\section{Proofs of the exact statements}
\label{app:proofs}

\subsection{Hermite closure}

\begin{proof}[Proof of \Cref{prop:finite-support}]
Each coordinate of \(h=Vz+b\) is affine in \(z\), hence each coordinate of
\(q=h^{\odot2}-\one\) has degree at most two.  Each coordinate of
\(r=Uz+Aq+c\) therefore has degree at most two, and each coordinate of
\(s=r^{\odot2}-\one\) has degree at most four.  The readout is a linear
combination of \(q,s\), and a constant, so \(f_\theta\) has degree at most
four.  The multivariate Hermite polynomials with \(|\gamma|\leq4\) form a basis
for polynomials of total degree at most four.
\end{proof}

\begin{proof}[Proof of \Cref{thm:exact-closure}]
By \Cref{prop:finite-support} and \eqref{eq:teacher},
\[
 f_\theta-f_\star=\sum_{\gamma\in\Gamma_4}e_\gamma\He_\gamma.
\]
Squaring, taking expectation, and using
\eqref{eq:hermite-orthogonality} gives
\[
 \cR=\frac12\sum_{\gamma\in\Gamma_4}\gamma!e_\gamma^2
 =\frac12e^\trans De.
\]
The chain rule yields
\[
 \nabla_\theta\cR=J^\trans De.
\]
Differentiating once more gives
\[
 \nabla_\theta^2\cR
 =J^\trans DJ+\sum_\gamma\gamma!e_\gamma\nabla_\theta^2c_\gamma.
\]
Along gradient flow,
\[
 \dot e=J\dot\theta=-JJ^\trans De.
\]
Multiplication by \(D^{1/2}\) yields \eqref{eq:weighted-flow}.  Finally,
\[
 \frac{\dd}{\dd t}\cR
 =\langle\nabla_\theta\cR,\dot\theta\rangle
 =-\|\nabla_\theta\cR\|^2
 =-u^\trans Bu.
\]
\end{proof}

\begin{proof}[Proof of \Cref{cor:interpolation}]
If \(e=0\), the residual-curvature sum in \eqref{eq:hessian-decomp} vanishes.
The agreement of nonzero spectra follows from the standard fact that
\(X^\trans X\) and \(XX^\trans\) have the same nonzero eigenvalues, applied to
\(X=D^{1/2}J\).
\end{proof}

\begin{proof}[Proof of \Cref{prop:quadrature}]
An \(n_q\)-point one-dimensional Gaussian quadrature rule is exact for
polynomials through degree \(2n_q-1\).  Tensorization preserves this
coordinatewise property.  For \(|\gamma|\leq4\),
\(f_\theta\He_\gamma\) has coordinatewise degree at most eight, as does
\((f_\theta-f_\star)^2\).  Hence \(n_q\geq5\) is sufficient.  Parameter
differentiation changes coefficients but does not increase polynomial degree
in \(z\), so differentiating the exact finite quadrature sum gives the exact
derivatives.
\end{proof}

\subsection{Schur hierarchy and clocks}

\begin{proof}[Proof of \Cref{prop:schur-hierarchy}]
The orthogonal projector onto \(\Ran(M_{<r})\) can be written
\[
 P_{<r}=M_{<r}(M_{<r}^\trans M_{<r})^{\pinv}M_{<r}^\trans.
\]
Therefore
\begin{align*}
 \overline M_r^\trans\overline M_r
 &=M_r^\trans(\Id-P_{<r})M_r\\
 &=M_r^\trans M_r
 -M_r^\trans M_{<r}
 (M_{<r}^\trans M_{<r})^{\pinv}
 M_{<r}^\trans M_r,
\end{align*}
which is \eqref{eq:schur}.  It is positive semidefinite because it is a Gram
matrix.  Expanding the earlier-sector blocks gives
\eqref{eq:explicit-schur}.
\end{proof}

\begin{proof}[Proof of \Cref{prop:curvature-certificate}]
Because \(Q_r\) is orthogonal to \(\Ran(M_{<r})\),
\[
 Q_r^\trans M_r=Q_r^\trans\overline M_r.
\]
The Gauss--Newton Hessian is a sum of positive semidefinite terms, so
\[
 Q_r^\trans H_{\mathrm{GN}}Q_r
 \succeq
 Q_r^\trans\overline M_r\overline M_r^\trans Q_r.
\]
The matrix on the right is positive definite on the columns of \(Q_r\), and
its eigenvalues are the strictly positive eigenvalues of
\(\overline M_r^\trans\overline M_r=S_r\).  Weyl's inequality and
\(\|Q_r^\trans H_{\mathrm{res}}Q_r\|_{\op}=\rho_r\) give
\eqref{eq:curvature-bound}.
\end{proof}

\begin{proof}[Proof of \Cref{prop:conditional-clock}]
Let \(\mathcal U(t,s)\) be the evolution operator generated by
\(-B_{rr}(t)\).  The lower bound on \(B_{rr}\) gives
\[
 \|\mathcal U(t,s)\|_{\op}
 \leq\exp\!\left(-\int_s^t\kappa_r(v)\dd v\right).
\]
Variation of constants yields
\[
 u_r(t)=\mathcal U(t,t_0)u_r(t_0)
 +\int_{t_0}^t\mathcal U(t,s)\zeta_r(s)\dd s.
\]
Taking norms proves \eqref{eq:duhamel-sector}.  If \(\zeta_r=0\), then
\[
 E_r(t)=\|u_r(t)\|^2
 \leq e^{-2\int_{t_0}^t\kappa_r(s)\dd s}E_r(t_0),
\]
which is at most \(E_r(t_0)/2\) under the stated integrated-curvature
condition.
\end{proof}

\subsection{Spectral update}

\begin{proof}[Proof of \Cref{prop:muon-descent}]
SVD orthogonality and \eqref{eq:muon-map} give
\[
 \langle G,\Phi_{a,\eps}(G)\rangle_\F
 =c_{a,\eps}(G)^{-1}\sum_i\sigma_iw_i^{(a,\eps)}.
\]
At least one \(\sigma_i\) is positive, and every \(w_i^{(a,\eps)}\) is
positive, proving strict positivity.  The standard descent lemma for an
\(L_F\)-smooth function gives
\[
 F(W-\eta\Phi)\leq
 F(W)-\eta\langle G,\Phi\rangle_\F
 +\frac{L_F\eta^2}{2}\|\Phi\|_\F^2.
\]
The unit-RMS normalization gives \(\|\Phi\|_\F^2=mn\).  The quadratic upper
bound is strictly below \(F(W)\) under \eqref{eq:sufficient-step}.
\end{proof}

\begin{proof}[Proof of \Cref{prop:muon-frechet}]
The derivative of \(S(G)=GG^\trans\) in direction \(\Delta\) is
\[
 DS(G)[\Delta]=G\Delta^\trans+\Delta G^\trans=E.
\]
The Daleckii--Krein formula for the spectral matrix function \(S^b\) gives
\[
 D(S^b)[E]
 =L\!\left[F_b^{[1]}\odot(L^\trans E L)\right]L^\trans.
\]
Applying the product rule to \(\Psi_a(G)=S(G)^bG\) proves
\eqref{eq:muon-frechet-unnormalized}.  Finally,
\[
 \Phi_a(G)=q(G)\Psi_a(G),\qquad
 q(G)=\sqrt{mn}\,\|\Psi_a(G)\|_\F^{-1},
\]
and
\[
 Dq(G)[\Delta]
 =-q(G)\,
 \frac{\langle\Psi_a(G),D\Psi_a(G)[\Delta]\rangle_\F}
 {\|\Psi_a(G)\|_\F^2}.
\]
Another product rule gives \eqref{eq:muon-frechet-normalized}.
\end{proof}

\begin{proof}[Proof of \Cref{prop:finite-muon-action}]
Since \(GG^\trans=L\Diag(\sigma_i^2)L^\trans\),
\[
 (GG^\trans)^{(a-1)/2}G
 =L\Diag(\sigma_i^{a-1})L^\trans
 L\Diag(\sigma_i)R^\trans
 =L\Diag(\sigma_i^a)R^\trans.
\]
Multiplication by \(\bar\sigma^{-a}\) gives
\eqref{eq:finite-muon-action}.  The pseudoinverse functional calculus gives
the same identity on the positive singular subspace.
\end{proof}

\subsection{Analytic and empirical Hessians}

\begin{proof}[Proof of \Cref{prop:link-hessian}]
Because
\[
 \frac{\partial f}{\partial q}
 =\alpha+2A^\trans(\beta\odot r)=d_h,
\qquad
 \frac{\partial q}{\partial h}=2\Diag(h),
\]
we obtain \(\nabla_hf=2h\odot d_h\).  Direct differentiation in \(u\) gives
\(\nabla_uf=2\beta\odot r\), proving
\eqref{eq:link-gradient}.

Next,
\[
 \frac{\partial d_h}{\partial h}
 =2A^\trans D_\beta A(2D_h)
 =4A^\trans D_\beta AD_h.
\]
Differentiating \(2h\odot d_h\) gives
\[
 K_{hh}=2\Diag(d_h)+
 2D_h(4A^\trans D_\beta AD_h),
\]
which is \eqref{eq:Khh}.  Since
\(\partial d_h/\partial u=2A^\trans D_\beta\),
\[
 K_{hu}=2D_h(2A^\trans D_\beta),
\]
and \(K_{uu}=2D_\beta\).  Finally, the Hessian chain rule for
\(\ell=\frac12(f-y)^2\) gives
\[
 \nabla_v^2\ell=(\nabla_vf)(\nabla_vf)^\trans+(f-y)\nabla_v^2f.
\]
\end{proof}

\begin{proof}[Proof of \Cref{thm:block-wishart-identity}]
For one observation, \(\dd v=(\dd W)x\).  Therefore the Hessian bilinear form
in direct-weight perturbations \(\Delta W,\widetilde{\Delta W}\) is
\[
 \langle \Delta Wx,T_\theta\widetilde{\Delta W}x\rangle,
\]
which is represented by \(T_\theta\kron xx^\trans\) under vectorization.
Projecting both ambient factors by \(\Theta_\perp\) replaces \(x\) by
\(\Theta_\perp^\trans x=\xi\), proving
\eqref{eq:exact-block-wishart}.  At the embedded reduced state, \(T_\theta\)
depends only on \(z\) and \(y=f_\star(z)\).  Gaussian orthogonal coordinates
make \(z\) and \(\xi\) independent, both within and across observations.
\end{proof}

\begin{proof}[Proof of \Cref{lem:clipping}]
By the triangle inequality and the multiplicativity of the operator norm under
Kronecker products,
\begin{align*}
 \|H_{\perp\perp}^{(n)}-H_{\perp\perp,L}^{(n)}\|_{\op}
 &\leq\frac1n\sum_\mu
 \|(T_\mu-\Pi_L(T_\mu))\kron\xi_\mu\xi_\mu^\trans\|_{\op}\\
 &=\frac1n\sum_\mu
 \|T_\mu-\Pi_L(T_\mu)\|_{\op}\,
 \|\xi_\mu\xi_\mu^\trans\|_{\op},
\end{align*}
and \(\|\xi\xi^\trans\|_{\op}=\|\xi\|^2\).
\end{proof}

\begin{proof}[Justification of \Cref{cor:MS}]
The clipped blocks are iid, self-adjoint, uniformly bounded, and independent
of iid Gaussian \(\xi_\mu\).  The block size \(r_b\) is fixed and the aspect
ratio converges to \(\alpha\).  These are the block-Wishart hypotheses of
\citet{MontanariSaeed2026}.  Their matrix Stieltjes fixed point and
variational edge theorems give
\eqref{eq:matrix-fixed-point}--\eqref{eq:left-edge} after matching the
normalization \(n/(d-k)=\alpha\).
\end{proof}

\begin{proof}[Proof of \Cref{prop:finite-schur}]
For \(\lambda\notin\spec(B_n)\), multiply \(H_n-\lambda\Id\) on the left by
\[
 \begin{bmatrix}
 \Id&0\\
 -C_n^\trans(B_n-\lambda\Id)^{-1}&\Id
 \end{bmatrix},
\]
whose determinant is one.  The product is block upper triangular with
diagonal blocks \(B_n-\lambda\Id\) and
\[
 D_n-\lambda\Id-C_n^\trans(B_n-\lambda\Id)^{-1}C_n.
\]
Taking determinants proves \eqref{eq:schur-determinant}.  Outside the bulk
spectrum, the first determinant is nonzero, so zeros are exactly those of
\(F_n\).
\end{proof}

\subsection{Hermite-tail transfer}

\begin{proof}[Proof of \Cref{thm:hermite-tail-transfer}]
Orthogonality of \(P_R\) gives
\[
 \|\delta\|_{\cH}^2
 =\|P_R\delta\|_{\cH}^2
  +\|(\Id-P_R)\delta\|_{\cH}^2,
\]
which proves \eqref{eq:risk-tail}.  Hilbert-space differentiation gives
\[
 D\cR[u]=\langle\delta,Dc[u]\rangle_{\cH},
\qquad
 D^2\cR[u,v]
 =\langle Dc[u],Dc[v]\rangle_{\cH}
  +\langle\delta,D^2c[u,v]\rangle_{\cH}.
\]
Subtracting the corresponding projected identities leaves only the
orthogonal tails.  Cauchy--Schwarz yields
\[
 |D\cR[u]-D\cR_R[u]|
 \leq\eps_0\eps_1\|u\|,
\]
and, for unit \(u,v\),
\[
 |D^2\cR[u,v]-D^2\cR_R[u,v]|
 \leq\eps_1^2+\eps_0\eps_2.
\]
Taking the appropriate suprema proves
\eqref{eq:gradient-tail}--\eqref{eq:hessian-tail}.  The pointwise argument
commutes with a supremum over a compact set.
\end{proof}

\begin{proof}[Proof of \Cref{cor:spectral-tail-stability}]
\Cref{thm:hermite-tail-transfer} bounds the operator-norm perturbation of the
two real symmetric \(P\times P\) Hessians by \(\eta_R\).  Weyl's eigenvalue
perturbation inequality gives the ordered-eigenvalue claim.  If a truncated
cluster has gap greater than \(2\eta_R\), the closed
\(\eta_R\)-neighborhoods of the cluster and of the remaining spectrum are
disjoint, so the ordered eigenvalue count in the cluster is unchanged.
\end{proof}

\begin{proof}[Proof of \Cref{prop:conditional-dynamic-closure}]
Assumption (i) makes \(\nu_t\) a deterministic functional of the limiting
state.  Assumption (ii) therefore makes the regular bulk edges deterministic
and time dependent.  By (iii), each finite-signal resolvent quadratic form in
\eqref{eq:Fn} converges locally uniformly outside the bulk, so the finite
determinants converge locally uniformly as well.  Simple separated zeros are
stable under locally uniform convergence.  Assumption (iv) licenses the same
conclusion for the raw rather than truncated blocks.  At contact, regularity
of the edge and transversality of the zero distinguish a genuine branch
crossing from tangency or spectral sticking.
\end{proof}
