The Geometry of Understanding: An Exploration of the Embedding Spaces of Large Language Models and the Search for the Locus of Consciousness

When we talk about artificial intelligence, especially large language models (LLMs), we often imagine something that thinks, understands, and even possesses the rudiments of consciousness. But is that really the case? I found myself wondering: where, exactly, is this intelligence "located"? In what space does it exist? And could something resembling the human mind arise within the modern architecture of neural networks?

In my investigation, I decided to look inside these models and study the geometry of their "inner world"—the space of embeddings into which words, phrases, and entire texts are transformed. My hypothesis is simple: if this space is static, linear, and subject to rigid laws, then no real dynamics, and therefore no genuine intelligence, can emerge there. The model merely glides along pre-laid tracks, accumulating context like a snowball gathers snow as it rolls down a mountain, until it bumps into a ready-made answer.

1. Relevance of the Research: From Narrow AI to General AI

The question of the possibility of creating artificial general intelligence (AGI) remains one of the most controversial in science. Some researchers, mainly from the engineering camp, believe in the achievability of AGI and are developing corresponding systems—from algorithms based on reinforcement learning (such as AIXI and BoMAI) to massive models like DeepMind's Gato, capable of performing hundreds of diverse tasks. Others, primarily philosophers, point to fundamental limitations of machine reason: the lack of bodily experience, non-formalizable background knowledge, and intuition in computers, which underlie human understanding of the world.

However, as Mostafa Chehreghani aptly notes in his work "The Embeddings World and Artificial General Intelligence," the very formulation of the question "is AGI possible?" requires clarification. Perhaps AGI should be viewed not as an end product or algorithm, but as a continuous process, akin to the development of human civilization. And a key role in this process is played by pre-trained embeddings—vector representations that could become the foundation for building an "intelligent world" accessible to machines.

The paradox is that these very embeddings, intended to be a bridge to AGI, upon closer inspection reveal properties that cast doubt on the very possibility of the emergence of reason within the current paradigm. To test this, I analyzed three popular embedding models (a monolingual model, a multilingual model, and LaBSE) on four different text collections—from "The Little Prince" in Russian and English to technical product descriptions and Wikipedia articles. The results were strikingly consistent and found confirmation in the latest research.

2. How Modern Embedding Models Work

First, let's understand what I was dealing with. All three models transform text into multi-dimensional vectors, but they do so slightly differently.

Model 1: Monolingual (all-MiniLM-L6-v2)

Model 2: Multilingual (paraphrase-multilingual-MiniLM-L12-v2)

Model 3: LaBSE (Language-agnostic BERT Sentence Embedding)

As can be seen from the tables, each model has a stated dimension (384 or 768), but the actual "intrinsic dimension"—that is, how many dimensions are actually used to store meaning—is much smaller: between 51 and 85. This is a first hint that the spaces are highly compressed and structured. This phenomenon is confirmed in the work of Mikhalchuk et al. (2024), who showed that during training, the intrinsic dimension first grows, expanding the representation space, and then shrinks, forming more compact representations.

3. Five Universal Regularities

Analyzing all the collections, I discovered five properties that repeat regardless of the model and text. I called them invariants—quantities that remain constant.

Invariant 1: Strong Non-Uniformity (Anisotropy)

All spaces turned out to be inhomogeneous—vectors are not scattered uniformly in all directions, but gathered in a narrow cone. The non-uniformity coefficient (the closer to 1, the stronger the clustering) is everywhere above 0.82, with an average value of 0.87. Here is sample data:

The scatter is very small (only 4% of the average), meaning this is not a coincidence but a fundamental property.

Interestingly, research by Chistyakova (2024) shows that different architectures can exhibit different patterns of anisotropy. For example, autoregressive models like GPT-2 are characterized by particularly high anisotropy and large mean vector values, indicating a fundamentally different way of encoding information compared to encoders like BERT. Mikhalchuk and colleagues (2024) refine this picture, finding that in transformer decoders, anisotropy is distributed in a bell-shaped curve with a peak in the middle layers, whereas in encoders it is more uniform.

Invariant 2: Self-Similarity at All Scales

If you take the space and look at it under different "magnifications," the structure remains the same. The consistency coefficient equals exactly 1 for all collections. The space, as it were, repeats itself—a sign of fractal organization.

Invariant 3: Reliability of Measurements

All the "fingerprints" of the spaces I constructed turned out to be of high quality: the reliability indicator never drops below 0.64. This means the observations are not caused by random fluctuations but reflect the actual structure of the models.

Invariant 4: Rich Structure

Although the spaces are non-uniform, they are not empty. Spectral entropy—a measure of how evenly information is distributed across different directions—exceeds 3.4 everywhere. This indicates that even within a narrow cone, there are many meaningful "channels."

Invariant 5: Uniform Normalization

All models bring vectors to unit length (norm = 1) with microscopic error (about 2.7·10⁻⁸). This means that the meaning of words is compared only by direction, not by magnitude. The angle between vectors becomes the sole measure of proximity.

4. Bases: A Common Coordinate System

I compared how different texts "lie" within the space of a single model. It turned out that all collections—both "The Little Prince," technical descriptions, and Wikipedia—use one and the same coordinate system. The bases coincide with precision up to one:

This is a very important result: the model does not rearrange its space for each new text, but projects everything into a pre-defined rigid grid.

However, between different models, the bases already differ. For example, the monolingual and multilingual models are only 70–80% similar, and to align them, a special rotation (rotation matrix) is needed. This is like two different languages: they describe the world in their own way, although they talk about the same things. Research by Chistyakova (2024) confirms that Procrustes alignment is an effective method for aligning embeddings from different models.

5. How Many Dimensions Are Actually Needed?

Intrinsic dimension is the number of independent directions that actually carry information. My measurements showed:

The most stable is LaBSE, its intrinsic dimension hardly changes from language to language. The Products collection gave the highest value, 85, probably due to the large diversity of categories.

These numbers agree well with recent research. Golden (2025), in his work "Large Language Models are Locally Linear Mappings," showed that modern LLMs (Llama 3, Gemma 3, Qwen 3, and others) operate in extremely low-dimensional subspaces, where many of the largest singular vectors are decoded into concepts related to the most probable output token. Moreover, Tsukagoshi and Sasano (2025) found that embeddings for classification and clustering tasks exhibit lower intrinsic dimension and less isotropy compared to embeddings for search and semantic similarity. This means the specific task influences how the model uses its space.

6. Symmetries Within the Space

I tried to find hidden symmetries in these spaces—for example, ones that reflect grammatical rules. It turned out they do exist, and they are more pronounced in the Russian text:

The Russian text yields clearer and more consistent symmetries (value >0.79). But it's important to understand: these symmetries are merely static imprints of rules embedded in the training texts. The model does not "know" the rule; it simply repeats a statistical regularity frozen into the geometry.

7. The Possibility of Knowledge Transfer

If within one model all texts lie in the same basis, then training on one collection should easily transfer to another. And this is indeed the case: the transfer is perfect. However, transfer between different models requires additional alignment (rotation of the space) and yields similarities from 50 to 80%.

For practical tasks, I suggest using the following universal settings:

• take 50 principal components (this covers almost all useful information),

• note that the intrinsic dimension is on average about 63,

• expect non-uniformity around 0.87,

• remember the unit norm of vectors.

8. Discussion: Linearity as a Fundamental Property

The key question that confronts us is: what do we gain for understanding the nature of language models? The answer, it seems to me, lies in the realm of linearity.

Golden (2025) made a startling discovery: the inference operations of several open-source large language models can be mapped to an exactly equivalent linear system for the input sequence without changing the model's weights or altering the output predictions. Using methods from the field of image diffusion models, the researcher showed that the model's Jacobian almost exactly reproduces the forward pass using a linear system.

Independently, another group of researchers developed the LAMP (Linear Attribution Mapping Probe) method, which allows extracting locally linear decision surfaces from the "world models" of LLMs. This method, applied to tasks of sentiment analysis, controversial topic detection, and safety auditing, showed that many LLMs exhibit locally linear decision landscapes.

These discoveries echo earlier work on anisotropy. If embeddings are clustered in a narrow cone and the model's behavior is locally linear, then the space of possible states is rigidly constrained. The model does not generate new dynamics, but merely moves along predetermined trajectories.

9. Main Conclusion: The Static Snowball vs. Living Mind
Visualization

So, what do we have? The embedding space is a rigidly structured, non-uniform, but very stable coordinate system. It is linear at every point (as shown by recent experiments by Golden and the LAMP group). Within it, there is no room for genuine dynamics, for self-change, for the emergence of the new.

The model works like a snowball rolling down a mountain. The ball has a shape (the structure of the ice—that's the basis of the embeddings), but its path is predetermined by the mountain's terrain (the latent space). It accumulates mass (context) until it reaches a lowland—the final answer. At zero temperature, or when choosing the most probable continuation (argmax), it will always stop in the deepest local hole, without the ability to change the landscape.

Altmeyer et al. (2024), in their principled article "Stop Making Unscientific AGI Performance Claims," warn: the discovery of meaningful patterns in the latent spaces of models cannot be considered evidence in favor of AGI. Models of any complexity—from random projections to deep autoencoders and transformers—successfully distill information that can be used to predict hidden variables. And yet, none of them have previously been associated with AGI. The authors remind us that people are prone to anthropomorphization and pattern-seeking, and the modern media context creates ideal conditions for distorting correlations as "understanding."

Returning to Chehreghani's question: can pre-trained embeddings become the basis for an "intelligent world" for AGI? My analysis and the cited research compel the answer: in their current form—no. Static geometry and local linearity make them excellent tools for compressing and reproducing information, but not an environment for the emergence of genuine intelligence.

For genuine intelligence, one needs not just to glide on ready-made rails, but to be able to rebuild the coordinate system itself, create new dimensions, and go beyond the predetermined. Modern embeddings and LLMs are merely grand, but static, atlases of human knowledge. They exist like a "snowball" just before the throw: the structure is defined, the mass is known, and the trajectory is predictable.

Therefore, to the question "Where can AI exist?" I answer: not in the current embedding spaces. For a genuine artificial mind, a fundamentally different architecture is needed—dynamic, self-modifying, capable of reflection and rebuilding its own basis. For now, we are dealing with brilliantly made but static models that imitate understanding, while remaining merely complex reflectors of our own texts.

References

1. Mikhalchuk, M., et al. (2024). The Shape of Learning: Anisotropy and Intrinsic Dimensions in Transformer-Based Models. Findings of EACL 2024.

2. Chistyakova, K. E. (2024). Issledovaniye razlichiy v vektornykh predstavleniyakh, poluchennykh s pomoshch'yu raznykh yazykovykh modeley [Investigation of Differences in Vector Representations Obtained Using Different Language Models]. Graduate Qualification Work, HSE University.

3. Golden, J. R. (2025). Large Language Models are Locally Linear Mappings. arXiv preprint arXiv:2505.24293.

4. Chehreghani, M. H. (2022). The Embeddings World and Artificial General Intelligence. arXiv preprint arXiv:2209.06569.

5. Chen, C., et al. (2025). LAMP: Extracting Locally Linear Decision Surfaces from LLM World Models. arXiv preprint arXiv:2505.11772.

6. Tsukagoshi, H., & Sasano, R. (2025). Redundancy, Isotropy, and Intrinsic Dimensionality of Prompt-based Text Embeddings. arXiv preprint arXiv:2506.01435.

7. Altmeyer, P., et al. (2024). Position: Stop Making Unscientific AGI Performance Claims. arXiv preprint arXiv:2402.03962.