# Dense spectral-time panel

## Object

One paired training instance is rerun with a dense checkpoint grid. A single Gaussian probe `(z, xi)` is drawn independently of training and then frozen across checkpoints and optimizers. This coupling makes spectral motion readable; it is not a same-data Hessian claim.

## How to read the spectral panels

- The blue band is the empirical spectrum of the orthogonal Hessian `sum T(z_mu) tensor xi_mu xi_mu^T / n`.
- Every point is an eigenvalue of the raw full direct-weight Hessian.
- Point color is the eigenvector mass in the teacher-coordinate subspace: dark means nearly orthogonal, yellow means teacher aligned.
- A green ring marks a point that is both outside the orthogonal bulk and above the pre-specified teacher-overlap threshold.
- Vertical dotted lines are the first recorded half-error events for Hermite degrees 2, 3, and 4.

## How to read `dmft_bbp_rmt_master_panel.png`

- Row A contains exact population quantities. The black curve is risk normalized by its initial value; the colored curves are exact Hermite-sector errors normalized in the same way.
- Row B is deterministic finite algebra at the trained state. A colored curve is the ratio `lambda_min_positive(S_r) / rho_r`; values above one certify positive curvature on the residualized degree-r tangent space. Values below one mean only that this sufficient certificate fails.
- Row C is a fresh-sample finite-size diagnostic. The band is an empirical orthogonal bulk, point color is teacher-coordinate mass, and green diamonds apply the declared detachment-and-overlap rule.
- Columns use the same initialization, training observations, and fresh Gaussian probe. Dotted lines repeat the sector half-error events across all three rows; visual proximity is not a causality test.

## Configuration

- seed: `424242`
- training steps: `1200`
- diagnostic interval: `20`
- spectral checkpoints: `25`
- train sample size: `1024`
- fresh probe size: `145`
- ambient dimension: `32`; latent dimension: `3`
- maximum step-zero spectral discrepancy across optimizers: `0.000e+00`

- exact geometry states evaluated: `100`

## Half-error times on the displayed instance

| optimizer | degree 2 | degree 3 | degree 4 |
|---|---:|---:|---:|
| Gradient descent | 120 | 260 | not reached |
| Polar Muon ($a=0$) | 80 | 120 | 140 |
| Muon ($a=1/7$) | 60 | 120 | 100 |
| Muon ($a=1/3$) | 60 | 380 | 80 |

## Scope

The orthogonal bulk identity is exact conditional on the latent probe coordinates. The plotted edges are empirical finite-sample edges, not Montanari--Saeed variational limits. Colored detached points are diagnostics; a BBP theorem still requires an anisotropic local law and a controlled finite Schur limit.