Muon for Phase Retrieval I — Fixed power, one spectral clock
This is the fixed-exponent layer of the story. We freeze the Muon exponent (a), solve one Paquette/Volterra state, and ask for a prediction stronger than “the loss goes down”. The prediction should give the learning curve and also the spectra seen during training.
The story is organized as follows: Part I fixes (a); Part II reads the three spectra from the same state; Part III lets (a) move. The proof appendix then separates the exact reductions from the genuinely hard RMT bridge.
The model is quadratic phase retrieval: [ f_\star(x)=\sum_i\mu_i(\theta_i^\top x)^2, \qquad f_W(x)=\frac1P\sum_{b=1}^P(w_b^\top x)^2 . ] The teacher strengths are ordered, often as a power law (\mu_i\simeq i^{-\alpha}). The useful variables are not the coordinates of all weights. They are the captured masses (q_i^2(t)), the residual masses, and the aggregate scale (Q(t)). Training is a front moving through the teacher spectrum.
The Volterra state
After inserting the Muon power filter into the Paquette projected-risk recursion, each mode obeys [ \dot q_i^2(t) = -2\eta\,\delta_i^{(a)}Q(t)^{(a-1)/2}q_i^2(t) + \eta^2\nu_i^{(a)}Q(t)^a . ] The drift coefficient (\delta_i^{(a)}) is the useful spectral motion of mode (i). The coefficient (\nu_i^{(a)}) is the volatility paid by applying the same power near the hard edge.
The proof of the Volterra equation is just integration of this linear mode equation in the right clock. Set [ d\tau=\eta Q(t)^{(a-1)/2}dt . ] Then [ \partial_\tau q_i^2 = -2\delta_i^{(a)}q_i^2 + \eta\nu_i^{(a)}Q(\tau)^{(a+1)/2}. ] Solving gives [ q_i^2(\tau)=e^{-2\delta_i^{(a)}\tau}q_i^2(0) +\eta\nu_i^{(a)}\int_0^\tau e^{-2\delta_i^{(a)}(\tau-u)}Q(u)^{(a+1)/2}du . ] Summing over modes with the deterministic risk weights gives [ Q(\tau)=F_a(\tau) + \eta\int_0^\tau K_a(\tau-u)Q(u)^{(a+1)/2}du . ] This is the fixed-(a) Volterra law. Everything spectral below is read from this same state.
The constant exponent
The exponent (a) changes two scales. Smaller (a) equalizes singular values more aggressively, so weak modes move earlier. But the hard edge becomes more expensive. In a local power-law window [ \mu_i\asymp i^{-\alpha}, \qquad q_i^2(0)\asymp i^{-\beta}, \qquad \alpha+\beta>1, ] the Volterra balance has the two powers [ e_{\rm bal}(a) = \max\left{ \frac{1-a}{2}, \frac{\alpha(1+a)}{2(\alpha+\beta-1)} \right}. ] The first term is hard-edge amplification. The second term is tail resolution. Minimizing the maximum gives [ a_{\rm bal,}(\alpha,\beta) = \left[ \frac{\beta-1}{2\alpha+\beta-1} \right]_{[0,1]} . ] For the near-zero phase-retrieval window, (\beta=\alpha), hence [ a_{\rm bal,}^{PR}(\alpha)=\frac{\alpha-1}{3\alpha-1}. ] This is not a magic decimal. It is the point where the two asymptotic errors have the same order. At the dimensions in the plots, nearby values such as (0.12) and (0.14) can be essentially indistinguishable.
Three spectra
The same Volterra state predicts three spectral readouts:
- the gradient spectrum, which tells us what the optimizer can denoise now;
- the weight spectrum, which tells us which teacher directions are represented;
- the Hessian spectrum, which tells us which residual or parallel directions are visible in local geometry.
The common algebra is finite-root elimination. Split a hermitized channel into a finite teacher block and a bulk block: [ \mathcal H_t^c= \begin{pmatrix} A_t^c&B_t^c
(B_t^c)^&\mathcal B_t^c \end{pmatrix}. ] For (z) outside the bulk spectrum, define [ \mathcal K_t^c(z)= A_t^c+B_t^c(zI-\mathcal B_t^c)^{-1}(B_t^c)^ . ] Outliers solve [ \det(zI-\mathcal K_t^c(z))=0 . ] If (x_+^c(t)) is the right bulk edge, the BBP margin is [ \Delta_i^c(t)=\lambda_i^c(t)-x_+^c(t). ] The exit time is the first time this margin becomes positive.
This is why sorted empirical eigenvalues are not the right primitive object. Close to a dense front, ranks switch. The reduced outlier root is labelled by the teacher mode, and the residue tells us whether the detached branch actually carries teacher mass: [ \Omega_i^c(t)= \frac1{(u_i^c)^\top (I-\partial_z\mathcal K_t^c(\lambda_i^c))u_i^c}. ]
The constant-(a) conclusion is simple: there is one deterministic Volterra state, and the gradient, weight and Hessian spectra are different projections of it. A good comparison is therefore not only predicted loss versus measured loss. It is the whole moving spectral diagram: bulk edges, finite roots, residues and BBP exit times.
