The Gibbs Phenomenon: Why LLM Hallucinations Are Mathematically Inevitable

When Large Language Models (LLMs) hallucinate, we often treat it as an engineering bug to be fixed. But what if hallucinations are not a flaw, but a mathematical inevitability? A recent interdisciplinary discussion revealed a profound connection between the Gibbs phenomenon in Fourier analysis and the fundamental limitations of neural networks.

Author: Koutian Wu; GitHub: ktwu01

The Gibbs Phenomenon: A Mathematical Truth

In mathematics, the Gibbs phenomenon describes what happens when you try to approximate a discontinuous function using a Fourier series (a sum of smooth sine and cosine waves). No matter how many terms you add, there’s always an overshoot of approximately 9% near the discontinuity point.

This isn’t an error in calculation—it’s a fundamental property of continuous approximations. When you use smooth, continuous functions to represent sharp, discontinuous jumps, the approximation necessarily oscillates around the discontinuity.

Neural Networks as Continuous Function Approximators

Here’s the key insight: neural networks are essentially sophisticated continuous function approximators. They map inputs to outputs through layers of smooth, differentiable transformations. This is what makes them trainable through backpropagation.

But the world contains discontinuities:

• Logical truths are binary (true/false)
• Many real-world boundaries are sharp (legal/illegal, safe/unsafe)
• Factual statements have discrete truth values

When we force a continuous probability distribution to fit these discrete realities, we encounter the same mathematical constraint as the Gibbs phenomenon. The 9% overshoot in Fourier analysis manifests in LLMs as hallucinations—confident but incorrect statements that oscillate around the truth.

The Black Hole Imaging Analogy

The discussion drew a fascinating parallel to black hole imaging. The Event Horizon Telescope created the first image of a black hole not by taking a photograph, but by reconstructing it from extremely sparse data using interferometry and Fourier transforms.

Similarly, LLMs reconstruct human knowledge from the sparse, noisy “time series” of human civilization’s written output. Both processes involve:

• Working with incomplete, sparse data
• Extracting signal from noise (signal-to-noise ratios of 1e-6 or worse)
• Reconstructing structure in high-dimensional spaces
• Using mathematical transforms to find patterns

The black hole image required filling in gaps in Fourier space. LLMs fill gaps in knowledge space. Both are fundamentally limited by the mathematics of reconstruction from incomplete information.

Why This Matters

Understanding hallucinations as a mathematical phenomenon rather than just an engineering problem has profound implications:

1. Hallucinations are unavoidable: As long as we use continuous approximators (neural networks) to model discrete truths, some degree of error is mathematically guaranteed.

2. The 9% rule: Just as Fourier series can’t eliminate the Gibbs overshoot, we may never fully eliminate hallucinations—only reduce their frequency and impact.

3. Architecture matters: The solution isn’t just more data or bigger models, but potentially new architectures that can better handle discontinuities.

Next Token Prediction and Manifold Geometry

The discussion also touched on why “next token prediction” works so remarkably well. Human knowledge, accumulated over 100,000 years, forms a highly structured, smooth manifold in high-dimensional space. Predicting the next token isn’t just statistical pattern matching—it’s finding geodesics (shortest paths) on this knowledge manifold.

This explains why LLMs can sometimes solve problems in seconds that took humans years to figure out. They’re not just memorizing—they’re discovering shortcuts through high-dimensional knowledge space that individual human brains cannot traverse.

Conclusion

The Gibbs phenomenon reminds us that some limitations are not bugs to be fixed, but fundamental mathematical constraints. Hallucinations in LLMs may be the price we pay for using continuous approximations in a world with discrete truths.

This doesn’t mean we should accept hallucinations passively. But it does suggest that eliminating them entirely may require fundamentally different approaches—perhaps hybrid architectures that combine continuous neural networks with discrete symbolic reasoning.

The conversation between mathematics, physics, and AI continues to reveal deep truths about the nature of intelligence itself.

Content was synthesized from multiple sources and rephrased for compliance with licensing restrictions. Key concepts drawn from research on Fourier analysis, neural network approximation theory, and LLM hallucination research.