\section{Fresh-sample Hessian, block Wishart, and the BBP interface}
\label{sec:rmt}

\subsection{Analytic preactivation Hessian}

At a fixed latent input \(z\), regard the two direct preactivations
\[
 h=Vz+b\in\R^p,\qquad u=Uz\in\R^q
\]
as independent variables and write
\[
 q_h=h^{\odot2}-\one_p,\qquad
 r=u+Aq_h+c,\qquad
 f=a_0+\alpha^\trans q_h+\beta^\trans(r^{\odot2}-\one_q).
\]
Define
\begin{equation}
 d_h=\alpha+2A^\trans(\beta\odot r),\qquad
 D_h=\Diag(h),\qquad D_\beta=\Diag(\beta).
 \label{eq:dh}
\end{equation}

\begin{proposition}[Link gradient and Hessian]
\label{prop:link-hessian}
With \(v=(h,u)\in\R^{p+q}\), the gradient of \(f\) is
\begin{equation}
 g_\theta(z)=\nabla_vf
 =\begin{bmatrix}2h\odot d_h\\2\beta\odot r\end{bmatrix}.
 \label{eq:link-gradient}
\end{equation}
The Hessian \(K_\theta(z)=\nabla_v^2f\) has blocks
\begin{align}
 K_{hh}
 &=2\Diag(d_h)+8D_hA^\trans D_\beta A D_h,\label{eq:Khh}\\
 K_{hu}
 &=4D_hA^\trans D_\beta,\label{eq:Khu}\\
 K_{uu}
 &=2D_\beta,\qquad K_{uh}=K_{hu}^\trans.
 \label{eq:Kuu}
\end{align}
For square loss \(\ell_\theta(z,y)=\frac12(f_\theta(z)-y)^2\), its
preactivation Hessian is
\begin{equation}
 T_\theta(z,y)
 :=\nabla_v^2\ell_\theta(z,y)
 =g_\theta(z)g_\theta(z)^\trans
 +(f_\theta(z)-y)K_\theta(z).
 \label{eq:T-block}
\end{equation}
\end{proposition}

\subsection{Direct-weight Kronecker structure}

Stack the ambient direct matrices as
\[
 W=\begin{bmatrix}W_1\\W_2\end{bmatrix}
 \in\R^{(p+q)\times d}.
\]
Since \(v=Wx+(b,0)\), the per-example direct-weight Hessian is, up to the
fixed permutation associated with vectorization,
\begin{equation}
 \nabla_{\mathrm{vec}(W)}^2\ell_\theta(x,y)
 =T_\theta(z,y)\kron xx^\trans.
 \label{eq:per-sample-kron}
\end{equation}
Let \(r_b=p+q\) denote the block size.

\begin{theorem}[Exact fresh orthogonal block-Wishart identity]
\label{thm:block-wishart-identity}
Embed a fixed reduced state as in \Cref{sec:model}.  Draw a fresh independent
sample
\[
 x_\mu=\Theta z_\mu+\Theta_\perp\xi_\mu,\qquad
 (z_\mu,\xi_\mu)\stackrel{\mathrm{iid}}{\sim}
 \mathcal N(0,\Id_k)\kron\mathcal N(0,\Id_{d-k}),
\]
and set \(y_\mu=f_\star(z_\mu)\).  Let \(H_{WW}^{(n)}\) be the empirical
direct-weight Hessian.  Its principal block on
\(\R^{r_b}\kron\Ran(\Theta_\perp)\) is exactly
\begin{equation}
 H_{\perp\perp}^{(n)}
 =\frac1n\sum_{\mu=1}^n
 T_\theta(z_\mu,y_\mu)\kron\xi_\mu\xi_\mu^\trans.
 \label{eq:exact-block-wishart}
\end{equation}
Conditional on \(\{z_\mu\}_{\mu\leq n}\), the Gaussian vectors
\(\{\xi_\mu\}_{\mu\leq n}\) are independent of the self-adjoint matrix weights
\(\{T_\theta(z_\mu,y_\mu)\}_{\mu\leq n}\).  Hence
\eqref{eq:exact-block-wishart} is a matrix-weighted block-Wishart matrix with
fixed block size \(r_b\).
\end{theorem}

This theorem is an identity, not an asymptotic approximation.  Freshness is
essential: if the same observations trained \(\theta_t\), then
\(T_{\theta_t}(z_\mu,y_\mu)\) would depend on \(\xi_\mu\) through the entire
trajectory.

\subsection{Bounded blocks and imported global spectrum}

For \(L>0\), define spectral clipping
\[
 \Pi_L(T)=Q\Diag(\max\{-L,\min\{\lambda_i,L\}\})Q^\trans
\]
when \(T=Q\Diag(\lambda_i)Q^\trans\).  Let
\[
 H_{\perp\perp,L}^{(n)}
 =\frac1n\sum_{\mu=1}^n\Pi_L(T_\mu)\kron\xi_\mu\xi_\mu^\trans.
\]

\begin{lemma}[Deterministic clipping perturbation]
\label{lem:clipping}
For every realized sample,
\begin{equation}
 \|H_{\perp\perp}^{(n)}-H_{\perp\perp,L}^{(n)}\|_{\op}
 \leq\frac1n\sum_{\mu=1}^n
 \|T_\mu-\Pi_L(T_\mu)\|_{\op}\,\|\xi_\mu\|_2^2.
 \label{eq:clip-bound}
\end{equation}
\end{lemma}

The bound explains why a small clipping fraction need not imply a small edge
perturbation: a rare polynomial block can have both a large excess norm and a
large \(\|\xi_\mu\|^2\).

Let \(\nu_{\theta,L}\) be the law of \(\Pi_L(T_\theta(Z,f_\star(Z)))\).
For \(S\succ0\), define the matrix \(K\)-transform
\begin{equation}
 \mathbb K_{\alpha,\nu}(S)
 =\int T(\Id_{r_b}+ST)^{-1}\nu(\dd T)
 -\alpha^{-1}S^{-1}.
 \label{eq:block-K}
\end{equation}
Its derivative in a self-adjoint direction \(\Delta\) is
\begin{align}
 D\mathbb K_{\alpha,\nu}(S)[\Delta]
 &=\alpha^{-1}S^{-1}\Delta S^{-1}\nonumber\\
 &\quad-\int
 T(\Id+ST)^{-1}\Delta(\Id+TS)^{-1}T\,\nu(\dd T).
 \label{eq:block-K-derivative}
\end{align}
Let \(\cP_-(\alpha,\nu)\) contain those \(S\succ0\) for which
\(S^{-1}+T\succ0\) on \(\supp\nu\) and
\(D\mathbb K_{\alpha,\nu}(S)\succ0\) on the self-adjoint matrix space.

\begin{corollary}[Imported block-Wishart law and edges]
\label{cor:MS}
Assume \(r_b\) is fixed, \(n/(d-k)\to\alpha\in(0,\infty)\), and use the clipped
blocks with fixed \(L\).  Then the hypotheses of
\citet{MontanariSaeed2026} hold.  The empirical spectral distribution of
\(H_{\perp\perp,L}^{(n)}\) converges almost surely to a deterministic compactly
supported law \(\mu_{\alpha,\nu_{\theta,L}}\).  Its matrix Stieltjes coordinate
\(S(z)\) solves
\begin{equation}
 \mathbb K_{\alpha,\nu_{\theta,L}}(S(z))=z\Id_{r_b}.
 \label{eq:matrix-fixed-point}
\end{equation}
The left edge is
\begin{equation}
 E_-=
 \sup_{S\in\cP_-(\alpha,\nu_{\theta,L})}
 \lambda_{\min}\big(\mathbb K_{\alpha,\nu_{\theta,L}}(S)\big),
 \label{eq:left-edge}
\end{equation}
and the right edge follows by reflection
\(E_+(\alpha,\nu)=-E_-(\alpha,\check\nu)\), where
\(\check\nu=\operatorname{Law}(-T)\).
\end{corollary}

\begin{remark}[What is and is not imported]
\Cref{cor:MS} supplies a global law, support convergence, variational edges,
and an exterior logarithmic potential.  It does not supply the anisotropic
local resolvent estimates needed to replace a finite signal-sector quadratic
form by its deterministic limit.
\end{remark}

\subsection{Exact finite signal-sector Schur equation}

Reorder all Hessian coordinates into
\[
 \underbrace{\R^{r_b}\kron\Ran(\Theta_\perp)}_{\text{extensive orthogonal block}}
 \oplus
 \underbrace{\left[
 \R^{r_b}\kron\Ran(\Theta)\right]\oplus
 \R^{P_{\mathrm{fin}}}}_{\text{finite signal sector}},
\]
where \(P_{\mathrm{fin}}=qp+2p+2q+1\) contains
\((A,b,c,\alpha,\beta,a_0)\).  Write the complete empirical Hessian as
\begin{equation}
 H_n=
 \begin{bmatrix}
 B_n&C_n\\
 C_n^\trans&D_n
 \end{bmatrix},
 \qquad B_n=H_{\perp\perp}^{(n)}.
 \label{eq:hessian-block}
\end{equation}

\begin{proposition}[Finite Schur determinant]
\label{prop:finite-schur}
For every \(\lambda\notin\spec(B_n)\),
\begin{equation}
 \det(H_n-\lambda\Id)
 =\det(B_n-\lambda\Id)
 \det\!\left[
 D_n-\lambda\Id-C_n^\trans(B_n-\lambda\Id)^{-1}C_n
 \right].
 \label{eq:schur-determinant}
\end{equation}
Thus every eigenvalue of \(H_n\) outside \(\spec(B_n)\) is a zero of the
finite-dimensional determinant
\begin{equation}
 F_n(\lambda)=
 \det\!\left[
 D_n-\lambda\Id-C_n^\trans(B_n-\lambda\Id)^{-1}C_n
 \right].
 \label{eq:Fn}
\end{equation}
\end{proposition}

\Eqref{eq:Fn} is the exact BBP interface.  To obtain a theorem along a
trajectory, one must prove that the matrix-valued resolvent quadratic form
inside \(F_n\) converges uniformly for \(\lambda\) outside the moving bulk and
that its limiting zeros cross a regular edge transversely.  This is the
anisotropic, vector-resolvent level of control familiar from local-law theory
\citep{KnowlesYin2017}; a convergent empirical spectral distribution alone
cannot supply it.

\begin{conjecture}[Dynamic BBP crossing]
\label{conj:dynamic-bbp}
Under a fresh-sample protocol, bounded link derivatives, proportional
asymptotics, and a regular time-uniform anisotropic local law, \(F_n(t,\lambda)\)
converges locally uniformly outside the bulk to a deterministic
\(F(t,\lambda)\).  A simple zero of \(F(t,\cdot)\) detaches from a regular
support edge exactly when the corresponding finite-signal branch satisfies the
edge contact and transversality equations.
\end{conjecture}

\subsection{Raw polynomial blocks}

For the quadratic composition, \(T_\theta(Z,f_\star(Z))\) is an unbounded
Gaussian polynomial matrix.  The clipped corollary therefore cannot be
silently applied to the raw Hessian.  Removing clipping requires at least:
\begin{enumerate}[leftmargin=*]
 \item a tail estimate for the maximum block norm on the relevant \(n,d\)
 scale;
 \item a comparison showing that discarded blocks do not move a regular bulk
 edge at the target scale;
 \item an anisotropic local law stable under this comparison;
 \item a separate analysis if rare blocks create their own extreme-value
 eigenvalues.
\end{enumerate}
The experiments consequently report raw empirical edges and eigenpairs in
separate columns from clipped variational and clipped empirical edges.
