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Same physical state, different collective dynamics: state encodings select synchronization outcomes in language-model agents

This paper demonstrates that state encodings are not neutral interfaces for language-model agents but function as model-dependent effective interaction laws that fundamentally determine collective synchronization outcomes, even when the underlying physical system remains fixed.

Original authors: Takahiro Ezaki, Naoto Imura, Katsuhiro Nishinari

Published 2026-08-10

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Takahiro Ezaki, Naoto Imura, Katsuhiro Nishinari

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 a world where thousands of tiny, invisible dancers are trying to move in perfect unison. They can't see each other directly; instead, they rely on a translator to describe the crowd's movements. If the translator says, "Everyone is moving slightly to the left," the dancer might step left. But what if the translator says, "The crowd's average position is 30 degrees north"? Or what if they say, "Here is a chart showing 24 different buckets of movement"? Even though the actual dance floor hasn't changed, the dancer's reaction might be totally different depending on how the information is described. This is the heart of a new study exploring how Artificial Intelligence (AI) agents behave when they are part of a group.

In the world of computer science, researchers are increasingly building "swarms" of AI agents—programs that talk to each other to solve problems, make decisions, or even simulate human societies. A key idea in physics and math is "synchronization," like fireflies blinking together or clocks ticking in time. Usually, scientists assume that if you give an AI the same facts, it will act the same way. But this new research asks a tricky question: Does the format of the facts matter? If you tell an AI "the crowd is 60% to the left" versus "here is a list of 24 numbers that add up to 60%," does the AI's brain process that differently? The answer could change how we build everything from traffic systems to financial markets, because if the "translation" changes the outcome, then the way we talk to AI is just as important as what we say.

The Great AI Dance-Off

The researchers set up a giant, digital dance floor with 17 AI agents. Each agent was a "phase oscillator," a fancy term for a dancer spinning in a circle. Their goal wasn't to win a prize; they were just told to look at their neighbors and decide whether to spin faster (advance), spin slower (retard), or stay put (stay). The computer engine that ran the dance floor was perfectly fair and identical for everyone. The only thing that changed was the "observation map"—the specific way the AI was told about its neighbors' positions.

The team tested three different ways to describe the same dance floor:

The "Moments" Map: This was like giving the AI a high-level summary, a "vibe check" that compressed the whole crowd's movement into just three numbers describing the overall direction and how tightly they were grouped.
The "Centers" Map: This was a detailed list of 24 buckets, telling the AI exactly how many dancers were in each slice of the circle, labeled by the center point of that slice.
The "Intervals" Map: This was the same detailed list of 24 buckets, but labeled by the edges of the slices instead of the centers.

Here is the twist: The "Centers" and "Intervals" maps contained the exact same amount of information (the same 24 numbers), just written with different labels. The "Moments" map had less raw data but a different kind of summary.

The Results: It's All About the Presentation

When the researchers ran the experiment with GPT (a specific AI model), the results were shocking. When the AI saw the "Moments" summary, the entire group of 17 dancers locked into perfect synchronization 100% of the time (6 out of 6 attempts). But when they saw the "Centers" or "Intervals" lists, the group never synchronized (0 out of 6 attempts). The AI agents acted as if they were in a completely different world, even though the physical dance floor was identical.

To make sure this wasn't just a fluke, they ran the same test with Claude and Gemini, two other AI models. The results were even more surprising. In Claude, the "Moments" summary caused the group to fail to synchronize, while the detailed "Centers" and "Intervals" lists actually made them lock together perfectly. The AI models didn't just react differently; they reacted in opposite directions.

This proves that the "observation map" isn't just a neutral window into reality. It is part of the AI's brain. Changing how you describe the world changes how the AI acts, and this effect survives even when the AI is part of a feedback loop where its actions change the world it sees next.

The Microscopic Mystery

To understand why this happened, the researchers looked at the AI's "microscopic response"—what the AI decided to do when shown a single, frozen snapshot of the dance floor. They found that the AI's decision-making "operator" (its internal rule for choosing advance, stay, or retard) shifted dramatically based on the encoding.

For example, when shown a specific pattern of dancers, the "Moments" AI might almost always choose to "advance," while the "Centers" AI might mostly choose to "stay." This happened even when the researchers used the exact same numbers, just rearranged. They even tested if it was just the length of the text or the position of the numbers that mattered. They found that adding "task-irrelevant" text (fluff that didn't change the numbers) changed the AI's behavior just as much as changing the data format itself.

The "Surrogate" Test: Can We Fake It?

Finally, the team asked a practical question: Could we train a simple, cheap computer program (a "surrogate") to predict what the expensive AI would do, so we don't have to run the real AI every time? They tried to build these surrogates for the three different maps.

The "Moments" surrogate could be trained to predict the AI's behavior well, but only after the researchers fixed a specific bug where the AI was supposed to "stay" but the surrogate predicted it would "act." However, the surrogates for the "Centers" and "Intervals" maps failed completely. Even though they could predict the AI's behavior on the training data, they couldn't handle the new, complex patterns that the AI generated when it was actually dancing in a group. The researchers found that for these maps, the AI's behavior was too inconsistent to be predicted by a simple model, suggesting that the way the information was serialized (written out) created a chaotic effect that simple math couldn't capture.

The Takeaway

The main finding is that for AI agents, how you describe the world is part of the rules of the world. There is no such thing as a "neutral" interface. If you change the format of the data, you change the collective outcome, and this change depends entirely on which AI model you are using. What works for GPT might fail for Claude, and what helps one group synchronize might break another.

This isn't just a theoretical curiosity. It means that when we build systems with multiple AI agents, we can't just assume that "more data" or "better data" is the solution. We have to be incredibly careful about the "observation map"—the specific way we translate reality into text for the AI. Because, as this study shows, the translation isn't just a mirror; it's a lens that distorts the reality it's supposed to show.

Technical Summary: State Encodings Select Synchronization Outcomes in Language-Model Agents

Problem Statement
Language-model (LM) agents act on encoded descriptions of their environment rather than the physical state itself. While it is established that LM outputs are sensitive to prompt formatting, information ordering, and labeling in single-turn interactions, it remains unknown whether these sensitivities survive in multi-agent feedback loops. Specifically, does a change in the "observation map" (the transformation of a physical state into a text string) alter the microscopic response operator of an agent, and does this microscopic shift propagate through feedback to produce qualitatively different macroscopic collective outcomes? The authors argue that if state encodings are not neutral interfaces but part of the effective policy, then identical physical systems driven by different encodings could converge to distinct social conventions or market states.

Methodology
The authors designed a minimal, stateless circular-synchronization assay using pretrained language models (GPT-5.4-mini, Claude Haiku 4.5, and Gemini 3.5 Flash) as agents in a coupled-oscillator system.

System Dynamics: A population of $N=17$ agents possesses phases $x_i(t)$ . At each step, a deterministic engine updates phases based on a natural increment $\omega_i$ and a coupling term $K f_i(t)$ , where $f_i(t) \in \{-1, 0, +1\}$ (retard, stay, advance) is the agent's action. The model receives no history, identity, or absolute phase; it only sees a text description of the relative phases of its peers.
Observation Maps (Intervention): The physical relative-phase field was encoded into text using three distinct maps, all derived from the same 24-bin histogram of the field:
1. Moments: The first three circular moments (real and imaginary components), summarizing mean direction and concentration.
2. Centers: A 24-bin histogram labeled by bin centers.
3. Intervals: The same 24-bin masses labeled by half-open bin intervals.
Experimental Controls:
- Matched Seeds: Collective experiments used identical physical seeds across encodings to isolate the effect of the encoding on the trajectory.
- Identical-Field Replay: To decouple the encoding from the feedback loop, the authors froze 48 physical fields generated during collective runs and re-presented them to the models under all three encodings. This isolated the "microscopic response operator" (the probability distribution of actions given a fixed field).
- Same-Information Controls: Variants of the "Moments" encoding were tested where the numerical values were identical, but the layout (narrative vs. table) or context (task-irrelevant padding) changed.

Key Results

Macroscopic Divergence: Changing the encoding alone systematically shifted collective synchronization outcomes. In GPT, the "Moments" encoding synchronized the population in 6/6 seeds at positive coupling ( $K=0.08, 0.15$ ), whereas both histogram encodings ("Centers" and "Intervals") failed to synchronize (0/6). Conversely, in Claude, the relationship reversed: "Moments" failed to synchronize (0/6), while the histogram encodings succeeded (6/6). This demonstrates that the encoding-to-outcome mapping is model-family specific and not an intrinsic property of the encoding.
Microscopic Operator Shift: The "Identical-Field Replay" confirmed that the encoding alters the microscopic response operator. For the same frozen physical field, the action distributions (advance/stay/retard probabilities) differed significantly across encodings. The mean pairwise total-variation distance between operators was 0.344, far exceeding the test-retest noise floor (0.092).
Presentation Sensitivity: Even when the task-relevant numerical information (the moments) was held fixed, changing the presentation (e.g., adding task-irrelevant context or changing layout) significantly altered the response operator. Adding irrelevant context shifted the operator by 0.414, a magnitude comparable to the difference between distinct information-rich encodings.
Surrogate Failure: The authors attempted to train surrogate models to predict agent actions from field descriptors to replace costly LM calls in simulations. While all encodings passed in-distribution compressibility tests, only the "Moments" branch passed the "closed-loop support" check (verifying that the training data covers the fields generated by the interacting population). The histogram branches failed this stage, indicating that their learned response laws were not robust to the state distribution of the closed loop.

Significance and Claims
The paper claims that the observation map is a constituent of the effective policy implemented by a language-model agent, not a neutral interface.

Effective Interaction Law: The state representation is part of the effective interaction law. Changing the serialization of the same physical state can select different collective outcomes (e.g., synchronization vs. partial order) and can even reverse the direction of this selection across different model families.
Feedback Survival: The sensitivity to prompt formatting and serialization is not limited to single-turn interactions; it survives and amplifies in feedback loops, leading to qualitatively different macroscopic states.
Evaluation Implications: The authors argue that validation of multi-agent systems cannot rely on analogy between models or encodings. An encoding that works for one model family may fail or produce opposite results in another. Furthermore, observation serializers must be treated as versioned components that require re-validation with the specific model and in the specific closed-loop context in which they operate.

The study concludes that "state encodings form part of a model-dependent effective interaction law," challenging the assumption that agents act on a consistent underlying reality regardless of how that reality is described to them.

The Great AI Dance-Off

The Results: It's All About the Presentation

The Microscopic Mystery

The "Surrogate" Test: Can We Fake It?

The Takeaway

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