The Lemonade Principle: Interpreting Meaning in Large Language Models

The interpretation of intermediate representations in transformer models has become a critical area of inquiry in artificial intelligence research. Like the gradual addition of lemon to water, these computational steps accumulate information—but their meaning is far from straightforward.

Intermediate Representations: A Fundamental Misinterpretation

Researchers frequently attempt to extract meaning from individual layers of transformer models, treating each intermediate representation as a potential source of understanding. This approach fundamentally misunderstands the nature of computational representation in large language models.

The Droplet Problem

Consider an intermediate layer as a single droplet of lemon in water. Attempting to determine the nature of the final solution by analyzing this droplet is not only premature but mathematically incorrect. The meaning emerges only through cumulative computational processes, with significance determined by final layer transformations.

Computational Dimensionality and Representation

Large language models operate across thousands of dimensions, rendering simplistic interpretations impossible. Each intermediate representation is:

• Partial and non-conclusive

• Dynamically reconfigured through subsequent computational steps

• Devoid of intrinsic, extractable meaning

The Fallacy of Fixed Representation

A token's representation is not a static entity but a dynamic, context-dependent approximation. Different models, even with similar architectures, may generate fundamentally different representations for identical inputs.

Tokens, Context, and Probabilistic Meaning

A token like "cat" exemplifies the arbitrary nature of representation. Contrary to intuitive understanding, this sequence of characters holds no inherent connection to a feline. Its representation is a proxy derived entirely from training data relationships—a mathematical definition based on token replaceability and contextual proximity.

The Myth of Universal Representation

Consider the concept of a "cat". There's no universal, canonical representation. Instead, we have a statistical profile constructed from:

• Frequency of token appearances

• Contextual usage patterns

• Relational characteristics with other tokens

This approach creates a "symbol profile" that is fundamentally flawed. It assumes knowledge convergence—a perfectly defined target—which is both optimistic and technically incorrect.

Symbol Profiles and Computational Limitations

The actual definition of meaning in large language models is riddled with gaps and disclaimers. The current state of data curation and lack of canonical definitions make it overly optimistic to pretend we can identify consistent representations across different contexts.

Challenges in Interpretation

Closed System Limitations

LLMs are inherently closed systems where external mental models cannot be directly mapped. Meaning must be expressed exclusively through the model's training data framework. This constraint reveals profound limitations in current interpretative approaches.

Incompatible Symbol Profiles

A critical, often-overlooked challenge is the fundamental incompatibility of token representations across different models. Even with identical architectures, the meaning assigned to a token can vary dramatically due to:

• Variations in training data

• Differences in computational dimensions

• Algorithmic nuances

Benchmark and Evaluation Misconceptions

Current AI benchmarks operate under a false assumption of universal knowledge. Take, for instance, claims of passing standardized tests like the Bar exam. Such interpretations fundamentally misunderstand the model's operation.

An LLM doesn't "understand" or "pass" a test. It statistically aligns with its training data. Given sufficient representational droplets, it will project information proportionally—including correct responses, incorrect answers, and occasional nonsensical statements.

Toward a New Interpretative Framework

Rather than seeking deterministic meaning, researchers must embrace:

• Probabilistic representation

• Contextual variability

• Computational uncertainty

The goal is not to extract fixed meanings but to comprehend the dynamic, context-dependent nature of computational representation.

Conclusion

Just as lemonade emerges through a specific concentration of ingredients, meaningful representation in large language models is a matter of computational percentage—not binary determination.

Our challenge is not to dissect individual droplets but to comprehend the entire computational process that transforms raw data into contextually relevant outputs.

Meaning in artificial intelligence is not discovered—it is probabilistically generated.​​​​​​​​​​​​​​​​