🧬 biology

Same Geometry, Opposite Noise: Transformer Magnitude Representations Lack Scalar Variability

This paper demonstrates that while transformer language models capture the log-compressive geometry of numerical magnitudes, they fail to exhibit the biologically observed scalar variability (constant coefficient of variation), instead showing representational noise that decreases as magnitude increases.

Original authors: Jon-Paul Cacioli

Published 2026-04-07

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Jon-Paul Cacioli

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ⚕️ This is an AI-generated explanation of a preprint that has not been peer-reviewed. It is not medical advice. Do not make health decisions based on this content. Read full disclaimer

The Big Question: Do AI Brains Work Like Human Brains?

Imagine you are trying to understand how a machine "thinks" about numbers. Scientists have known for a long time how human brains handle numbers. They follow two specific rules:

The Logarithmic Rule: We see the difference between 1 and 2 as huge, but the difference between 100 and 101 as tiny. Our internal "ruler" squishes big numbers together.
The "Scalar Variability" Rule: This is the tricky part. In humans, the more uncertain we are about a number, the bigger the number is.
- Analogy: Imagine you are guessing the weight of a feather. You might be off by a gram. But if you are guessing the weight of an elephant, you might be off by 100 pounds. The percentage of your error stays the same. This is called Scalar Variability. It's like a blurry photo that gets fuzzier the further away you zoom out.

The researchers wanted to know: Do Large Language Models (LLMs) like Llama and Mistral follow these same rules?

The Experiment: Taking a "X-Ray" of the AI

The researchers took three popular AI models and looked inside their "brains" (specifically, the hidden layers where they process numbers). They asked the AI to think about 26 different numbers (from 1 to 1,000) inside 5 different sentences.

They measured how much the AI's internal representation of a number "wobbled" or varied depending on the sentence.

If the AI were human: The wobble should get bigger as the numbers get bigger (just like the elephant vs. the feather).
If the AI is different: The wobble might stay the same, or get smaller.

The Surprise: The AI Does the Exact Opposite

The results were shocking. The AI did not behave like a human brain.

The Finding:
Instead of getting "fuzzier" with big numbers, the AI's representation of big numbers became sharper and more precise.

Human Brain: Big numbers = High noise (fuzzy).
AI Brain: Big numbers = Low noise (crystal clear).

The Analogy of the Library:
Imagine a library where books are arranged by how popular they are.

Small numbers (1, 2, 3) are like bestsellers. They appear in millions of different stories, genres, and contexts. Because they are everywhere, the AI sees them in a chaotic, noisy environment.
Big numbers (500, 1000) are like rare, obscure books. They only appear in very specific, technical contexts (like "500 years ago" or "1000 dollars"). Because they appear in fewer, more predictable places, the AI has a very clear, rigid idea of what they mean.

The AI isn't "guessing" big numbers; it's actually more confident about them because it has seen them in very specific, repetitive patterns.

Why Does This Happen?

The paper suggests that the AI learned the shape of numbers (the logarithmic rule) just by reading text, because numbers in real life follow a "power law" (small numbers are common, big numbers are rare).

However, the AI lacks the biological constraints that force humans to be fuzzy.

Human Brains: We have a limited energy budget. To save energy, our brains accept that big numbers will be a bit fuzzy. It's a trade-off.
AI Brains: They have infinite energy (metaphorically). They don't need to save space or energy. So, when they encounter a rare, big number, they just memorize it perfectly because it's easy to do so. They don't have a "noise budget" to force them to be sloppy.

The Takeaway

This study proves that reading a lot of text is not enough to make an AI think like a human.

Geometry: The AI successfully learned the shape of how we see numbers (logarithmic compression).
Noise: The AI failed to learn the uncertainty of how we see numbers.

In simple terms: The AI has the right map, but it doesn't have the same shaky hands as a human. It is too perfect, too precise, and too confident about big numbers, whereas our brains are naturally a bit messy and fuzzy when dealing with the large scale of things.

This tells us that to build AI that truly mimics human cognition, we might need to give them artificial "limitations" or "energy budgets" to force them to be a little bit uncertain, just like us.

1. Problem Statement

The paper investigates whether Large Language Models (LLMs) exhibit scalar variability, a hallmark of biological magnitude systems.

Biological Context: In biological systems (e.g., human perception of time, number, or space), representational noise scales proportionally with the magnitude of the stimulus. This results in a constant coefficient of variation (CV), known as Weber's Law.
Theoretical Framework: Under the Efficient Coding hypothesis, two properties emerge jointly when a system with a fixed noise budget (metabolic constraint) encodes power-law distributed inputs:
1. Log-compressive geometry: Internal distances follow a logarithmic function of magnitude.
2. Scalar noise: Variability scales with magnitude (constant CV).
The Gap: Previous work (Cacioli, 2026) established that transformers encode numerical magnitude with log-compressive geometry (consistent with efficient coding under power-law input distributions). However, transformers lack biological metabolic constraints (no firing-rate bounds or fixed noise budgets).
Research Question: Do transformers exhibit the second property (scalar noise) despite lacking the metabolic constraints that theoretically drive it in biological systems? Or are geometry and noise dissociable?

2. Methodology

The study utilized a pre-registered analysis of hidden-state representations from three 7–8B parameter models: Llama-3-8B-Instruct, Mistral-7B-Instruct-v0.3, and Llama-3-8B-Base.

Data Source: Hidden-state vectors were extracted at the magnitude token position (pre-RMSNorm) across all 33 layers.
Stimuli: 26 numerical magnitudes ranging from 1 to 1000 (e.g., 1, 2, ..., 90, 100, 500, 1000) embedded in 5 distinct carrier sentences.
Variability Measures:
1. $V_{eucl}$ : Mean Euclidean distance of the 5 carrier-sentence vectors from their centroid at each magnitude and layer.
2. $V_{residual}$ : Sentence-identity corrected variance (subtracting the mean vector per sentence across all magnitudes to remove systematic sentence shifts).
3. $V_{proj}$ : Variance of vectors projected onto the first principal component (PC1) of the 26 centroids, isolating dispersion along the magnitude axis.
Analysis:
- Focused on 16 "primary" layers (16–31) where magnitude geometry peaks.
- Regressed $\log(V)$ on $\log(n)$ (magnitude) using Ordinary Least Squares (OLS) and Theil–Sen estimators to calculate the scaling exponent ( $\alpha$ ).
- Hypotheses:
  - $H_1$ : $\alpha > 0$ (variability increases with magnitude).
  - $H_2$ : $\alpha \approx 1$ (constant CV, scalar property).
  - $H_3$ : $\alpha < 1$ (sub-scalar, reflecting lack of metabolic constraints).
  - $H_4$ : $\alpha$ varies with layer depth.

3. Key Results

The results contradicted the hypothesis that transformers exhibit scalar variability. Instead, they demonstrated an "anti-scalar" pattern.

Negative Scaling Exponent ( $\alpha < 0$ ):
- Contrary to biological systems where noise increases with magnitude, transformer representational variability decreased as magnitude increased.
- Across all 16 primary layers and all three models, $\alpha$ was consistently negative.
- Mean $\alpha$ values:
  - Raw ( $V_{eucl}$ ): Ranged from $-0.031$ (Llama-Base) to $-0.046$ (Llama-Instruct).
  - Sentence-corrected ( $V_{residual}$ ): Ranged from $-0.005$ to $-0.009$.
  - Magnitude-axis projection ( $V_{proj}$ ): Strongest effect, ranging from $-0.085$ to $-0.210$.
Rejection of Scalar Hypothesis ( $H_2$ ): No layer in any model produced an $\alpha$ within the scalar range $[0.8, 1.2]$ .
Stability Across Depth ( $H_4$ ): The anti-scalar pattern was stable across all layers; there was no significant correlation between layer depth and $\alpha$ .
Magnitude Axis Specificity (Exploratory E4): The anti-scalar effect was 3–5 times stronger along the magnitude axis (PC1) than in orthogonal dimensions, indicating the phenomenon is concentrated in the behaviorally relevant subspace.
Instruction Tuning Effect (Exploratory E6): Instruction-tuned models (Llama-Instruct) showed a more negative $\alpha$ than base models, suggesting fine-tuning further compresses representations for large numbers.
Corpus Frequency Correlation (Exploratory E5): Per-magnitude variability was strongly correlated with log corpus frequency ( $\rho \approx 0.84$ ). Frequent (small) numbers appeared in diverse contexts (high dispersion), while rare (large) numbers appeared in stereotyped contexts (low dispersion).

4. Key Contributions

Dissociation of Geometry and Noise: The paper provides empirical evidence that log-compressive geometry and scalar noise are theoretically and empirically dissociable. Transformers can learn the geometric structure of magnitude from distributional statistics alone without acquiring the noise scaling properties.
Identification of "Anti-Scalar" Variability: It characterizes a novel representational property in LLMs where representational uncertainty decreases for larger magnitudes, the opposite of biological scalar variability.
Mechanism of Variability: It attributes representational variability in transformers to contextual diversity (driven by corpus frequency) rather than a fixed noise budget. Large numbers are represented more precisely because they appear in fewer, more specific linguistic contexts.
Limit of Distributional Learning: The study demonstrates that distributional learning (gradient-based optimization on text) is sufficient to produce the geometry of efficient coding but insufficient to produce the noise structure without metabolic constraints.

5. Significance

Theoretical Implications: The findings challenge the assumption that efficient coding in AI mirrors biological efficient coding in all aspects. It suggests that metabolic constraints (or equivalent capacity limits) are a necessary condition for the emergence of scalar variability (constant CV).
Understanding LLM Representations: It reveals that LLMs represent large numbers with higher precision (lower noise) than small numbers, a counter-intuitive finding compared to human psychophysics. This implies that LLM "understanding" of magnitude is driven by statistical regularities in training data rather than an internalized, biologically plausible magnitude system.
Future Directions: The results suggest that to replicate biological magnitude systems fully, AI architectures may need to incorporate explicit capacity constraints or noise budgets, rather than relying solely on scaling laws and distributional learning.

Conclusion:
While transformers successfully replicate the log-compressive geometry of biological magnitude systems, they fail to replicate the scalar noise signature. Instead, they exhibit an anti-scalar pattern where representational variability decreases with magnitude, driven by the distributional frequency of numbers in the training corpus rather than metabolic constraints. This confirms that geometry and noise scaling are distinct phenomena in neural representation.