Equilibrium Reasoners

Learning Attractors Enables Scalable Reasoning

Benhao Huang · Zhengyang Geng · Zico Kolter
CMU

Code for reproducing EqR experiments on Sudoku-Extreme and Maze-Unique.

Setup

uv venv
source .venv/bin/activate
uv pip install -r requirements.txt

adam-atan2 must install its CUDA backend (adam_atan2_backend); a Python-only install is not sufficient for training. Build it in an environment where CUDA and Python headers are available:

# If CUDA is not already configured, point CUDA_HOME at the active CUDA
# toolkit. For example:
# export CUDA_HOME="$(dirname "$(dirname "$(which nvcc)")")"
# export PATH="$CUDA_HOME/bin:$PATH"

# Optional: set this when building without visible GPUs or when cross-compiling.
# Examples: H100=9.0, A100=8.0, RTX 4090/L40=8.9.
# export TORCH_CUDA_ARCH_LIST=9.0

python -m pip install --no-build-isolation --no-cache-dir --force-reinstall \
  adam-atan2==0.0.3
python - <<'PY'
from adam_atan2 import AdamATan2
import adam_atan2_backend
print(AdamATan2, adam_atan2_backend.__file__)
PY

Maze training also requires FlashAttention on CUDA.

FlashAttention install notes

EqR follows the HRM attention import pattern: prefer FlashAttention-3 via flash_attn_interface when available, and fall back to FlashAttention-2 via flash_attn. On NVIDIA Hopper GPUs, FlashAttention-3 is recommended. If a local wheel is available:

python -m pip install --no-deps <path-to-flash-attn-3-wheel.whl>
python - <<'PY'
import flash_attn_interface
print(flash_attn_interface.__file__)
PY

If FlashAttention-3 is unavailable, install FlashAttention-2:

python -m pip install flash-attn --no-build-isolation
python - <<'PY'
from flash_attn import flash_attn_func
print(flash_attn_func)
PY

For W&B defaults, copy config/secrets.example.yaml to config/secrets.yaml and fill in your entity/project. config/secrets.yaml is ignored by git.

Data

Default dataset paths:

Dataset	Path
Sudoku-Extreme	data/sudoku-extreme-1k-aug-1000
Maze-Unique	data/maze-30x30-unique-1k

Download datasets:

bash scripts/download_artifacts.sh

Download datasets and pretrained checkpoints:

bash scripts/download_artifacts.sh --with-ckpts

The helper downloads Hugging Face snapshots under downloads/eqr-artifacts, copies datasets into data/, and copies checkpoints into downloaded_checkpoints/.

Checkpoint paths:

downloaded_checkpoints/sudoku-extreme/eqr.pth
downloaded_checkpoints/maze-unique/eqr.pth

Optional: rebuild datasets

Dataset rebuild commands

Use these commands only if you want to regenerate the datasets instead of downloading locuslab/EqR-data.

Sudoku-Extreme follows the sapientinc/HRM setup. The builder downloads train.csv and test.csv from sapientinc/sudoku-extreme, converts them to HRM/EqR-compatible NumPy arrays, and augments only the training split with Sudoku-preserving digit, row, column, and transpose transformations.

Default EqR/HRM small-sample dataset:

python -m dataset.build_sudoku_dataset \
  --output-dir data/sudoku-extreme-1k-aug-1000 \
  --subsample-size 1000 \
  --num-aug 1000 \
  --seed 42

Full Sudoku-Extreme dataset:

python -m dataset.build_sudoku_dataset \
  --output-dir data/sudoku-extreme-full \
  --seed 42

The generated directory should contain train/, test/, and identifiers.json. Each split contains dataset.json plus all__inputs.npy, all__labels.npy, all__puzzle_identifiers.npy, all__puzzle_indices.npy, and all__group_indices.npy.

Additional Sudoku-Extreme options:

python -m dataset.build_sudoku_dataset --help

Build Maze-Unique:

python dataset/build_maze_unique_dataset.py \
  --output-dir data/maze-30x30-unique-1k \
  --grid-size 30 \
  --train-samples 1000 \
  --test-samples 1000 \
  --maze-mode perfect \
  --length-distribution uniform \
  --min-path-length 100 \
  --max-path-length 140 \
  --require-unique \
  --dedupe

Training

scripts/train.sh launches torchrun --standalone with one local process by default. Override NPROC_PER_NODE only when you want multi-GPU training:

bash scripts/train.sh eqr_sudoku
bash scripts/train.sh trm_sudoku
bash scripts/train.sh eqr_maze_unique
bash scripts/train.sh trm_maze_unique

NPROC_PER_NODE=2 bash scripts/train.sh eqr_maze_unique

Dataset	Config	Steps
Sudoku-Extreme	config/train/eqr_sudoku.yaml	50k
Sudoku-Extreme TRM baseline	config/train/trm_sudoku.yaml	50k
Maze-Unique	config/train/eqr_maze_unique.yaml	100k
Maze-Unique TRM baseline	config/train/trm_maze_unique.yaml	100k

Evaluation

scripts/eval.sh uses config/eval/depth_breadth.yaml. By default it runs depth 16 with breadth 1.

bash scripts/eval.sh /path/to/checkpoint.pth
bash scripts/eval.sh /path/to/checkpoint.pth halt_max_steps=64
bash scripts/eval.sh /path/to/checkpoint.pth \
  halt_max_steps=64 different_init=128 convergence_top_k=1 global_batch_size=16

After downloading checkpoints, evaluate the pretrained EqR models:

bash scripts/eval.sh downloaded_checkpoints/sudoku-extreme/eqr.pth
bash scripts/eval.sh downloaded_checkpoints/maze-unique/eqr.pth

Evaluation config:

config/eval/depth_breadth.yaml

TODO

• Release XLA code for large-scale inference.

Acknowledgements

This codebase builds on the HRM and TRM repositories.

License

This project is released under the Apache License 2.0. See LICENSE for details.

Citation

@article{huang2026equilibrium,
  title={Equilibrium Reasoners: Learning Attractors Enables Scalable Reasoning},
  author={Huang, Benhao and Geng, Zhengyang and Kolter, Zico},
  journal={arXiv preprint arXiv:2605.21488},
  year={2026}
}