The Condensate Theorem: Transformers are O(n), Not $O(n^2)$

TL;DR

This paper introduces the Condensate Theorem, showing that attention sparsity in transformers is a learned topological property, enabling lossless, linear-time attention computation with significant speedups and reduced inference costs.

Contribution

It demonstrates that attention can be computed losslessly in linear time by projecting onto a learned topological manifold, challenging the quadratic complexity assumption.

Findings

Attention mass concentrates on a topological manifold in trained models

Projection onto the Condensate Manifold achieves lossless $O(n)$ attention

Significant speedups in inference performance across multiple models

Abstract

We present the Condensate Theorem: attention sparsity is a learned topological property, not an architectural constraint. Through empirical analysis of trained language models, we find that attention mass concentrates on a distinct topological manifold -- and this manifold can be identified dynamically without checking every position. We prove a general result: for any query, projecting attention onto the Condensate Manifold (Anchor + Window + Dynamic Top-k) achieves 100% output equivalence with full $O(n^2)$ attention. This is not an approximation -- it is lossless parity. We validate this across GPT-2, Pythia, Qwen2, TinyLlama, and Mistral, demonstrating bit-exact token matching on 1,500+ generated tokens. By mapping this topology to hardware, our Topological Attention kernel achieves a 159x measured speedup at 131K tokens (3.94ms vs 628ms) and a projected >1,200x speedup at 1M tokens, reducing inference costs by >99.9% compared to Flash Attention. We conclude that the quadratic bottleneck is an artifact of naive implementation, not intelligence.