# Smith-normal-form Cayley positional encoding MVP

Run:

```bash
/home/maxwelhelp/main/bin/python3 experiment.py
```

`experiment.py` implements:

- all-pairs shortest paths and the paper's metric-parallel edge relation;
- connected components as candidate generator classes;
- a spanning-tree fundamental-cycle basis;
- the integer cycle-class matrix and Smith normal form;
- exact edge-integral consistency checks for the resulting labels.

The output reports the number of generator classes, cycle generators, nontrivial SNF torsion factors, free factors, and raw-versus-quotient coordinate dimensions. `path_failures=0` certifies that each fundamental cycle is represented by the integer cycle lattice; `edge_failures=0` checks that tree and non-tree edge increments are consistent with the integrated vertex labels.

The toy check is deliberately mathematical rather than a claim of transformer performance. It does not train a graph transformer, compare against Laplacian eigenvectors, measure runtime, or establish distance preservation for arbitrary coarsened partitions. The metric-parallel partition is exact for these tiny graphs, but larger-graph scalability and ML task accuracy remain open.