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Do Language Models Converge to Themselves? Recursive Self-Refinement as Textual Relaxation

This paper demonstrates that recursive self-refinement in large language models acts as a dynamical process where text rapidly converges to a stable, model-preferred equilibrium characterized by exponential decay in edit magnitude, rather than undergoing indefinite optimization.

Original authors: Xuening Wu, Qianya Xu, Yanlan Kang, Zeping Chen, Yubin Liu, Shenqin Yin

Published 2026-07-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Xuening Wu, Qianya Xu, Yanlan Kang, Zeping Chen, Yubin Liu, Shenqin Yin

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a magical, super-smart editor who can rewrite anything you write to make it clearer and better. You give it a draft, it spits out a polished version, and you think, "Great! Let's do it again!" So you feed the new version back to the same editor, who tweaks it again. You keep doing this, over and over, hoping the text gets infinitely perfect. This is the world of Large Language Models (LLMs), the AI brains behind many modern chatbots and writing tools. Scientists call this process "recursive self-refinement." But here's the big question: Does this endless loop keep making things better forever, like a video game character leveling up infinitely? Or does the text eventually hit a wall where the AI just starts making tiny, pointless changes, like a car engine idling in neutral? To understand this, we need to look at dynamical systems, a branch of science that studies how things change over time, and the idea of a fixed point, which is a state where a system stops changing because it has found its "comfort zone."

A team of researchers decided to treat this AI editing process like a science experiment to see what happens when you let an AI edit its own work repeatedly. They didn't just ask, "Is the final result good?" They asked, "How does the text move as it gets edited?" They treated the text like a ball rolling down a hill. Does it keep rolling forever, or does it eventually settle into a valley?

The researchers used a powerful AI model to rewrite 50 scientific abstracts (short summaries of research papers) ten times in a row. They watched closely to see how much the text changed with each rewrite. They found that the AI didn't keep improving the text forever. Instead, the changes happened fast at the beginning, and then the text quickly settled down. It's like when you first start tuning a guitar; you make big, obvious adjustments to get the strings in the right place. But after a few turns of the peg, the string is so close to the right note that you're just making tiny, almost invisible tweaks. The researchers call this settling down "textual relaxation."

Here is what they discovered in detail:

The Fast Stop: The AI made most of its changes in the first few rounds. By the time it reached the 3rd or 4th rewrite, the text was already very stable. After that, the AI was mostly just fiddling with small details like commas or hyphenation.
The "Soft" Landing: The text didn't stop changing completely (which would be an "exact" stop). Instead, it entered a "soft fixed-point region." This means the text was so good that the AI couldn't really find a better version, so it just made tiny, surface-level wiggles.
The Math of Settling: The size of the changes followed a very predictable pattern called an "exponential relaxation." Imagine a cup of hot coffee cooling down; it cools fast at first, then slows down as it approaches room temperature. The text changes did the exact same thing. The amount of editing dropped sharply and then flattened out near a tiny "floor" of small changes.
Temperature Matters: The researchers tested two settings. When they set the AI to be very strict and deterministic (temperature 0), it stopped changing completely and found an "exact" fixed point very quickly (on average, in just 2.6 rounds). When they let the AI be a bit more creative and random (default temperature), it still settled down, but it kept wiggling around a bit more, taking longer to stop moving.
It Actually Gets Better: A common worry is that if the text stops changing, the AI is just stuck and not improving. But the researchers checked this with an outside judge (another AI). They found that the final versions were indeed clearer, more concise, and better written than the originals, even though the AI stopped making big changes. The "stopping" wasn't because the AI gave up; it was because it had found the best version it could make.

The researchers also tried this on older papers from 2020, and the same thing happened. The AI settled down quickly no matter what year the paper was from.

So, what does this mean for us? It suggests that when we use AI to rewrite our work, we don't need to ask it to edit a hundred times. The magic happens in the first few rounds. After that, the AI is just spinning its wheels. The paper suggests that we can use the size of the changes as a signal: once the changes get tiny and stop getting any smaller, we should just stop. We've reached the "textual equilibrium," and any more editing is just noise. This turns a mysterious, endless loop of AI editing into a predictable process with a clear finish line.

Technical Summary: Do Language Models Converge to Themselves?

Problem Statement

Large language models (LLMs) are increasingly deployed in recursive self-refinement workflows, where an initial draft is repeatedly revised by the same model. While these workflows are common in scientific writing, coding, and agentic systems, their long-term dynamical behavior remains poorly understood. The central question addressed is whether repeated refinement leads to indefinite improvement or if the process relaxes toward a stable textual form. Existing evaluations typically focus on single-step quality improvements rather than the dynamics induced by repeated self-application, leaving open questions regarding convergence, stability, attractors, and appropriate stopping criteria.

Methodology

The authors frame recursive self-refinement as a discrete dynamical process over text, defined by the iterative application of a refinement operator $R_\theta$ :
$V_{t+1} = R_\theta(V_t)$
where $V_t$ represents the text at iteration $t$ .

Experimental Setup:

Datasets: The study utilizes 50 abstracts from ICML 2025 and a cross-year comparison set of 15 abstracts from ICML 2020.
Model: GPT-5.5 is used as the refinement operator.
Prompting: The model is instructed to act as an expert scientific editor, improving clarity, coherence, and conciseness while strictly preserving all technical claims, methods, results, and numerical values.
Decoding Conditions: Experiments are conducted under two settings: default-temperature and deterministic decoding (temperature 0).
Trajectory Length: Each abstract undergoes $T=10$ refinement iterations.

Metrics and Analysis:

Transition Magnitude ( $\Delta_t$ ): Measured using normalized Levenshtein edit distance between consecutive versions, $d(V_t, V_{t+1})$ , normalized by the maximum length of the two texts.
Fixed-Point Definitions:
- Exact Fixed Point: $V_{t+1} = V_t$ .
- Approximate Fixed Point: Defined by $\Delta_t < \epsilon_{edit}$ (0.02) and relative word-count change $< \epsilon_{wc}$ (0.03) for two consecutive transitions.
Relaxation Modeling: The authors fit an exponential decay model to the average transition magnitude: $\Delta_t \approx Ae^{-kt} + c$ , where $c$ represents a residual fluctuation floor.
External Evaluation: A subset of trajectories is evaluated by an external LLM-as-a-judge (Gemini 3.5 Flash) to assess clarity, conciseness, scientific style, and technical fidelity, ensuring convergence does not equate to textual stagnation.

Key Contributions

Theoretical Framing: The paper introduces the concept of "textual relaxation," framing recursive self-refinement as a discrete dynamical process that drives text toward a model-preferred "soft fixed-point region" rather than an open-ended optimization procedure.
Empirical Evidence of Saturation: The study provides empirical evidence that refinement trajectories rapidly saturate. Most edits occur within the first few iterations, after which the system enters a region characterized by small residual surface-level changes.
Decoding Temperature Effects: The authors demonstrate that decoding temperature controls the nature of the fixed-point region. Deterministic decoding (temperature 0) reaches exact fixed points more rapidly with lower residual fluctuations, while default-temperature decoding exhibits larger residual fluctuations around a similar soft fixed-point region.
Practical Stopping Criteria: The work motivates the use of edit-magnitude saturation as a practical stopping criterion for recursive workflows, suggesting that iterative refinement can be terminated once changes fall below a specific threshold.

Results

Rapid Convergence: On 50 ICML 2025 abstracts, GPT-5.5 achieved approximate convergence for all trajectories under both decoding settings. Exact fixed points were reached by 100% of trajectories under temperature 0 (mean convergence time: 2.6 iterations) and 94% under default temperature (mean: 4.66 iterations).
Exponential Relaxation: The average edit magnitude $\Delta_t$ follows a consistent exponential relaxation pattern ( $R^2 = 0.990$ for the default-temperature condition). This indicates a rapid decay in edit pressure followed by a small residual floor, supporting the relaxation hypothesis over indefinite optimization.
Quality Improvement: External evaluation by the Gemini judge confirmed that converged abstracts ( $V_{10}$ ) showed significant improvements in clarity, conciseness, and scientific style compared to the original ( $V_0$ ) and first-refined ( $V_1$ ) versions, while maintaining perfect technical fidelity and meaning preservation.
Cross-Year Consistency: The convergence behavior observed in ICML 2020 abstracts was qualitatively similar to ICML 2025, suggesting the phenomenon is not unique to contemporary LLM-era writing, though 2020 abstracts exhibited slightly larger initial edit distances and later exact convergence.

Significance and Claims

The paper argues that recursive self-refinement is better understood as textual relaxation than as open-ended optimization. The authors claim that:

Repeated revision drives text toward a model-preferred equilibrium of scientific writing style.
Once the text approaches this region, further refinements yield diminishing returns, primarily introducing minor surface-level fluctuations rather than substantive improvements.
The distinction between exact and approximate fixed points is crucial; approximate convergence better captures the practical stopping point where additional iterations become redundant.
Viewing LLM self-refinement through the lens of dynamical systems (convergence, stability, attractors) offers a compact framework for analyzing iterative AI behaviors and establishing robust stopping criteria.

The authors remain modest regarding their claims, noting that their convergence metrics are surface-based and that the exponential relaxation model is an empirical description rather than a proof of physical laws or gradient descent mechanisms. They acknowledge limitations in scope (focusing on scientific abstracts and a single model) and call for future work to analyze stability in continuous representation spaces.