Reasoning on the Simplex: Geometric Fixed-Point Models

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 $\arg \max$ is the answer at every step. Because the state is a belief, task structure can be imposed on it directly by replacing the product of simplices with an instance-dependent compact convex set, such as a Sudoku consistency polytope, a unit-flow polytope, or the Birkhoff polytope; the update remains a continuous self-map of that set, 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.