\section{Spectral optimization and Muon geometry}
\label{sec:muon}

Let \(G\in\R^{m\times n}\) be a nonzero matrix gradient and write a thin SVD
\[
 G=L\Diag(\sigma_1,\ldots,\sigma_\ell)R^\trans,\qquad
 \ell=\min(m,n),\qquad \sigma_i\geq0.
\]
Set \(\bar\sigma=\ell^{-1}\sum_i\sigma_i>0\).  For \(a\geq0\) and numerical
floor \(\eps\geq0\), define
\begin{align}
 w_i^{(a,\eps)}
 &=\max\{\sigma_i/\bar\sigma,\eps\}^{a},\label{eq:muon-weights}\\
 c_{a,\eps}(G)
 &=\left(\frac1{mn}\sum_{i=1}^{\ell}
 (w_i^{(a,\eps)})^2\right)^{1/2},\nonumber\\
 \Phi_{a,\eps}(G)
 &=c_{a,\eps}(G)^{-1}
 L\Diag(w_1^{(a,\eps)},\ldots,w_\ell^{(a,\eps)})R^\trans.
 \label{eq:muon-map}
\end{align}
Thus \(\|\Phi_{a,\eps}(G)\|_\F^2=mn\), matching the unit-entry-RMS
normalization in the implementation.  At \(a=0\),
\[
 \Phi_{0,\eps}(G)=\sqrt{\frac{mn}{\ell}}\,LR^\trans,
\]
the normalized polar or SignSVD direction.  Ignoring the floor, \(a=1\)
preserves the gradient singular-value profile up to a scalar.

\begin{proposition}[Muon powers are descent directions]
\label{prop:muon-descent}
For every nonzero \(G\), \(a\geq0\), and \(\eps>0\),
\begin{equation}
 \langle G,\Phi_{a,\eps}(G)\rangle_\F
 =\frac{\sum_{i=1}^{\ell}\sigma_iw_i^{(a,\eps)}}
 {c_{a,\eps}(G)}>0.
 \label{eq:muon-inner}
\end{equation}
Let \(F\) have an \(L_F\)-Lipschitz gradient in a neighborhood of \(W\), with
\(G=\nabla F(W)\).  Then
\begin{equation}
 F(W-\eta\Phi_{a,\eps}(G))
 \leq F(W)-\eta\langle G,\Phi_{a,\eps}(G)\rangle_\F
 +\frac{L_F\eta^2}{2}mn.
 \label{eq:descent-lemma}
\end{equation}
Consequently the update strictly decreases \(F\) whenever
\begin{equation}
 0<\eta<
 \frac{2\langle G,\Phi_{a,\eps}(G)\rangle_\F}{L_Fmn}.
 \label{eq:sufficient-step}
\end{equation}
\end{proposition}

The experiment uses backtracking, so it enforces actual loss decrease rather
than assuming \eqref{eq:sufficient-step} with an unknown local smoothness
constant.

\begin{proposition}[Fr\'echet derivative of the spectral channel]
\label{prop:muon-frechet}
Let \(G\in\R^{m\times n}\), \(m\leq n\), have full row rank, set
\(S=GG^\trans=L\Diag(\lambda_1,\ldots,\lambda_m)L^\trans\), and put
\(b=(a-1)/2\).  Ignoring the numerical floor, define
\[
 \Psi_a(G)=S^bG,\qquad
 \Phi_a(G)=\sqrt{mn}\,\frac{\Psi_a(G)}{\|\Psi_a(G)\|_\F}.
\]
For a perturbation \(\Delta\), let
\[
 E=G\Delta^\trans+\Delta G^\trans
\]
and let \(F_b^{[1]}\) be the Loewner matrix of \(x^b\):
\[
 (F_b^{[1]})_{ij}=
 \begin{cases}
 \dfrac{\lambda_i^b-\lambda_j^b}{\lambda_i-\lambda_j},
 &\lambda_i\neq\lambda_j,\\[6pt]
 b\lambda_i^{b-1},&\lambda_i=\lambda_j.
 \end{cases}
\]
Then
\begin{align}
 D\Psi_a(G)[\Delta]
 &=L\!\left[
 F_b^{[1]}\odot(L^\trans E L)
 \right]L^\trans G+S^b\Delta,
 \label{eq:muon-frechet-unnormalized}\\
 D\Phi_a(G)[\Delta]
 &=q\,D\Psi_a(G)[\Delta]
 -q\,\Psi_a(G)
 \frac{\langle\Psi_a(G),D\Psi_a(G)[\Delta]\rangle_\F}
 {\|\Psi_a(G)\|_\F^2},
 \label{eq:muon-frechet-normalized}
\end{align}
where \(q=\sqrt{mn}/\|\Psi_a(G)\|_\F\).
\end{proposition}

\Cref{prop:muon-frechet} is the object needed for a spectral Onsager reaction:
an AMP correction uses an averaged Jacobian or response contraction of the
denoising channel.  The derivative is matrix valued and couples singular
directions; replacing it by the scalar derivative of
\(\sigma\mapsto\sigma^a\) would discard the rotations of the singular vectors
and the unit-RMS normalization.

\subsection{Blockwise update used in the experiments}

For \(B\in\{V,U,A\}\), let \(G_B=\nabla_B F\).  A spectral run updates
\begin{equation}
 B^+=B-\eta_B\,\Phi_{a_B,\eps}(G_B).
 \label{eq:block-muon}
\end{equation}
Biases and readout vectors use ordinary gradient descent.  The reported
constant-power controls impose \(a_V=a_U=a_A=a\), with
\[
 a\in\left\{0,\frac17,\frac13\right\}.
\]
The values \(1/7\) and \(1/3\) come from distinct normalizations in the
power-law phase-retrieval program and are \emph{transfer controls}.  They are
not derived minimizers of a three-layer objective.

\begin{remark}[Different blocks have different spectra]
\(V\) controls the first quadratic feature, \(U\) injects a direct linear
coordinate into the second block, and \(A\) mixes already nonlinear features.
Even when \(p=q=k\), their gradients have different factorizations.  A
theoretically natural control is therefore
\[
 (a_V(t),a_U(t),a_A(t),\eta_V(t),\eta_U(t),\eta_A(t)),
\]
not a single universal exponent.
\end{remark}

\subsection{Why spectral flattening may change sector clocks}

Write the degree-\(r\) coefficient derivative with respect to block \(B\) as
\(J_{r,B}\).  Under ordinary gradient flow, the block contributes
\[
 B_{rs}^{(B)}
 =D_r^{1/2}J_{r,B}J_{s,B}^\trans D_s^{1/2}
\]
to the coefficient kernel.  Muon is nonlinear in the full gradient, so it
does not correspond to replacing \(J_{r,B}J_{s,B}^\trans\) by a fixed
preconditioner.  Nevertheless, at one state its progress is exactly
\begin{equation}
 -\left\langle\nabla_BF,\Phi_{a_B,\eps}(\nabla_BF)\right\rangle_\F,
 \label{eq:block-progress}
\end{equation}
which weights singular channels by
\(\sigma_i^{1+a_B}\) up to normalization.  Lower \(a_B\) flattens the channel
speeds and may expose weak late-sector directions sooner; it can also
over-amplify noisy or unhelpful channels.  This tradeoff explains why the pilot
contains both large gains and paired failures.

\subsection{Relation to the two-layer spectral theory}

In matrix least squares, the spectral-optimizer dynamics can be reduced to
deterministic spectral observables \citep{PaquetteEtAl2026}.  In a low-rank
two-layer factorization, invariance can reduce Muon to a finite right action.
For \eqref{eq:network}, the gradient of \(V\) appears both directly through
\(q\) and indirectly through \(Aq\); the action on \(A\) changes the basis seen
by the \(U\)-block.  The exact Hermite map remains finite, but the Muon update
is not a function of coefficient error alone.  This is the precise extra state
that a three-layer controlled DMFT must retain.

\begin{conjecture}[Block-specific spectral control]
\label{conj:block-control}
In a proportional growing-width and growing-rank limit, the asymptotically
optimal power schedule is block specific and state dependent.  Each
\(a_B(t)\) is determined by a local projection of the corresponding
signal-to-noise spectral filter onto the power family
\(\sigma\mapsto\sigma^{a_B}\).
\end{conjecture}

This conjecture is falsifiable by measuring useful and bulk singular-vector
overlaps separately for \(G_V,G_U,G_A\).  Endpoint risk alone cannot identify
the proposed mechanism.
