A real trained Hierarchical Reasoning Model's recurrent dynamics develop multiple fixed points rather than a single one, and the model 'guesses' by converging to whichever fixed point it reaches first -- which can be wrong even on trivially simple puzzles

- Ren & Liu mechanistically analyze a real Hierarchical Reasoning Model (HRM) trained from scratch on Sudoku-Extreme, finding (largely from qualitative PCA-projected trajectories) it can fail on extremely simple puzzles (e.g. those with only one unknown cell), tracing this to violation of the fixed-point property that is a fundamental assumption of HRM's recurrent-convergence design. [ren-liu-2026-reasoning-or-guessing-hrm] - The model's recurrent dynamics are found to possess multiple fixed points rather than a single one; HRM converges to whichever fixed point its recurrent iteration reaches first, and this first-reached fixed point can be an incorrect answer even when the correct answer is trivial to determine -- a "guessing," rather than "reasoning," dynamical signature. [ren-liu-2026-reasoning-or-guessing-hrm] - This multiple-fixed-point structure co-occurs with a "grokking"-like training dynamic in which answer correctness improves non-uniformly, tied to specific critical reasoning steps; based on this diagnosis, the authors propose data augmentation, input perturbation, and model-bootstrapping strategies (an "Augmented HRM") that raise Sudoku-Extreme accuracy to 96.9% for the Augmented HRM, versus 55.0% for the baseline HRM. [ren-liu-2026-reasoning-or-guessing-hrm]