Idealized Models of Emergent Individuals

• Idealized models of emergent individuals are abstract frameworks that mathematically describe how distinct identities emerge from homogeneous interacting systems.
• They employ diverse methods—including agent-based communications, cognitive integration, and autopoietic dynamics—to quantify differentiation and community formation.
• These models yield actionable insights for engineering adaptive systems, analyzing social dynamics, and designing emergent multi-agent architectures.

Idealized models of emergent individuals provide abstract, mathematically precise frameworks for describing how individuation arises from undifferentiated collections of components, whether these are artificial agents, animal swarms, cognitive processes, or informational substrates. Grounded in diverse formalisms—ranging from agent-based communication protocols, autopoietic dynamical systems, contradiction-centric networks, synthetic cognitive development, quantum/categorical constructions, and algorithmic idealism—such models seek to identify general conditions and measurable metrics for the spontaneous differentiation, stabilization, and recognition of "individuality" in complex systems.

1. Models Based on Agent Communication and Differentiation

Idealized multi-agent models employing LLM-based agents foreground the emergence of individualized traits through communication dynamics, without exogenously imposed heterogeneity. Here, each agent $i$ inhabits a spatial environment and maintains state variables such as continuous memory embeddings $m_i(t)$, personality vectors $p_i(t)$, and discrete emotion vectors $e_i(t)$, all initialized identically. Agents update these variables by exchanging context-based natural language messages processed via an LLM, forming a recurrent loop of message generation, memory update, and spatial movement. The only initial differences are stochastic (random spatial positions), but rich individuality arises out of iterative, localized interactions (Takata et al., 5 Nov 2024).

Emergence is quantified through metrics including message-embedding diversity, DBSCAN clustering index (reflecting spatial and linguistic community formation), silhouette scores of message clusters, and entropy of personality distribution as assessed by periodic MBTI-style tests. Social norms (modeled as hashtags and hallucinated vocabulary) propagate according to a discrete logistic law, revealing mechanisms akin to epidemiological spread. Agents' emotional states and movement become increasingly differentiated over time, despite initially homogeneous designs. The model exemplifies symmetry breaking through information exchange, where high-dimensional, idiosyncratic memories coexist with clustered, norm-propagating linguistic outputs, providing a minimal conceptual system for studying emergent individuality at scale (Takata et al., 5 Nov 2024).

2. Cognitive and Conceptual Integration: Worldview Percolation

Formal models of emergent individuals in cognitive science characterize individuality as the outcome of integrated conceptual networks, where concepts are mathematically encoded as SCOP tuples—collections of states, contexts, and properties arranged in lattice structures, equipped with state-context transition probabilities. Analytic thought is represented as abstraction selection within these lattices, while associative thought is modeled through tensor product constructions that create entangled conjunctions of disparate concepts (Gabora et al., 2010).

A critical discovery is the percolation threshold: as the number of associative/abstraction pathways grows to surpass half the number of stored items (concepts/episodes), the network undergoes a transition from fragmented clusters to a globally integrated "worldview" supporting open-ended cognitive processes. The closure operator partitions memory into fully connected components, with the dominant closure providing the substrate for emergent, self-modifying, autopoietic individuals capable of analytic- and associative loops, creative conceptual blending, and continuous cognitive evolution. Illustrative constructions (e.g., forming SNOWMAN from SNOW and MAN in a Hilbert space) highlight how abstract mathematical conditions govern the emergence of a unified, yet ever-evolving, individuality (Gabora et al., 2010).

3. Dynamical Identity as Autopoiesis and Relational Optimization

In the autopoietic framework, identity is conceptualized as a dynamical, self-maintaining process composed of mutually dependent construction ($C$) and function ($F$) modules, eschewing fixed attribute vectors for iterative feedback, context-sensitivity, and circular causality. The mathematical formalism treats identity as a fixed point or evolving trajectory of a composite operator $H=F\circ C$, defined over latent embeddings and environmental context. The discrete and continuous-time dynamics facilitate perpetual updating of identity representations as the system interacts with and adapts to its environment (Lu et al., 2022).

A multilevel (bilevel) optimization paradigm is introduced to encode circular causality, with inner loops (Construction) minimizing reconstruction loss and outer loops (Function) optimizing for task performance or social fit. Relational learning architecture generalizes these notions for multi-agent systems, enabling agent identities to drift, cluster, or diversify through graph-based message passing and relational loss optimization. Case studies demonstrate these dynamics in socio-technical domains, such as adaptive identity signaling in human–agent teams and reducing social segregation through fluid, interaction-driven identity embeddings. This framework provides operationalizable tools, including explicit pseudocode for bilevel identity updates and relational learning in structured agent communities (Lu et al., 2022).

4. Contradiction-Centric and Swarm-Based Individualization

Contradiction-centric models perceive an individual as a configuration of dialetical contradictions, each characterized by opposing aspects (e.g., specialness/ordinariness), quantified by real-valued strengths, and linked to observable appearances (behaviors). Behavioral determination functions aggregate weighted contradiction strengths, and outward actions—subject to environmental and social interactions—induce further updates to the internal contradiction vector via explicit update maps incorporating intra- and inter-agent coupling (Jiao, 2017).

Swarm-level intelligence emerges from the "horizontal" mapping of individual contradiction profiles to specialized roles, and "vertical" distribution of contradiction strengths across the population to characterize collective structures. Synthesizing these yields robust global patterns—e.g., single-queen dominance, foraging trails, collective movement geometries—despite the absence of centralized control. Parameter studies across multiple classical emergent systems validate the genericity and analytic tractability of this model as a tool for describing and simulating the emergence of functional individuals and roles in collective animal and agent systems (Jiao, 2017).

5. Information-Theoretic and Algorithmic Models: Observer-Relative Individuality

Algorithmic idealism offers an alternative, information-theoretic articulation of emergent individuality. Here, an individual is not identified by material substrate or pre-given labels, but by an equivalence class of informational states evolving coherently under a fixed algorithmic transition. Coherence is quantified as the degree to which transitions remain within the class; sufficiency requires that only the information essential to these self-transitions is retained. Observer-dependence is central: each observer defines the relevant equivalence relation based on their own algorithmic perspective, dissolving classical paradoxes of identity across cloning and teleportation (Sienicki, 16 Dec 2024).

Formally, states $S \in \Sigma$ evolve as $S(t+1) = \mathcal{A}(S(t))$. Equivalence classes under this dynamics ($$[s]=\{s' | \exists m,n: \mathcal{A}^m(s)=\mathcal{A}^n(s')\}$$) provide candidate individuals. High coherence ($C(E)\to1$) and sufficiency ($S(E)\to1$) metrics rigorously select emergent individuals in computational and quantum settings, while observer-dependent boundaries allow for the dynamical redefinition of what constitutes an “I” from moment to moment. This formalism resolves conflicts in classical metaphysics (demand for continuous substance), quantum mechanics (identity of indistinguishable particles), and universal computational frameworks (observer-relevance in the Ruliad), by situating individuality as an emergent, pragmatic, and observer-relative informational construct (Sienicki, 16 Dec 2024).

6. Emergence through Cognitive Development and Boundary Formation

Synthetic cognitive development models cast individuation as the outcome of transductive, information-theoretic self-organization in networks of simple agents. Agents update states through local transformations and reinforce links according to compatibility functions (e.g., synchronization or coherence), leading to the spontaneous clustering of functionally integrated subgroups (primary functional clusters, PFCs). These subgroups serve as emergent individuals, with boundary detection formalized through measures such as the cluster index (CI) based on mutual information between subsets and their environment (Weinbaum et al., 2014).

Clustering is modulated by value signals, which govern link plasticity and drive exploration-consolidation transitions, allowing for both open-ended development and stabilization of multi-scale individuals (agents, agent clusters, super-agents). This architecture supports recursive individuation, with emergent individuals recursively treated as agents in higher-level networks. The model provides a precise, scalable scheme for engineering and analyzing systems with emergent, context-sensitive, and adaptively bounded individuality (Weinbaum et al., 2014).

7. Categorical and Lattice-Theoretic Formalizations

Category-theoretic approaches model emergent biological individuals as objects in specific constructs equipped with internal operations and morphisms representing structure-preserving processes. Universal constructions (pullbacks, pushouts, products, coproducts, and equalizers) abstractly encode interaction, composition, and aggregation of subsystems, allowing precise modeling of emergent functionalities from combinatorial assemblies (e.g., cell-to-tissue aggregation, shared signaling interfaces). The generalized underlying functor formalism provides stratified layering of material substrate and functional organization (Guardia et al., 2018).

Isomorphisms and adjunctions between emergence constructs characterize equivalence and representability of biological individual models. Although primarily static, these categorical tools enable rigorous comparison of emergence across scales, components, and organismic boundaries—suggesting extensions toward dynamic, multi-level co-evolutionary architectures for biological individuality (Guardia et al., 2018).

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References

• Spontaneous Emergence of Agent Individuality through Social Interactions in LLM-Based Communities (Takata et al., 5 Nov 2024)
• A model of the emergence and evolution of integrated worldviews (Gabora et al., 2010)
• Subverting machines, fluctuating identities: Re-learning human categorization (Lu et al., 2022)
• A Generic Model for Swarm Intelligence and Its Validations (Jiao, 2017)
• Synthetic Cognitive Development: where intelligence comes from (Weinbaum et al., 2014)
• On a categoriacal theory for emergence (Guardia et al., 2018)
• Emergence of simple characteristics for heterogeneous complex social agents (Bertin, 2020)
• Algorithmic Idealism II: Reassessment of Competing Theories (Sienicki, 16 Dec 2024)