\section{Exact Hermite population theory}
\label{sec:hermite}

\begin{proposition}[Finite Hermite support]
\label{prop:finite-support}
For every parameter \(\theta\), the function \(f_\theta\) in
\eqref{eq:network} is a polynomial in \(z\) of total degree at most four.
Consequently,
\[
 f_\theta(z)=\sum_{\gamma\in\Gamma_4}c_\gamma(\theta)\He_\gamma(z)
\]
as an exact polynomial identity.
\end{proposition}

The point is elementary but decisive.  The first feature \(q_\theta\) has
degree two, \(r_\theta\) has degree two, and squaring \(r_\theta\) creates
degrees at most four.  No uncontrolled Hermite tail exists in the reduced
model.

\begin{theorem}[Exact coefficient risk, flow, and Hessian]
\label{thm:exact-closure}
Let \(f_\star\) have degree at most four.  Then
\begin{align}
 \cR(\theta)
 &=\frac12 e(\theta)^\trans D e(\theta),
 \label{eq:risk-coeff}\\
 \nabla_\theta\cR(\theta)
 &=J(\theta)^\trans D e(\theta),
 \label{eq:gradient-coeff}\\
 \nabla_\theta^2\cR(\theta)
 &=J(\theta)^\trans D J(\theta)
   +\sum_{\gamma\in\Gamma_4}
     \gamma!\,e_\gamma(\theta)\nabla_\theta^2c_\gamma(\theta).
 \label{eq:hessian-decomp}
\end{align}
Under population gradient flow
\(\dot\theta_t=-\nabla_\theta\cR(\theta_t)\), the coefficient error obeys the
exact, generally non-autonomous equation
\begin{equation}
 \dot e_t=-J_tJ_t^\trans D e_t.
 \label{eq:coeff-flow}
\end{equation}
Equivalently, for the weighted error \(u_t=D^{1/2}e_t\),
\begin{equation}
 \dot u_t=-B_tu_t,\qquad
 B_t=D^{1/2}J_tJ_t^\trans D^{1/2}\succeq0.
 \label{eq:weighted-flow}
\end{equation}
In particular,
\begin{equation}
 \frac{\dd}{\dd t}\cR(\theta_t)
 =-\|\nabla_\theta\cR(\theta_t)\|_2^2
 =-u_t^\trans B_tu_t\leq0.
 \label{eq:risk-dissipation}
\end{equation}
\end{theorem}

We write
\begin{equation}
 H_{\mathrm{GN}}=J^\trans DJ=\sum_{r=0}^4M_rM_r^\trans,\qquad
 H_{\mathrm{res}}=\sum_{\gamma\in\Gamma_4}
 \gamma!\,e_\gamma\nabla_\theta^2c_\gamma,
 \label{eq:gn-res}
\end{equation}
so \(H=H_{\mathrm{GN}}+H_{\mathrm{res}}\).
The first term is positive semidefinite.  The second term is the source of
negative curvature away from interpolation and vanishes whenever the entire
coefficient error vanishes.

\begin{corollary}[Geometry at interpolation]
\label{cor:interpolation}
If \(c(\theta)=c_\star\), then
\[
 \nabla_\theta^2\cR(\theta)=J(\theta)^\trans D J(\theta)\succeq0.
\]
Its nonzero eigenvalues agree, including multiplicity, with those of
\(B(\theta)=D^{1/2}JJ^\trans D^{1/2}\).
\end{corollary}

\begin{remark}[Closure in parameters versus coefficients]
\Eqref{eq:coeff-flow} is exact, but it is not autonomous in \(e_t\) because
\(J_t\) depends on the current factorization
\((V_t,U_t,A_t,\alpha_t,\beta_t,\ldots)\).  Calling
\eqref{eq:coeff-flow} a ``DMFT'' would hide precisely the feature-learning
problem.  It is a finite deterministic pullback flow.
\end{remark}

\subsection{Exact Gauss--Hermite evaluation}

Let \(\{x_i,w_i\}_{i=1}^{n_q}\) be the one-dimensional Gauss--Hermite rule
scaled to \(\mathcal N(0,1)\), and use its \(k\)-fold tensor product.

\begin{proposition}[Quadrature exactness]
\label{prop:quadrature}
If \(n_q\geq5\), the tensor Gauss--Hermite rule evaluates exactly:
\begin{enumerate}[label=(\roman*),leftmargin=*]
  \item every coefficient \(c_\gamma(\theta)\), \(|\gamma|\leq4\);
  \item the population risk \(\cR(\theta)\);
  \item every derivative obtained by differentiating these finite polynomial
  sums with respect to \(\theta\).
\end{enumerate}
\end{proposition}

Indeed, \(f_\theta\He_\gamma\) and
\((f_\theta-f_\star)^2\) have coordinatewise degree at most eight, while an
\(n_q\)-point Gaussian rule integrates degree \(2n_q-1\).  The implementation
uses \(n_q=7\), giving a safety margin through degree thirteen.

\subsection{Degree-wise error observables}

Define
\begin{equation}
 E_r(\theta)=\sum_{|\gamma|=r}\gamma!\,e_\gamma(\theta)^2,
 \qquad
 \cR(\theta)=\frac12\sum_{r=0}^4E_r(\theta).
 \label{eq:sector-errors}
\end{equation}
These observables are exact and orthogonal in function space.  They are not
independent under training because the coefficient kernel \(B_t\) has
off-diagonal degree blocks.  The next section uses Schur residualization to
separate genuinely new sector directions from parameter directions already
used by earlier degrees.

\subsection{Relation to general-link and alternating theories}

For a general multi-index link \(g(Wz)\), Hermite projection gives the same
risk identity but an infinite coefficient hierarchy unless the link is
polynomial.  The present architecture is therefore a finite exact testbed for
the general-link closure.  Joint direction/link convergence in the single-index
setting can be proved for an alternating RKHS scheme
\citep{PillaudVivienSchertzer2025}; our simultaneous factorized flow is
different, but \eqref{eq:coeff-flow} makes its additional coupling explicit.
