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  pdftitle={Sequential Feature Recovery in a Three-Layer Network},
  pdfauthor={Janis Heran Aiad}
}

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\title{\bfseries Sequential Feature Recovery in a Three-Layer Network:\\
Exact Hermite Geometry, a Conditional DMFT--RMT--BBP Closure,\\
and Muon Experiments}
\author{
  Janis Heran Aiad\\
  \texttt{janis.aiad@polytechnique.edu}
}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We develop a proof-level bridge between hierarchical feature learning, spectral
optimization, dynamical mean-field theory, and random-matrix transitions in a
three-layer quadratic network.  The model composes two quadratic Hermite
feature maps and therefore produces a degree-four polynomial of a latent
Gaussian subspace.  At fixed latent dimension we prove an exact population
closure: the risk is a weighted Euclidean distance between multivariate
Hermite coefficients, the parameter gradient is the pullback of this
coefficient error, and the Hessian decomposes into a positive semidefinite
Gauss--Newton term plus residual curvature.  Residualized degree-$2$, $3$, and
$4$ coefficient Jacobians yield positive semidefinite Schur matrices that
quantify access to successively new feature sectors.

We then analyze a singular-value power family containing polar Muon
(\(\mathrm{SignSVD}\)) and prove that every nonzero update in the tested family
is a strict first-order descent direction, while emphasizing that no universal
power is implied for the three trainable matrix blocks.  We also derive the
exact Fr\'echet derivative of its nonseparable spectral channel.  This is the
input required by a matrix Onsager correction: ordinary Muon-\(a\) is
AMP-like as a denoiser, but the tested optimizer is not AMP because it has no
Onsager reaction.  To connect the finite
population theory to empirical spectra, we embed a reduced state in ambient
dimension and evaluate a fresh Gaussian Hessian.  Conditional on the latent
coordinates, its orthogonal direct-weight block is \emph{exactly} matrix
weighted block Wishart.  This identity permits a rigorous import of the
bounded-block global law and variational support-edge formulas of Montanari and
Saeed; an exact finite Schur determinant isolates the remaining signal sector.
We state separately the additional local-law, leave-one-out, and tail
conditions needed for a genuine dynamic BBP theorem and for a reused-sample
DMFT.

Beyond the polynomial testbed, we prove a Hermite-tail transfer theorem:
degree-\(R\) truncation errors in the coefficient map and its first two
derivatives give explicit errors for population risk, gradient, and Hessian.
This supplies a controlled extension to general links while making clear that
data reuse, growing ranks, and unbounded empirical blocks still require
model-specific analysis.  We assemble these ingredients into a conditional
DMFT--RMT--BBP closure whose four hypotheses are stated individually, and
separate the MSRJD, dynamic-cavity, replica, and spectral-AMP routes.  We also
distinguish population, fresh-sample, same-data trajectory, and
critical-point Hessian laws, placing solvable transient-BBP and
BBP--Kac--Rice results as controls rather than silently importing them.

A pre-specified confirmatory campaign compares gradient descent, polar Muon,
and two transferred singular-value powers on twenty paired datasets.  It finds
no endpoint dominance: the spectral rules win \(9/20\), \(8/20\), and \(10/20\)
pairs, and every paired bootstrap interval for the median log-risk difference
contains zero.  Their sector clocks are nevertheless earlier: all three halve
degree two at Kaplan--Meier median step \(80\), versus \(160\) for gradient
descent, and move the degree-four median from \(480\) to \(240\)--\(320\),
without uniformly improving terminal event coverage.  A nested sample-size
panel and fresh-Hessian diagnostics remain exploratory.  The paper thus
supplies an exact deterministic core, a theorem-compatible random-matrix lift,
an empirical speed--coverage tradeoff, and a falsifiable roadmap for the
still-open joint three-layer DMFT.
\end{abstract}

\paragraph{Keywords.}
Hierarchical learning; three-layer neural networks; Hermite expansion; Muon;
SignSVD; dynamical mean-field theory; block-Wishart matrices; BBP transition;
approximate message passing; cavity method; replica method; sequential feature
recovery; scaling laws.

\paragraph{Proof-status convention.}
Unqualified theorems and propositions are proved in this paper.  Results
imported from another source are labeled ``imported corollary'' and retain the
source assumptions.  Statements requiring a new high-dimensional argument are
labeled assumption, conditional proposition, or conjecture.  This convention
is essential because the deterministic population model, the fresh-sample
random-matrix lift, and reused-data training are different probability spaces.

\tableofcontents

\input{sections/01_introduction}
\input{sections/02_model}
\input{sections/03_exact_hermite}
\input{sections/04_sector_hierarchy}
\input{sections/05_muon}
\input{sections/06_hessian_rmt}
\input{sections/07_dmft_scaling}
\input{sections/08_universality}
\input{sections/08_experiments}
\input{sections/09_discussion}
\appendix
\input{sections/appendix_proofs}
\input{sections/appendix_reproducibility}

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