# Hierarchical three-layer Muon/BBP empirical report

The run separates exact population dynamics, fixed-sample training, and a fresh-sample ambient Hessian. The orthogonal Hessian block is conditionally block-Wishart; reported variational edges therefore concern the fresh bulk only.

## Finite-sample endpoint

| optimizer | mean exact test risk | median | std | wins vs GD | order-2 half time | order-3 half time | order-4 half time |
|---|---:|---:|---:|---:|---:|---:|---:|
| gd | 0.581553 | 0.756893 | 0.473 | 0/3 | 120 | 160 | None |
| muon_a0 | 0.146298 | 0.0871199 | 0.108 | 2/3 | 120 | 120 | 680 |
| muon_a1_7_transfer | 0.365472 | 0.411379 | 0.28 | 2/3 | 80 | 120 | 1160 |
| muon_a1_3_normalized | 0.145714 | 0.0888746 | 0.133 | 2/3 | 80 | 80 | 800 |

## Fresh-sample spectral audit

| optimizer | checkpoint | risk | clipped variational bulk | clipped empirical bulk | clipped informative | raw empirical bulk | raw informative | raw max overlap | truncation |
|---|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| gd | 0 | 2.57 | [-1.15, 1.11] | [-0.954, 0.886] | 2 | [-0.954, 0.886] | 4 | 0.519 | 0 |
| gd | 400 | 0.763 | [-10.4, 35.1] | [-9.35, 38.5] | 1 | [-9.34, 51.6] | 1 | 0.337 | 0.0069 |
| gd | 800 | 0.754 | [-2.39, 27.7] | [-1.29, 14.1] | 0 | [-1.29, 14.1] | 3 | 0.512 | 0 |
| gd | 1200 | 0.757 | [-4.21, 34.2] | [-2.24, 22.5] | 0 | [-2.24, 32.7] | 2 | 0.411 | 0.0138 |
| muon_a0 | 0 | 2.57 | [-1.15, 1.11] | [-0.954, 0.886] | 2 | [-0.954, 0.886] | 4 | 0.519 | 0 |
| muon_a0 | 400 | 0.648 | [-9.64, 41.2] | [-5.71, 46.6] | 1 | [-5.66, 176] | 0 | 0.214 | 0.0345 |
| muon_a0 | 800 | 0.288 | [-3.66, 35.6] | [-1.78, 28.4] | 1 | [-1.77, 49.6] | 1 | 0.289 | 0.0069 |
| muon_a0 | 1200 | 0.271 | [-22.3, 37] | [-24.3, 34.9] | 1 | [-24.2, 63.7] | 2 | 0.391 | 0.0276 |
| muon_a1_7_transfer | 0 | 2.57 | [-1.15, 1.11] | [-0.954, 0.886] | 2 | [-0.954, 0.886] | 4 | 0.519 | 0 |
| muon_a1_7_transfer | 400 | 0.641 | [-12.7, 42.9] | [-7.36, 46.8] | 1 | [-7.27, 161] | 0 | 0.204 | 0.0345 |
| muon_a1_7_transfer | 800 | 0.524 | [-5.28, 43.3] | [-3.56, 28.8] | 1 | [-3.54, 87.4] | 1 | 0.304 | 0.0276 |
| muon_a1_7_transfer | 1200 | 0.411 | [-19.5, 46.5] | [-22.1, 38.8] | 1 | [-21.8, 86.6] | 1 | 0.362 | 0.0621 |
| muon_a1_3_normalized | 0 | 2.57 | [-1.15, 1.11] | [-0.954, 0.886] | 2 | [-0.954, 0.886] | 4 | 0.519 | 0 |
| muon_a1_3_normalized | 400 | 0.656 | [-3.72, 46.4] | [-2.9, 47.4] | 1 | [-2.9, 92.3] | 1 | 0.353 | 0.0414 |
| muon_a1_3_normalized | 800 | 0.362 | [-3.43, 41.5] | [-1.86, 42.5] | 1 | [-1.86, 64.1] | 2 | 0.348 | 0.0414 |
| muon_a1_3_normalized | 1200 | 0.0889 | [-2.83, 34] | [-2.77, 24.9] | 1 | [-2.77, 39.2] | 3 | 0.386 | 0.0345 |

## Scope

- `muon_a0` is exact SignSVD/polar Muon on matrix blocks; vectors and biases use gradient descent.
- `a=1/7` is a raw phase-retrieval transfer control and `a=1/3` a normalized-branch control, not L3 optima.
- The training plot shows the median, individual trajectories, and the min--max envelope over paired repetitions.
- Hessian samples are fresh and independent of training. Same-sample DMFT/BBP needs a leave-one-out theorem and is not claimed here.
- The variational edge controls the bulk. A theorem for detached eigenvalues additionally needs an anisotropic local law and finite Schur determinant.
- The polynomial link has unbounded Hessian weights. Variational edges are therefore reported for explicitly clipped blocks; raw empirical bulk/outlier columns are kept separate.