## Summary of Installation Issues with AlphaGeometry and Meliad

I am trying to run the full `alphageometry` solver by following the instructions in the `README.md`. While the `ddar` solver works perfectly after some modifications, I am unable to get the full `alphageometry` mode (which requires the Meliad library) to run.

I have followed all installation steps, including manually cloning the Meliad repository and setting the `PYTHONPATH`. However, this leads to a series of persistent import errors.

### Environment
* **Python Version**: 3.10.9 (in a virtual environment)
* **System**: Remote server (Beagle3)

### Sequence of Errors and Fixes Attempted

1. **Initial Error**: `ImportError: cannot import name 'decoder_stack' from 'transformer'`.
* This occurred because Python could not find the Meliad library.
* **Fix**: I confirmed that I had cloned Meliad into `meliad_lib/meliad` and correctly set the `PYTHONPATH` using `export PYTHONPATH=$PYTHONPATH:$(pwd)/meliad_lib/meliad`.

2. **Circular Import Error**: After fixing the path, a new error appeared: `ImportError: cannot import name 'decoder_stack' from partially initialized module 'transformer' (most likely due to a circular import)`.
* This seemed to be caused by a file name conflict, as both AlphaGeometry and Meliad have a file named `decoder_stack.py`.
* **Fix**: I renamed the AlphaGeometry file to `ag_decoder_stack.py` and updated the corresponding import statement in `models.py`.

3. **Module Not Found Error**: The previous fix led to `ModuleNotFoundError: No module named 'transformer.decoder_stack'`.
* To diagnose this, I ran a direct import test in a Python shell (`import transformer.decoder_stack`), which also failed. This confirmed the issue was with the Meliad library itself, not the AlphaGeometry script.
* **Fix**: I installed Meliad's own dependencies by running `pip install -r meliad_lib/meliad/requirements.txt`.

4. **NumPy Version Conflict**: After installing Meliad's dependencies, a new error occurred: `ImportError: numpy.core.multiarray failed to import`.
* This was because Meliad's requirements installed NumPy 2.x, which conflicted with `matplotlib` and other packages that required NumPy 1.x.
* **Fix**: I downgraded NumPy by running `pip install --force-reinstall "numpy<2.0"`.

5. **Persistent `ModuleNotFoundError`**: After all these fixes, the error has reverted back to `ModuleNotFoundError: No module named 'transformer.decoder_stack'`.

### Current Status
The purely symbolic **DDAR solver works correctly**. However, the full `alphageometry` mode is unusable due to what appears to be a fundamental issue with the way the Meliad library's modules are structured or imported. Despite all troubleshooting, the core problem of importing `transformer.decoder_stack` persists.

Any help or guidance on how to resolve this final import issue would be greatly appreciated.