\section{Residualized Hermite sectors and sequential recovery}
\label{sec:sectors}

The target-bearing hierarchy is degree \(2\to3\to4\).  For
\(r\in\{2,3,4\}\), concatenate the earlier weighted Jacobians
\[
 M_{<r}=[M_2,\ldots,M_{r-1}],
\]
with \(M_{<2}\) empty, and let
\[
 P_{<r}=M_{<r}M_{<r}^{\pinv}
\]
be the orthogonal projector onto their parameter-space range.  Define the
new-sector Jacobian
\begin{equation}
 \overline M_r=(\Id_P-P_{<r})M_r.
 \label{eq:residualized-M}
\end{equation}

\begin{proposition}[Schur hierarchy]
\label{prop:schur-hierarchy}
Let \(G_{rs}=M_r^\trans M_s\) and let \(G_{<r,<r}\) denote the block Gram matrix
of degrees \(2,\dots,r-1\).  Then
\begin{equation}
 S_r
 :=G_{rr}-G_{r,<r}G_{<r,<r}^{\pinv}G_{<r,r}
 =\overline M_r^\trans\overline M_r
 \succeq0.
 \label{eq:schur}
\end{equation}
In particular,
\begin{align}
 S_2&=G_{22},\nonumber\\
 S_3&=G_{33}-G_{32}G_{22}^{\pinv}G_{23},\label{eq:explicit-schur}\\
 S_4&=G_{44}
 -\begin{bmatrix}G_{42}&G_{43}\end{bmatrix}
 \begin{bmatrix}G_{22}&G_{23}\\G_{32}&G_{33}\end{bmatrix}^{\pinv}
 \begin{bmatrix}G_{24}\\G_{34}\end{bmatrix}.\nonumber
\end{align}
\end{proposition}

\begin{remark}[Interpretation]
\(S_r\) does not measure the raw sensitivity of degree \(r\).  It measures the
part of that sensitivity that cannot be synthesized from parameter directions
already used by earlier target-bearing sectors.  A zero eigenvalue can mean
either a symmetry/redundancy or a genuine failure to expose a new feature
direction.
\end{remark}

Let \(Q_r\) have orthonormal columns spanning
\(\Ran(\overline M_r)\), and define the projected residual-curvature radius
\begin{equation}
 \rho_r(\theta)=\|Q_r^\trans H_{\mathrm{res}}(\theta)Q_r\|_{\op}.
 \label{eq:rho-r}
\end{equation}
Write \(\lambda_{\min}^+(S_r)\) for the smallest strictly positive eigenvalue,
when it exists.

\begin{proposition}[Active curvature certificate]
\label{prop:curvature-certificate}
At every fixed state with \(\Ran(\overline M_r)\neq\{0\}\),
\begin{equation}
 \lambda_{\min}\!\left(
 Q_r^\trans\nabla_\theta^2\cR(\theta)Q_r
 \right)
 \geq \lambda_{\min}^+(S_r)-\rho_r.
 \label{eq:curvature-bound}
\end{equation}
Hence \(\lambda_{\min}^+(S_r)>\rho_r\) is a sufficient certificate of positive
curvature throughout the active degree-\(r\) tangent space.
\end{proposition}

The certificate separates two mechanisms.  \(S_r\) is a Gauss--Newton access
term: it records whether a parameter perturbation can change the new
coefficient sector independently of earlier ones.  \(\rho_r\) is a
non-interpolating correction: it can erase that access through negative
residual curvature.

\subsection{Conditional sector clocks}

An instantaneous Schur matrix is not, by itself, a theorem that sectors learn
in order.  The coefficient equation \eqref{eq:weighted-flow} contains
time-dependent cross blocks.  The following elementary result states exactly
which dynamical decoupling is needed to turn curvature into a clock.

\begin{proposition}[Conditional error clock]
\label{prop:conditional-clock}
Let \(u_r=D_r^{1/2}e_r\).  Suppose that on an interval \([t_0,t_1]\) the
weighted coefficient dynamics restricted to sector \(r\) satisfy
\[
 \dot u_r(t)=-B_{rr}(t)u_r(t)+\zeta_r(t),
 \qquad B_{rr}(t)\succeq\kappa_r(t)\Id,
\]
where \(\zeta_r\) collects cross-sector forcing.  Then
\begin{equation}
 \|u_r(t)\|
 \leq e^{-\int_{t_0}^t\kappa_r(s)\dd s}\|u_r(t_0)\|
 +\int_{t_0}^t
 e^{-\int_s^t\kappa_r(v)\dd v}\|\zeta_r(s)\|\dd s.
 \label{eq:duhamel-sector}
\end{equation}
In particular, if \(\zeta_r=0\), the half-error time is reached once
\(\int_{t_0}^t\kappa_r(s)\dd s\geq(\log2)/2\).
\end{proposition}

The factor \(1/2\) in the last statement appears because
\(E_r=\|u_r\|^2\).  In a staged algorithm, freezing or fitting earlier blocks
can make \(\zeta_r\) small.  In joint backpropagation this must be verified; it
cannot be assumed from a plot.

\subsection{Match to hierarchical depth theory}

The controlled mechanism of
\citet{DandiPesceZdeborovaKrzakala2025} reduces the effective latent dimension
at successive layers.  The layerwise guarantees of
\citet{NichaniDamianLee2025,WangNichaniLee2023} first recover a lower-degree
feature and then learn its outer polynomial.  In our finite model, the ranks
\[
 d_r^{\mathrm{new}}=\rank(\overline M_r)=\rank(S_r)
 \label{eq:effective-rank}
\]
are local, parameter-dependent analogues of ``new effective dimension.''  This
is a mathematical match at the level of tangent geometry, not an import of
their sample-complexity theorem.

The experiment tracks \(E_2,E_3,E_4\), the Schur spectra, and the projected
residual radii.  A publishable sequential-recovery claim should demonstrate all
three:
\begin{enumerate}[leftmargin=*]
 \item earlier errors become small before later ones;
 \item \(\lambda_{\min}^+(S_r)-\rho_r\) becomes positive in the same order;
 \item the forcing term implicit in \eqref{eq:duhamel-sector} is small enough
 to predict the measured crossing times.
\end{enumerate}
The checked-in pilot establishes the first item descriptively and verifies the
Schur algebra, but it does not yet establish a uniform dynamical bound for the
third.
