Muon for Phase Retrieval II — BBP exits in the spectra
This is the spectral readout layer. Once the Part I state is known, the question is no longer only how fast the risk moves. The question is where the gradient, weight and Hessian spectra place the informative branches as training time passes.
This part is not conceptually new optimization. It is deterministic spectral bookkeeping: bulk edges, reduced roots and residues are computed from the same state. The hard probability question is postponed to the proof appendix.
The model is [ f_W(x)=\frac1P\sum_{a=1}^P(w_a^\top x)^2, \qquad f_\star(x)=\sum_{i=1}^k\mu_i(\theta_i^\top x)^2 . ] The population summaries are [ Q=W^\top W, \qquad M=W^\top\Theta, \qquad C=M^\top Q^{-1}M . ] The matrix (C) measures how much of the teacher subspace is already in the student span. In a separated scalar window, the captured overlap of mode (i) obeys [ \dot r_i=\frac{8\mu_i}{P}r_i(1-r_i), \qquad r_i(t)= \frac{r_i(0)e^{8\mu_i t/P}} {1-r_i(0)+r_i(0)e^{8\mu_i t/P}} . ] If (r_i(0)\asymp d^{-1}), then a fixed amount of capture takes [ T_i=\frac{P}{8\mu_i}\log d+O(1). ] This is the population clock. A power-law spectrum turns it into a hierarchy of capture times.
Hessian as weighted sample covariance
For square loss, a Hessian block has the form [ H_{bc}(W)=\frac1n\sum_{\ell=1}^n \Phi_{bc}(W^\top x_\ell,\Theta^\top x_\ell)x_\ell x_\ell^\top , ] with [ \Phi_{bc}(h,y) = \frac4{P^2}h_bh_c + \frac2P \left(\frac1P|h|^2-y^\top\Lambda y\right)\delta_{bc}. ] After smoothing the loss, the weights depend only on finitely many Gaussian projections. That puts the Hessian in the finite-summary random-matrix setting: the bulk is deterministic, and informative teacher directions enter through finite resolvent matrices.
In a scalar reduction, the bulk edge can be parametrized by a Stieltjes coordinate (g): [ z(g)=-\frac1g+\alpha\,\mathbb E \frac{\phi}{\alpha+g\phi}. ] The outlier kernel for a normalized teacher coordinate (\xi) is [ K_\xi(z)=\alpha\,\mathbb E \frac{\xi^2\phi}{\alpha+g(z)\phi}. ] Edges are turning points of (z(g)). Outliers are solutions of the finite root equation. The full multi-index version is matrix-valued, but this scalar window shows the mechanism.
The large-sample edge calculation
For large Hessian sample ratio (\alpha), write [ m_\ell=\mathbb E[\phi^\ell], \qquad \theta_\ell=\mathbb E[\xi^2\phi^\ell]. ] Solving the edge equation gives [ x_\pm = m_1\pm 2\sqrt{\frac{m_2}{\alpha}} +\frac{m_3}{m_2\alpha} +O(\alpha^{-3/2}), ] and the finite kernel at the edge expands as [ K^\pm(x_\pm) = \theta_1 \pm\frac{\theta_2}{\sqrt{m_2}\sqrt\alpha} + \frac1\alpha \left( \frac{\theta_3}{m_2} - \frac{m_3\theta_2}{m_2^2} \right) +O(\alpha^{-3/2}). ] This is where the Hessian BBP threshold becomes computable. On a residual plateau (J), define [ \rho_J=\sum_{j\in J}\mu_j,\qquad S_{2,J}=\sum_{j\in J}\mu_j^2,\qquad a_J=\frac{\rho_J}{|J|+2}. ] The first left transition occurs, to leading order, when [ \mu_i^2>\frac{B_J}{2\alpha}, \qquad B_J=S_{2,J}+(|J|+8)a_J^2 . ] For a power-law teacher this creates a moving visibility front.
Residual and parallel branches
There are two Hessian stories per mode. A residual left branch measures error still outside the student span. As the mode is captured, this branch returns to the bulk and its residue disappears. A right branch measures alignment in the student span. It becomes informative only after it separates enough from the bulk.
The overlap is the BBP residue: [ \Omega(\lambda_\star) = \frac1{1-\partial_zK(\lambda_\star)} +o(1). ] At contact the residue vanishes. So a branch can be visible as an eigenvalue before it is a useful teacher eigenvector. The macroscopic alignment appears after a fixed gap opens from the edge.
The optimization interpretation is concrete. A preconditioner changes the training clock. That changes when residual branches dissolve and when parallel branches detach. If Muon equalizes the spectral front, the Hessian BBP exits become more regular in time. This couples the optimizer, the weight spectrum and the empirical Hessian through one deterministic state.
