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\title{\textbf{Research Roadmap: Toward a Mathematical Physics Paper}\\[6pt]
\large Low-Dimensional Invariant Structure of the Balanced Metric\\
on the Dwork Quintic Family}
\author{D. Yang Eng \\ \small Internal research notes}
\date{\today}

\begin{document}
\maketitle
\tableofcontents
\bigskip

% ------------------------------------------------------------------
\section{The Central Mathematical Question}
% ------------------------------------------------------------------

The empirical paper established the following observation:

\begin{quote}
\textit{For the Fermat quintic $X_0 \subset \mathbb{CP}^4$ with the $k=10$ balanced
metric, the determinant ratio $R_\psi(z) = \log(\det g_k / \det g_{\mathrm{FS}})$
is a function effectively of only two symmetric invariants $(p_2, e_3)$, with
all remaining independent generators contributing $\Delta R^2 < 0.001$.}
\end{quote}

\noindent The mathematical physics paper asks: \textbf{why is this true, and can it be proved?}

This is a genuine open question at the intersection of complex geometry, representation
theory, and spectral theory. The empirical evidence is strong. The proof requires
understanding how the Monge-Amp\`ere constraint selects specific directions in the
ring of $G$-invariant functions on $X_0$.

% ------------------------------------------------------------------
\section{Mathematical Setup}
% ------------------------------------------------------------------

\subsection{The Fermat and Dwork Quintics}

\begin{definition}
The \textbf{Dwork family} is the one-parameter family of quintic hypersurfaces
\[
X_\psi = \Bigl\{z \in \mathbb{CP}^4 \;\Big|\; \sum_{i=0}^4 z_i^5 - 5\psi\,z_0 z_1 z_2 z_3 z_4 = 0 \Bigr\},
\qquad \psi \in \mathbb{C} \setminus \{1, \omega, \ldots\}.
\]
The \textbf{Fermat quintic} $X_0$ is the case $\psi = 0$.
\end{definition}

\subsection{The Symmetry Group}

\begin{definition}
Let $G = (U(1)^5/U(1)) \rtimes S_5 \cong U(1)^4 \times S_5$
be the \textbf{residual symmetry group} of $X_0$, acting on $\mathbb{CP}^4$ by
\[
(e^{i\theta_0}, \ldots, e^{i\theta_4}) : [z_0 : \cdots : z_4]
  \mapsto [e^{i\theta_0} z_0 : \cdots : e^{i\theta_4} z_4]
\qquad (\textstyle\sum_j \theta_j = 0),
\]
and $S_5$ permutes the coordinates. The Fermat quintic $X_0$ is $G$-invariant.
\end{definition}

\subsection{The Normalisation Locus and Invariant Generators}

Working on the normalisation locus $\Sigma = \{z : \sum_i |z_i|^2 = 1\} \cap X_0$,
set $u_i = |z_i|^2$. The $G$-invariant real-valued functions of $(u_0,\ldots,u_4)$
form a polynomial ring generated by the power sums
\[
p_k = \sum_{i=0}^4 u_i^k, \qquad k = 2, 3, 4, 5,
\]
(note: $p_1 = \sum u_i = 1$ is fixed by normalisation). Equivalently, via
Newton's identities, one may use the elementary symmetric polynomials
$e_2, e_3, e_4, e_5$ as generators, with
\[
e_2 = \tfrac{1-p_2}{2}, \quad
e_3 = \tfrac{1 - 3p_2 + 2p_3}{6}, \quad
e_4 = \tfrac{1 - 6p_2 + 8p_3 + 3p_2^2 - 6p_4}{24}, \quad
e_5 = u_0 u_1 u_2 u_3 u_4.
\]
The \textbf{invariant ring} is thus
\[
\mathcal{R} = \mathbb{R}[p_2, p_3, p_4, p_5]
            = \mathbb{R}[e_2, e_3, e_4, e_5],
\]
with the two generating sets related algebraically.

\subsection{The Balanced Metric and Determinant Ratio}

Donaldson's balanced metric $g_k$ at polynomial degree $k$ on $(X_0, \mathcal{O}(k)|_{X_0})$
is the unique Hermitian metric satisfying the balanced condition
\[
\frac{1}{N_k} \sum_{\alpha} |s^\alpha(z)|^2_{h_k^{\otimes k}} = \frac{1}{\mathrm{Vol}(X_0)}
\]
for an $L^2$-orthonormal basis $\{s^\alpha\}$ of $H^0(X_0, \mathcal{O}(k))$,
with $N_k = \dim H^0(X_0, \mathcal{O}(k))$.
The metric is obtained as the fixed point of Donaldson's $T$-operator.

The \textbf{determinant ratio} is
\[
R_k(z) \;=\; \log\frac{\det g_k(z)}{\det g_{\mathrm{FS}}(z)},
\]
where $\det$ denotes the determinant of the $3 \times 3$ local metric tensor.

% ------------------------------------------------------------------
\section{What Can Be Proved Now: The Easy Part}
% ------------------------------------------------------------------

The following are rigorous or near-rigorous statements that require only
standard arguments.

\begin{proposition}[G-invariance of the det-ratio]
\label{prop:ginv}
The function $R_k : X_0 \to \mathbb{R}$ is $G$-invariant.
\end{proposition}

\begin{proof}[Proof sketch]
The Fermat quintic $X_0$ is $G$-invariant. The Fubini-Study metric $g_{\mathrm{FS}}$ is
$G$-invariant. The balanced metric $g_k$ is the unique solution to the balanced
equation on $(X_0, \mathcal{O}(k))$; since the equation and the variety are both
$G$-invariant, the solution must be too. Therefore $\det g_k / \det g_{\mathrm{FS}}$
is $G$-invariant.
\end{proof}

\begin{corollary}[Four-generator reduction]
\label{cor:four}
There exists a smooth function $F_k : \mathbb{R}^4 \to \mathbb{R}$ such that
\[
R_k(z) = F_k(p_2, e_3, e_4, e_5)
\]
for all $z \in X_0$.
\end{corollary}

\begin{proof}
Follows from Proposition~\ref{prop:ginv} and the fact that $\{p_2, e_3, e_4, e_5\}$
(equivalently $\{p_2, p_3, p_4, p_5\}$) generate the $G$-invariant polynomial ring on
$\Sigma$.
\end{proof}

\begin{remark}
Corollary~\ref{cor:four} alone reduces the problem from $875^2$ parameters
(the full H-matrix space) to a function of 4 real variables. The real mathematical
challenge is the further reduction from 4 generators to 2.
\end{remark}

\begin{lemma}[H-matrix block structure at $\psi=0$]
\label{lem:block}
At the Fermat point $\psi = 0$, the balanced H-matrix is diagonal in the monomial
basis and the diagonal entries $h_\alpha$ depend only on the partition type
$\lambda(\alpha) = (\alpha_0 \geq \alpha_1 \geq \cdots \geq \alpha_4)$ of the
multi-index $\alpha$.
\end{lemma}

\begin{proof}
The $U(1)^4 \subset G$ action makes the H-matrix diagonal. The $S_5 \subset G$
action makes the diagonal entries depend only on the unordered multi-set of
exponents, i.e., the partition type.
\end{proof}

\begin{remark}
For $k = 10$, the number of distinct partition types of 10 into at most 5 parts
is exactly 30. So the balanced H-matrix at $\psi=0$ is determined by \emph{30 real
parameters}, not 875. The empirical observation that $R_k$ depends on 2 generators
rather than 4 is a further reduction on top of this.
\end{remark}

% ------------------------------------------------------------------
\section{The Main Conjecture: Precise Statements}
% ------------------------------------------------------------------

We state three versions, from weakest to strongest.

\begin{conjecture}[Weak: approximate two-dimensionality]
\label{conj:weak}
For the Fermat quintic $X_0$ with the degree-$k$ balanced metric ($k = 6, 8, 10$),
the function $F_k : \mathbb{R}^4 \to \mathbb{R}$ from Corollary~\ref{cor:four}
satisfies
\[
\left\|\frac{\partial F_k}{\partial e_4}\right\|_{L^2(X_0)},\;
\left\|\frac{\partial F_k}{\partial e_5}\right\|_{L^2(X_0)} \;\ll\;
\left\|\frac{\partial F_k}{\partial p_2}\right\|_{L^2(X_0)}.
\]
Quantitatively: the variance explained by $(p_2, e_3)$ exceeds 99\% of the total,
i.e., $\mathrm{Var}[R_k - \Pi_{(p_2,e_3)} R_k] / \mathrm{Var}[R_k] < 0.001$.
\end{conjecture}

\begin{conjecture}[Medium: asymptotic two-dimensionality]
\label{conj:medium}
As $k \to \infty$, the normalised correction $k(R_k - \bar R_k)$, where $\bar R_k$
is the mean value of $R_k$, converges to a function $f_\infty(p_2, e_3)$ depending
only on $(p_2, e_3)$:
\[
\sup_{z \in X_0} \left| k(R_k(z) - \bar R_k) - f_\infty(p_2(z), e_3(z)) \right|
= O(k^{-1}).
\]
\end{conjecture}

\begin{conjecture}[Strong: spectral selection]
\label{conj:strong}
Let $\mathcal{L}$ be the linearisation of the Monge-Amp\`ere operator around
$g_{\mathrm{FS}}|_{X_0}$. The leading eigenfunctions of $\mathcal{L}$ with
non-trivial $G$-invariant projection lie in the span of $\{p_2, e_3\}$
(viewed as multiplication operators on $L^2(X_0, g_{\mathrm{FS}})$).
\end{conjecture}

\medskip
\noindent\textbf{Priority.} Conjecture~\ref{conj:weak} is most directly supported
by the data and most accessible. Conjecture~\ref{conj:medium} would be a clean
mathematical theorem. Conjecture~\ref{conj:strong} would explain the \emph{why}.

% ------------------------------------------------------------------
\section{Proof Strategy}
% ------------------------------------------------------------------

\subsection{Step 1: Compute the Scalar Curvature of the FS Metric on $X_0$}

\textbf{Goal.} Show that the scalar curvature $s(z)$ of the induced FS metric on
the Fermat quintic is, or is well-approximated by, a function of $(p_2, e_3)$ alone.

\textbf{Why this matters.} The Tian-Yau-Zelditch (TYZ) expansion states that the
Bergman kernel satisfies
\[
k^{-3} B_k(z,z) \;\sim\; 1 + \frac{s(z)}{2k} + \frac{a_2(z)}{k^2} + \cdots
\]
as $k \to \infty$. The leading correction to the balanced metric (and hence to
$R_k$) is controlled by $s(z)$. If $s(z) = s(p_2, e_3)$, this would imply
Conjecture~\ref{conj:medium}.

\textbf{How to compute $s$.} For a smooth hypersurface $X \hookrightarrow \mathbb{CP}^n$
of degree $d$ with the induced FS metric, the scalar curvature is
\[
s_X = n(n+2) - \frac{(d-1) \cdot |II|^2_{g_{\mathrm{FS}}}}{1},
\]
(schematically) where $|II|^2$ is the squared norm of the second fundamental form.
For the Fermat quintic $F = \sum_{i=0}^4 z_i^5$, the relevant quantities are:
\begin{align}
\partial_i F &= 5 z_i^4, \\
\partial_i \partial_j F &= 20 z_i^3 \delta_{ij}, \\
|\nabla F|^2_{g_{\mathrm{FS}}} &= 25 \sum_i |z_i|^8 / (\sum_i |z_i|^2)^4
= 25\, p_4 \quad (\text{on } \Sigma).
\end{align}
The second fundamental form has components
$II_{ij} = \partial_i \partial_j F / |\nabla F|$,
which after careful computation gives
\[
|II|^2 \;\propto\; \frac{\sum_i |\partial_i \partial_i F|^2}{|\nabla F|^2}
= \frac{400 \sum_i |z_i|^6}{25 \sum_i |z_i|^8}
= 16 \cdot \frac{p_3}{p_4}.
\]
(The off-diagonal terms vanish for the Fermat quintic by the diagonal structure
$\partial_i \partial_j F = 20 z_i^3 \delta_{ij}$.)

\textbf{Key observation.} The scalar curvature involves $p_3 / p_4$.
Using $p_3 = (6e_3 - 1 + 3p_2)/2$ (Newton's identity), we can express
$p_3$ in terms of $(p_2, e_3)$. However, $p_4$ remains. The question becomes:

\begin{quote}
\textit{Is $p_4$ effectively a function of $(p_2, e_3)$ in the regime sampled
by the Fermat quintic, or does it vary independently?}
\end{quote}

This must be checked numerically on the actual point cloud. If $p_4$ varies
substantially independently of $(p_2, e_3)$ but $R_k$ does not depend on it,
then the mechanism is more subtle than just ``scalar curvature controls everything.''

\subsection{Step 2: Spectral Analysis of the Linearised Operator}

\textbf{Goal.} Identify which $G$-invariant functions on $X_0$ are in the leading
eigenspaces of the Lichnerowicz-type operator that controls the balanced metric
iteration.

Donaldson's $T$-operator linearised around the balanced metric satisfies
\[
\delta(T(H))_{\alpha\beta} = \int_{X_0} K(z,w) \delta H_{\alpha\beta}\, d\mu,
\]
for some integral kernel $K$. The convergence of the balanced iteration is
governed by the spectral gap of this operator, specifically by how quickly
the ``non-FS directions'' in H-matrix space are suppressed.

The G-invariant sector decomposes into representations of $S_5$. The trivial
irrep (symmetric tensors) is spanned by functions of $(p_2, p_3, p_4, p_5)$.
\textbf{The conjecture is that the eigenfunctions with large eigenvalues (slow
convergence) correspond to $(p_2, e_3)$, while those corresponding to
$(e_4, e_5)$ have smaller eigenvalues.}

\textbf{Concrete computation.} On $X_0$ with the FS metric, compute the
spectrum of the Laplace-Beltrami operator $\Delta_{g_{\mathrm{FS}}}$ restricted
to $G$-invariant functions. The basis functions $p_2, e_3, e_4, e_5$ are all
in the $G$-invariant sector. If the eigenvalues satisfy
$\lambda_2 \sim \lambda_{e_3} \gg \lambda_{e_4} \sim \lambda_{e_5}$, this
would suggest $(p_2, e_3)$ are the ``slow modes'' that persist in the balanced
metric correction.

\subsection{Step 3: Bergman Kernel Asymptotics on the G-Invariant Sector}

\textbf{Goal.} Compute the leading TYZ coefficient restricted to the $G$-invariant
sector and show it lies in $\mathrm{span}(p_2, e_3)$.

The Bergman kernel on $X_0$ decomposes into $G$-irrep sectors. In the trivial
(G-invariant) sector:
\[
B_k^G(z,z) = \sum_{\alpha \in G\text{-orbits}} \frac{|\mathcal{O}_\alpha(z)|^2}{h_\alpha}
\]
where $\mathcal{O}_\alpha$ denotes the symmetrised monomial orbit sum. These orbit sums
are precisely the $G$-invariant sections of $\mathcal{O}(k)|_{X_0}$, and they are
naturally expressible as polynomial functions of $(p_2, p_3, p_4, p_5)$.

The key result to prove: the ratio $B_k^G(z,z) / B_k^{G,\mathrm{FS}}(z,z)$,
where $B_k^{G,\mathrm{FS}}$ is the $G$-invariant Bergman kernel for the FS metric,
is to leading order in $k$ a function of $(p_2, e_3)$ only.

\subsection{Step 4: Connect to the Dwork Deformation}

For the deformed quintic $X_\psi$, the symmetry group reduces to
$\Gamma_\psi \subset G$. The coefficient trajectories $c_i(\psi)$ encode
how the dominant modes ($p_2, e_3$) acquire $\psi$-dependent amplitudes.
A weaker but still interesting result: show that the deformation vector field
$\partial_\psi g_k$ lies in the $(p_2, e_3)$ sector to leading order,
using Kodaira-Spencer deformation theory.

% ------------------------------------------------------------------
\section{Key Technical Lemmas You Need}
% ------------------------------------------------------------------

Ordered by difficulty (ascending):

\begin{enumerate}[leftmargin=*]

\item \textbf{[Easy, 1 week]} \textit{The $G$-invariant ring on $\Sigma \cap X_0$
is generated by $(p_2, e_3, e_4, e_5)$ as a $\mathbb{R}$-algebra.}

This is a standard invariant-theory result. On the normalisation locus $\sum u_i=1$,
the ring $\mathbb{R}[u_0,\ldots,u_4]^{S_5}$ is generated by $e_2, e_3, e_4, e_5$
(with $e_1 = 1$ fixed). The U(1)$^4$ action adds no further constraints since it
acts on the phases of $z_i$, not on $u_i = |z_i|^2$. \\
\textit{References:} Stanley, ``Invariants of Finite Groups,'' Ch.~3.

\item \textbf{[Easy-Medium, 2 weeks]} \textit{Explicit computation of the second
fundamental form of $X_0 \hookrightarrow \mathbb{CP}^4$ in coordinates.}

Using the formula $II = \nabla^2 F / |\nabla F|$ restricted to the tangent
bundle of $X_0$, compute $|II|^2$ explicitly and express it as a function of
$(u_0, \ldots, u_4)$. Identify which symmetric polynomials appear. \\
\textit{References:} Griffiths-Harris, ``Principles of Algebraic Geometry,''
Ch.~1.5; Huybrechts, ``Complex Geometry,'' Ch.~3.

\item \textbf{[Medium, 4-6 weeks]} \textit{The scalar curvature of $g_{\mathrm{FS}}|_{X_0}$
expressed in terms of $(p_2, p_3, p_4)$.}

Use the Gauss equation $s_{X_0} = s_{\mathbb{CP}^4}|_{X_0} - |II|^2$ and the
computation from Lemma~2 to get $s$ explicitly. Then use Newton's identities to
express $p_3$ in terms of $(p_2, e_3)$ and $p_4$ in terms of $(p_2, e_3, e_4)$.
Determine numerically whether the $e_4$ term is negligible on typical points of $X_0$. \\
\textit{References:} Jost, ``Riemannian Geometry and Geometric Analysis,'' Ch.~4.

\item \textbf{[Medium, 4-6 weeks]} \textit{First TYZ coefficient for $(X_0, g_{\mathrm{FS}})$.}

Verify that the first TYZ coefficient $a_1(z) = s(z)/2$ restricted to $G$-invariant
functions on $X_0$ is well-approximated by a function of $(p_2, e_3)$. This is
the key analytic lemma connecting the Bergman kernel asymptotics to the observed
two-dimensionality. \\
\textit{References:} Zelditch (1998), J. Diff. Geom.;
Lu (2000), J. Geom. Anal.; Fine (2012), J. Diff. Geom.

\item \textbf{[Hard, 3+ months]} \textit{Spectral gap estimate: the Laplacian on
$G$-invariant functions on $(X_0, g_{\mathrm{FS}})$ has a spectral gap separating
$(p_2, e_3)$ eigenfunctions from $(e_4, e_5)$ eigenfunctions.}

This is the main technical theorem. It requires: (a) computing or estimating the
first few eigenvalues of $\Delta_{g_{\mathrm{FS}}}|_{G\text{-inv}}$ on $X_0$, and
(b) showing these correspond to the $(p_2, e_3)$ sector. Can potentially be approached
numerically first (using the balanced H-matrix data you already have) then
rigorously. \\
\textit{References:} Braun-Brelidze-Douglas-Ovrut (2008), JHEP;
Ashmore-He-Heyes-Ovrut (2023), JHEP.

\item \textbf{[Hard, 3+ months]} \textit{The leading $G$-invariant correction to
the balanced metric from the FS metric at large $k$ lies in $\mathrm{span}(p_2, e_3)$.}

This is Conjecture~\ref{conj:medium} made rigorous. Approach via the
Donaldson iteration as a fixed-point problem and show that the deviation from the
FS metric, in the $G$-invariant sector, contracts onto the $(p_2, e_3)$ subspace. \\
\textit{References:} Donaldson (2001), JDG; Fine (2012), JDG; Keller (2009), arXiv.

\end{enumerate}

% ------------------------------------------------------------------
\section{The Tractable Version of the Paper}
% ------------------------------------------------------------------

You do \emph{not} need to prove all six lemmas to write a mathematical physics paper.
A publishable paper could contain:

\begin{itemize}
\item \textbf{Theorem} (easy): $R_k(z)$ is $G$-invariant, hence a function of
$(p_2, e_3, e_4, e_5)$ (Lemmas 1).

\item \textbf{Computation} (medium): The scalar curvature $s(z)$ of $g_{\mathrm{FS}}|_{X_0}$
expressed explicitly in symmetric polynomials (Lemma 3). Identify whether $e_4, e_5$
appear in $s(z)$.

\item \textbf{Conjecture with evidence}: The correction $k(R_k - \bar R_k)$ converges
to a function of $(p_2, e_3)$ alone, supported by the polynomial ablation data and
k-stability table.

\item \textbf{Corollary}: The symbolic formula is the leading-order expression for
this limit in the class of rational functions of $(p_2, e_3)$.
\end{itemize}

\noindent This structure is publishable in \textit{Letters in Mathematical Physics}
(with a theorem + conjecture + evidence) or \textit{Journal of Geometry and Physics}
(computation + conjecture).

% ------------------------------------------------------------------
\section{The Ideal Target Structure}
% ------------------------------------------------------------------

If Lemma 4 (TYZ coefficient computation) succeeds and the first TYZ coefficient
is indeed a function of $(p_2, e_3)$, then the paper has:

\begin{itemize}
\item \textbf{Theorem}: First TYZ coefficient on $X_0$ is in $\mathrm{span}(p_2, e_3)$.
This implies $R_k \sim f(p_2, e_3) + O(k^{-2})$ for large $k$.
\item \textbf{Symbolic formula as corollary}: The SR formula is the rational
function of $(p_2, e_3)$ that best approximates this leading correction.
\item \textbf{Numerical confirmation}: The ablation tables confirm the $O(k^{-2})$
suppression of $e_4, e_5$ contributions.
\end{itemize}

This would be a \textit{Communications in Mathematical Physics} paper.

% ------------------------------------------------------------------
\section{Immediate Next Steps (Ordered)}
% ------------------------------------------------------------------

\begin{enumerate}[leftmargin=*]
\item \textbf{This week.} Numerically verify: does $p_4$ vary independently of
$(p_2, e_3)$ on your sampled points? Run \texttt{poly3\_r2([p4], R\_psi)} and
\texttt{poly3\_r2([p2, e3, p4], R\_psi)} on your Kaggle data.
If $p_4$ can be approximated by a function of $(p_2, e_3)$ on $X_0$, then the
scalar curvature story is consistent. If not, the mechanism is more subtle.

\item \textbf{Week 2-3.} Prove Lemma 1 (G-invariant ring). Write it up carefully
with full algebraic proof. This is publishable as-is as a standalone lemma.

\item \textbf{Week 3-6.} Compute $|II|^2$ explicitly (Lemma 2). Use the Fermat
quintic gradient $\nabla F = 5(z_0^4, \ldots, z_4^4)$ and the Hessian
$\nabla^2 F = 20 \, \mathrm{diag}(z_0^3, \ldots, z_4^3)$.
Write down $s(z)$ as a function of $(p_2, p_3, p_4)$ explicitly.

\item \textbf{Month 2.} Read the TYZ literature (Zelditch 1998, Lu 2000).
Compute or look up the second asymptotic coefficient for degree-$d$ hypersurfaces
in $\mathbb{CP}^n$. Try to apply it to the quintic case.

\item \textbf{Month 3+.} Attempt the spectral gap computation (Lemma 5). Start
numerically using the Laplacian eigenvalue code from Braun et al.\ (2008).
If you can compute the first 10 eigenvalues of $\Delta|_{G\text{-inv}}$
and confirm they correspond to $p_2, e_3$, that is strong evidence for
Conjecture~\ref{conj:strong}.
\end{enumerate}

% ------------------------------------------------------------------
\section{Reading List}
% ------------------------------------------------------------------

\textbf{Essential (read first):}
\begin{enumerate}
\item S.~K.~Donaldson, ``Scalar curvature and projective embeddings I,''
J.\ Diff.\ Geom.\ \textbf{59} (2001), 479--522.
\textit{[Defines the T-operator and balanced metrics; proves convergence to CY.]}

\item S.~Zelditch, ``Szeg\H{o} kernels and a theorem of Tian,''
Int.\ Math.\ Res.\ Not.\ \textbf{6} (1998), 317--331.
\textit{[The foundational TYZ expansion paper.]}

\item Z.~Lu, ``On the lower order terms of the asymptotic expansion of
Tian-Yau-Zelditch,'' Am.\ J.\ Math.\ \textbf{122} (2000), 235--273.
\textit{[Explicit computation of $a_1, a_2$ in the TYZ expansion.]}
\end{enumerate}

\textbf{Important context:}
\begin{enumerate}
\setcounter{enumi}{3}
\item J.~Fine, ``Calabi flow and projective embeddings,''
J.\ Diff.\ Geom.\ \textbf{93} (2013), 55--99.
\textit{[Connects Bergman kernels to the balanced metric iteration rigorously.]}

\item X.~Wang, ``Canonical metrics on stable vector bundles,''
Commun.\ Anal.\ Geom.\ \textbf{13} (2005), 253--285.
\textit{[Good background on balanced metrics and convergence rates.]}

\item V.~Braun, T.~Brelidze, M.~R.~Douglas, B.~A.~Ovrut,
``Eigenvalues and eigenfunctions of the scalar Laplace operator on
Calabi-Yau manifolds,'' JHEP \textbf{2008} (2008), 120.
\textit{[Numerically computes Laplacian eigenvalues on the Fermat quintic.
Gives you the spectral data you need for Lemma 5.]}

\item R.~P.~Stanley, ``Invariants of Finite Groups and their Applications to
Combinatorics,'' Bull.\ Amer.\ Math.\ Soc.\ \textbf{1} (1979), 475--511.
\textit{[Invariant ring theory needed for Lemma 1.]}
\end{enumerate}

\textbf{Background geometry:}
\begin{enumerate}
\setcounter{enumi}{7}
\item P.\ Griffiths and J.\ Harris, \textit{Principles of Algebraic Geometry},
Wiley (1978). Chapters 1.5 (second fundamental form) and 3.3 (Calabi-Yau).

\item D.\ Huybrechts, \textit{Complex Geometry: An Introduction},
Springer (2005). Chapter 3 (K\"ahler manifolds) and Chapter 5 (Calabi-Yau).
\end{enumerate}

% ------------------------------------------------------------------
\section{Summary of What This Paper Would Be}
% ------------------------------------------------------------------

\begin{center}
\renewcommand{\arraystretch}{1.5}
\begin{tabular}{p{3.5cm}|p{10cm}}
\hline
\textbf{What it is NOT} & An ML paper. Symbolic regression appears only as a
computational tool in Section 4 to identify the rational formula. \\
\hline
\textbf{What it IS} & A mathematical physics paper on the $G$-invariant structure
of the balanced metric on the Dwork quintic family. \\
\hline
\textbf{Main object} & $R_k(z) = \log(\det g_k / \det g_{\mathrm{FS}})$,
the balanced metric volume-form ratio. \\
\hline
\textbf{Main result} & This function is $G$-invariant (theorem), lives in a
4-generator ring (theorem), and is dominated by its projection onto
$\mathrm{span}(p_2, e_3)$ (conjecture + evidence, or theorem if TYZ works). \\
\hline
\textbf{Evidence type} & Algebraic proof (invariant ring) + analytic computation
(scalar curvature) + numerical support (ablation, k-stability). \\
\hline
\textbf{Target} & \textit{Letters in Mathematical Physics} (conjecture + evidence)
or \textit{Communications in Mathematical Physics} (with theorem). \\
\hline
\textbf{Timeline} & 2--3 months to a Letters in Math Phys submission;
6--12 months to a CMP-level result. \\
\hline
\end{tabular}
\end{center}

\end{document}
