Reasoning on the Simplex: Geometric Fixed-Point Models

arxiv:2609.33540

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Reasoning and the "physics" of language models · 2026-09-27

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1. Part 1
2. Part 2
3. Part 3
4. Part 4

Contents

• Reasoning on the Simplex: Geometric Fixed-Point Models
• Abstract
• 1 Introduction
• 2 Background: Looped Reasoning as Fixed-Point Iteration
• Existence and reachability.
• Are FPRM’s halted states fixed points?
• 3 Method
• 3.1 Belief state and update
• Why a simplex.
• 3.2 Training and inference
• Which fixed point is learned.
• 3.3 Language models: wavefront decode
• 3.4 Task-specific convex constraints
• Instantiations.
• Fenchel–Young readout.
• 4 Experiments
• 4.1 Tasks
• Sudoku-Extreme.
• Maze-Hard.
• S 5 S_{5} state tracking.
• Language modeling.
• 4.2 Baselines
• 4.3 Results
• What the belief state shows.
• 4.4 Language-model results
• 5 Related Work
• 6 Conclusion
• Appendix A FPRM and GFPR side by side
• Appendix B Model and training settings
• Appendix C Evaluation protocol
• Sudoku-Extreme.
• Maze-Hard.
• S5.
• Parent-field target.
• Appendix D Task-specific convex sets
• Sudoku.
• Maze.
• Fenchel–Young training.
• Appendix E Simplex language-model decode
• Appendix F FineWeb-Edu language-model evaluation
• Appendix G FPRM diagnostic protocol
• Local amplification probe (power iteration).
• Instructions for reporting errors