# Geometric Dynamics of Agentic Loops in Large Language Models

arXiv: https://arxiv.org/abs/2512.10350 · HTML: https://mlsanity.com/paper/2512.10350/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 8 of its 63 references that are in the map.
- First posted on arXiv 11 Dec 2025.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)
- LLM agents and tool use / [General LLM agent frameworks and benchmarks](https://mlsanity.com/agents/general-llm-agent-frameworks-and-benchmarks/)

## Builds on (most cited first, 8 of 8)

- [Chain-of-Thought Prompting Elicits Reasoning in Large Language Models](https://mlsanity.com/paper/2201.11903/) (2022)
- [ReAct: Synergizing Reasoning and Acting in Language Models](https://mlsanity.com/paper/2210.03629/) (2022)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [Generative Agents: Interactive Simulacra of Human Behavior](https://mlsanity.com/paper/2304.03442/) (2023)
- [Self-Refine: Iterative Refinement with Self-Feedback](https://mlsanity.com/paper/2303.17651/) (2023)
- [Understanding Contrastive Representation Learning through Alignment and Uniformity on the Hypersphere](https://mlsanity.com/paper/2005.10242/) (2020)
- [How Contextual are Contextualized Word Representations? Comparing the Geometry of BERT, ELMo, and GPT-2 Embeddings](https://mlsanity.com/paper/1909.00512/) (2019)
- [Assessing and Understanding Creativity in Large Language Models](https://mlsanity.com/paper/2401.12491/) (2024)

## Built on by (most cited first, 4 of 4)

- [Attractor States Emerge in Multi-Turn LLM Conversations](https://mlsanity.com/paper/2606.30571/) (2026)
- [QUIVER: A Formal Framework for Quantifying Perturbation Propagation and Bifurcation in Compound AI Systems](https://mlsanity.com/paper/2605.23956/) (2026)
- [Perturbation Dose Responses in Recursive LLM Loops: Raw Switching, Stochastic Floors, and Persistent Escape under Append, Replace, and Dialog Updates](https://mlsanity.com/paper/2605.02236/) (2026)
- [Towards Better Exploration in Sequential Test-Time Scaling](https://mlsanity.com/paper/2609.39632/) (2026)

## Abstract (arXiv)

Iterative LLM systems(self-refinement, chain-of-thought, autonomous agents) are increasingly deployed, yet their temporal dynamics remain uncharacterized. Prior work evaluates task performance at convergence but ignores the trajectory: how does semantic content evolve across iterations? Does it stabilize, drift, or oscillate? Without answering these questions, we cannot predict system behavior, guarantee stability, or systematically design iterative architectures. We formalize agentic loops as discrete dynamical systems in semantic space. Borrowing from dynamical systems theory, we define trajectories, attractors and dynamical regimes for recursive LLM transformations, providing rigorous geometric definitions adapted to this setting. Our framework reveals that agentic loops exhibit classifiable dynamics: contractive (convergence toward stable semantic attractors), oscillatory (cycling among attractors), or exploratory (unbounded divergence). Experiments on singular loops validate the framework. Iterative paraphrasing produces contractive dynamics with measurable attractor formation and decreasing dispersion. Iterative negation produces exploratory dynamics with no stable structure. Crucially, prompt design directly controls the dynamical regime - the same model exhibits fundamentally different geometric behaviors depending solely on the transformation applied. This work establishes that iterative LLM dynamics are predictable and controllable, opening new directions for stability analysis, trajectory forecasting, and principled design of composite loops that balance convergence and exploration.