A Gauge Theory of Superposition: Toward a Sheaf-Theoretic Atlas of Neural Representations
This addresses interpretability challenges in large language models for AI researchers, though it appears incremental as it builds on existing gauge theory and sheaf concepts.
The paper tackles the problem of superposition in large language models by developing a discrete gauge-theoretic framework that replaces a single global dictionary with a sheaf-theoretic atlas of local semantic charts, resulting in measurable obstructions to global interpretability and non-vacuous certified bounds with high coverage and zero violations across seeds/hyperparameters.
We develop a discrete gauge-theoretic framework for superposition in large language models (LLMs) that replaces the single-global-dictionary premise with a sheaf-theoretic atlas of local semantic charts. Contexts are clustered into a stratified context complex; each chart carries a local feature space and a local information-geometric metric (Fisher/Gauss--Newton) identifying predictively consequential feature interactions. This yields a Fisher-weighted interference energy and three measurable obstructions to global interpretability: (O1) local jamming (active load exceeds Fisher bandwidth), (O2) proxy shearing (mismatch between geometric transport and a fixed correspondence proxy), and (O3) nontrivial holonomy (path-dependent transport around loops). We prove and instantiate four results on a frozen open LLM (Llama~3.2~3B Instruct) using WikiText-103, a C4-derived English web-text subset, and \texttt{the-stack-smol}. (A) After constructive gauge fixing on a spanning tree, each chord residual equals the holonomy of its fundamental cycle, making holonomy computable and gauge-invariant. (B) Shearing lower-bounds a data-dependent transfer mismatch energy, turning $D_{\mathrm{shear}}$ into an unavoidable failure bound. (C) We obtain non-vacuous certified jamming/interference bounds with high coverage and zero violations across seeds/hyperparameters. (D) Bootstrap and sample-size experiments show stable estimation of $D_{\mathrm{shear}}$ and $D_{\mathrm{hol}}$, with improved concentration on well-conditioned subsystems.