\section{Beyond the quadratic link: a controlled universality layer}
\label{sec:universality}

The degree-four model is exactly closed, but ``three layers'' is not a
universality class by itself.  Changing the activation, teacher spectrum,
width scaling, data covariance, or training-data protocol can change both the
effective state and the random-matrix ensemble.  This section gives a precise
extension that is valid for any link with controlled Hermite tails, and then
states the additional hypotheses needed to connect that extension to DMFT and
BBP.

\subsection{Hilbert-space coefficient calculus}

Let \(\cH=L^2(\gamma_k)\), with its orthonormal Hermite basis, and identify a
function with its weighted coefficient vector.  Let
\(c:\Theta\subset\R^P\to\cH\) be the coefficient map of a general three-layer
network and \(c_\star\in\cH\) the teacher.  Write \(P_R\) for orthogonal
projection onto total Hermite degree at most \(R\), set
\[
 \delta=c-c_\star,\qquad
 \cR=\frac12\|\delta\|_{\cH}^2,\qquad
 \cR_R=\frac12\|P_R\delta\|_{\cH}^2,
\]
and define, at a fixed parameter value,
\begin{align}
 \eps_0(R)&=\|(\Id-P_R)\delta\|_{\cH},\label{eq:eps0}\\
 \eps_1(R)&=\|(\Id-P_R)Dc\|_{\op(\R^P,\cH)},\label{eq:eps1}\\
 \eps_2(R)&=\sup_{\|u\|=\|v\|=1}
 \|(\Id-P_R)D^2c[u,v]\|_{\cH}.
 \label{eq:eps2}
\end{align}

\begin{theorem}[Hermite-tail transfer]
\label{thm:hermite-tail-transfer}
Suppose \(c\) is twice Fr\'echet differentiable as an \(\cH\)-valued map.
Then
\begin{align}
 \cR-\cR_R&=\frac12\eps_0(R)^2,\label{eq:risk-tail}\\
 \|\nabla\cR-\nabla\cR_R\|_2
 &\leq\eps_0(R)\eps_1(R),\label{eq:gradient-tail}\\
 \|\nabla^2\cR-\nabla^2\cR_R\|_{\op}
 &\leq\eps_1(R)^2+\eps_0(R)\eps_2(R).
 \label{eq:hessian-tail}
\end{align}
The same bounds hold uniformly on a compact parameter set \(K\) after taking
the supremum of each \(\eps_j\) over \(K\).
\end{theorem}

The two terms in \eqref{eq:hessian-tail} have different meanings.
\(\eps_1^2\) is the omitted Gauss--Newton tail; \(\eps_0\eps_2\) is the
omitted residual curvature.  For the quadratic architecture,
\(\eps_j(R)=0\) for \(R\geq4\), so
\Cref{thm:exact-closure} is recovered exactly.  For a smooth non-polynomial
activation, \Cref{thm:hermite-tail-transfer} turns a Hermite approximation
into an explicit \(C^2\) approximation rather than an informal appeal to
universality.

\begin{corollary}[Stability of finite Hessian branches]
\label{cor:spectral-tail-stability}
At a fixed state, let
\(\eta_R=\eps_1(R)^2+\eps_0(R)\eps_2(R)\).  Every ordered eigenvalue of
\(\nabla^2\cR_R\) differs from the corresponding eigenvalue of
\(\nabla^2\cR\) by at most \(\eta_R\).  Consequently, an isolated finite
eigenvalue cluster separated from the rest of the truncated spectrum by a gap
larger than \(2\eta_R\) has the same multiplicity after restoring the Hermite
tail.
\end{corollary}

This corollary concerns the finite population Hessian.  It does not remove
unbounded per-example blocks from a high-dimensional empirical Hessian:
moment, truncation, and local-law estimates remain separate requirements.

\paragraph{Distributional universality is a separate interface.}
\citet{MontanariSaeedUniversality2026} prove universality of the minimum value,
and under additional hypotheses train and test error, for fixed-index
empirical-risk problems when the relevant minimizers satisfy a delocalization
condition.  This justifies Gaussian replacement in the models covered by
their theorem.  It does not imply universality of a training trajectory,
Hessian edge, BBP residue, or Muon response kernel.  Extending the present
Gaussian theory to non-Gaussian covariates therefore requires both their
optimization-level hypotheses and new derivative/local-law control for the
spectral observables.

\subsection{The DMFT--RMT--BBP closure diagram}

The physical link can be stated without identifying objects that live in
different probability spaces.  At time \(t\), a reused-data DMFT would
determine the law of the effective preactivations and residuals.  Through
\eqref{eq:T-block}, this law induces a time-dependent block law
\[
 \nu_t=\operatorname{Law}\big(T_{\theta_t}(Z,Y)\big).
\]
For a fresh probe, \(\nu_t\) is the input to the block-Wishart matrix Dyson
equation and therefore determines the orthogonal bulk.  The finite
teacher-sector Schur kernel then tests whether a signal branch lies outside
that bulk.  In short,
\begin{equation}
\begin{aligned}
 \boxed{\text{DMFT state }(Q,M,R,\varphi)}
 &\ \Longrightarrow\
 \boxed{\text{block law }\nu_t}
 \ \Longrightarrow\
 \boxed{\text{RMT bulk }[E_-(t),E_+(t)]}\\
 &\ \xrightarrow{\text{finite Schur equation}}\
 \boxed{\text{BBP branch}}.
\end{aligned}
\label{eq:dmft-rmt-bbp-chain}
\end{equation}
The first arrow is conjectural for jointly trained, reused observations; the
fresh conditional block-Wishart identity behind the second arrow is exact;
the bounded global edge theorem is imported; and the final outlier arrow
requires an anisotropic local law.

\begin{proposition}[Conditional dynamic spectral closure]
\label{prop:conditional-dynamic-closure}
Fix a compact time interval.  Assume:
\begin{enumerate}[label=(\roman*),leftmargin=*]
 \item the joint training state converges uniformly to a deterministic causal
 DMFT solution and induces a deterministic block law \(\nu_t\);
 \item the fresh orthogonal Hessian obeys a time-uniform matrix Dyson equation
 with regular support edges;
 \item an anisotropic local law holds uniformly away from those edges and the
 finite signal-sector matrices converge;
 \item either the blocks are uniformly bounded or truncation removal is
 controlled in operator norm at the local-law scale.
\end{enumerate}
Then the limiting bulk edges and every simple separated teacher-sector branch
are deterministic functions of the DMFT state.  A branch contact is a BBP
transition only when the limiting Schur zero reaches a regular edge and the
contact is transverse.
\end{proposition}

The proposition is deliberately conditional: it packages the four analytic
locks into one reusable statement but does not claim that the locks have been
proved for reused-data Muon.

\subsection{What ``many three-layer models'' requires}

The framework extends by substitution only after the following objects have
been identified for the new model:
\begin{center}
\small
\begin{tabular}{p{0.20\linewidth}p{0.31\linewidth}p{0.38\linewidth}}
\toprule
Model choice & Object that changes & Proof obligation\\
\midrule
Activation/link & coefficient map \(c(\theta)\), block \(T_\theta\) &
Hermite \(C^2\) tails and per-example moments\\
Teacher spectrum & target coefficients and active ranks &
signal-strength scaling and sector forcing\\
Data covariance & Hermite basis or orthogonal polynomials &
population closure and anisotropic bulk law\\
Widths/ranks & dimension of the finite signal sector &
uniform rank control or a growing-block RMT theorem\\
Optimizer & drift and response kernels &
descent, stability, and DMFT well-posedness\\
Data reuse & dependence of \(T_{\theta_t}\) on its own sample &
dynamic leave-one-out and time-uniform local law\\
\bottomrule
\end{tabular}
\end{center}
Thus the present paper gives a master reduction, not a theorem for every
possible three-layer network.  The exact finite algebra survives broadly;
the asymptotic closure is model- and protocol-dependent.

This dependence is physical rather than cosmetic.  In the non-lazy Bayesian
theory of \citet{VanMeegenSompolinsky2025}, changing the activation changes
the symmetry-breaking pattern and the internal coding scheme even when
kernel-level observables can look similar.  The Hermite-tail quantities
\(\eps_0,\eps_1,\eps_2\) therefore control approximation to a declared link;
they do not imply an activation-independent representation law.
