match locations, or resulting graph state encodings). This ensures a unique trajectory for a given configuration and initial state, facilitating debugging and analysis, although a truly realistic model of fundamental reality might require inherent non-determinism. * **Incorporating Exploration (DC4 Implementation in P3):** The Exploration Drive (DC4) is implemented through the selection mechanism, introducing a non-deterministic or probabilistic element that replaces or augments deterministic greedy selection (3.1.2 in AUTX-M1). This allows the system to explore the $L_A$ landscape effectively, escaping local minima and discovering potentially more fruitful areas of the state space. Examples of selection functions $f(\cdot, \cdot)$ implementing this include Probabilistic Selection (e.g., Boltzmann distribution, allowing lower-$L_A$ states to be selected with non-zero probability, with parameters like $\beta$ potentially dynamic), $\epsilon$-Greedy Selection (selecting randomly with probability $\epsilon$), Rank-Based Selection (probabilistic selection favoring higher-$L_A$ states), Multi-Objective Optimization (optimizing for novelty, diversity, etc., alongside $L_A$), Rule-Specific Selection Bias (some rules having inherent propensity), and Context-Dependent Stochasticity (level of randomness depending on local/global state). The specific mechanism for handling ties and incorporating exploration is a critical parameterizable component of the Autaxys Configuration under investigation (4.1.4.4 in AUTX-M1). This selection process, repeated at each step, forms the deterministic (or probabilistic) time evolution $G_t \to G_{t+1}$. The selection principle is the "engine" that drives the system through the state space according to the landscape defined by $L_A$ and the rules. This process provides the fundamental 'time' dimension of the universe, where each step is a discrete unit of change. The probabilistic selection mechanism offers a concrete computational model for the origin of quantum probability and fluctuations at the fundamental level, where the universe "chooses" among possible next states with probabilities related to their "fitness." **Derived Concepts (Internal to the Formalism):** 1. **DC1 (Emergent Patterns - $P_{ID}$s):** These are attributed subgraph configurations that exhibit persistence, recurrence, and structural integrity over significant durations within the global graph dynamics defined by P2 and P3. They achieve a degree of dynamic stability and self-maintenance against the background of continuous transformation. They are identified as distinct pattern types ($P_{ID}$s - Pattern Identifiers) if instances of the subgraph configuration and its associated proto-properties recurrently emerge and are maintained across different simulations or regions, exhibiting a degree of *Ontological Closure* (DC3). They act as dynamic units, "quasi-particles," composite structures, or emergent objects within the relational substrate, exhibiting properties and behaviors distinct from the underlying distinctions and relations. Their emergence signifies a spontaneous self-organization of the substrate into semi-autonomous, stable entities, akin to dissipative structures or solitons, but with precise, quantifiable properties (AQNs). Their identity is defined by their stable attributed graph structure, their characteristic statistical distribution of AQNs (DC2), and their emergent interaction behaviors ($I_R$). * **Formal Identification:** Involves sophisticated computational algorithms for detecting frequent, persistent, or structurally invariant attributed subgraphs in simulation trajectories and final states (DC1 in `D-P6.2-4`, 5.2 in AUTX-M1). This requires techniques robust to minor structural variations or 'noise'. Methods include: frequent attributed subgraph mining algorithms (e.g., gSpan, MoFa, GADEM adapted for attributes and dynamic graphs, SUBDUE), graph clustering algorithms (e.g., based on graph kernels, spectral properties, or attributed graph metrics, community detection adapted for attributed graphs, clustering based on node or edge embeddings), topological data analysis (identifying persistent topological features), and quantitatively assessing their *Ontological Closure* ($S$) and relative *Complexity* ($C$). Pattern identification can be done via template matching or unsupervised clustering. Patterns can be simple or complex, hierarchical composites built from instances of simpler $P_{ID}$s. Criteria for distinguishing "fundamental" $P_{ID}$s can include minimal complexity, maximal stability-to-complexity ratio, or being irreducible under $\mathcal{R}$. Patterns can be localized in the graph or potentially distributed/non-local. Identifying non-local patterns is a significant computational challenge. Pattern identification must be efficient enough to be integrated into the $L_A$ calculation loop for potential successor states (P3). Formal, algorithmic criteria for defining persistence and recurrence in dynamic graphs are essential (5.2 in AUTX-M1). * $P_{ID}$s are the candidates for the fundamental constituents of observed reality (particles, fields, composite structures, potentially even emergent spacetime structures or topological defects). Their identity is defined by their stable attributed graph structure, their characteristic statistical distribution of AQNs (DC2), and their emergent interaction behaviors ($I_R$). The discovery and characterization of these patterns is the primary output of the Autaxys generative modeling process and forms the basis of the Autaxic Table (H1.2). 2. **DC2 (Autaxic Quantum Numbers - AQNs):** These are quantifiable, intrinsic properties of emergent patterns ($P_{ID}$s), derived computationally from the structure, proto-properties, and dynamic resilience of their corresponding attributed subgraphs ($G_{P_{ID}}$) within the AGE dynamics (DC2 in `D-P6.2-4`). They are hypothesized to map directly to observable physical properties of fundamental particles, composite structures, and forces. AQNs provide the quantitative link between the abstract relational substrate and empirical measurements. They are not fundamental inputs but are computed from the emergent structure and dynamics, representing the 'measurable' characteristics of the emergent entities. The specific set of AQNs and their mapping to physical properties is part of the hypothesis and refined through comparison with observation. The statistical distributions of AQNs for each pattern type, calculated across ensembles (MVU-2+), provide the quantitative predictions of the theory, including uncertainties and correlations (H1.2, H1.3). * **Complexity ($C$):** The AQN corresponding to observed mass and energy content ($E=mc^2$) of a pattern instance. Formalized using **Algorithmic Information Theory (AIT)**. $C(P_{ID}) \approx K_{\mathcal{R}}(G_{P_{ID}})$, the Kolmogorov Complexity of the pattern's attributed subgraph structure and proto-property assignment *relative to the Cosmic Algorithm rule set $\mathcal{R}$* used as the reference programming language/Turing machine (DC2 in `D-P6.2-4`). This is the length of the shortest valid sequence of rule applications from $\mathcal{R}$, starting from a minimal initial state, required to deterministically generate an instance of $G_{P_{ID}}$ in its specific form and attributed state, relative to its context. This measures the irreducible generative "cost," resource accumulation (distinctions, relations, properties), or intrinsic information content of the pattern within the system's own grammar. * **Computational Approximation:** Exact $K$ is uncomputable. Practical computation relies on robust, computable approximations suitable for dynamic attributed graph structures and scalable for ensemble analysis (DC2 in `D-P6.2-4`, 3.1.3.1 in AUTX-M1): shortest observed derivation sequence, compression algorithms (Lempel-Ziv, NCD, gzip) applied to canonical encodings, Minimum Description Length (MDL) principle, or structural proxies (easily computable attributed graph metrics empirically correlating with complexity, such as number of nodes/edges, unique proto-properties, complex motif counts, graph depth, structural interconnectedness measures, entropy measures, spectral properties, or GNN features). Relative complexities between patterns are often more meaningful than absolute values. The distribution of these approximated $C$ values across many instances provides a statistical prediction for the pattern's mass/energy analogue. * **Implication:** Observed mass/energy is an emergent property reflecting the irreducible informational/structural 'cost', resource accumulation, or inherent organizational complexity required to instantiate and maintain a stable configuration within the generative system. The $E=mc^2$ relation might emerge as a fundamental relationship between the stability ($S$) required to maintain a certain complexity ($C$), potentially related to the energy cost ($\Delta E_{OC}$) of breaking ontological closure. * **Topology ($T$):** The AQN corresponding to observed internal quantum numbers such as electric charge, spin, color charge, weak isospin, baryon number, lepton number, parity, and other discrete conserved quantities. Formalized using concepts from **Group Theory** and **Attributed Graph Invariants**. $T(P_{ID})$ is a set of descriptors derived from the symmetry properties, structural invariants, and stable proto-property configurations of $G_{P_{ID}}$ (DC2 in `D-P6.2-4`). It captures the pattern's intrinsic qualitative nature and how it transforms under internal rearrangements or interactions. * **Attributed Automorphism Group ($Aut(G_{P_{ID}}, f_{P_{ID}}, R_{P_{ID}}^{multiset})$):** The group of all isomorphisms of the underlying graph structure of $G_{P_{ID}}$ that *also* preserve the assigned proto-properties $f_{P_{ID}}$ and respect the multiset nature of relations. These symmetries represent internal invariances. The algebraic structure of this group (order, generators, relations, subgroups, quotient groups, center), its irreducible representations, and its relationship to the proto-property algebra are hypothesized to correspond directly to observed conservation laws and charges under fundamental forces and the emergent gauge symmetries (DC2 in `D-P6.2-4`). Specific subgroups might reveal symmetries corresponding to $U(1)$, $SU(2)$, or $SU(3)$. Patterns transforming according to particular irreducible representations would correspond to particles exhibiting those charges. Computing automorphism groups is computationally challenging; approximations based on local symmetries or specific feature vectors are used (DC2 in `D-P6.2-4`, 3.1.3.2 in AUTX-M1). * **Other Attributed Invariants**: Additional computable graph invariants and stable proto-property configurations provide further topological, qualitative, and potentially conserved characterization (DC2 in `D-P6.2-4`): Spectral Graph Theory (eigenvalue distributions of attributed matrices, relating to connectivity, cycles, diffusion, energy levels, global structure, and potentially spin or other quantum numbers), Topological Data Analysis (persistent homology, identifying persistent topological features like cycles, voids, Betti numbers, potentially relating to topological charge or emergent internal dimensions), Proto-property Sums/Configurations (counts, sums, or invariant relative arrangements of proto-properties conserved under dynamics, potentially mapping to electric charge, color charge, baryon/lepton number, spin), Knot Theory Analogs (for complex cyclic structures or embeddings, potentially relating to topological quantum numbers), Algebraic Invariants (invariants of the proto-property algebra under rule-induced transformations or internal pattern symmetries), and Representations of Abstract Algebras (if proto-properties form representations of specific algebras, the representation space could define $T$). * These invariants and symmetries are hypothesized to map to quantum numbers like spin, parity, and conserved numbers. The specific mapping is refined through empirical comparison (H1.3). The statistical distribution of these invariants across ensembles provides the quantitative prediction for the pattern's quantum numbers. * **Stability ($S$):** The AQN corresponding to observed lifetime, decay rates, and interaction cross-sections (quantifying resilience to disruption). Formalized using concepts from **Dynamical Systems Theory**, **Perturbation Analysis**, and **Statistical Analysis of Ensembles**. * A stable pattern instance corresponds to a robust **attractor** (fixed point, limit cycle, or more complex attractor) in the graph state space under the AGE dynamics defined by $\mathcal{R}$ and $L_A$. Meta-stable patterns correspond to less robust attractors or limit cycles with finite escape paths. Unstable "patterns" are transient states outside of significant attractor basins. * The stability $S(P_{ID})$ is quantitatively proportional to the "depth" or resilience of this attractor basin. This depth is measured by the minimum "escape energy" ($\Delta E_{OC}$) required to break the pattern's *Ontological Closure* (DC3) (DC2 in `D-P6.2-4`). $\Delta E_{OC}$ is the minimal "cost" or "effort" associated with a sequence of rule applications (a "perturbation") required to transform an instance of $G_{P_{ID}}$ into a state where it is no longer identifiable as that $P_{ID}$. This cost is defined in terms of the cumulative impact on the Autaxic Lagrangian along the escape path. > $ S(P_{ID}) \propto \Delta E_{OC}(P_{ID}) = \min_{\text{perturbation seq } \sigma} \left( \sum_{(r_i, \mu) \in \sigma} \max(0, L_A(G_{next}) - L_A(G_{current})) \right) $ where $G_{current}$ is the graph state before applying $(r_i, \mu)$ and $G_{next}$ is the resulting state, and the minimum is taken over all sequences of rule applications $\sigma$ that, when applied sequentially starting from a graph containing an instance of $G_{P_{ID}}$ in its stable configuration and relational context, result in a state where the pattern instance is destroyed or fundamentally altered. The $\max(0, \cdot)$ term ensures that only increases in $L_A$ are summed, representing the energetic "cost" or "effort" required to push the system out of the pattern's stable attractor basin. * **Computational Approximation:** $\Delta E_{OC}$ is computationally challenging. Practical computation relies on approximations (DC2 in `D-P6.2-4`, 3.1.3.3, 4.1.2.2.4 in AUTX-M1): simulating targeted perturbations (measuring $L_A$ changes and structural deviations), analyzing simulation trajectories (empirical decay rate/lifetime across many runs), multiverse/ensemble analysis (frequency of pattern recurrence/persistence), and identifying minimal destructive rule sequences (finding shortest/lowest-$L_A$-cost rule sequences). The distribution of lifetimes/decay rates across ensembles provides the statistical prediction for pattern lifetimes and branching ratios. * **Implication:** A pattern's observed lifetime, decay probability, and interaction strength (cross-section) are direct measures of its inherent resilience to the dynamic transformations encoded in the Cosmic Algorithm and guided by $L_A$, reflecting the strength of its self-maintaining ontological closure. The $S/C$ ratio in $L_A$ suggests that systems favor patterns that are maximally stable for their generative complexity/resource cost, potentially explaining why observed fundamental particles are highly stable or meta-stable relative to their complexity. * **Emergent Interaction Rules ($I_R$):** These are not fundamental AQNs of individual patterns but are emergent, effective descriptions of how instances of different $P_{ID}$ types transform each other or the surrounding graph structure when they co-occur in specific relational configurations (DC2 in `D-P6.2-4`). $I_R(P_{ID\_A}, P_{ID\_B} \to P_{ID\_C}, ...)$ describe processes analogous to binding, scattering, decay, annihilation, creation, and mediation of forces observed in physics. They are *derived computationally* by observing and statistically characterizing sequences of Cosmic Algorithm rule applications ($\{r_i\}$) that occur when instances of $P_{ID\_A}$ and $P_{ID\_B}$ are present in relational "proximity" (defined relationally, not spatially, e.g., by graph distance, shared neighborhood, specific connecting relations or motifs, or co-occurrence within a small, highly active graph region, or proximity in emergent spacetime - H1.5) or specific relational contexts, and which result in identifiable changes to these patterns (e.g., fusion, decay, state change, relation modification). Observed fundamental forces (Electromagnetism, Weak Nuclear, Strong Nuclear, and potentially Gravity) are hypothesized to be these emergent interaction patterns between specific fundamental $P_{ID}$s (analogues of leptons, quarks), potentially mediated by other specific $P_{ID}$s (analogues of force-carrying bosons) which facilitate specific rule applications between distant patterns by creating mediating relational structures, carrying proto-properties/information, or modifying the local graph dynamics. This requires a formal process of *coarse-graining* the underlying fine-grained GRS dynamics into effective interaction potentials, cross-sections, or reaction rules between emergent patterns (4.3.3 in AUTX-M1). Techniques from complex systems science (symbolic dynamics, process mining, information flow analysis), statistical physics (deriving effective potentials/Lagrangians, coarse-graining field theories on graphs), machine learning (training models to predict interaction outcomes/rates, learning effective potentials), or network science (analyzing information flow/influence, defining effective distance/coupling) could be used. The emergent $I_R$ should predict outcomes, rates, probabilities, and context-dependencies of interactions between $P_{ID}$s that match experimental data (H1.3, H1.4). Conservation laws (like charge, energy, momentum analogues, baryon/lepton number) are hypothesized to emerge from symmetries in these emergent $I_R$ and the underlying dynamics (e.g., conservation of proto-property sums, symmetries of $\mathcal{R}$ or $L_A$). Different rule types or specific proto-properties could mediate different force types. The derived $I_R$ should form a consistent effective theory at the emergent level, potentially approximating a quantum field theory or a force law like electromagnetism or gravity in the appropriate limits. 3. **DC3 (Ontological Closure - OC):** A state achieved by a pattern ($P_{ID}$) or a larger subgraph when its internal attributed structure and its relational context within the global graph $G_t$ constitute a dynamically self-consistent and self-maintaining configuration (DC3 in `D-P6.2-4`). This means the pattern is a robust local optimum or attractor under the $L_A$-driven AGE dynamics. Rule applications from the Cosmic Algorithm in the vicinity of the pattern instance tend to preserve, reinforce, or quickly restore its structure and properties, or deviations are counteracted by the dynamics, potentially via specific "repair" or "stabilization" rules implicitly or explicitly favored by $L_A$. OC is the qualitative state of being a dynamically stable, self-maintaining entity within the relational substrate, resilient to the system's inherent dynamism and the continuous application of rules. High $S$ is the quantitative measure of the strength of this OC, representing the depth of its attractor basin ($\Delta E_{OC}$) or its empirical persistence. Patterns with strong OC are the dynamically stable building blocks from which more complex, hierarchical structures can emerge and persist. OC is not a static property but a dynamic process of continuous self-constitution against the background of change, driven by the local $L_A$ optimization which favors states where these coherent structures are maintained. It is analogous to a self-repairing, homeostatic, or autocatalytic system, where the pattern's structure and properties enable the very rule applications needed to maintain or restore it by making the resulting states high in $L_A$. Emergent entities exist by virtue of their ability to self-maintain their relational identity against the background dynamics, effectively carving out a region of stability in the turbulent state space. Ontological Closure implies that the pattern is a fixed point or limit cycle not just structurally, but also in terms of its attributed state and its ability to resist typical perturbations from the surrounding dynamics, quantified by $\Delta E_{OC}$. The degree of OC can vary, leading to a spectrum of stabilities from highly stable fundamental patterns to transient, unstable configurations. v2.0 expands v1.0's structural consistency filter to dynamic stability under a rich rule set driven by $L_A$ optimization and incorporating attributes as a *generative principle*. OC is the mechanism by which stable "objects" emerge from a dynamic, relational process. It is the computational realization of the philosophical idea that existence is tied to self-constitution and resilience. It represents the self-organization of the substrate into persistent, identifiable forms. The concept of OC also extends to larger structures or even the entire graph, representing a form of global stability or cosmic coherence. 4. **DC4 (The Exploration Drive):** A hypothesized emergent meta-principle or a specific set of mechanisms implemented within the $L_A$ function or the rule selection process (P3) designed to counteract the natural tendency for purely greedy local $L_A$ maximization to quickly converge to simple, low-$L_A$, homogeneous, or stagnant local optima, especially in early or low-complexity states of the graph (DC4 in `D-P6.2-4`). This drive is necessary to explain the emergence of the universe's observed complexity, diversity, and potentially large-scale structure, which may require transient decreases in $L_A$ or non-obvious state transitions to discover and reach higher-peak, more complex, globally favorable stable patterns and regions in the vast $L_A$ fitness landscape. Without it, the system might get stuck in a trivial "dead" state (e.g., a simple, highly ordered but low-$L_A$ graph with no potential for further complex evolution), failing to explore the potential for richer, more complex existence. The Exploration Drive introduces a mechanism for escaping local minima and discovering new, potentially more fruitful areas of the state space. This drive could potentially be linked to the observed quantum fluctuations or the probabilistic nature of quantum mechanics, providing a mechanism for the universe to explore potential futures beyond the classically deterministic path of maximal fitness increase. It ensures the system doesn't get stuck in simple, uninteresting configurations and can discover the complex, highly fit configurations that correspond to observed reality. This is crucial for explaining the universe's observed richness and the existence of complex structures like life. It provides a mechanism for "creativity" or "innovation" in the cosmic evolution. It biases the system towards novelty and complexity, while still being guided by the fitness function. This can be viewed as a computational implementation of a principle of "sufficient reason" or a drive towards realizing potential complexity. The specific mechanisms for implementing the Exploration Drive are detailed as part of the state selection principle in P3 (e.g., stochastic selection, $\epsilon$-greedy, rank-based selection, multi-objective optimization, rule-specific biases, context-dependent stochasticity). **Central Hypothesis (H1):** > **H1:** The central hypothesis of Autaxys v2.0 is that a sufficiently simple, minimal, and well-defined Autaxys Configuration – comprising finite Proto-property spaces ($\Pi_D, \Pi_R$) with defined semantics and potential algebraic structures, a minimal, finite Cosmic Algorithm ($\mathcal{R}$) of attributed graph rewrite rules, and a computable Autaxic Lagrangian ($L_A$) embodying the Economy of Existence and incorporating an Exploration Drive mechanism (DC4 via P3) – acts as a generative 'seed' or 'algorithm' for reality. The iterative application of the $L_A$-guided dynamic (P3) to an initially simple or random graph state ($G_0$) will spontaneously and robustly generate a non-trivial diversity of stable, meta-stable, and interacting emergent patterns ($P_{ID}$s) characterized by computationally derivable Autaxic Quantum Numbers (AQNs: $C, T, S$) and emergent Interaction Rules ($I_R$), leading to an increasingly complex, organized, and ordered macroscopic state. Furthermore, through a systematic process of computational simulation, analysis, and iterative refinement guided by rigorous empirical comparison with observed physical reality, it is hypothesized that a specific, parsimonious Autaxys Configuration exists and can be identified whose generated pattern landscape and emergent large-scale structure and dynamics (H1.5) quantitatively and qualitatively resemble the observed physical universe, including its fundamental particles, forces, and spacetime. **Corollaries (Testable Implications of H1):** * **H1.1 (Emergence of Order and Structure):** The system's evolution, driven by local $L_A$ maximization (and influenced by the Exploration Drive), will exhibit a macroscopic trend towards states of higher average and aggregate $L_A$ over sufficiently long durations, despite potential transient local decreases (facilitated by the Exploration Drive). This implies a form of emergent self-organization and complexity science where $L_A$ acts as a potential function or fitness landscape guiding the system's trajectory from less structured, lower $L_A$ states towards more structured, higher $L_A$ states characterized by the persistent presence, coherent interaction, and hierarchical organization of stable $P_{ID}$s. This emergent trend suggests an intrinsic arrow of time tied to the increase in organizational fitness and complexity, potentially explaining the thermodynamic arrow of time as a coarse-grained manifestation of the underlying $L_A$ dynamics, where the increase in thermodynamic entropy in spacetime might be coupled to the increase in structural complexity, pattern organization, and information content in the underlying graph, or to the rate of information processing within the system. The dynamics are hypothesized to naturally move towards states of greater information content, organization, and coherence, counteracting simple entropic decay towards structural homogeneity or maximal undirected entropy, driven by the $L_A$ function's preference for complexity, stability, diversity, and structured relationships. This emergent ordering is a key prediction of the framework, suggesting a fundamental principle of self-organization inherent in the substrate. The rate of $L_A$ increase or the rate of state space exploration might also be linked to cosmological parameters like the Hubble constant, the scale of inflation, or the rate of dark energy expansion. The direction of time's arrow could also be tied to the breaking of time-reversal symmetry in $\mathcal{R}$ or $L_A$, or to the probabilistic nature of the selection process (P3/DC4), or to the emergent causal structure (H1.5). The emergence of distinct phases in the universe's evolution (e.g., inflationary epoch, symmetry breaking phase transitions, structure formation era) should be observed in simulation trajectories as periods with distinct dynamical behaviors, dominant rule applications/pattern types, or changes in global graph metrics, linked to changes in global $L_A$, transitions between attractor basins in the $L_A$ landscape, or changes in the parameters of the Exploration Drive. The framework predicts that the universe's history is a trajectory through the $L_A$ landscape, guided by optimization and exploration, leading to the observed complex structures and thermodynamic arrow of time. * **H1.2 (Derivation of the Autaxic Table):** The set of all robustly emergent $P_{ID}$s identified across a sufficient exploration of the system's state space under a specific Autaxys Configuration (e.g., via extensive multiverse simulation as defined in MVU-2+), along with their statistically characterized, computationally derived AQNs ($C, T, S$) and emergent Interaction Rules ($I_R$), will form a self-generated "Autaxic Table of Patterns." This table is the framework's analogue to the Standard Model of particle physics and its associated force interactions, but its entries are *derived* from the generative process, not postulated as fundamental. The statistical distributions of AQNs for each $P_{ID}$ type observed across the ensemble provide the predicted quantitative properties (e.g., mean mass, variance in charge, distribution of spin values, range of lifetimes, correlations between properties, decay branching ratios, effective coupling constants, scattering cross-sections) for the emergent particles/entities, including potential relationships or correlations between different AQNs for a given pattern type (e.g., a relationship between $C$ and $S$ that resembles mass-lifetime relationships, or correlations between $C$ and $T$ like mass-charge ratios, or relations between $T$ and $I_R$ like charge determining interaction type/strength, or relations between $C$ and $I_R$ like mass influencing interaction cross-section). The table will also include the derived $I_R$ describing how these patterns interact (e.g., reaction probabilities, effective coupling constants, decay modes, branching ratios, scattering cross-sections, conservation laws associated with specific interactions, symmetry breaking patterns, force carrier identities and properties, range and form of effective potentials derived from relational dynamics). This provides a theoretical table of fundamental constituents and their properties that is testable against empirical observation. The statistical nature of the derived properties is a key feature, aligning with the probabilistic nature of quantum mechanics, and provides a basis for quantitative statistical comparison with experimental data. The table should ideally predict the hierarchy of masses, the spectrum of charges and spins, the relative stabilities, and the interaction strengths of observed fundamental particles and forces, and potentially predict new ones. The table will also include information on composite patterns and their constituent fundamental patterns, revealing the hierarchical structure of emergent reality. The emergence of fundamental constants (like particle masses, coupling constants, mixing angles, potentially even $\hbar, c, G$ as ratios of emergent properties or parameters of the configuration) as specific values derived from the configuration's dynamics and parameters is a key prediction captured in the table. The table should also include predictions for predicted but unobserved patterns (e.g., dark matter candidates, new force carriers, exotic composite states, signatures of emergent fields) along with their predicted AQNs and $I_R$. The process of populating this table involves computationally identifying patterns, classifying them into types, computing their AQNs and $I_R$ by analyzing their structure and behavior across many simulation steps and ensemble runs, and statistically characterizing their properties. This requires sophisticated data analysis pipelines (DC1, DC2 in `D-P6.2-4`, 4.1.2.3 in AUTX-M1). The table serves as the direct bridge between the abstract Autaxys configuration and the observable universe, providing the specific predictions to be validated against empirical data. * **H1.3 (Connection to Physics - Empirical Validation):** Through a systematic, iterative process of computational simulation (MVU-n, progressing from MVU-1 and MVU-2 to more complex simulations with richer $\Pi, \mathcal{R}, L_A$), detailed analysis of emergent phenomena (patterns, AQNs, $I_R$, emergent large-scale structure, emergent spacetime properties, emergent quantum phenomena, emergent conservation laws, emergent thermodynamic properties), and rigorous quantitative comparison with empirical data from physics—including observed particle properties (masses, charges, spins, lifetimes, decay modes, scattering cross-sections, mixing angles), fundamental forces (strength, range, symmetry properties, mediated particles), cosmological structure (LSS, CMB, expansion rate, dark matter/energy), spacetime properties (metric, curvature, causality, dimensionality, gravitational phenomena), and specific experimental results testing fundamental symmetries and quantum phenomena—the Autaxys Configuration ($\Pi_D, \Pi_R, \mathcal{R}, L_A(G), G_0$, and parameters of the Exploration Drive) will be iteratively refined (4.1.4 in AUTX-M1). The hypothesis predicts that a specific, parsimonious configuration exists such that the properties (statistical distributions of AQNs) and emergent interaction rules ($I_R$) of the most stable, frequent, and energetically favorable $P_{ID}$s generated by the framework will quantitatively and qualitatively correspond to the observed properties and interactions of fundamental particles and forces in the physical universe. Furthermore, the emergent large-scale structure and dynamics of the graph should match observed cosmology. This provides a direct path for empirical validation and potential falsification of specific configurations and the overall framework. The ability to reproduce known physics with a parsimonious configuration (measured by the complexity of $\Pi, \mathcal{R}, L_A$ - see 5.2 in AUTX-M1) and high statistical fidelity is the primary benchmark for scientific viability. Parsimony in this context means a configuration with minimal complexity in its definition: small $\Pi_D, \Pi_R$, small number of rules in $\mathcal{R}$, simple structure of rules, and a simple functional form for $L_A$ with few parameters. This aligns with the philosophical preference for Ockham's Razor – the simplest generative principles that produce the observed complexity are preferred. This involves comparing statistical distributions of emergent properties to experimental measurements, including error bars and confidence intervals, using standard statistical tests. The framework must explain the values of fundamental constants as emergent properties or parameters derived from the configuration's dynamics and parameters, which should ideally be few in number and simple in form in the fundamental configuration. It must also reproduce emergent conservation laws and symmetries observed in physics, hypothesizing their emergence from the algebraic structure of proto-properties and the symmetry properties of $\mathcal{R}$ and $L_A$. The goal is to find the minimal generative principles that predict the observed universe. This requires defining clear mappings between computational outputs (graph metrics, AQN distributions, rule application statistics, emergent relational structures, emergent causal structure, emergent metric properties, statistical properties of emergent pattern interactions) and empirical observables (particle masses, cross-sections, cosmological parameters, spacetime curvature, etc.). The statistical comparison should be quantitative and rigorous. Failure to find such a configuration after extensive, systematic search within a reasonable space of simple configurations would constitute strong evidence against the Autaxys hypothesis. * **H1.4 (Predictive Power):** Once an Autaxys configuration is identified through the iterative refinement process that successfully reproduces a significant portion of known physics (H1.3) with high fidelity and parsimony, the framework becomes a predictive scientific tool. It can be used to predict the existence and properties (AQNs) of new, as yet unobserved stable or meta-stable patterns ($P_{ID}$s) that are robust attractors of the configuration but haven't been detected experimentally (e.g., new particles beyond the Standard Model, dark matter candidates, signatures of emergent fields). It can predict the outcomes and rates of interactions ($I_R$) at energy scales or under conditions not yet probed experimentally (e.g., high-energy scattering outcomes, behavior in extreme environments, interactions between known particles and predicted new patterns, precise values for coupling constants or mixing angles, specific decay chains and branching ratios). It can predict novel emergent phenomena ($I_R$, spacetime properties, cosmological features, phase transitions in the structure of reality, specific features of emergent gravity or quantum behavior, violations or modifications of assumed principles like Lorentz invariance or locality at fundamental scales, specific signatures in gravitational wave data or CMB) that differ from current predictions. These predictions provide concrete, testable targets for future experiments in particle physics, cosmology, astrophysics, and potentially other fields. Falsification of these predictions (e.g., failure to discover a predicted particle with specified properties within experimental reach, observation of phenomena incompatible with predicted emergent spacetime properties or cosmological features, experimental results that contradict predicted interaction outcomes or rates) would rule out the specific configuration and provide guidance for further refinement or potential rejection of the overall hypothesis if no simple, reality-matching configuration can be found after extensive search. The framework must predict the *statistical distributions* of properties and interactions, reflecting the probabilistic nature of quantum mechanics, and these distributions must match experimental observations within statistical uncertainties. The ability to make successful *novel* predictions beyond the data used for tuning is the hallmark of a successful scientific theory. The framework should also be able to provide postdictions for existing puzzles not explicitly used in tuning (e.g., the hierarchy problem, neutrino masses and mixing, the matter-antimatter asymmetry, the cosmological constant problem, the nature of dark matter and dark energy, the origin of inflation) by showing that these are natural, emergent consequences of the identified configuration's dynamics. The goal is to move from fitting existing data to making new, falsifiable predictions that distinguish Autaxys from other theoretical frameworks. * **H1.5 (Emergence of Spacetime):** Spacetime itself, with its observed dimensionality (3 spatial, 1 temporal), metric structure (defining distances and durations), causal relationships, and dynamics (e.g., curvature, expansion, speed of light, gravitational phenomena), is hypothesized to be an emergent, macroscopic, and coarse-grained property of the relational graph dynamics, not a pre-existing arena or fundamental entity (H1.5 in `D-P6.2-4`). Properties analogous to spatial distance, temporal duration, causality, and metric coefficients are expected to arise from the large-scale structure, connectivity patterns, dynamics of rule application (including its locality, sequence, potential for parallel application, and rate), and specific types of relations and their transformations within the evolving ARG, particularly as mediated by specific $P_{ID}$s (potentially gravitational analogues) or global graph properties. The structure and dynamics of spacetime are thus intimately tied to the collective behavior of emergent patterns and their interactions, potentially exhibiting discrete, non-manifold, fractal, or fundamentally relational properties at the most fundamental scales before coarse-graining into the continuum spacetime of General Relativity and Quantum Field Theory at larger scales. Spacetime is a property of the organization of the fundamental relational substrate, not a separate container. * **Emergent Causality/Time:** The discrete, sequential nature of rule applications (P2, P3) provides a fundamental ordering, defining a discrete "cosmic time step." The directed nature of relations (P1) and the way rules propagate influence across the graph can give rise to an emergent causal structure. Macroscopic, continuous time emerges from the aggregate effect, ordering, and rate of these microscopic causal steps across the entire graph. The direction of the emergent arrow of time might be linked to the macroscopic increase in $L_A$ (H1.1), the increase in complexity ($C$), or the increasing number/stability of $P_{ID}$s, suggesting a connection between cosmic evolution, complexity, and the perception of time's flow. Time might be better understood as a measure of accumulated change or computational steps in the substrate, or as a dimension orthogonal to the state space defined by rule application sequences. The rate of emergent time might vary across different regions of the graph depending on local activity, rule application rates, or structure, potentially linking to concepts of time dilation. The causal structure of the graph could converge to a causal set structure at large scales. The speed of light might emerge as a maximum rate of information propagation or causal influence across the graph, limited by the discrete steps and the "length" of relational paths or by the rate of rule application. The arrow of time could also be tied to the breaking of time-reversal symmetry in $\mathcal{R}$ or $L_A$, or to the probabilistic nature of the selection process, or to the emergent causal structure. The emergence of time as a one-dimensional flow from an underlying discrete, relational process is a key prediction. * **Emergent Space/Distance:** Spatial distance is hypothesized to emerge not as a fundamental property of a background manifold, but from the relational structure of the graph (H1.5 in `D-P6.2-4`). Relational "proximity" or "distance" between two distinctions or patterns might be defined by various graph-theoretic or dynamic measures (e.g., Graph Distance, Number/Type of Relations, Strength/Frequency of Interaction, Shared Context, Information Distance, Resistance Distance, Effective Metrics from Dynamics). Discrete curvature measures can provide local geometric information. A graph metric could be defined on the set of nodes or patterns based on these relational measures, approximating a continuous spatial metric at large scales. The "dimensionality" of emergent space could arise from graph dimensions (e.g., embedding dimension, fractal dimension, spectral dimension, growth rate of graph volume, number of independent propagation directions, causal set structure, persistent homology). Specific relation types or patterns might act as 'metric carriers'. The perceived 3+1 dimensions of spacetime are hypothesized to be the dimensionality of the dominant, stable emergent structures or relation types that form the effective background for emergent particle interactions, potentially arising from specific algebraic properties of proto-properties or symmetries of $\mathcal{R}$ or $L_A$ that favor the formation of structures with these dimensional characteristics. The topology of emergent space would be determined by the graph's large-scale structure and dynamics. The emergence of specific spatial dimensions (e.g., 3+1) as a preferred emergent property is a key prediction. * **Emergent Spacetime Dynamics & Gravity:** Gravity is hypothesized to be an emergent force ($I_R$) arising from the collective dynamics of specific patterns (potentially gravitational analogues) and their influence on the large-scale relational structure, which is perceived as spacetime curvature (H1.5 in `D-P6.2-4`). Changes in the density or configuration of these patterns could alter the local 'relational metric' defined by the graph structure/dynamics, leading to effects analogous to gravitational attraction or spacetime warping. The dynamics of the graph at large scales (e.g., overall growth rate, connectivity changes, phase transitions, formation of dense or sparse regions, changes in effective dimensionality) could correspond to cosmological phenomena like expansion or contraction. The speed of light might emerge as a maximum rate of information propagation or causal influence across the graph. Entanglement and non-locality might be naturally represented by specific, non-local relational structures or proto-property correlations that persist across large graph distances or are established by specific rules or favored by the $L_A$ principle, suggesting the underlying substrate is fundamentally non-spatial or exists in an emergent space with more complex topology or higher dimensionality at the fundamental level. Deriving the specific properties of observed spacetime (Lorentz invariance, the form of the metric tensor, Einstein's field equations as effective equations) from the underlying graph dynamics and emergent patterns is a major long-term goal requiring sophisticated techniques for coarse-graining discrete graph structures and dynamics into continuum field theories or effective force laws. This requires identifying the macroscopic observables of the graph that correspond to metric components and stress-energy tensors and showing that their dynamics approximate Einstein's equations or their quantum gravity counterparts. This could involve applying techniques from statistical mechanics of networks, coarse-graining of graph rewriting systems, or deriving effective field theories from the underlying discrete dynamics. The emergence of Lorentz invariance as a symmetry of the emergent $I_R$ and spacetime dynamics at high effective speeds is a key prediction. The framework aims to explain gravity as an emergent phenomenon arising from the dynamics of the relational substrate and the distribution of emergent patterns within it. Einstein's field equations might emerge as effective equations describing the dynamics of this emergent metric as influenced by the distribution and dynamics of other emergent patterns (matter/energy analogues). Black holes and gravitational waves would be emergent phenomena in the large-scale graph dynamics. The framework should explain the geometric nature of gravity and its link to energy/momentum. The relationship between the emergent spacetime properties and the emergent quantum properties ($L_A$ terms $\Lambda_{spacetime}, \Lambda_{quantum}$) is critical, aiming to show that quantum mechanics and gravity are not fundamentally separate forces but different emergent aspects of the same underlying relational dynamics, potentially unifying them at the most fundamental level via the structure of the graph, rules, and $L_A$. The challenge includes deriving the quantum behavior of emergent gravity and the gravitational influence on emergent quantum particles.