Generative Recursive Reasoning — part 5/5

Cleaned-text code points [62753, 67174). Bibliography omitted here; source.md includes it.

As shown in Table 9 , GRAM achieves 99.05% validity, outperforming all D3PM baselines. The strongest D3PM baseline, D3PM-Uniform (Big), reaches 91.33% validity while using 55.1M parameters and 1000 denoising steps. In contrast, GRAM uses fewer parameters and only 16 inference steps. In all cases, the valid samples are unique under exact board matching, indicating that the reported validity is not due to simple repetition of a small set of boards. These results show that GRAM can generate highly constrained symbolic structures from an empty input, supporting its potential as a generator beyond conditional puzzle solving.

Figure 17 illustrates the unconditional Sudoku generation setup. Starting from an empty board, the task is to generate complete boards, and validity is determined by whether the generated board satisfies all Sudoku constraints. Figure 7 shows qualitative examples of boards generated by GRAM.

### D.6 Visualizing Latent Recursion Process

To understand how stochastic guidance shapes reasoning, we visualize latent trajectories during recursive computation. Specifically, we track the high-level state h h at each supervision step throughout the recursion process. For visualization, we project these latent vectors into 2D using PCA [ 59 ] and interpolate unobserved states via K-D tree [ 60 ] to construct a continuous loss landscape.

Figures 18 and 19 compare TRM and GRAM on the same Sudoku puzzle. TRM follows a single deterministic path from initialization to solution, offering no mechanism to escape if the trajectory enters a suboptimal region. In contrast, GRAM samples diverse trajectories that explore different regions of latent space before converging. While some trajectories become trapped in local minima (bright yellow regions), others successfully navigate toward the global optimum (dark blue regions). This diversity enables GRAM to discover valid solutions more reliably through parallel exploration.

## Licenses

Asset Use in this paper License / terms Source link MNIST Binarized MNIST generation experiments Creative Commons Attribution-Share Alike 3.0 https://keras.io/api/datasets/mnist/ ARC-AGI-1 / original ARC ARC-AGI reasoning benchmark Apache License 2.0 https://github.com/fchollet/ARC-AGI ARC-AGI-2 ARC-AGI-2 reasoning benchmark / reference results Apache License 2.0 https://github.com/arcprize/ARC-AGI-2 HRM repository HRM baseline and Sudoku-Extreme-related reference implementation Apache License 2.0 https://github.com/sapientinc/HRM TinyRecursiveModels / TRM repository TRM baseline and recursive reasoning reference implementation MIT License https://github.com/SamsungSAILMontreal/TinyRecursiveModels MDLM repository Masked diffusion baseline reference implementation, if public code is used Apache License 2.0 https://github.com/kuleshov-group/mdlm Google Research D3PM implementation D3PM image-generation baseline reference implementation, if public code is used Apache License 2.0 https://github.com/google-research/google-research/blob/master/d3pm/images/diffusion_categorical.py Looped Transformer repository Looped Transformer baseline reference implementation, if public code is used MIT License https://github.com/Leiay/looped_transformer N-Queens Synthetic multi-solution constraint satisfaction task generated by the authors Not an external asset N/A Graph Coloring Synthetic multi-solution constraint satisfaction task generated by the authors Not an external asset N/A

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