# Reasoning on the Simplex: Geometric Fixed-Point Models

arXiv: https://arxiv.org/abs/2609.33540 · HTML: https://mlsanity.com/paper/2609.33540/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Test-time compute and efficient thinking, one of 17 directions in LLM reasoning.
- Builds on 3 of its 32 references that are in the map.
- First posted on arXiv 27 Sep 2026.

## Placed in

- LLM reasoning / [Test-time compute and efficient thinking](https://mlsanity.com/lr/test-time-compute-and-efficient-thinking/)

## Builds on (most cited first, 3 of 3)

- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [Adaptive Computation Time for Recurrent Neural Networks](https://mlsanity.com/paper/1603.08983/) (2016)
- [Sparse Sequence-to-Sequence Models](https://mlsanity.com/paper/1905.05702/) (2019)

## Abstract (arXiv)

Looped reasoners spend test-time compute by iterating a weight-tied map, but a small residual does not mean the state is a fixed point when that map lives in unconstrained latent space. We propose Geometric Fixed-Point Reasoning (GFPR), in which the iterated state is the prediction itself: a field of categorical beliefs on a product of simplices, whose argmax is the answer at every step. Because the state is a belief, task structure can be imposed through compact convex relaxations, either as structured readouts or directly in the recurrent state; in the latter case the update remains a continuous self-map, so a fixed point exists for any parameters. At about 7M parameters, GFPR reaches 95.1% exact match on Sudoku-Extreme, 92.0% on Maze-Hard, and 100% sequence accuracy on S_5 length 128, above the published FPRM numbers at the same scale. The same update also trains a 201M language model on FineWeb-Edu in which each site is a distribution over the vocabulary; with 24 Picard steps it is above GPT-2 small on four zero-shot multiple-choice tasks and above GPT-2 medium on ARC-Easy.