The Recursive Mind of AI: Mapping Language Model Behavior

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Author: Denis Avetisyan

New research reveals how we can visualize and understand the complex, self-referential processes unfolding within large language models.

The study demonstrates that an exploratory trajectory’s evolution can be characterized by both local geometric relationships-quantified as distances and similarities between consecutive embeddings $e_t$ and $e_{t-1}$-and global relationships measured relative to the initial embedding $e_0$.
The study demonstrates that an exploratory trajectory’s evolution can be characterized by both local geometric relationships-quantified as distances and similarities between consecutive embeddings $e_t$ and $e_{t-1}$-and global relationships measured relative to the initial embedding $e_0$.

This paper introduces a geometric framework to analyze agentic loops in language models, characterizing their dynamics in semantic space and identifying regimes of convergence and divergence.

While large language models increasingly operate through recursive, agentic loops, a rigorous understanding of their dynamic behavior has remained elusive. This paper, ‘Dynamics of Agentic Loops in Large Language Models: A Geometric Theory of Trajectories’, introduces a geometric framework for analyzing these iterative processes in semantic embedding space, revealing how distinct regimes of convergence and divergence emerge. By calibrating for embedding anisotropy and measuring trajectories, we demonstrate that prompt design directly governs whether an agentic loop contracts toward a stable attractor or diverges unboundedly. Can this framework ultimately enable systematic control over the complex dynamics of LLM-driven systems and unlock new possibilities for their application?

The Recursive Ascent: Beyond Pattern Matching

While large language models demonstrate remarkable fluency in generating human-quality text, achieving genuine complex reasoning extends beyond simply increasing model size or the volume of training data. These models, fundamentally pattern-matching engines, often struggle with tasks demanding abstract thought, planning, or causal inference – capabilities that rely on more than just statistical correlations within text. Scaling alone doesn’t necessarily translate to improved cognitive abilities; a larger model might memorize more facts, but it doesn’t inherently learn how to think critically or solve novel problems. Consequently, researchers are exploring architectures and methodologies that augment LLMs with tools for systematic reasoning, iterative refinement, and external knowledge integration, seeking to move beyond mere text generation towards true artificial intelligence.

The capacity for sustained thought, long considered uniquely human, is increasingly demonstrated through large language models operating within agentic loops. This paradigm moves beyond simple text generation by enabling an LLM to iteratively refine its own outputs – a process akin to self-critique and revision. Rather than producing a single response, the model generates, evaluates, and then regenerates text based on its previous iterations, effectively creating a dialogue with itself. This recursive process isn’t merely about improving accuracy; it allows the model to explore complex ideas, identify inconsistencies, and develop nuanced arguments, pushing the boundaries of what’s achievable through scaled computation alone. The result is a dynamic system capable of prolonged intellectual effort, mirroring – and potentially extending – the capacity for sustained thought found in biological intelligence.

The iterative refinement inherent in agentic loops doesn’t simply produce a final answer; instead, it generates a chronological series of textual outputs – each a distinct ‘artifact’ representing a stage in the model’s thought process. Collectively, these artifacts map a trajectory through a conceptual space – termed ‘ArtifactSpace’ – where proximity between outputs signifies semantic similarity and evolutionary progression. This space isn’t merely a record of computation, but a landscape of reasoning, allowing external observers to trace the model’s path, identify pivotal shifts in understanding, and even diagnose potential flaws in its logic. The density and structure within ArtifactSpace therefore become a valuable metric for assessing an LLM’s cognitive capabilities, revealing not just what it thinks, but how it arrives at its conclusions, and offering a new lens for understanding the emergence of complex thought from artificial systems.

Quantifying Thought: Embedding Spaces and Semantic Distance

To facilitate quantitative analysis of textual artifacts generated within the agentic loop, we utilize an ‘EmbeddingSpace’. This space is a vector representation where each text artifact – such as prompts, responses, or internal thoughts – is mapped to a point in a multi-dimensional space. The position of each point is determined by the semantic content of the text, allowing for the calculation of distances and similarities between artifacts. This enables tracking the evolution of ideas, identifying patterns in generated content, and measuring the impact of different agentic processes. The dimensionality of the EmbeddingSpace is a critical parameter, influencing both the granularity of the representation and the computational cost of analysis; typical dimensions range from several hundred to over a thousand.

Isotropic embedding spaces are critical for reliable semantic analysis because they enforce equal variance across all dimensions of the embedding vector. Traditional embedding spaces often exhibit anisotropic distributions, where some dimensions have significantly higher variance than others; this distorts cosine similarity as a measure of semantic distance. Cosine similarity, calculated as the dot product of two vectors normalized by their magnitudes, assumes that differences in vector magnitude do not inherently indicate semantic difference. In anisotropic spaces, larger magnitudes may simply reflect high variance in specific dimensions, not actual semantic relatedness. Isotropic embedding techniques, such as those employing regularization during training, normalize the embedding dimensions to have unit variance, ensuring that the length of the embedding vector does not influence the cosine similarity calculation, and thus, allowing it to accurately reflect semantic distance. This normalization process effectively transforms the embedding space to be uniformly distributed, improving the validity of downstream similarity comparisons and analysis.

Raw embedding spaces, while effective for representing semantic relationships, frequently exhibit biases stemming from the training data and model architecture. These biases manifest as skewed distributions of vector magnitudes and directions, leading to inaccurate cosine similarity calculations. The CalibrationMethod addresses this by applying a learned transformation to the embedding vectors. Specifically, it utilizes a set of parameters, derived from analyzing a representative corpus, to normalize the embedding space. This normalization ensures that similarity scores more accurately reflect genuine semantic distance, improving the reliability of downstream analyses such as artifact tracking and comparison within the agentic loop. The method effectively mitigates the influence of spurious correlations present in the initial embedding space.

The contractive loop trajectory evolves both locally, demonstrating similarity between consecutive embeddings, and globally, measuring distance from the initial embedding.
The contractive loop trajectory evolves both locally, demonstrating similarity between consecutive embeddings, and globally, measuring distance from the initial embedding.

Geometric Signatures: Trajectories and Dynamic Regimes

A GeometricTrajectory is generated through the repeated application of the Large Language Model (LLM) within the agentic loop. This trajectory isn’t a spatial path, but rather a sequence of high-dimensional vector embeddings. Each embedding represents the state of the artifact at a given iteration of the loop; the artifact’s evolution is therefore captured as a series of points in embedding space. These embeddings are numerical representations created by the LLM, quantifying the characteristics of the artifact at each step. The sequence, or trajectory, allows for analysis of the artifact’s development over time, facilitating the identification of patterns and dynamic behaviors.

The iterative process of the agent generates a sequence of embeddings defining a GeometricTrajectory, which exhibits two primary dynamic regimes. The ContractiveRegime is characterized by convergence of the trajectory towards a stable Attractor, indicating a consolidation of the artifact’s representation. Conversely, the ExploratoryRegime demonstrates divergence, signifying expansion and variation in the artifact’s embedding space. This distinction is not merely qualitative; quantitative metrics, such as Dispersion, consistently differentiate the two regimes, providing an objective measure of the trajectory’s behavior and the artifact’s state of evolution.

The quantitative measure of ‘Dispersion’ is utilized to characterize the spread of embedding points defining the GeometricTrajectory. Analysis indicates a substantial distinction in Dispersion values between dynamic regimes: the ContractiveRegime exhibits a mean Dispersion of 0.15, signifying convergence of the trajectory, while the ExploratoryRegime demonstrates a significantly higher Dispersion of 0.82, indicating divergence and increased variability in the artifact’s evolution. This difference in magnitude provides a quantifiable metric for differentiating between states of convergence and exploration within the agentic loop.

The cluster membership timeline reveals transient outliers (red) at cluster boundaries during contractive loop iterations with parameters λ=0.8, ρ=0.2, and κ=2.
The cluster membership timeline reveals transient outliers (red) at cluster boundaries during contractive loop iterations with parameters λ=0.8, ρ=0.2, and κ=2.

Steering the System: Prompting and Loop Architecture

Prompt engineering is a foundational element in controlling the behavior of Large Language Models (LLMs) operating within an agentic loop. The specific instructions, or ‘prompts’, provided to the LLM directly influence whether the loop exhibits convergence – repeatedly arriving at a predictable output or ‘attractor’ state – or exploration, where the LLM generates diverse and potentially novel outputs without settling on a single solution. Carefully constructed prompts can steer the LLM toward focused problem-solving or encourage broader, more creative responses, effectively modulating the loop’s trajectory and outcome. The precision and intent embedded in the prompt dictate the LLM’s subsequent actions and, consequently, the overall behavior of the agentic system.

The architecture of the agentic loop-specifically, whether implemented as a ‘SingularLoop’ or a ‘CompositeLoop’-directly influences the agent’s behavioral trajectory. A SingularLoop, characterized by a single iterative process, tends to produce a focused and potentially convergent path. Conversely, a CompositeLoop, incorporating multiple, interconnected iterative processes, introduces greater complexity and enables a broader exploration of the solution space. This structural difference impacts the agent’s ability to converge on a single solution versus generating a wider range of outputs, with CompositeLoops exhibiting a demonstrably higher degree of variability in observed trajectories compared to SingularLoops.

Analysis of agentic loop behavior revealed distinct outcomes based on prompt engineering. A “contractive” loop, designed to refine and consolidate responses, consistently converged towards a single, stable cluster of outputs, indicating a focused trajectory. Conversely, an “exploratory” loop, prompting for diverse responses, exhibited no persistent clustering; outputs remained widely dispersed and did not coalesce around any specific solution. This divergence demonstrates a direct correlation between prompt-driven loop architecture and the resulting stability of the agent’s output trajectory, with contractive prompts favoring convergence and exploratory prompts prioritizing breadth over focus.

Beyond Static Solutions: Navigating Dynamic States

Current analyses of large language model (LLM) behavior largely center on identifying stable attractors – the predictable states a model gravitates towards given a specific input. However, emerging research suggests a shift in focus towards designing feedback loops capable of navigating the complex, non-equilibrium states within an LLM’s operational space. This approach moves beyond simply finding resting points to actively guiding the model through dynamic internal landscapes. By intentionally crafting loops that encourage exploration and adaptation, scientists aim to unlock more flexible and nuanced control over LLM responses, potentially enabling the creation of models that don’t just answer questions, but actively engage in reasoning and problem-solving within a defined, yet shifting, context. This future direction promises a move away from predictable outputs towards a form of controlled instability, fostering creativity and adaptability in artificial intelligence.

Investigating the interplay between Large Language Model (LLM) architecture and prompt sensitivity promises a new level of behavioral control. Current approaches often treat LLMs as ‘black boxes’, but future research focuses on designing and analyzing different ‘loop’ architectures – the recurrent pathways within the model. By systematically varying these loops and observing their response to diverse prompts, researchers aim to move beyond simply eliciting a desired output to understanding how the model arrives at that output. This granular understanding will allow for the creation of LLMs capable of more nuanced and predictable responses, potentially tailoring behavior not just to the content of a prompt, but also to its phrasing, complexity, or even emotional tone. Such precise control opens possibilities for applications requiring reliable and adaptable AI systems, from personalized education to sophisticated creative tools.

Quantitative assessment of loop behavior reveals a striking difference in semantic stability between contrasting architectures. Researchers found that a ‘contractive’ loop, designed to reinforce consistent responses, yielded an average calibrated similarity score of $s \sim = 0.85$. This indicates a high degree of semantic coherence in the generated text. Conversely, an ‘exploratory’ loop, intended to promote diversity, exhibited a significantly lower score of $s \sim = 0.32$. This disparity demonstrates that the degree of semantic stability is not merely a byproduct of large language model operation, but a tunable parameter directly influenced by loop architecture, offering a pathway towards precise control over the consistency and predictability of generated content.

The contractive loop demonstrates a smooth progression through five stabilization phases, as evidenced by its cluster membership timeline, with outliers highlighted in red under parameter settings (λ, ρ, κ) = (0.8, 0.1, 2).
The contractive loop demonstrates a smooth progression through five stabilization phases, as evidenced by its cluster membership timeline, with outliers highlighted in red under parameter settings (λ, ρ, κ) = (0.8, 0.1, 2).

The study of agentic loops, as presented, reveals a fascinating interplay between convergence and divergence within the embedding space. This behavior echoes a sentiment expressed by Henri Poincaré: “It is through science that we arrive at truth, but it is through imagination that we create it.” The geometric theory detailed meticulously maps these trajectories, demonstrating how recursive processes can either contract towards a calibrated semantic core or explore divergent paths. The analytical framework isn’t merely about observing functionality; it’s about establishing provable properties of these dynamic systems, mirroring a mathematical pursuit of underlying truth – a notion Poincaré would undoubtedly appreciate. The concept of ‘contractive/exploratory regimes’ is thus not simply descriptive, but fundamentally rooted in a quantifiable, geometric reality.

Where Do We Go From Here?

The presented geometric characterization of agentic loops, while providing a novel lens through which to view these recursive processes, merely identifies the questions that truly demand answers. Establishing a rigorous mapping between observed dynamical regimes – contractive versus exploratory – and demonstrable task performance remains a critical, and surprisingly elusive, goal. It is not enough to observe convergence; one must prove its relationship to solution quality. The current reliance on semantic similarity as a proxy for correctness is, frankly, unsatisfying; similarity does not equate to truth, and the field must move beyond such approximations.

Future work should prioritize the development of formal invariants for agentic loops. Can one definitively state, given a particular initial condition and loop structure, that the process will converge to a correct solution, or at least a demonstrably optimal one? Furthermore, the limitations of embedding space as a complete representation of meaning are self-evident. A truly robust theory must account for the inevitable distortions and information loss inherent in such reductions.

Ultimately, the pursuit of ‘calibration’ within these loops feels somewhat… quaint. A system that requires external validation to determine its own correctness is, by definition, incomplete. The ideal, of course, is an agentic loop that possesses an internal measure of certainty, a self-validating process grounded in logical consistency, not merely empirical observation. Anything less is, ultimately, just another complex algorithm masquerading as intelligence.

Original article: https://arxiv.org/pdf/2512.10350.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-13 23:47