A Topology of Cognition

Suppose abstract meaning could be geometrically mapped.

What if every concept, from the affective weight of “love” to the perceptual crispness of an “apple”, could be represented as a point in a vast, high-dimensional semantic manifold?

In this model, identity arises not from labels, but from spatial relationships: from proximity, orientation, and alignment within a distributed field of meaning.

This is not a philosophical conjecture. It's the operational reality of Large Language Models (LLMs), whose internal representations exist within what is called latent space, a multidimensional embedding space shaped by statistical patterns in language.

In this space, prompting an LLM is not merely a query; it is a vector operation. It is semantic navigation: a traversal from one region of meaning to another, guided by learned relational geometry.

I'm aiming to articulate this latent space beyond simply a computational artifact, but as an epistemic landscape, one that we as advanced users and prompters, must learn to map and navigate. We will examine its foundational structures, governing mechanisms, and the pragmatic strategies that enable precise movement through conceptual terrain.

Conceptual Embeddings and Vector Semantics

Understanding latent space begins with understanding how language is embedded in vector form.

Principle 1: Words as Vectors in a High-Dimensional Space

LLMs represent tokens (words or subwords) as vectors, coordinates in a high-dimensional space (typically hundreds or thousands of dimensions).

The vector for a word like "king" does not encode a literal image or definition, but rather its statistical co-occurrence relationships within language. Thus, its proximity to "queen," "monarch," or "power" reflects how often and in what contexts these words appear together.

One of the canonical demonstrations of this phenomenon is vector arithmetic:

vector("king") - vector("man") + vector("woman") ≈ vector("queen")

This works because these vectors capture static meanings, and, relational semantics.

The relationships between words are preserved as geometric transformations. This arithmetic implies that analogical reasoning is a form of geometric translation within latent space.

Principle 2: Measuring Semantic Relationships

The structure of this space is shaped by relevance, not randomness.

There are two dominant metrics for measuring relationships between vectors:

• Euclidean Distance: Captures absolute difference in position. Useful for identifying direct similarity, but can be misleading in cases where antonyms (e.g., “hot” and “cold”) share contextual domains.
• Cosine Similarity: Measures angular difference, independent of magnitude. This is particularly important in high-dimensional spaces, where direction (i.e., conceptual alignment) carries more semantic weight than distance.

In practice, cosine similarity is often used to identify semantically aligned tokens, as it emphasizes orientation in latent space over raw magnitude. These two metrics, taken together, form the basic tools for probing semantic topology.

Principle 3: Contextual Fluidity via the Attention Mechanism

Crucially, token representations are dynamic. The vector for “bank” is context-sensitive: it shifts in meaning depending on its neighboring words. For example:

• “He sat by the river bank” invokes a geo-spatial context.
• “She deposited funds in the bank” invokes a financial context.

This disambiguation is made possible through self-attention, which allows every token in a sequence to condition its representation on every other token. As a result, meaning is not static, it emerges through contextual entanglement. The final interpretation of a token is its equilibrium point within the attention-mediated interaction of the entire sequence.

The Mechanics of Semantic Navigation

With the conceptual structure in place, we now examine the mechanisms by which LLMs navigate this space.

The Role of Attention: Dynamic Context Alignment

The core computational unit of the Transformer architecture, as introduced in the seminal paper "Attention Is All You Need" by Vaswani et al. (2017), is the scaled dot-product attention function.

This mechanism enables dynamic alignment of contextual information across input sequences, allowing each token to weigh the relevance of others in a data-driven manner.

$$\text{Attention}(Q, K, V) = \operatorname{softmax}\left( \frac{Q K^{T}}{\sqrt{d_{k}}} \right) V$$

Figure 1: Scaled dot-product attention mechanism from the Transformer model.

In this equation:

• Q (query), K (key), and V (value) matrices are learned linear projections derived from the input token embeddings, typically using separate weight matrices for each (W_Q, W_K, W_V). These projections transform the input into subspaces that facilitate similarity computation and value aggregation.

• The dot product $QK^T$ computes pairwise similarity scores between every query and key, capturing how relevant each key (representing a token's context) is to the query. This results in an attention matrix of shape [sequence_length, sequence_length].

• The scaling factor $\sqrt{d_k}$ (where d_k is the dimensionality of the keys) prevents the dot products from growing too large in magnitude, which could lead to vanishing gradients during training due to the softmax function's sensitivity to extreme values. Preventing information from being "forgotten" too soon.

• The softmax operation normalizes these similarity scores row-wise into a probability distribution, yielding attention weights that sum to 1 for each query.

• The resulting weighted sum aggregates the values (V) according to these weights, producing a contextually enriched representation for each token. This output effectively modulates the token's embedding based on relevant parts of the sequence, enabling the model to handle dependencies regardless of distance.

This self-attention mechanism is inherently permutation-invariant without additional positional encodings, but in practice, absolute or relative positional embeddings are added to the input to incorporate sequential order.

To enhance representational capacity, the mechanism is implemented as multi-head attention (MHA). In MHA, h independent attention heads (typically h=8 or more) operate in parallel on subspaces of the projected inputs (each head processes d_model / h dimensions).

Each head can specialize in capturing different aspects of linguistic or relational features, such as syntactic dependencies (e.g., subject-verb agreement), semantic relationships (e.g., coreference resolution), or pragmatic cues (e.g., discourse structure). The outputs from all heads are concatenated along the feature dimension and passed through a final linear transformation (W_O) to produce the aggregated result.

This parallelism allows the model to encode multifaceted dependencies and hierarchical structures within the data, contributing to the Transformer's superior performance on tasks like machine translation, text generation, and question answering.

However, the standard full self-attention has quadratic computational complexity O(n²) with respect to sequence length n, which becomes prohibitive for long contexts. Empirical scaling laws, as outlined in Kaplan et al. (2020) scaling laws, demonstrate that model performance improves predictably with increased scale—including larger parameters, datasets, and context lengths, but this necessitates efficiency innovations to manage resource constraints.

One such innovation is Sliding Window Attention (SWA), a sparse attention variant popularized in models like Longformer (Beltagy et al., 2020) and later adopted in efficient LLMs such as Mistral, Gemma, and the new GPT-OSS by OpenAI. SWA applies a fixed-size window (e.g., w=4096 tokens) that "slides" across the sequence, restricting each token's attention to only the w/2 tokens before and after it (or similar asymmetric configurations). This reduces complexity to O(n * w), enabling longer effective contexts (up to millions of tokens in some implementations) while maintaining linear scaling.

In multi-head setups, SWA can be combined with global attention in select heads for critical tokens (e.g., [CLS] or special markers), but in pure form, it relies on stacked layers to propagate information across windows indirectly, each layer's window allows distant tokens to influence each other through intermediate hops.

While clever for efficiency, SWA is double-edged: it excels in local dependency modeling and reduces memory footprint, but it may degrade performance on tasks requiring very long-range interactions (e.g., document-level reasoning), as tokens outside the window are ignored, potentially leading to information loss or suboptimal global coherence.

Approaches, like hybrid sparse-dense attention or dynamic window sizing, are emerging to mitigate these trade-offs, with contrarian views from experts suggesting that fully global attention remains essential for certain high-fidelity applications, despite the computational cost.

Prompt Engineering as Geometric Control

Okay, back to practicality. Knowing just enough of the inner-workings is important for the next bit ... the how.

How do we make use of this information? The majority of you reading this are consumers of LLMs, users wanting to leverage their raw power for your own tasks.

The magic words that some love, and others ridicule ... Prompt Engineering.

Prompt engineering is the practical application of latent space manipulation. Some will tell you that it is a "fad", or "cope". However, we just learned that an LLM is really just a big token transformer. A well-constructed prompt (token sequence input) functions as an initial condition: it sets the model's trajectory through conceptual space. The location you enter the pool at, so to speak. Thus, all further transformations will come from this root locality.

Technique 1: Directional Priming via Lexical Anchors

Certain tokens exert strong directional pull in latent space. For instance:

"Write a visionary pitch for a groundbreaking technology."

Words like “visionary” and “groundbreaking” activate high-magnitude vectors in positive evaluative dimensions. I.e, semantically affirmative vocabulary. These serve as semantic priors, biasing subsequent generations toward innovation, excitement, and futurism relative to the context - which in our example is "technology".

At this point, you will know that you are getting this because a key thought should arise ... "Isn't technology too broad?" And you would be right. Indeed, the usage of the broad scope here is intentional, I want you to begin to see how to guide the text transformations.

This, fundamentally, is the "art" of prompt engineering.

Prompting, in this sense, becomes a form of semantic vector priming.

Technique 2: Layered Prompting (Token Surfing)

Let's dig a little deeper. For precise output, it's often necessary to guide the model across intermediate regions of the concept-space, reducing the scope of our question to be highly precise. The greater your semantic precision, the better the output transformation will be.

This technique, akin to wayfinding, structures prompts as a sequence of conceptual anchors:

“Let’s begin with Artificial Intelligence. Within that, we focus on Machine Learning. More specifically, we examine Neural Networks.”

Each clause acts as a coordinate transformation, narrowing the vector field. This layered approach reduces ambiguity and increases output determinacy.

This is sometimes referred to as "snowballing" or "snowball prompting" and is a fantastic way to curate robust context before asking the real questions.

"What are stocks and how do they work?" -> "What are exchanges and brokerages?" -> "What defines a good brokerage or exchange?" -> "List the top 10 best brokerages to purchase ETFs within the US"

"What are design practices -> What design practices are used in Website design -> What are Material UI style guidelines? What design practices are common in womens apparel -> Provide an outline for a modern website design for a womens apparel company"

Effectively, guiding the attention-heads to give us a more accurate higher-quality output.

We are leveraging the vast nature of the concept space and "navigating it" to perform transformations that will give us the output we are after.

Now, some of you reading this may consider this "a lot of work", and this is sort of a fair assessment from a pure UX point of view. However, the truth is these models just aren't yet capable of decomposing and navigating the vector space on their own. Work is being done in this direction, with OpenAI's GPT-5 and xAI's Grok 4 being the leaders in this area.

Perhaps, one day, we will be able to ask arbitrarily vague questions and the model can "deduce" where to navigate in the space properly. Still, even in that hypothetical I still find that prompt engineering will be a strong foundation. Just as you don't go to your contractor and say "Build me a fantastic new kitchen" and expect it to be exactly what you were envisioning. It would, however, be closer to a true "conversation".

Technique 3: Symbolic Compression and Abstract Hyperlinks

This next technique isn't very intuitive, but it is one of the most powerful. The term "LLM" is a bit of a misnomer. We tend to attribute that the use of the term language here means we should "communicate" with these systems like we would speak to a colleague or friend.

However, this is fundamentally a flawed point of view.

Natural language (like english, german, or cantonese) are themselves structured with rules, however the way that we use them is highly unstructured and full of connotation, vernacular, dialects. Conveying complex concepts and ideas is inherently prone to be low-resolution and of low-determinism. We all know how hard it can be to convey a point, or describe our feelings.

The good news is LLMs are not people. They are algorithms design specifically to perform computation across symbolic token sets. Put simply, we do not need to "have a conversation" with an LLM. We need only input the tokens we want to transform and perform our operations.

What does that look like?

Modern LLMs are trained heavily on structured symbolic data (e.g., code, documentation, JSON) and thus internalize symbolic protocols.

Prompts that build-on/invoke these protocols act as high-efficiency instructions:

/refine: 1) 🎓 Draft 3 versions. 2) 🕵🏻 Evaluate critically. 3) 🧑💼 Synthesize final output.

This prompt does not describe the process; it encodes it.

Here, we are using control-flow statements and information compression to encode more complex meaning.

The slash command and emoji sequence map directly to known patterns in the model's training data. This forms a hyperlane, a symbolic shortcut across large semantic distances within the concept space.

The use of contextual compression and control-flow statements in semantic contexts is a super power that makes little sense from the perspective of a "conversation", however LLM's are not chatbots, even though that is how people tend to use them.

The goal of any prompt should be to place the context exactly at the location in the attention-heads that has the highest impact for our desired output.

In our example above, this allows us to re-use "/refine" later, or it can simply act as structure in the latent-space of the model's attention representation, biasing in the direction of meaning that it represents.

You will have probably seen this hyped-up on social platforms like X as "JSON Prompting" where people have begun to use raw JSON strings to force more structured interactions with LLMs.

What I would like to impart onto you is that any structured control-flow semantics/grammar works here.

Limits, Anomalies, and Epistemic Boundaries

Even with these tools, latent space is not fully understood. Several anomalies and limitations remain:

• Non-Euclidean Curvature: Empirical studies suggest that latent space exhibits non-linear geometry. Isometric projections distort clusters, implying the true manifold may be curved or warped.

• Cultural Dependency: The embedding structure is not universal. As shown in “Should We Respect LLMs?,” concepts like politeness are situated differently across linguistic corpora. Semantic fields are culturally contingent. Meaning, in our example of politeness - it is encoded and acted upon differently for different languages. Example, to be polite in Japanese will result in different attention-head behaviour than to be polite in English, or Italian.

• Reflexivity: Prompting alters the vector landscape by influencing token expectations. Each interaction leaves a statistical trace. Thus, prompting is not merely observational, it is participatory. The map changes with each journey. This is often referred to as "context poisoning," where previous messages/tokens within the attention-heads context window can "poison" or decrease the quality of the response, often leading to undesirable behaviour and outputs, or even an increase in hallucinations.

This, of course, goes both ways. A well curated and constructed context window can yield incredibly effective and long-running interactions with LLMs. This very concept has given rise to a new "field" within the prompt engineering landscape called "Context Engineering", and it is exactly what it sounds like. An attempt to optimize the return on investment of prompt inputs.

Toward a Praxis of Conceptual Navigation

Latent space provides a very useful lens for understanding language, as a symbolic system, but more as a dynamic topological structure. LLMs are more than simple search engines, they are geometrical operators over symbolic relationships. To use them well, one must learn to navigate the space they inhabit.

In this view, the prompter is much more than a passive user. Prompt engineering is an act of cartography, crafting charts of semantic gradients, constructing symbolic bridges, and calibrating vectors with intention.

The act of prompting becomes a form of conceptual engineering: designing the trajectory through meaning space to elicit desired outputs.

The frontier of this work is beyond simple conversations, it is geometry and cognition.

So, the journey continues, across expanding maps, through dynamic constellations, towards an ever-deeper understanding of language as space.

I hope that you enjoyed this piece, perhaps even has an aha moment! wouldn't that be nice. If you like this type of content, please like, share, and follow for more to come.