Large Language Models and Gaps in Meaning (Theory)
Original
At the Tel Aviv AI conference, I watched a presenter build an AI music video with Suno in real time. The presenter nudged prompts, regenerated, tweaked, and regenerated again, not because they could fully explain what was missing, but because they could feel it. The groove was slightly off. The texture was too glossy. The difference between “close” and “right” was obvious to a musician and frustratingly hard to name.
That moment highlighted something artists know well. Musicians often operate on feel, and they are frequently at a loss for words when asked to describe the feeling they are trying to deliver. They navigate by small edits, guided by an internal objective that is stable enough to steer their work, but not always easy to compress into explicit language. There are meaningful distinctions we can reliably perceive and act on even when we cannot cleanly articulate them.
I wanted to take that observation and generalize it beyond music. In many domains, there are ideas we cannot define crisply in language, even when we can recognize them, compare them, or move toward them by iterative refinement. We may have a sense that an argument is stronger before we can specify why. We may know a conversation is off in tone without being able to formalize the defect. We may detect a contradiction in a narrative “shape” before we can point to the sentence that caused it. The gap between what humans can mean and what humans can precisely say is directly connected to the structure of large language models, because those models are trained and operated through discrete language.
My paper is an attempt to describe that gap mathematically.
The core idea
Large language models manipulate discrete token sequences, but human meanings behave like points in a continuous space. The mismatch forces any model to represent meaning as a sparse, distorted sampling of what humans can mean. Some regions of meaning become unreachable, unstable, or hard to verify, even if they are obvious to people.
Why it’s important
A lot of discussion about LLM limitations stays at the surface level: hallucinations, brittleness, prompt sensitivity, or lack of grounding. These symptoms often get treated as engineering glitches rather than structural constraints.
My claim is stronger. Some failures are not bugs that disappear with better prompts. Some failures are not even problems of insufficient intelligence. They are consequences of mapping a continuous target, human meaning, onto a discrete interface, tokens, with a finite internal representation.
If you want to build systems that are reliable, steerable, or safe, you need a vocabulary for these structural gaps.
Plain language explanation
The paper builds a three-part picture and then uses it to explain where LLM behavior breaks.
1) Token space: what the model literally sees
An LLM is trained on sequences drawn from a finite vocabulary. Even though the number of possible strings is enormous, the set is still fundamentally combinatorial. In any single call, the context window limits the model to a finite set of possible inputs.
This gives the first constraint: the model’s interface is discrete and bounded.
2) Meaning manifold: what humans navigate
Humans experience meaning differently than token strings. Meanings have neighborhoods and smooth variation. You can soften a claim slightly, make an instruction more urgent, shift an emotional tone, or refine an aesthetic feel.
These are not naturally modeled as jumps between discrete symbols. They behave more like motion in a continuous space, which the paper calls the meaning manifold.
The music example is a particularly vivid case. “More intimate” or “less glossy” is meaningful and actionable, but often hard to define precisely in words.
3) Discrete semantic manifold: what the model can actually represent
Inside the model, we talk about embeddings as vectors. But the model is still a finite machine with finite parameters. That means it cannot realize a perfect continuous image of meaning. What it realizes is a discrete cloud of representable internal states.
The paper calls that cloud the discrete semantic manifold. It is the set of internal states the model can actually visit and use.
Separating representation from endorsement
A common mistake is to treat “the model produced it” as “the model knows it.” This paper splits that into two steps.
First is representation. Did the model land in a state that corresponds to a stable human meaning at all?
Second is endorsement. Even if it corresponds to a meaning, is it a meaning the system should accept under verification?
That distinction becomes a formal tool in the paper.
A guided tour of the argument
The paper begins with discreteness. Token strings form a countable space, and per call the context window makes the relevant input set finite. Even before we talk about intelligence, there is already an interface mismatch with human meaning.
It then introduces the meaning manifold as an ideal target. The manifold captures semantic neighborhoods, smooth variation, and fuzzy boundaries.
Next, it defines the model’s realized semantic space as discrete. For a fixed architecture and context limit, the set of internal states the model can produce is effectively finite. This makes the tension between discrete representation and continuous meaning explicit.
To connect model states to human meanings, the paper introduces a conceptual projection from internal states to points on the meaning manifold. This projection can be many to one, partial, and non-surjective. Those three properties correspond to redundancy, incoherence, and unreachable meanings in practice.
The paper then defines an operational information space using a generator and a verifier. This separates what the model can produce from what the system can produce and endorse under checks. Many hallucinations live in the gap between those two sets.
Returning to music
Music provides a concrete anchor for why the framework matters. Musicians often know exactly what they want and can move toward it through incremental edits, even when they cannot describe it precisely. This shows that humans can navigate meaning spaces that are only partially expressible in language.
Music also highlights the difficulty of verification. For aesthetic and cultural meaning, verification is often subjective, community-dependent, and unstable over time. That makes the boundary of what counts as acceptable output fuzzy and expensive to define.
This is the broader gap the paper is trying to formalize. It is not only a representational gap between tokens and meaning, but also a verification gap between what can be generated and what can be reliably endorsed.
What to watch for in the full paper
As you read the full draft, keep three distinctions in view.
First, the difference between a discrete interface and a continuous target. Tokens are discrete, but meaning behaves continuously.
Second, the difference between representable and unreachable meanings. Some meanings are structurally missing from the model’s sampling, not just difficult to reach.
Third, the difference between meaningful and endorsed outputs. A sentence can express a clear meaning and still fail verification.
Closing thought
The Tel Aviv demo made visible something easy to miss when thinking only in terms of language. Many of the most important human judgments are not discrete propositions. They are directions, gradients, and neighborhoods in a space we can feel our way through.
Large language models can be extremely powerful within the regions of that space they sample well. But the structure of tokens, finite context, and finite representation means they will always leave gaps. The goal of this paper is to describe those gaps clearly enough that we can reason about them, measure them, and design around them.