The Geometry of Thought: Disclosing the Transformer as a Tropical Polynomial Circuit

Faruk Alpay; Bilge Senturk

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arXiv · 2601.09775

The Geometry of Thought: Disclosing the Transformer as a Tropical Polynomial Circuit

Faruk Alpay·Bilge Senturk

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Abstract

We prove that the Transformer self-attention mechanism in the high-confidence regime ($β\to \infty$, where $β$ is an inverse temperature) operates in the tropical semiring (max-plus algebra). In particular, we show that taking the tropical limit of the softmax attention converts it into a tropical matrix product. This reveals that the Transformer's forward pass is effectively executing a dynamic programming recurrence (specifically, a Bellman-Ford path-finding update) on a latent graph defined by token similarities. Our theoretical result provides a new geometric perspective for chain-of-thought reasoning: it emerges from an inherent shortest-path (or longest-path) algorithm being carried out within the network's computation.

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Faruk Alpay, Bilge Senturk. 2026-01-14. The Geometry of Thought: Disclosing the Transformer as a Tropical Polynomial Circuit. https://arxiv.org/abs/2601.09775

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