\begin{tikzpicture}[samples=100, variable=\t]
  \def\a{4}
  \def\b{1.5}
  \draw[domain=0:2*pi]             plot ({\a*cos(\t r)},{\b*sin(\t r)});
  \draw[domain=pi/6:5*pi/6]        plot ({\a*cos(\t r)},{\b*sin(\t r) - 0.7});
  \draw[domain=-0.3+5*pi/4:0.3+7*pi/4]
                                   plot ({\a*cos(\t r)},{\b*sin(\t r) + 0.8});
  \pgfmathsetmacro{\R}{0.8/(2*pi)}
  \draw[domain=0:2*pi, smooth, gray] plot ({-\R*(1-cos(\t r))}, {-\R*(\t - sin(\t r)) - 0.7});
  \draw[domain=0:2*pi, smooth, gray, dotted]
    plot ({\R*(1 - cos(\t r))},
        {-\R*(\t - sin(\t r)) - 0.7});

ll[blue] (0,-0.7) circle(0.05);
ll[blue] (0,-1.5) circle(0.05);
ll[red] (-0.25,-1.2) circle(0.05);
\end{tikzpicture}
\end{subfigure}
\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}
\label{ellipse_torus}
\end{figure}

To formalize the above discussion, let $\iota : M \to \mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \emph{scaled oriented Grassmannian}  $(\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\mathbf{Gr}_c^+(D,d)} = c \cdot g^{\mathbf{Gr}^+(D,d)}$. Let $\bar{\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map
$$\bar{\mathbf{g}}^+_c : M \xrightarrow{\bar{\mathbf{g}}^+} (\mathbf{Gr}^+(D,d), g^{\mathbf{Gr}^+(D,d)}) \xrightarrow{ \cdot c } (\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)}).$$
We consider the graph embedding $\iota \times \bar{\mathbf{g}}_c^+ : M \to (\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d),\, g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\iota \times \bar{\mathbf{g}}^+_c)^*(g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\mathbf{II}$ for the second fundamental form of $M \subset \mathbb{R}^D$ with the induced Euclidean metric and $\mathbf{II}_c$ for that of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\mathrm{vol}(M)$ for the Euclidean volume of $M \subset \mathbb{R}^D$.
\begin{definition}\label{grassmannian_distance}
For $c>0$, we define a distance $d_c$ on $M$ by
\[
  d_{c}(q_1,q_2) = d_{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((q_1, \bar{\mathbf{g}}_c^+(q_1)),(q_2, \bar{\mathbf{g}}_c^+(q_2))\bigr)
  = \sqrt{\|q_1-q_2\|_{\mathbb{R}^D}^2
      + c\, d_{\mathbf{Gr}^+(D,d)}\bigl(\bar{\mathbf{g}}^+(q_1),\bar{\mathbf{g}}^+(q_2)\bigr)^2},
\]
where $\|\cdot\|_{\mathbb{R}^D}$ denotes the Euclidean norm and
$d_{\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian
$\mathbf{Gr}^+(D,d)$.
\end{definition}

We first show that, by embedding $M$ into $\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \in M$ and a nonzero vector $v \in T_pM$, denote by $\kappa_c(v)$ and $\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ at $(p,\bar{\mathbf{g}}_c^+(p))$ in the direction $(v, d\bar{\mathbf{g}}_c^+(v))$ and of $M \subset \mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.
\begin{theorem}\label{principal_curvature}
  Let $M \subset \mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. Fix $p \in M$ and a unit vector $v \in T_p M$. Assume that the normal curvature at $p$ in the direction $v$ is sufficiently large so that
  \begin{equation*}
    \mathrm{vol}_c(M) \leq \bigl(1 + c \,\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2\bigr)^{d/2} \mathrm{vol}(M),
    \qquad
    \kappa(v) > \frac{\|(\nabla_v^{M} \mathbf{II}) (v,\cdot) \|_{\operatorname{HS}}}{\|\mathbf{II} (v,\cdot)\|_{\operatorname{HS}}},
  \end{equation*}
  where $\|\cdot\|_{\operatorname{HS}}$ denotes the Hilbert--Schmidt norm. Then, for every $c>0$,
  \[
    \kappa_c(v)\, \mathrm{vol}_c(M)^{1/d}
    \;\leq\;
    \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^{M} \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}\,
    \mathrm{vol}_c(M)^{1/d}
    \;<\;
    \kappa(v)\, \mathrm{vol}(M)^{1/d},
  \]
and the middle term in the inequalities is strictly decreasing as $c$ grows.
\end{theorem}
In directions with large normal curvatures, even a small displacement along the curve already produces a large change in the tangent space, so the Grassmannian component of $g_c$ contributes an increasingly large portion of the speed. Since the normal curvature is defined as the size of the normal acceleration divided by the square of this speed, reparameterizing the curve to have unit speed with respect to $g_c$ makes the normal curvature decrease as $c$ grows.

Next, to quantify how distances between bottleneck pairs change with respect to $d_c$, we set
\begin{align*}
  L_c(M)
  &= \min\Bigl\{\frac{1}{2}d_c(q_1,q_2) \;\Big|\; (q_1,q_2) \text{ is a bottleneck of } M \subset \mathbb{R}^D\Bigr\} \\
  &= \min\Bigl\{\frac{1}{2}d_c(q_1,q_2) \;\Big|\; \overline{q_1q_2} \perp T_{q_1}M,\ \overline{q_1q_2} \perp T_{q_2}M\Bigr\}.
\end{align*}
Similarly, in the Euclidean case, we define
\[
  L(M)
  = \min\Bigl\{\frac{1}{2}\|q_1-q_2\|_{\mathbb{R}^D} \;\Big|\; (q_1,q_2) \text{ is a bottleneck of } M \subset \mathbb{R}^D\Bigr\}.
\]
Under a mild curvature hypothesis, we show that for a smooth closed orientable hypersurface $M$, the bottlenecks are separated when distances are measured with $d_c$ from Definition~\ref{grassmannian_distance}.
\begin{theorem}\label{normalized_bottleneck}
  Let $M \subset \mathbb{R}^D$ be a smooth closed orientable hypersurface. Suppose that $L(M) \leq \frac{1}{\|\mathbf{II}\|_2}$.
  Then for every $c \in \bigl(0, \tfrac{12L(M)^2}{\pi^2}\bigr]$,
    \[
    \frac{L_c(M)}{\mathrm{vol}_c(M)^{1/(D-1)}}
    \;\geq\;
    \frac{\sqrt{4L(M)^2 + c\pi^2}}{2\,\mathrm{vol}_c(M)^{1/(D-1)}}
    \;>\;
    \frac{L(M)}{\mathrm{vol}(M)^{1/(D-1)}},
  \]
  and the middle term in the inequalities is strictly increasing in $c$ on the interval $\bigl(0, \tfrac{12L(M)^2}{\pi^2}\bigr]$.
\end{theorem}

We obtain an explicit range of radii $r$ for which the ambient Čech complex $\check{C}(M, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to $M$. In particular, this yields a lower bound on the lengths of the barcodes in the associated persistent homology, from which the homology of $M$ is recovered from the distance $d_c$.
\begin{theorem}\label{length_barcodes}
Let $M \subset \mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold. For $c>0$, the ambient Čech complex $\check{C}(M, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d);r)$ is homotopy equivalent to \(M\) for all
\[
  r <
  \min\!\left(
    \sqrt{\frac{c}{2}}\arctan\sqrt{\frac{2}{c\|\mathbf{II}_c\|_2}},\,
    \frac{\sqrt{c}\,\pi}{2},\,
    L'_c(M)
  \right),
\]
where
\[
  L'_c(M)
  = \min \Bigl\{\frac{1}{2}d_c(q_1,q_2) \;\Bigm|\;
  \overline{(q_1,\bar{\mathbf{g}}_c^+(q_1))(q_2,\bar{\mathbf{g}}_c^+(q_2))}
  \perp T_{(q_i,\bar{\mathbf{g}}_c^+(q_i))}(M,g_c),\ i=1,2 \Bigr\}.
\]
\end{theorem}

Fix a field $\mathbb{k}$. Since the metric $d_c$ depends both on the ambient positions and on the oriented tangent spaces of $M$, the persistence module $H_j(\mathbb{\check{C}}(M, \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d));\mathbb{k})$ is not expected to be stable with respect to the Hausdorff distance on $\mathbb{R}^D$. Instead, we work with a stronger notion of convergence that also takes into account the distance between tangent spaces.
\begin{theorem}\label{stability}
  Let $\{M_i\}_{i \in \mathbb{N}}$ be a sequence of smooth closed orientable $d$-dimensional submanifolds of $\mathbb{R}^D$ converging to a smooth closed orientable submanifold $M_\infty \subset \mathbb{R}^D$ in the Whitney $C^1$ topology (see Definition~\ref{Whitney}). Fix $c>0$. Then for every field $\mathbb{k}$ and every $j \in \mathbb{N} \cup \{0\}$,
  \[
    \lim_{i \to \infty}
    d_I\Bigl(
      H_j(\mathbb{\check{C}}(M_i, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d));\mathbb{k}),
      H_j(\mathbb{\check{C}}(M_\infty, \mathbb{R}^D\times\mathbf{Gr}_c^+(D,d));\mathbb{k})
    \Bigr) = 0,
  \]
  where $d_I$ denotes the interleaving distance between persistence modules.
\end{theorem}

Based on the above theorems, we describe a method for computing the distance matrix for a finite subset of $M$ with respect to the distance $d_c$ from Definition~\ref{grassmannian_distance}. To illustrate the performance of this method, we display three computational examples: a time-delay embedded attractor, an approximate quasi-halo orbit in the Saturn--Enceladus system, and a classification of three-dimensional image shapes.

\section{Theoretical background}

\subsection{Persistence theory}

A \emph{filtered space (or filtration)} $\mathbb{X}$ is a collection of topological spaces $\{X_i\}_{i \in \mathbb{R}_{\geq 0}}$ equipped with inclusion maps $\iota_i^j : X_i \to X_j$ for all $i \leq j$. Let \( (X, d) \) be a compact metric space and let \( \varepsilon \in \mathbb{R}_{\geq 0} \). The \emph{Vietoris--Rips complex} \( \mathrm{VR}(X;\varepsilon) \) is the abstract simplicial complex whose simplices are finite subsets \( \sigma \subset X \) such that $d(x, y) < \varepsilon$ for all $x, y \in \sigma$. For a non-decreasing sequence of parameters $\varepsilon_0 \leq \varepsilon_1 \leq \varepsilon_2 \leq \dots$, these complexes form a filtered space
\[
\mathrm{VR}(X;\varepsilon_0) \hookrightarrow \mathrm{VR}(X;\varepsilon_1) \hookrightarrow \mathrm{VR}(X;\varepsilon_2) \hookrightarrow \cdots,
\]
called the \emph{Vietoris--Rips filtration}, denoted by $\mathbb{V}R(X)$. The (intrinsic) \emph{Čech complex} $\check{C}(X;r)$ is the nerve of $r$-balls centered at points of $X$, and the corresponding \emph{Čech filtration} $\mathbb{\check{C}}(X)$ is defined analogously. Let $Y$ be a subset of a metric space $(Z,d_Z)$. The \emph{ambient Čech complex} $\check{C}(Y, Z; r)$ is defined as the nerve of the family of $r$-balls $\{B_Z(y, r)\}_{y \in Y}$ taken in the ambient space $Z$. The corresponding filtration is denoted $\mathbb{\check{C}}(Y, Z)$.

For $n \in \mathbb{N} \cup \{0\}$, a field $\mathbb{k}$ and a filtration $\mathbb{X}= \{X_i\}_{i \in \mathbb{R}_{\geq 0}}$, the \emph{$n$-th persistent homology} $ H_n(\mathbb{X}; \mathbb{k}) $ of $\mathbb{X}$ is a family of vector spaces $H_n(X_i; \mathbb{k})$ with induced linear maps $H_n(\iota_i^j) : H_n(X_i; \mathbb{k}) \to H_n(X_j; \mathbb{k})$ for $i\leq j$. If \( H_n(\mathbb{X}; \mathbb{k}) \) is pointwise finite-dimensional and decomposes as a direct sum of interval modules, then it corresponds to a multiset of points \( (b,d) \subset \mathbb{R}_{\geq 0}^2 \), called the \emph{persistence diagram} in degree \( n \).

For a field $\mathbb{k}$, a \emph{persistence module (over $\mathbb{R}_{\geq 0}$)} $\mathbb{V}$ is a family of vector spaces $\{V_{i}\}_{i \in \mathbb{R}_{\geq 0}}$ equipped with linear maps $v_i^j :V_i \to V_j$ for every $0 \leq i \leq j$, satisfying $v_j^k \circ v_i^j = v_i^k$ whenever $i \leq j \leq k$ and $v_i^i$ is the identity map on $V_i$. A \emph{morphism of degree $\varepsilon$} between two persistence modules $\mathbb{V}$ and $\mathbb{W}$ is a family of linear maps $\Phi = \{\phi_i:V_i\to W_{i+\varepsilon}\}_{i\in\mathbb{R}_{\geq 0}}$ such that for all $i\le j$, it holds that
\[
w_{i+\varepsilon}^{j+\varepsilon}\circ \phi_i \;=\; \phi_j\circ v_i^j.
\]
For $\varepsilon\ge 0$, the \emph{shift} $\Sigma^\varepsilon$ of $\mathbb{V}$ by $\varepsilon$ is
\[
(\Sigma^\varepsilon \mathbb{V})_i=V_{i+\varepsilon},\qquad
(\Sigma^\varepsilon)(v_i^j)=v_{i+\varepsilon}^{j+\varepsilon}:V_{i+\varepsilon} \to V_{j+\varepsilon}.
\]
Two persistence modules $\mathbb{V}$ and $\mathbb{W}$ are \emph{$\varepsilon$-interleaved} if there exist morphisms $\Phi=\{\phi_i : V_i \to W_{i+\varepsilon}\}_{i \in \mathbb{R}_{\geq 0}}$ and $\Psi = \{\psi_i : W_i \to V_{i+\varepsilon}\}_{i \in \mathbb{R}_{\geq 0}}$ of degree $\varepsilon$ such that
\[
\psi_{\,i+\varepsilon}\circ \phi_i \;=\; v_{i}^{i+2\varepsilon},
\qquad
\phi_{\,i+\varepsilon}\circ \psi_i \;=\; w_{i}^{i+2\varepsilon},
\]
for every $i \in \mathbb{R}_{\geq 0}$. The \emph{interleaving distance} $d_I$ between $\mathbb{V}$ and $\mathbb{W}$ is
\[
d_I(\mathbb{V},\mathbb{W})\;=\;\inf\bigl\{\varepsilon\ge 0 \,\big|\, \mathbb{V} \text{ and } \mathbb{W} \text{ are } \varepsilon\text{-interleaved}\bigr\}.
\]
For more details, see e.g.\ \cite{oudot2015persistence, chazal2021introduction}.

\subsection{Basic Riemannian geometry}

In this subsection, we refer to \cite{do1992riemannian} and \cite{lee2018introduction}. Let $M$ be a smooth compact $n$-dimensional manifold. For a choice of a smooth section $g \in \Gamma(M; \mathrm{Sym}^2 T^*M)$, a pair $(M,g)$ is called a \emph{Riemannian manifold} if for each $p \in M$, the fiberwise bilinear map $g_p : T_pM \times T_pM \to \mathbb{R}$ is positive definite. The section $g$ is called a \emph{(Riemannian) metric} on $M$.  We call the map
$$\nabla : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M); \quad (X,Y) \mapsto \nabla_X Y$$
an \emph{affine connection} if $\nabla_X Y$ is linear over $C^\infty (M)$ in $X$ and linear over $\mathbb{R}$ in $Y$, satisfying $\nabla_X(fY) = f(\nabla_X Y) + (Xf)Y$ for any $f \in C^\infty(M)$. The \emph{Levi--Civita connection} is the unique affine connection $\nabla$ on $(M,g)$ that satisfies $Z(g(X,Y)) =g(\nabla_Z X, Y) + g(X,\nabla_Z Y)$ and $\nabla_X Y - \nabla_Y X = [X,Y]$ for any $X,Y,Z \in \mathfrak{X}(M).$

For a Riemannian manifold $(M,g)$ with Levi--Civita connection $\nabla$, the \emph{Riemann curvature tensor} $R$ is defined by $R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z- \nabla_{[X,Y]}Z$ for $X,Y,Z \in \mathfrak{X}(M)$. For a tangent $2$-plane $\Sigma_pM \subset T_pM$ with an orthonormal basis $\{u,v\}$, the \emph{sectional curvature} of $\Sigma_p M$ is defined by $K_p(u,v)=g_p(R(u,v)v,u).$

\subsection{Convergence theory}
We refer to \cite{gromov2007metric, hirsch2012differential} in this subsection. Let $(Z,d_Z)$ be a metric space and let $A,B\subset Z$ be nonempty compact subsets. The \emph{Hausdorff distance} between $A$ and $B$ is defined by
\[
d_H^Z(A,B)
=\max\Big\{\,\sup_{a\in A}\inf_{b\in B} d_Z(a,b)\;,\;
\sup_{b\in B}\inf_{a\in A} d_Z(a,b)\,\Big\}.
\]
The following stability theorem establishes the continuity of persistent homology of the ambient Čech complex with respect to the Hausdorff distance.
\begin{theorem}[{\cite[Theorem~5.6]{chazal2014persistence}}]\label{stability_theorem} Let $A,B$ be compact subsets of a metric space $Z$. Then for every $j \in \mathbb{N} \cup \{0\}$ and every field $\mathbb{k}$,
\[
d_I\bigl(H_j(\mathbb{\check{C}}(A, Z); \mathbb{k}), H_j(\mathbb{\check{C}}(B, Z); \mathbb{k})\bigr)
\leq d_H^Z(A,B).
\]
\end{theorem}

Let $(N,g_N)$ be a fixed Riemannian manifold, and let $\{M_i\}_{i\in\mathbb{N}}$ be a family of smooth closed submanifolds of $N$ such that $M_i = F_i(M)$ for every $i \in \mathbb{N}$, where $M$ is a fixed smooth closed manifold and each $F_i : M \to N$ is a smooth embedding.
\begin{definition}[{\cite[Chapter~2]{hirsch2012differential}}]\label{Whitney}
We say that the family of submanifolds $\{M_i\}_{i\in\mathbb{N}}$ \emph{converges to $M_\infty$ in the (Whitney) $C^k$ topology} if there exists an embedding
$$
F_\infty : M \to N, \qquad M_\infty = F_\infty(M),
$$
such that $F_i \to F_\infty$ in the Whitney $C^k$ topology on $C^k(M,N)$. That is, there exist locally finite atlases
$\{(U_\alpha,\varphi_\alpha)\}_{\alpha}$ of $M$ and $\{(V_\alpha,\psi_\alpha)\}_{\alpha}$ of $N$ with $F_\infty(U_\alpha) \subset V_\alpha$ such that, for each $\alpha$ and each $\varepsilon_\alpha > 0$, there exists $n_\alpha \in \mathbb{N}$ with the property that for every $i \geq n_\alpha$,
\[
\max_{|\beta|\le k}
\sup_{x \in \varphi_\alpha(U_\alpha)}
\left\|
D^\beta\big(\psi_\alpha \circ F_i \circ \varphi_\alpha^{-1}\big)(x)
-
D^\beta\big(\psi_\alpha \circ F_\infty \circ \varphi_\alpha^{-1}\big)(x)
\right\|_2
< \varepsilon_\alpha.
\]
\end{definition}

\subsection{Geometry of an embedded submanifold}

Let $M \subset \mathbb{R}^D$ be a smooth closed $d$-dimensional submanifold with $d < D$. Two distinct points $q_1,q_2 \in M$ form a \emph{bottleneck} in $\mathbb{R}^D$ if the line segment $\overline{q_1 q_2}$ is orthogonal to both tangent spaces $T_{q_1}M$ and $T_{q_2}M$. The \emph{width} of such a bottleneck is defined as $\tfrac{1}{2}\|q_1-q_2\|_{\mathbb{R}^D}$. This notion should not be confused with the bottleneck distance between persistence modules
or with bottlenecks in graph theory.

Define the (Euclidean) \emph{reach} of $M \subset \mathbb{R}^D$ by
\[
\textsf{rch}_{\mathbb{R}^D}(M)
= \sup \Bigl\{ r \geq 0 \,\Big|\,
\text{every } p \in \mathbb{R}^D \text{ with } d_{\mathbb{R}^D}(p,M)< r
\text{ admits a unique nearest point projection onto } M \Bigr\}.
\]
\begin{theorem}[{\cite[Theorem~3.4]{aamari2019estimating}, \cite[Theorem~7.8]{breiding2024metric}}]\label{reach}
    Suppose that the reach of a closed submanifold $M \subset \mathbb{R}^D$ is $\tau>0$. Then at least one of the following holds.
    \begin{itemize}
        \item There exist a point $q \in M$ and a unit-speed geodesic $\gamma : (-\varepsilon, \varepsilon) \to M$ for some $\varepsilon>0$ such that $\gamma(0)=q$ and $|\gamma''(0)| = \frac{1}{\tau}$.
        \item There exist distinct points $q_1, q_2 \in M$ forming a bottleneck of $M \subset \mathbb{R}^D$ such that $\|q_1 - q_2\|_{\mathbb{R}^D} = 2\tau$.
    \end{itemize}
    In particular, the reach $\tau$ of $M$ is realized either as the reciprocal of the normal curvature of a geodesic or as the width of a bottleneck pair.
\end{theorem}

Suppose a $d$-dimensional smooth compact manifold $M$ is embedded in a closed Riemannian manifold $(N,g)$. Denote the Levi--Civita connection of $N$ by $\nabla$ and the normal bundle of $M$ by $\nu$. The \emph{second fundamental form} of $M$ is defined to be the tensor $\mathbf{II} : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \Gamma(\nu)$ given by $\mathbf{II}(X,Y) = (\nabla_X Y)^{\perp}$, where the map $(\cdot)^{\perp}$ denotes the orthogonal projection onto $\nu$. The \emph{normal exponential map} $\exp_\nu : \nu \to N$ is defined by
$$ \exp_\nu(q,v) = \exp_q(v),\quad (q,v) \in \nu.$$
For $p \in M$ and a nonzero vector $v \in T_pM$, the \emph{(norm of the) normal curvature} $\kappa(v)$ at $p$ in the direction $v$ is
\[
\kappa(v)
=
\sup_{\substack{w \in \nu_p \\ \|w\|_2=1}}
\frac{g_p\bigl(\mathbf{II}_p(v,v),\, w\bigr)}{\|v\|_2^2}.
\]
The \emph{operator norm} (or \emph{2-norm}) of $\mathbf{II}_p$ is
\[
\|\mathbf{II}_p\|_2
=
\sup_{\substack{v \in T_pM \\ \|v\|_2=1}}
\kappa(v)
=
\sup_{\substack{v \in T_pM \\ \|v\|_2=1}}
\bigl\|\mathbf{II}_p(v,v)\bigr\|_2.
\]
Denote $\|\mathbf{II}\|_2 = \sup_{p \in M} \|\mathbf{II}_p\|_2$. For two linear operators $A_p,B_p : T_pM \to \nu_p$ and an orthonormal basis $\{e_i\}_{i=1}^d$ of $T_pM$, the \emph{Hilbert--Schmidt inner product} between them is
$$\langle A_p, B_p \rangle_{\operatorname{HS}} =\sum_{i=1}^d g_p\bigl(A_p(e_i), B_p(e_i)\bigr).$$
Denote by $\|\cdot\|_{\operatorname{HS}}$ the induced norm.

\begin{definition}[{\cite[Definition~2.5]{prasad2023cut}, \cite[Definitions~12 and 13]{attali2022tight}}]
Let $S$ be a closed subset embedded in a closed Riemannian manifold $N$. Denote the distance function on $N$ by $d_N$ and the length of a curve $\gamma \subset N$ by $\mathrm{len}_N(\gamma)$. We define a geodesic $\gamma : [0,T] \to N$ to be ($S$-)\emph{distance-minimal} if $\mathrm{len}_N(\gamma|_{[0,t]}) = d_N(S,\gamma(t))$ for all $t \in [0,T]$. The \emph{cut locus} of $S$, denoted by $\mathsf{Cu}_N(S)$, is the set of points $p \in N$ for which there exists a distance-minimal geodesic $\gamma$ from $S$ to $p$ such that any extension of $\gamma$ beyond its endpoint $p$ is not distance-minimal. The \emph{(cut locus) reach} (or \emph{normal injectivity radius}) of $S$ in $N$ is
$$\mathsf{rch}_N(S) = \inf\{ d_N(p,q) \mid q \in S,\ p \in \mathsf{Cu}_N(S)\}.$$
\end{definition}
If $S$ is a smooth closed submanifold, then $\mathsf{rch}_N(S)$ is the supremum of all $\varepsilon>0$ such that the restriction of the normal exponential map
\[
\exp_\nu : \{(p,v)\in \nu \mid \|v\|_2<\varepsilon\} \longrightarrow N
\]
is an embedding. We give an analogue of Theorem~\ref{reach} below.
\begin{theorem}[{\cite[Section~2]{singh1988closest}, \cite[Lemma~A.2]{basu2023connection}}]\label{cutlocus}
    Let $M$ be a smooth closed submanifold of a Riemannian manifold $(N,g)$, and suppose $\mathsf{rch}_N(M)=T>0$. Denote the normal bundle of $M$ by $\nu$ and its unit normal bundle by $S(\nu)$. Then one of the following holds (see Figure~\ref{cutlocus_figure}):
    \begin{itemize}
        \item[\textnormal{(focal)}] There exists a pair $(p,v) \in S(\nu)$ such that the differential of the normal exponential map $d(\exp_\nu)_{(p,Tv)}$ is not of full rank, whereas $d(\exp_\nu)_{(p',tv')}$ has full rank for every $(p',v') \in S(\nu)$ and every $t$ with $0 <t<T$. In this case, the point $\exp_\nu(Tv)$ is called a \emph{first focal point} and $T$ a \emph{first focal time}.
        \item[\textnormal{(non-focal)}] There exist at least two distinct unit-speed distance-minimal geodesics $\gamma_1, \gamma_2 : [0,T] \to N$ from (not necessarily distinct) two points in $M$ satisfying $\gamma_1(T) = \gamma_2(T)$ and $ \gamma_i'(0) \perp M$ for $i=1,2$.
    \end{itemize}
\end{theorem}

\begin{figure}[ht]
\begin{center}
\begin{tikzpicture}[scale=1.5]

  \begin{scope}[yshift=-0.8cm]
    \draw[thick,black,domain=-1.1:1.1,samples=200] plot (\x,{\x*\x});
ll[red] (0,0.5) circle(0.03);
ll[blue] (0,0) circle(0.03);
    \draw[gray,->] (0,0) -- (0,0.5);

    \pgfmathsetmacro{\aone}{0.12}
    \pgfmathsetmacro{\atwo}{-0.10}
    \pgfmathsetmacro{\xone}{\aone}
    \pgfmathsetmacro{\yone}{\aone*\aone}
    \pgfmathsetmacro{\xtwo}{\atwo}
    \pgfmathsetmacro{\ytwo}{\atwo*\atwo}

ll[blue] (\xone,\yone) circle(0.03);
ll[blue] (\xtwo,\ytwo) circle(0.03);
    \draw[gray,->] (\xone,\yone) -- (0,0.5);
    \draw[gray,->] (\xtwo,\ytwo) -- (0,0.5);
  \end{scope}

  \begin{scope}[xshift=3.2cm]
    \def\az{25}\def\el{35}
    \pgfmathsetmacro{\caz}{cos(\az)} \pgfmathsetmacro{\saz}{sin(\az)}
    \pgfmathsetmacro{\cel}{cos(\el)} \pgfmathsetmacro{\sel}{sin(\el)}

    \draw[lightgray] (0,0) circle (1);

    \def\ang{30}\def\hw{16}

    \pgfmathsetmacro{\cang}{cos(\ang)} \pgfmathsetmacro{\sang}{sin(\ang)}
    \pgfmathsetmacro{\PX}{\cang*\caz - \sang*\saz}
    \pgfmathsetmacro{\PY}{(\cang*\saz + \sang*\caz)*\cel}
    \pgfmathsetmacro{\QX}{-\PX} \pgfmathsetmacro{\QY}{-\PY}
ll[blue] (\PX,\PY) circle (0.03);
ll[red]  (\QX,\QY) circle (0.03);

    \draw[gray,domain=0:360,samples=360,variable=\x]
      plot ({cos(\x)*(\cang*\caz - \sang*\saz)},
            {cos(\x)*(\cang*\saz + \sang*\caz)*\cel - sin(\x)*\sel});

    \draw[gray,->]
      ({cos(70)*(\cang*\caz - \sang*\saz)},
       {cos(70)*(\cang*\saz + \sang*\caz)*\cel - sin(70)*\sel})
      --
      ({cos(100)*(\cang*\caz - \sang*\saz)},
       {cos(100)*(\cang*\saz + \sang*\caz)*\cel - sin(100)*\sel});

    \draw[gray,->]
      ({cos(270)*(\cang*\caz - \sang*\saz)},
       {cos(270)*(\cang*\saz + \sang*\caz)*\cel - sin(270)*\sel})
      --
      ({cos(240)*(\cang*\caz - \sang*\saz)},
       {cos(240)*(\cang*\saz + \sang*\caz)*\cel - sin(240)*\sel});
  \end{scope}

  \begin{scope}[xshift=6.4cm,scale=0.7]
    \draw[thick,black,domain=-1.3:1.3,samples=300]
      plot ({ sqrt(1+\x*\x) },{\x});
    \draw[thick,black,domain=-1.3:1.3,samples=300]
      plot ({-sqrt(1+\x*\x) },{\x});

ll[blue] ( 1,0) circle(0.03);
ll[blue] (-1,0) circle(0.03);
ll[red] (0,0) circle(0.03);

    \draw[gray,->] ( 1,0) -- (0,0);
    \draw[gray,->] (-1,0) -- (0,0);
  \end{scope}
\end{tikzpicture}

\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \emph{\v{C}ech complex} \(\check{C}(X;r)\) is the nerve of the open cover of $r$-balls $\{B_X(x,r)\}_{x\in X}$, and the \emph{Vietoris--Rips complex} $\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\sigma\subset X$ with $\operatorname{diam}(\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \cite{virk20201, lim2024vietoris, balitskiy2025geometric}.

The \emph{reach} (or \emph{normal injectivity radius}) $\mathsf{rch}_{\mathbb{R}^D}(M)$ of a smooth submanifold $M \subset \mathbb{R}^D$ is the supremum of $\tau>0$ such that every point $x \in \mathbb{R}^D$ with $d_{\mathbb{R}^D}(x,M)<\tau$ has a unique nearest point in $M$ \cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \cite{niyogi2008finding, kim2019homotopy}, and the \emph{metric thickening} of $M$ \cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.

Aamari et al. \cite{aamari2019estimating} showed that the reach of a submanifold $M \subset \mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \emph{maximal normal curvature} of $M$ and the minimal width of its \emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\mathbb{R}^2$ or a thin $2$-torus with small systole in $\mathbb{R}^3$ as in Figure~\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.

In both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \subset \mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \emph{oriented Grassmannian bundle} $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\mathbf{Gr}^+(D,d)$.

\begin{figure}
\centering
\begin{subfigure}[t]{0.47\textwidth}
\centering
\begin{tikzpicture}
  \draw[thick] (0,0) ellipse [x radius=3.0cm, y radius=0.5cm];
ll[blue] (0,0.5) circle(0.05);
ll[blue] (0,-0.5) circle(0.05);
ll[red] (3.0,0) circle(0.05);
\end{tikzpicture}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.47\textwidth}
\centering
\begin{tikzpicture}[samples=100, variable=\t]
  \def\a{4}
  \def\b{1.5}
  \draw[domain=0:2*pi]             plot ({\a*cos(\t r)},{\b*sin(\t r)});
  \draw[domain=pi/6:5*pi/6]        plot ({\a*cos(\t r)},{\b*sin(\t r) - 0.7});
  \draw[domain=-0.3+5*pi/4:0.3+7*pi/4]
                                   plot ({\a*cos(\t r)},{\b*sin(\t r) + 0.8});
  \pgfmathsetmacro{\R}{0.8/(2*pi)}
  \draw[domain=0:2*pi, smooth, gray] plot ({-\R*(1-cos(\t r))}, {-\R*(\t - sin(\t r)) - 0.7});
  \draw[domain=0:2*pi, smooth, gray, dotted]
    plot ({\R*(1 - cos(\t r))},
        {-\R*(\t - sin(\t r)) - 0.7});

ll[blue] (0,-0.7) circle(0.05);
ll[blue] (0,-1.5) circle(0.05);
ll[red] (-0.25,-1.2) circle(0.05);
\end{tikzpicture}
\end{subfigure}
\caption{A highly eccentric ellipse (left) and a thin 2-torus (right). Points of large curvature are marked in red, and the narrowest bottlenecks are indicated in blue.}
\label{ellipse_torus}
\end{figure}

To formalize the above discussion, let $\iota : M \to \mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \emph{scaled oriented Grassmannian}  $(\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\mathbf{Gr}_c^+(D,d)} = c \cdot g^{\mathbf{Gr}^+(D,d)}$. Let $\bar{\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map
$$\bar{\mathbf{g}}^+_c : M \xrightarrow{\bar{\mathbf{g}}^+} (\mathbf{Gr}^+(D,d), g^{\mathbf{Gr}^+(D,d)}) \xrightarrow{ \cdot c } (\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)}).$$
We consider the graph embedding $\iota \times \bar{\mathbf{g}}_c^+ : M \to (\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d),\, g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\iota \times \bar{\mathbf{g}}^+_c)^*(g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\mathbf{II}$ for the second fundamental form of $M \subset \mathbb{R}^D$ with the induced Euclidean metric and $\mathbf{II}_c$ for that of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\mathrm{vol}(M)$ for the Euclidean volume of $M \subset \mathbb{R}^D$.
\begin{definition}\label{grassmannian_distance}
For $c>0$, we define a distance $d_c$ on $M$ by
\[
  d_{c}(q_1,q_2) = d_{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((q_1, \bar{\mathbf{g}}_c^+(q_1)),(q_2, \bar{\mathbf{g}}_c^+(q_2))\bigr)
  = \sqrt{\|q_1-q_2\|_{\mathbb{R}^D}^2
      + c\, d_{\mathbf{Gr}^+(D,d)}\bigl(\bar{\mathbf{g}}^+(q_1),\bar{\mathbf{g}}^+(q_2)\bigr)^2},
\]
where $\|\cdot\|_{\mathbb{R}^D}$ denotes the Euclidean norm and
$d_{\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian
$\mathbf{Gr}^+(D,d)$.
\end{definition}

We first show that, by embedding $M$ into $\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \in M$ and a nonzero vector $v \in T_pM$, denote by $\kappa_c(v)$ and $\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ at $(p,\bar{\mathbf{g}}_c^+(p))$ in the direction $(v, d\bar{\mathbf{g}}_c^+(v))$ and of $M \subset \mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.

\emph{Persistent homology} computes barcodes of filtered chain complexes constructed from a metric space $X$ \cite{zomorodian2004computing, oudot2015persistence}. For $r>0$, the \emph{\v{C}ech complex} \(\check{C}(X;r)\) is the nerve of the open cover of $r$-balls $\{B_X(x,r)\}_{x\in X}$, and the \emph{Vietoris--Rips complex} $\mathrm{VR}(X;r)$ is the abstract simplicial complex whose simplices are the finite subsets $\sigma\subset X$ with $\operatorname{diam}(\sigma)<r$. If $M$ is a Riemannian manifold, then the Čech complex $\check{C}(M;r)$ is homotopy equivalent to $M$ for $r<r_{\operatorname{conv}}(M)$ by the nerve lemma, while the systole and the filling radius bound the range of $r$ for which the Vietoris--Rips complex $\mathrm{VR}(M;r)$ has the same first and top-dimensional homology groups as $M$ \cite{virk20201, lim2024vietoris, balitskiy2025geometric}.

The \emph{reach} (or \emph{normal injectivity radius}) $\mathsf{rch}_{\mathbb{R}^D}(M)$ of a smooth submanifold $M \subset \mathbb{R}^D$ is the supremum of $\tau>0$ such that every point $x \in \mathbb{R}^D$ with $d_{\mathbb{R}^D}(x,M)<\tau$ has a unique nearest point in $M$ \cite{federer1959curvature, aamari2019estimating}. For smooth embedded submanifolds, this agrees with the supremum of radii for which the normal exponential map is injective. The reach of $M$ gives bounds on the scales $r$ where the Vietoris--Rips or Čech complex built from sufficiently dense points of $M$ in the ambient Euclidean metric \cite{niyogi2008finding, kim2019homotopy}, and the \emph{metric thickening} of $M$ \cite{adams2019metric, adams2022metric, adams2024persistent}, the 1-Wasserstein analogue of the Vietoris--Rips complex, are homotopy equivalent to $M$.

Aamari et al. \cite{aamari2019estimating} showed that the reach of a submanifold $M \subset \mathbb{R}^D$ can be expressed as the minimum of the reciprocal of the \emph{maximal normal curvature} of $M$ and the minimal width of its \emph{bottlenecks}, pairs of points in $M$ whose connecting line segment is orthogonal to both tangent spaces. For example, for a highly eccentric ellipse in $\mathbb{R}^2$ or a thin $2$-torus with small systole in $\mathbb{R}^3$ as in Figure~\ref{ellipse_torus}, the normal exponential map ceases to be injective at a radius comparable to the reach, since the points of large curvature (marked in red) induce focal points and the bottlenecks (marked in blue) lie close to each other in the Euclidean metric.

In both examples, the indicated bottlenecks have opposite normal directions. This suggests that, for an orientable submanifold $M \subset \mathbb{R}^D$, embedding $M$ into an augmented ambient space, such as the \emph{oriented Grassmannian bundle} $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, can enlarge the normal injectivity radius of $M$. In the product metric on $\mathbb{R}^D \times \mathbf{Gr}^+(D,d)$, the contribution from the Grassmannian factor enlarges the lengths of curves near points of large normal curvature, so that unit-speed geodesics have smaller normal curvature, and bottleneck pairs with opposite normal directions are pushed farther apart because their oriented tangent spaces are far from each other in $\mathbf{Gr}^+(D,d)$.

To formalize the above discussion, let $\iota : M \to \mathbb{R}^D$ be an embedding of a smooth closed orientable $d$-dimensional submanifold. Fix $c>0$ and consider the \emph{scaled oriented Grassmannian} $(\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)})$, whose underlying manifold is the oriented Grassmannian $\mathbf{Gr}^+(D,d)$ equipped with the rescaled metric $g^{\mathbf{Gr}_c^+(D,d)} = c \cdot g^{\mathbf{Gr}^+(D,d)}$. Let $\bar{\mathbf{g}}^+$ denote the Gauss map, and define the rescaled Gauss map
$$\bar{\mathbf{g}}^+_c : M \xrightarrow{\bar{\mathbf{g}}^+} (\mathbf{Gr}^+(D,d), g^{\mathbf{Gr}^+(D,d)}) \xrightarrow{ \cdot c } (\mathbf{Gr}_c^+(D,d), g^{\mathbf{Gr}_c^+(D,d)}).$$
We consider the graph embedding $\iota \times \bar{\mathbf{g}}_c^+ : M \to (\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d),\, g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)}),$ and define $g_c = (\iota \times \bar{\mathbf{g}}^+_c)^*(g^{\mathbb{R}^D} \oplus g^{\mathbf{Gr}_c^+(D,d)})$. Throughout, we write $\mathbf{II}$ for the second fundamental form of $M \subset \mathbb{R}^D$ with the induced Euclidean metric and $\mathbf{II}_c$ for that of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ with the product metric. Likewise, we write $\mathrm{vol}_c(M)$ for the Riemannian volume of $(M,g_c)$ and $\mathrm{vol}(M)$ for the Euclidean volume of $M \subset \mathbb{R}^D$.
\begin{definition}\label{grassmannian_distance}
For $c>0$, we define a distance $d_c$ on $M$ by
\[
 d_{c}(q_1,q_2) = d_{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((q_1, \bar{\mathbf{g}}_c^+(q_1)),(q_2, \bar{\mathbf{g}}_c^+(q_2))\bigr)
 = \sqrt{\|q_1-q_2\|_{\mathbb{R}^D}^2
 + c\, d_{\mathbf{Gr}^+(D,d)}\bigl(\bar{\mathbf{g}}^+(q_1),\bar{\mathbf{g}}^+(q_2)\bigr)^2},
\]
where $\|\cdot\|_{\mathbb{R}^D}$ denotes the Euclidean norm and
$d_{\mathbf{Gr}^+(D,d)}$ is the Riemannian distance on the oriented Grassmannian
$\mathbf{Gr}^+(D,d)$.
\end{definition}

We first show that, by embedding $M$ into $\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, we can reduce the normal curvature of $M$ along directions where the original normal curvature is large. For $p \in M$ and a nonzero vector $v \in T_pM$, denote by $\kappa_c(v)$ and $\kappa(v)$ the (operator norms of the) normal curvatures of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$ at $(p,\bar{\mathbf{g}}_c^+(p))$ in the direction $(v, d\bar{\mathbf{g}}_c^+(v))$ and of $M \subset \mathbb{R}^D$ at $p$ in the direction $v$, respectively, where both geodesics are parametrized by arc length.

\begin{proof}[Proof of Theorem~\ref{principal_curvature}]
Denote by $\nabla^c$ and $\nu_c$ the Levi--Civita connection on $\mathbb{R}^D \times \mathbf{Gr}^+_c(D,d)$ and the normal bundle of $(M,g_c) \subset \mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)$, respectively. Suppose that there exists a unit vector $(w - cS(Q),Q)$ of $\nu_c$ at $p$ such that
\[
\|w\|_{2}^2 + c^2\|S(Q)\|_{2}^2 + c\|Q\|_{\operatorname{HS}}^2 = 1,
\]
as in Lemma~\ref{normal_bundle}, where $w \perp T_pM$. By Lemma~\ref{grass_connection},
\[
\nabla^c_{(v,d\bar{\mathbf{g}}_c^+(v))}(v,d\bar{\mathbf{g}}_c^+(v))
= \bigl(\nabla_{v}^{\mathbb{R}^D} v,\,
\nabla^{\mathbf{Gr}_c^+(D,d)}_{d\bar{\mathbf{g}}_c^+(v)} d\bar{\mathbf{g}}_c^+(v)\bigr)
= \bigl(\nabla_{v}^M v + \mathbf{II}(v,v),\, (\nabla_{v}^M \mathbf{II})(v,\cdot) + \mathbf{II}(\nabla_{v}^M v,\cdot)\bigr).
\]
Since
\[
d(\iota \times \bar{\mathbf{g}}_c^+)_p(\nabla_{v}^M v)
= \bigl(\nabla_{v}^M v,\, \mathbf{II}(\nabla_{v}^M v,\cdot)\bigr),
\]
the projections of $\nabla^c_{(v,d\bar{\mathbf{g}}_c^+(v))}(v,d\bar{\mathbf{g}}_c^+(v))$ and $\bigl(\mathbf{II}(v,v),(\nabla_{v}^M \mathbf{II})(v,\cdot)\bigr)$ onto $\nu_c$ coincide.
On the other hand,
\[
g^{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((w,0),(\mathbf{II}(v,v),(\nabla_{v}^M \mathbf{II})(v,\cdot))\bigr)
= \bigl\langle w,\mathbf{II}(v,v)\bigr\rangle_{\mathbb{R}^D},
\]
and since $S(Q) \perp T_pM$,
\[
g^{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}\bigl((-cS(Q),Q),(\mathbf{II}(v,v),(\nabla_{v}^M \mathbf{II})(v,\cdot))\bigr)
= c \bigl\langle Q,(\nabla_{v}^M \mathbf{II})(v,\cdot)\bigr\rangle_{\operatorname{HS}}.
\]
Therefore,
\begin{flalign*}
 |g^{\mathbb{R}^D \times \mathbf{Gr}_c^+(D,d)}((w-cS(Q),Q), (\mathbf{II}({v},{v}) ,(\nabla_{v}^M \mathbf{II})({v},\cdot)))| &= |\langle w, \mathbf{II}(v,v) \rangle_{\mathbb{R}^D} + c\langle Q, (\nabla_{v}^M \mathbf{II})({v},\cdot) \rangle_{\operatorname{HS}}|
 \\ &\leq (\kappa(v) \|w\|_2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}\|Q\|_{\operatorname{HS}})
 \\ &\leq \sqrt{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}\sqrt{\|w\|_2^2 + c\|Q\|_{\operatorname{HS}}^2}
 \\ &\leq \sqrt{\kappa(v)^2 + c \|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2},
\end{flalign*}
where we used $\|w\|_2^2 + c^2\|S(Q)\|_2^2 + c\|Q\|_{\operatorname{HS}}^2 = 1$ in the last inequality. Hence,
$$\kappa_c(v)
\leq \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^M\mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{\|(v,d\bar{\mathbf{g}}_c^+(v))\|_2^2}
= \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}.$$
By the volume assumption in Theorem~\ref{principal_curvature}, we obtain
\begin{flalign*}
\kappa_c(v)\,(\mathrm{vol}_c(M))^{1/d}
&\leq \frac{\sqrt{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}}{\sqrt{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}}\,
(\mathrm{vol}(M))^{1/d}.
\end{flalign*}
We compute
\begin{flalign*}
\frac{d}{dc}\left(\frac{\kappa(v)^2 + c\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2}{1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}\right)
&= \frac{\|(\nabla_v^M \mathbf{II})(v, \cdot)\|_{\operatorname{HS}}^2 - \kappa(v)^2\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2}
{(1 + c\|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}^2)^2} < 0,
\end{flalign*}
by the assumption. This completes the proof.
\end{proof}

\begin{remark}
In codimension one, if $\{e_i\}_{i=1}^{D-1}$ is an orthonormal eigenbasis of $T_qM$ with principal curvatures
$\lambda_1(q),\dots,\lambda_{D-1}(q)$, then in this basis, the metric $g_c$ satisfies
\[
 [(g_c)_q] = I + c H_q,\qquad
 H_q = \operatorname{diag}(\lambda_1(q)^2,\dots,\lambda_{D-1}(q)^2),
\]
so that $\operatorname{tr}H_q = \sum_{i=1}^{D-1} \lambda_i(q)^2$. Therefore, for small $c>0$, we obtain
\[
 d\mathrm{vol}_c
 = \sqrt{\det(I + cH_q)}\,d\mathrm{vol}
 = \Bigl(1 + \frac{c}{2}\,\sum_{i=1}^{D-1} \lambda_i(q)^2 + O(c^2)\Bigr)\,d\mathrm{vol}.
\]
In particular, regions where many principal curvatures are simultaneously large
contribute most to the first-order increase of $\mathrm{vol}_c(M)$. On the other hand, for any $p\in M$ and any unit vector $v \in T_pM$, the volume condition in Theorem~\ref{principal_curvature} states
\[
 \mathrm{vol}_c(M) \;\leq\; \bigl(1 + c \,\|\mathbf{II}_p(v,\cdot)\|_{\operatorname{HS}}^2\bigr)^{(D-1)/2} \mathrm{vol}(M) = \Bigl(1 + \frac{c\,(D-1)}{2}\|\mathbf{II}_p(v,\cdot)\|_{\operatorname{HS}}^2 + O(c^2)\Bigr)\,\mathrm{vol}(M).
\]
Thus, for small $c$, one may interpret this condition as requiring that at points $p$ and directions $v$ the normal curvature is significantly larger than the average curvature of $M$. In higher codimension, the analogous condition may require such a largeness assumption for all sections of the normal bundle.
\end{remark}

The condition
$\kappa(v) > \frac{\|(\nabla_v^M \mathbf{II})(v,\cdot) \|_{\operatorname{HS}}}{\|\mathbf{II} (v,\cdot)\|_{\operatorname{HS}}}$ in Theorem~\ref{principal_curvature}
indicates that the normal curvature of $M$ in the direction $v$ dominates the logarithmic variation of the second fundamental form in the same direction.
\begin{theorem}\label{log_II_derivative_bound}
 Let $M \subset \mathbb{R}^D$ be a smooth $d$-dimensional submanifold. For a unit-speed geodesic $\gamma \subset M$ such that $\gamma'(t)=v(t)$, suppose that the tensor $\mathbf{II}_{\gamma(t)}(v(t),\cdot)$ does not vanish along $\gamma$. Then
\[
\left|
\frac{d}{dt}\log \|\mathbf{II}(v,\cdot)\|_{\operatorname{HS}}
\right|
\le
\frac{
\bigl\|(\nabla_v^M \mathbf{II})(v,\cdot)\bigr\|_{\operatorname{HS}}
}{
\bigl\|\mathbf{II}(v,\cdot)\bigr\|_{\operatorname{HS}}
}.
\]
\end{theorem}
The proof is given in Appendix~\ref{proof_log_II}.

For each $x \in M$, fix an ordered orthonormal basis for $T_xM$ compatible with the given
orientation on $M$, and let $B_x$ be the matrix whose columns are this basis. Let
$\widehat{T_xM}$ be the estimated tangent space at $x$, and choose an ordered orthonormal basis for
$\widehat{T_xM}$ so that it induces the same orientation as $T_xM$. Denote the corresponding basis matrix by
$\widehat{B_x}$. To establish that such orientations can be chosen consistently across tangent spaces, we prove the following theorem.
\begin{theorem}\label{orientation}
 Let $M \subset \mathbb{R}^D$ be a smooth closed connected orientable $d$-dimensional submanifold with $\mathsf{rch}_{\mathbb{R}^D}(M) = \tau >0$. If two points $p,q \in M$ satisfy $\|p-q\|_{\mathbb{R}^D} < \frac{\tau}{2}$, then
 $$ \det (B_p^\top B_q ) > 0.$$
\end{theorem}
We postpone the proof to Appendix~\ref{orientation_proof}. If two points $x_1$ and $x_2$ in $\mathbf{X}$ are adjacent and the basis matrix $\widehat{B_{x_1}}$ is oriented, then we orient $\widehat{B_{x_2}}$ by flipping the sign of its last row whenever $\det (\widehat{B_{x_1}}^\top \widehat{B_{x_2}})<0$. This yields a consistent orientation of the estimated tangent spaces over $\mathbf{X}$. We choose $c > 0$ so that the diameters of $M$ and $\mathbf{Gr}_c^+(D,d)$ coincide.
Approximating the diameter of $M$ by $\widehat{\mathrm{diam}}(M) = \mathrm{diam}(\mathbf{Y})$ and using $\mathrm{diam}(\mathbf{Gr}^+(D,d)) = \max\bigl(\pi, \frac{\pi}{2}\,\sqrt{\min(d,D-d)}\bigr)$ from \cite[Theorem 12.6]{kozlov2000geometry}, we set
\begin{equation*}\label{parameter}
c = \frac{\mathrm{diam}(\mathbf{Y})^2}{\max(\pi, \frac{\pi}{2}\sqrt{\min(d,D-d)})^2}.
\end{equation*}
We then compute the distance matrix \(\mathbf{D}\) with entries
\begin{equation*}\label{grass_distance}
\bigl(\mathbf{D}\bigr)_{ij} = \sqrt{\|y_i - y_j\|_{\mathbb{R}^D}^2 + c \, d_{\mathbf{Gr}^+(D,d)}(\widehat{T_{y_i}M},\widehat{T_{y_j}M})^2}
\end{equation*}
for $1\leq i,j \leq |\mathbf{Y}|$ and perform Vietoris--Rips persistent homology with respect to \(\mathbf{D}\).
We compute persistent homology using the Ripser library~\cite{bauer2021ripser}, with coefficients in $\mathbb{Z}/2$.

Question:
Let $M\subset \mathbb{R}^D$ be a smooth closed orientable $d$-dimensional submanifold, let $p\in M$, and let $v\in T_pM$ be a unit tangent vector. For $c>0$, let $\bar{\mathbf g}^+:M\to \mathbf{Gr}^+(D,d)$ be the Gauss map to the oriented Grassmannian of oriented $d$-planes in $\mathbb{R}^D$, equip $\mathbf{Gr}^+(D,d)$ with the rescaled metric $c\,g^{\mathbf{Gr}^+(D,d)}$, and define the pullback metric
$$
g_c=(\iota\times \bar{\mathbf g}_c^+)^*\bigl(g^{\mathbb{R}^D}\oplus g^{\mathbf{Gr}_c^+(D,d)}\bigr)
$$
on $M$, where $\iota:M\hookrightarrow \mathbb{R}^D$ is the embedding. Let $\mathrm{vol}(M)$ be the Euclidean volume of $M$, let $\mathrm{vol}_c(M)$ be the Riemannian volume of $(M,g_c)$, let $\mathbf{II}$ be the second fundamental form of $M\subset \mathbb{R}^D$, and write $\|\cdot\|_{\mathrm{HS}}$ for the Hilbert--Schmidt norm. Let $\kappa(v)$ denote the normal curvature of $M$ at $p$ in the direction $v$, and let $\kappa_c(v)$ denote the corresponding normal curvature for the graph embedding $(M,g_c)\subset \mathbb{R}^D\times \mathbf{Gr}_c^+(D,d)$. Assume
$$
\mathrm{vol}_c(M)\le \bigl(1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2\bigr)^{d/2}\,\mathrm{vol}(M),
\qquad
\kappa(v)>\frac{\|(\nabla_v^M\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}}{\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}}.
$$
Under these assumptions, which quantitative estimate holds?

For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}\,\mathrm{vol}_c(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}(M)^{1/d},
$$
and the middle expression is strictly decreasing as $c$ grows.

Choices:
B.
For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{\sqrt{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}}\,\mathrm{vol}_c(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}(M)^{1/d},
$$
and the middle expression is strictly decreasing as $c$ grows.
C.
For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}(M)^{1/d}.
$$
D.
There exists $c_0>0$ such that for every $c\ge c_0$,
$$
\kappa_c(v)\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}\,\mathrm{vol}_c(M)^{1/d}
\;\le\;
\kappa(v)\,\mathrm{vol}(M)^{1/d},
$$
and the middle expression is nonincreasing as $c$ grows.
E.
For every $c>0$,
$$
\kappa_c(v)\,\mathrm{vol}(M)^{1/d}
\;\le\;
\frac{\sqrt{\kappa(v)^2+c\,\|(\nabla_v^{M}\mathbf{II})(v,\cdot)\|_{\mathrm{HS}}^2}}{1+c\,\|\mathbf{II}(v,\cdot)\|_{\mathrm{HS}}^2}\,\mathrm{vol}(M)^{1/d}
\;<\;
\kappa(v)\,\mathrm{vol}_c(M)^{1/d},
$$
and the middle expression is strictly decreasing as $c$ grows.

speed-squared denominator from reparameterization

explicit intermediate upper bound and monotonicity statement

for every c and strict decrease replaced by eventual/nonstrict version

volume factor attached to the wrong side of the comparison

The stem does not state the conclusion or uniquely telegraph the correct option. It gives hypotheses and definitions only; the exact inequality, denominator, volume placement, and monotonicity claim must be chosen from the options.

This is essentially a theorem-recall item: the stem lists the hypotheses and asks which quantitative estimate holds, and the correct choice is the theorem-style conclusion. That makes it close to a direct restatement rather than a fresh inferential task.

There is some reasoning pressure because the options differ in subtle but meaningful ways: denominator form, strict vs nonstrict monotonicity, universal vs eventual quantification, and placement of volume factors. However, the item mainly rewards recognizing the exact stated result rather than generating a conclusion from first principles.

The distractors are strong and mathematically plausible. They target realistic failure modes: confusing a linear factor with a square root, selecting a weaker true statement, relaxing quantifiers/monotonicity, and misplacing the volume term.

A mathematically careful MCQ with high-quality distractors and little answer leakage, but it is primarily a theorem-recognition/restatement question rather than a genuinely generative reasoning task.


For a positive integer $n$, let $[n]:=\{1,2,\ldots,n\}$. A {\em set composition} $\phi$ of $[n]$,
denoted by $\phi\vDash[n]$, is a list of mutually disjoint nonempty subsets
$\phi_1/\phi_2/\ldots/\phi_{\ell}$ of $[n]$ whose union is $[n]$.
Each $\phi_i$ is called a {\em block} of $\phi$ and the number of blocks of $\phi$ is denoted
by $\ell(\phi)$.
A set composition $\phi$ is said to be {\em odd} if each block of $\phi$ consists of odd number
of elements. For instance, $\phi=7/148/9/6/235$ is an odd set composition of $[9]$ with $\ell(\phi)=5$.
Let
$$
O(n,k):=|\{\phi\vDash[n]: \phi\text{ odd}, \ell(\phi)=k\}|.
$$
Note that $O(n,k)=0$ if $n$ and $k$ have different parity. By the compositional formula for
the exponential generating functions (see~\cite[Proposition~5.1.3]{St}), we have
\begin{align*}
&\quad\sum_{n,k} O(n,k)t^k\frac{x^n}{n!}=\frac{t\sinh(x)}{1-t\sinh(x)}\\
&=tx+2t^2\frac{x^2}{2!}+(t+6t^3)\frac{x^3}{3!}+(8t^2+24t^4)\frac{x^3}{3!}+
(t+60t^3+120t^5)\frac{x^5}{5!}+\cdots.
\end{align*}

The {\em Catalan numbers}
$$
\left(C_n=\frac{1}{n+1}{2n\choose n}\right)_{n\geq0}=(1,1,2,5,14,42,132,\ldots)
$$
and the {\em tangent numbers} $\{T_{2n+1}\}_{n\geq0}$ defined by the Taylor expansion
$$
\tan(x)=\sum_{n\geq0}T_{2n+1}\frac{x^{2n+1}}{(2n+1)!}=
x+2\frac{x^3}{3!}+16\frac{x^5}{5!}+272\frac{x^7}{7!}+7936\frac{x^9}{9!}+\cdots
$$
are well-known combinatorial sequences in enumerative combinatorics.

The following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,
was found by Aliniaeifard and Li~\cite{AL} in their study of non-commutative peak algebra
and later proved by Zhao, Lin and Zang~\cite{ZLZ} using generating functions.

The following identity, which connects Catalan numbers with tangent numbers via $O(n,k)$,
was found by Aliniaeifard and Li~\cite{AL} in their study of non-commutative peak algebra
and later proved by Zhao, Lin and Zang~\cite{ZLZ} using generating functions.

For instance, when $n=2$ identity~\eqref{eq:Cat-Tan} reads
$$
1\cdot2^4C_0-60\cdot2^2C_1+120\cdot2^0C_2=16.
$$
Although Catalan numbers and tangent numbers have been extensively studied in combinatorics
(see~\cite[Exercise~6.19]{St} and~\cite{St10}), there seems to be no combinatorial identity relating
them other than~\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\cite{AL},
combining~\cite[Corollary~11.5]{AL} and
Theorem~\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers
\begin{equation}\label{eq:Genocchi}
\sum_{k=0}^{n-1}(-1)^{k}{2n\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.
\end{equation}
A generating function proof of Theorem~\ref{thm:Cat-Tan} and a combinatorial proof of~\eqref{eq:Genocchi},
together with its surprising arithmetic applications, were given in~\cite{ZLZ}.
However, a combinatorial proof of Theorem~\ref{thm:Cat-Tan} remains elusive.

The main objective of this paper is to provide a combinatorial involution proof of
Theorem~\ref{thm:Cat-Tan}.
In the course, we find a new combinatorial identity similar to~\eqref{eq:Genocchi}:
\begin{equation}\label{eq:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.
\end{equation}
This identity is also reminiscent of the one proved by Andrews and Gessel~\cite{AG}:
\begin{equation}\label{eq:TanAG}
T_{2n+1}+\sum_{k=1}^{n}(-1)^{k}{2n+1\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.
\end{equation}

Let $\LB_{2n+1}$ be the set of all labeled binary trees on $[2n+1]$ and let $\IB_{2n+1}$ be the set of
all complete increasing binary trees on $[2n+1]$. It is clear that $\IB_{2n+1}\subseteq\LB_{2n+1}$.
For a labeled binary tree $T\in\LB_{2n+1}$, let $\h(T)$ be half the number of edges of $T$.
The following lemma shows that the left-hand side of~\eqref{eq:Cat-Tan} is
a signed counting of labeled binary trees on $[2n+1]$.
\begin{lemma}\label{lem:sign}
For $n\geq0$, we have
\begin{equation}\label{eq:sign}
\sum_{T\in\LB_{2n+1}}(-1)^{\h(T)}=\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k.
\end{equation}
\end{lemma}
\begin{proof}
Each labeled binary tree in $\LB_{2n+1}$ with $2k$ edges can be constructed by
\begin{itemize}
\item choosing a complete binary tree $T$ with $2k+1$ nodes,
\item choosing a set composition $\phi$ of $[2n+1]$ with $2k+1$ blocks,
\item forming an odd unimodal permutation using letters in $\phi_i$ for the label of the $i$-th node
(under the in-order traversal) of $T$ for each $1\leq i\leq 2k+1$.
\end{itemize}
There are $C_k$ choices for $T$, $O(2n+1,2k+1)$ choices for $\phi$, and $2^{2n+1-(2k+1)}$ ways to form
the labels of $T$ from $\phi$, whence~\eqref{eq:sign} follows.
\end{proof}

A permutation $\pi=\pi_1\pi_2\cdots\pi_n$ of $[n]$ is {\em down-up} (or {\em alternating}) if
$$
\pi_1>\pi_2<\pi_3>\pi_4<\cdots.
$$
If the above inequalities are reversed, then $\pi$ is said to be {\em up-down}.
Let $\A_n$ be the set of all down-up permutations of $[n]$.
A result attributed to Andr\'e~\cite{An} asserts that
\begin{equation}\label{eul:egf}
1+\sum_{n\geq1}|\A_n|\frac{x^n}{n!}=\sec(x)+\tan(x).
\end{equation}
Thus, $T_{2n+1}=|\A_{2n+1}|$.
It is a classical result (see~\cite{St10}) that $T_{2n+1}(q)$ has the combinatorial interpretation
\begin{equation}\label{int:qtan}
T_{2n+1}(q)=\sum_{\pi\in\A_{2n+1}}q^{\inv(\pi)},
\end{equation}
where $\inv(\pi):=|\{(i,j)\in[n]^2: i<j,\pi_i>\pi_j\}|$ is the {\em inversion number} of a permutation $\pi$.
The following is our first $q$-analog of~\eqref{eq:Tan2}.
\begin{theorem}\label{thm:q-analog1}
For $n\geq0$, we have
\begin{equation}\label{q1:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k}\widetilde T_{2k+1}(q)=(-1)^nT_{2n+1}(q),
\end{equation}
where $\widetilde T_{1}(q)=1$ and
\begin{equation}
\widetilde T_{2k+1}(q)=\sum_{i=0}^{k-1} {2k\brack 2i+1} T_{2i + 1}(q) T_{2k - 2i - 1}(q) \quad (k \geq 1).
\end{equation}
\end{theorem}

For a labeled tree $T\in\LB_{2n+1}$ with $2k+1$ nodes, consider the word concatenation
$w(T)=\alpha_1\alpha_2\cdots\alpha_{2k+1}$, where $\alpha_i$ is the labeling of the unimodal
permutation of the $i$-th node (under the in-order traversal) of $T$. For instance,
if $T$ is the first tree in Fig.~\ref{ex:lbt}, then $w(T)=467381952$.
It is clear that $w(T)$ is a permutation of $[2n+1]$. The {\em inversion number of $T$},
denoted by $\inv(T)$, is thus defined as
$$
\inv(T):=\inv(w(T)).
$$
The following interpretation of signed $q$-tangent numbers is a consequence of
Lemma~\ref{lem:invo}.
\begin{lemma}\label{lem:newint}
For $n\geq0$, we have
\begin{equation}\label{new:tan}
\sum_{T\in\LB_{2n+1}}(-1)^{\h(T)}q^{\inv(T)}=(-1)^nT_{2n+1}(q).
\end{equation}
\end{lemma}
\begin{proof}It is plain to check that our involution $\kappa$ has the following feature:
$$
\inv(T)=\inv(\kappa(T)) \quad\text{for any $T\in\LB_{2n+1}$},
$$
that is, it preserves the inversion numbers of labeled trees. The result then follows from Lemma~\ref{lem:invo}, interpretation~\eqref{int:qtan} and the known fact that $T\mapsto w(T)$ establishes a one-to-one correspondence between $\IB_{2n+1}$ and $\A_{2n+1}$ preserving inversion numbers.
\end{proof}
 We are ready for the proof of Theorem~\ref{thm:q-analog1}.
\begin{proof}[{\bf Proof of Theorem~\ref{thm:q-analog1}}]
Recall the well-known combinatorial interpretation for the $q$-binomial coefficients:
\begin{equation}\label{eq:qmul}
{n\brack k}=\sum_{({\mathcal A}, {\mathcal B})}q^{\inv({\mathcal A}, {\mathcal B})},
\end{equation}
summed over all set compositions $({\mathcal A}, {\mathcal B})$ of $[n]$ such that $|{\mathcal A}|=k$, and
$$\inv({\mathcal A}, {\mathcal B}):=|\{(a,b)\in{\mathcal A}\times{\mathcal B}: a>b\}|.$$
A labeled binary tree $T\in\LB_{2n+1}$ can be decomposed as $(T_g,r,T_d)$, where $r$ is the root and $T_g$ and $T_d$ (possibly empty) are left branch and right branch of $r$, respectively. It follows from this decomposition, Lemma~\ref{lem:unimodal}, the interpretation~\eqref{new:tan} of $q$-tangent numbers and the interpretation~\eqref{eq:qmul} of $q$-binomial coefficients that
$$
(-1)^nT_{2n+1}(q)=(-q;q)_{2n}+\sum_{k=1}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k}\sum_{i=0}^{k-1}{2k\brack 2i+1}T_{2i+1}(q)T_{2k-2i-1}(q),
$$
where $(-q;q)_{2n}$ counts single-node labeled binary trees on $[2n+1]$ by inversion numbers.
This completes the proof of the theorem.
\end{proof}

In view of~\eqref{eul:egf}, the number $|\A_{2n}|:=S_{2n}$ is known as a {\em secant number}.
Consider the {\em$q$-secant number }
$$
S_{2n}(q):=\sum_{\pi\in\A'_{2n}}q^{\inv(\pi)},
$$
where $\A'_n$ denotes the set of all up-down permutations of $[n]$. For convenience, set $S_0(q)=1$. It was known (see~\cite{FH10}) that
$$
\sum_{n\geq0}S_{2n}(q)\frac{x^{2n}}{(q;q)_{2n}}=\frac{1}{\cos_q(x)}.
 $$
 The following is our second $q$-analog of~\eqref{eq:Tan2}.
\begin{theorem}\label{thm:q-analog2}
For $n\geq0$, we have
\begin{equation}\label{q2:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}(-q;q)_{2n-2k} T_{2k+1}(q)=(-1)^n\widehat T_{2n+1}(q),
\end{equation}
where
\begin{equation}
(-1)^n\widehat T_{2n+1}(q)=\sum_{k=0}^{n} (-1)^{k} {2n+1\brack 2k}q^{2k} S_{2k}(q).
\end{equation}
Equivalently,
$$
\sum_{k=0}^{n}(-1)^{k}{2n+1\brack 2k}\bigl((-q;q)_{2n-2k} T_{2k+1}(q)-q^{2k} S_{2k}(q)\bigr)=0.
$$
\end{theorem}

\begin{theorem}\label{thm:q-sec:tan}
For $n\geq0$, we have
\begin{equation}\label{eq:q-sec:tan}
\sum_{k=0}^n(-1)^k{2n+1\brack 2k}S_{2k}(q)=(-1)^nT_{2n+1}(q).
\end{equation}
\end{theorem}

\begin{equation}\label{eq:Cat-Tan}
\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}.
\end{equation}

\begin{theorem}\label{thm:Cat-Tan}
For $n\geq0$, we have
\begin{equation}\label{eq:Cat-Tan}
\sum_{k=0}^n (-1)^kO(2n+1,2k+1)2^{2n-2k}C_k=(-1)^nT_{2n+1}.
\end{equation}
\end{theorem}

For instance, when $n=2$ identity~\eqref{eq:Cat-Tan} reads
$$
1\cdot2^4C_0-60\cdot2^2C_1+120\cdot2^0C_2=16.
$$
Although Catalan numbers and tangent numbers have been extensively studied in combinatorics
(see~\cite[Exercise~6.19]{St} and~\cite{St10}), there seems to be no combinatorial identity relating
them other than~\eqref{eq:Cat-Tan}. As observed by Aliniaeifard and Li~\cite{AL},
combining~\cite[Corollary~11.5]{AL} and
Theorem~\ref{thm:Cat-Tan} results in the following intriguing identity for tangent numbers
\begin{equation}\label{eq:Genocchi}
\sum_{k=0}^{n-1}(-1)^{k}{2n\choose 2k+1}2^{2n-2k}T_{2k+1}=2^{2n+1}.
\end{equation}
A generating function proof of Theorem~\ref{thm:Cat-Tan} and a combinatorial proof of~\eqref{eq:Genocchi},
together with its surprising arithmetic applications, were given in~\cite{ZLZ}.
However, a combinatorial proof of Theorem~\ref{thm:Cat-Tan} remains elusive.

The main objective of this paper is to provide a combinatorial involution proof of
Theorem~\ref{thm:Cat-Tan}.
In the course, we find a new combinatorial identity similar to~\eqref{eq:Genocchi}:
\begin{equation}\label{eq:Tan2}
\sum_{k=0}^{n}(-1)^{k}{2n+1\choose 2k}2^{2n-2k}T_{2k+1}=(-1)^nT_{2n+1}.
\end{equation}
This identity is also reminiscent of the one proved by Andrews and Gessel~\cite{AG}:
\begin{equation}\label{eq:TanAG}
T_{2n+1}+\sum_{k=1}^{n}(-1)^{k}{2n+1\choose 2k}2^{2k-1}T_{2n-2k+1}=(-1)^n2^{2n}.
\end{equation}

The rest of this paper is organized as follows. In Section~\ref{sec:2}, we introduce the combinatorial
structure of complete binary trees labeled by odd unimodal permutations and then construct
a sign-reversing involution on them in Section~\ref{sec:3} to prove Theorem~\ref{thm:Cat-Tan}
combinatorially. In Section~\ref{sec:4}, we prove two different $q$-analogs of~\eqref{eq:Tan2}
using our involution on labeled binary trees and some other involutions on permutation pairs.

To prove Theorem~\ref{thm:Cat-Tan} combinatorially, the paper (i) introduces in Section~\ref{sec:2} “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in Section~\ref{sec:3} to prove Theorem~\ref{thm:Cat-Tan} combinatorially.”

To prove the main theorem combinatorially, the paper (i) introduces in the next section “the combinatorial structure of complete binary trees labeled by odd unimodal permutations,” and then (ii) “construct[s] a sign-reversing involution on them in a later section to prove the main theorem combinatorially.”

Question:
For a positive integer \(m\), let \([m]=\{1,2,\ldots,m\}\). A set composition \(\phi\vDash [m]\) is an ordered list of mutually disjoint nonempty subsets whose union is \([m]\); it is called odd if every block has odd cardinality. Let \(O(m,r)\) denote the number of odd set compositions of \([m]\) with exactly \(r\) blocks. Also let the Catalan numbers be \(C_k=\frac{1}{k+1}\binom{2k}{k}\), and let the tangent numbers \(T_{2j+1}\) be defined by
\[
\tan x=\sum_{j\ge 0} T_{2j+1}\frac{x^{2j+1}}{(2j+1)!}.
\]
For every integer \(n\ge 0\), which explicit identity holds?

\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = (-1)^n T_{2n+1}.
\]

Choices:
B.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = T_{2n+1}.
\]
C.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_k = \pm T_{2n+1}.
\]
D.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n+1-2k}C_k = (-1)^n T_{2n+1}.
\]
E.
\[
\sum_{k=0}^{n}(-1)^k\,O(2n+1,2k+1)\,2^{2n-2k}C_{2k+1} = (-1)^n T_{2n+1}.
\]

global sign from sign-reversing involution

exact parity-dependent sign factor (-1)^n

factor 2^{2n-2k} from odd unimodal labelings

Catalan index tied to complete binary trees with 2k+1 nodes

The stem defines the objects and asks which identity holds, but it does not reveal the exact sign, power of 2, or Catalan index. There is no explicit or trivial leakage of choice A.

This is close to theorem recall: the task is essentially to recognize the exact stated identity among small perturbations. It is not a pure verbatim restatement because the options compete on sign and indexing, but it is only mildly non-tautological.

Distinguishing A from B, D, and E requires some reasoning or checking small cases, but not substantial generative work. The presence of option C, a weaker true statement implied by A, further lowers the pressure to identify the full exact formula.

B, D, and E are plausible and target common formula-confusion errors. However, C is not a clean distractor: it is a weaker true statement, which introduces ambiguity and weakens the single-best-answer structure.

Moderate-quality MCQ: no answer leakage and some plausible distractors, but it mainly tests recognition of a known identity, and the weaker-true option makes the item less clean and less discriminating.


\begin{corollary}\label{CorollaryOnSheaves}
    Let $F$ be a torsion-free $H$-slope-semistable sheaf on $S_5$, then the numerical Chern characters of $F$ satisfy \eqref{Ch1Ch2Inequal}.
\end{corollary}

\begin{corollary}\label{MainCorollary}
    Let $F$ be a torsion-free $H$-slope-semistable sheaf on $X$, then the numerical Chern characters of $F$ satisfy \eqref{InequalityOnThreefolds}.
\end{corollary}

\begin{theorem}\label{MainTheorem}
     Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic $3$-fold and $H: = c_1(\mathcal{O}_X(1))$, $F$ be a $\nu_{\alpha,0,H}$-tilt-stable object in $\Coh^{0,H}(X)$ for some $\alpha >0$. Suppose $\mu_H(F)\in [-1,1]$. then
    \begin{equation}\label{InequalityOnThreefolds}
        \xi_H(F)\leq
    \begin{cases}
    \frac{-1}{2}|\mu_H(F)| & \text{when } 0\leq |\mu_H(F)| \leq \frac{1}{4}; \\[0ex]
    \frac{1}{2}|\mu_H(F)| - \frac{1}{4} & \text{when } |\mu_H(F)|\in[\frac{1}{4},\frac{5}{13}]\cup[\frac{8}{13},\frac{3}{4}]; \\[0ex]
    \frac{-3}{20}|\mu_H(F)|  & \text{when } \frac{5}{13} \leq |\mu_H(F)|\leq \frac{6}{13}; \\[0ex]
    \frac{1}{2}|\mu_H(F)|-\frac{3}{10} & \text{when } \frac{6}{13}\leq |\mu_H(F)| \leq \frac{7}{13}; \\[0ex]
    \frac{23}{20}|\mu_H(F)| - \frac{13}{20} & \text{when } \frac{7}{13}\leq |\mu_H(F)| \leq \frac{8}{13}; \\[0ex]
    \frac{3}{2}|\mu_H(F)| - 1 & \text{when } \frac{3}{4}\leq |\mu_H(F)| \leq 1.
\end{cases}
    \end{equation}
\end{theorem}

\begin{proposition}\label{ChInequalityOnSurface}
    Let $S_5\subset \mathbb{P}^3$ be a smooth irreducible quintic surface, $H = c_1(\mathcal{O}_{S_5}(1))$, and let $F$ be an object in $\Db(S_5)$ such that $\mu_H(F)\in (0,1)$. Suppose $F$ is $\nu_{\alpha,0,H}$-tilt-stable (or $\nu_{\alpha',1,H}$-tilt-stable) for some $\alpha > 0$ (resp. or $\alpha' > \frac{1}{2}$), then
    \begin{equation}\label{Ch1Ch2Inequal}
        \xi_H(F)\leq
    \begin{cases}
    \frac{17}{26}\mu_H(F) - \frac{2}{13} & \text{when } 0 < \mu_H(F) \leq \frac{7}{46}; \\[0ex]
    -\frac{5}{14}\mu_H(F)  & \text{when } \frac{7}{46} \leq  \mu_H(F)\leq \frac{7}{20}; \\[0ex]
    \frac{1}{2}\mu_H(F)-\frac{3}{10} & \text{when } \frac{7}{20}\leq \mu_H(F) \leq \frac{13}{20}; \\[0ex]
    \frac{19}{14}\mu_H(F) - \frac{6}{7} & \text{when } \frac{13}{20}\leq \mu_H(F)< \frac{39}{46}; \\[0ex]
    \frac{9}{26}\mu_H(F) & \text{when } \frac{39}{46}\leq \mu_H(F)< 1.
\end{cases}
    \end{equation}
\end{proposition}

The classical result of Bogomolov and Gieseker provides a fundamental inequality for stable sheaves on smooth projective varieties.

\maketitle

The Bogomolov--Gieseker (BG) inequality plays an important role in the study of stability conditions and moduli spaces of sheaves. It is a natural question to seek refinements of the BG inequality for special varieties; see for instance \cite[Question 5.1]{BM23}.

Motivated by Gepner type stability conditions on graded matrix factorizations, Toda proposed in \cite{Toda2017GepnerPoint} a stronger inequality on quintic threefolds, and proved it for $\rk(E) = 2$ case.
\begin{theorem}[{\cite[Conjecture $1.1$]{Toda2017GepnerPoint}}]\label{TodaConj0}
 Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\mathcal{O}_X(1))$. Then for any torsion-free $H$-slope-stable sheaf $E$ on $X$ with $\ch{1}(E)/\rk (E) = -H/2$, we have
 \begin{equation}\label{TodaConj1}
 \frac{H\cdot \Delta(E)}{\rk(E)^2} > 1.5139\cdots,
 \end{equation}
 where the right-hand side of is an irrational real number lying in $\mathbb{Q}(e^{2\pi \sqrt{-1}/5})$. Equivalently,
 \begin{equation}\label{TodaConj2}
 \frac{H\ch{2}(E)}{H^3\rk(E)} < -0.02639\cdots
 \end{equation}
\end{theorem}

Along the way, we also obtain a stronger Bogomolov--Gieseker type inequality for quintic surfaces.
\begin{theorem}[See \Cref{ChInequalityOnSurface} and \Cref{CorollaryOnSheaves}]
 Let $S_5\subset \mathbb{P}^3_{\mathbb{C}}$ be a smooth quintic surface and $H\coloneqq c_1(\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\Coh(X)$ with
 \begin{equation*}
 \mu_H(F)\in (0,1).
 \end{equation*}
 Then the following piecewise inequality holds:
 \begin{equation}\notag
 \xi_H(F)\leq
 \begin{cases}
 \frac{17}{26}\mu_H(F) - \frac{2}{13} & \text{when } 0 < \mu_H(F) \leq \frac{7}{46}; \\[0ex]
 -\frac{5}{14}\mu_H(F) & \text{when } \frac{7}{46} \leq \mu_H(F)\leq \frac{7}{20}; \\[0ex]
 \frac{1}{2}\mu_H(F)-\frac{3}{10} & \text{when } \frac{7}{20}\leq \mu_H(F) \leq \frac{13}{20}; \\[0ex]
 \frac{19}{14}\mu_H(F) - \frac{6}{7} & \text{when } \frac{13}{20}\leq \mu_H(F)< \frac{39}{46}; \\[0ex]
 \frac{9}{26}\mu_H(F) & \text{when } \frac{39}{46}\leq \mu_H(F)< 1.
\end{cases}
 \end{equation}
\end{theorem}

The classical BG inequality extends to tilt-semistable objects.
\begin{theorem}[{\cite[Theorem 7.3.1]{bayer_bridgeland_2013}}, {\cite[Proposition 2.21]{piyaratne_moduli_2019}}]
 Let $X$ be a smooth projective variety, and $H\in \mathrm{NS}(X)_{\mathbb{R}}$ an ample class. If $E$ is $\nu_{\alpha,\beta,H}$-tilt-semistable for some $\alpha > \frac{1}{2}\beta^2$, then $\overline{\Delta}_H(E) \geq 0$.
\end{theorem}

\subsection{Useful Lemmas}
Let $X$ be a smooth projective variety of dimension $n$ and $H\in \text{NS}(X)_{\mathbb{R}}$ be a real ample divisor class. For $E\in \Db(X)$, define
\begin{equation*}
 \overline{v}_H(E) \coloneqq (H^n\ch{0}(E),H^{n-1}\ch{1}(E),H^{n-2}\ch{2}(E)),
\end{equation*}
and, when $H^n\ch{0}(E) \neq 0$,
\begin{equation*}
 \text{and } p_H(E) \coloneqq \left(\frac{H^{n-1}\ch{1}(E)}{H^n\ch{0}(E)}, \frac{H^{n-2}\ch{2}(E)}{H^n\ch{0}(E)}\right) = (\mu_H(E),\xi_H(E)).
\end{equation*}
\begin{remark}
 For a smooth hypersurface $Y\in |mH|$ for $m > 0$, and for $0\leq i\leq n - 1$, we have
 \begin{equation*}
 mH^{n-i}\ch{i}(F) = H^{n-i-1}_Y\ch{i}(F|_Y).
 \end{equation*}
 Hence, $\mu_H$ and $\xi_H$ are invariant under restriction to hypersurfaces. In particular, we have
 \begin{equation*}
 p_H(F) = p_{H_Y} (F|_Y).
 \end{equation*}

\begin{lemma}\label{lem:convexfunction}
 Let $(X,H)$ be a polarized smooth variety such that every $H$-slope-semistable torsion-free sheaf $E$ satisfies
 \begin{align*}
 \frac{h^0(E)}{\rk(E)}\leq h(\mu_H(E))
 \end{align*}
 for some function $h$. Then for every torsion-free sheaf $E$, we have
 \begin{align*}
 \frac{h^0(E)}{\rk(E)} \leq \overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}(\mu_H(E)).
 \end{align*}
\end{lemma}
\begin{proof}
 Let
 \begin{equation*}
 0=E_0\hookrightarrow E_1\hookrightarrow\cdots\hookrightarrow E_n = E,
 \end{equation*}
 be the Harder--Narasimhan filtration of $E$.
 For each quotient $E_i/E_{i-1}$, semi-stability and our assumption yield
 \begin{equation*}
 \frac{h^0(E_i/E_{i-1})}{\rk(E_i/E_{i-1})} \leq h(\mu_H(E_i/E_{i-1})).
 \end{equation*}
 Using these inequalities and additivity of rank and degree, we have
 \begin{align*}
 \frac{h^0(E)}{\rk(E)}&\leq \sum\limits_{i=1}^n\frac{1}{\rk(E)}h^0(E_i/E_{i-1}) = \sum\limits_{i=1}^n\frac{\rk(E_i/E_{i-1})}{\rk(E)}\frac{h^0(E_i/E_{i-1})}{\rk(E_i/E_{i-1})} \\
 &\leq \sum\limits_{i=1}^n\frac{\rk(E_i/E_{i-1})}{\rk(E)}h(\mu_H(E_i/E_{i-1})) \\
 &\leq \sum\limits_{i=1}^n\frac{\rk(E_i/E_{i-1})}{\rk(E)}\overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}(\mu_H(E_i/E_{i-1})) \\
 &\leq \overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}(\mu_H(E)),
 \end{align*}
 where the last line follows from convexity of $\overline{h}_{\mu^-_{H}(E),\mu^+_H(E)}$.
\end{proof}

We will use the following variant of \cite[Lemma 3.3]{Naoki23_BG_hypersurfaces}, which allows us to assume $F$ is everywhere stable:
\begin{lemma}\label{LemmaReductionToStableObjects}
 Let $d\geq 1$ be an integer, and $f:(0,1)\rightarrow \mathbb{R}$ be a star-shaped function along the lines $\beta = 0, d$, satisfying
 \begin{equation*}
 f(t) \leq \tfrac{1}{2}t^2
 \end{equation*}
 for every $t\in (0,1)$. Assume that there exists objects $F'\in \Db(X)$ satisfying the following conditions:
 \begin{enumerate}[{\ \ }(a)]
 \item $F'$ is either $\nu_{0,\alpha}$-tilt-semistable for some $\alpha > 0$, or $\nu_{d,\alpha'}$-tilt-semistable for some $\alpha' > d^2/2$.
 \item $\mu_H(F')\in (0,1)$, $\xi_H(F') > f(\mu_H(F'))$.
 \end{enumerate}
 Then we can choose such an object $F$ so that:
 \begin{enumerate}
 \item $F$ is a reflexive coherent sheaf.
 \item $F\in\Coh^{0,H}(X)$ is $\nu_{\alpha,0,H}$-tilt-stable for any $\alpha > 0$.
 \item $F[1] \in \Coh^{d,H}(X)$ is $\nu_{\alpha',d,H}$-tilt-stable for any $\alpha' > d^2/2$.
 \end{enumerate}
\end{lemma}
\begin{proof}
 Take such an $F'$. Since the value of discriminant is a nonnegative integer, we may assume that $F'$ is an object with the minimal $\overline{\Delta}_H$ of all such objects. Suppose $F'$ becomes strictly $\nu_{\alpha,0,H}$-tilt-semistable for some $\alpha > 0$ (or strictly $\nu_{\alpha',1,H}$-tilt-semistable), then we consider the Jordan--H\"older filtration of $F'$. Since $f$ is star-shaped along the line $\beta = 0$ and $\beta = d$, there is at least one Jordan--H\"older factor $F'_i$ with $\mu_H(F'_i)\in(0,1)$ also satisfying $\xi_H(F'_i) > f(\mu_H(F'_i))$. By \Cref{RemarkOnDiscriminats}, we have $\overline{\Delta}_H(F') > 0$. Hence $\overline{\Delta}_H(F'_i) < \overline{\Delta}_H(F')$ by \Cref{LemmaDiscriminant}, and this violates the minimum assumption on $\overline{\Delta}_H(F')$. As a consequence, we may assume $F'$ is $\nu_{\alpha,0,H}$-tilt-stable (or $\nu_{\alpha',1,H}$-tilt-stable) for any $\alpha > 0$ (or $\alpha'>\frac{1}{2}$).

The Bogomolov--Gieseker (BG) inequality plays an important role in the study of stability conditions and moduli spaces of sheaves. It is a natural question to seek refinements of the BG inequality for special varieties; see for instance \cite[Question 5.1]{BM23}.

Motivated by Gepner type stability conditions on graded matrix factorizations, Toda proposed in \cite{Toda2017GepnerPoint} a stronger inequality on quintic threefolds, and proved it for $\operatorname{rk}(E) = 2$ case.
\begin{theorem}[{\cite[Conjecture $1.1$]{Toda2017GepnerPoint}}]\label{TodaConj0}
    Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\mathcal{O}_X(1))$. Then for any torsion-free $H$-slope-stable sheaf $E$ on $X$ with $\ch{1}(E)/\operatorname{rk} (E) = -H/2$, we have
    \begin{equation}\label{TodaConj1}
        \frac{H\cdot \Delta(E)}{\operatorname{rk}(E)^2} > 1.5139\cdots,
    \end{equation}
    where the right-hand side of is an irrational real number lying in $\mathbb{Q}(e^{2\pi \sqrt{-1}/5})$. Equivalently,
    \begin{equation}\label{TodaConj2}
    \frac{H\ch{2}(E)}{H^3\operatorname{rk}(E)} < -0.02639\cdots
    \end{equation}
\end{theorem}

In this paper, we establish a stronger Bogomolov--Gieseker type inequality on quintic threefolds, which in particular implies Toda's conjecture.

For a torsion-free coherent sheaf $E$ on an $n$-dimensional polarized smooth projective variety $(X,H)$, we denote
\begin{align*}
    \mu_H(E)\coloneqq\frac{H^{n-1}\ch{1}(E)}{H^n\operatorname{rk}(E)};\quad\xi_H(E)\coloneqq\frac{H^{n-2}\ch{2}(E)}{H^n\operatorname{rk}(E)}.
\end{align*}

\begin{theorem}[See \Cref{MainTheorem} and \Cref{MainCorollary}]
    Let $X\subset \mathbb{P}^4_{\mathbb{C}}$ be a smooth quintic threefold and $H: = c_1(\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\Coh(X)$. Assume that
    \begin{equation*}
        \mu_H(F)\in [-1,1].
    \end{equation*}
    Then
    \begin{equation}\notag
         \xi_H(F)\leq
    \begin{cases}
    -\frac{1}{2}|\mu_H(F)| & \text{when } 0\leq |\mu_H(F)| \leq \frac{1}{4}; \\[0ex]
    \frac{1}{2}|\mu_H(F)| - \frac{1}{4} & \text{when } |\mu_H(F)|\in[\frac{1}{4},\frac{5}{13}]\cup[\frac{8}{13},\frac{3}{4}]; \\[0ex]
    -\frac{3}{20}|\mu_H(F)|  & \text{when } \frac{5}{13} \leq |\mu_H(F)|\leq \frac{6}{13}; \\[0ex]
    \frac{1}{2}|\mu_H(F)|-\frac{3}{10} & \text{when } \frac{6}{13}\leq |\mu_H(F)| \leq \frac{7}{13}; \\[0ex]
    \frac{23}{20}|\mu_H(F)| - \frac{13}{20} & \text{when } \frac{7}{13}\leq |\mu_H(F)| \leq \frac{8}{13}; \\[0ex]
    \frac{3}{2}|\mu_H(F)| - 1 & \text{when } \frac{3}{4}\leq |\mu_H(F)| \leq 1.
\end{cases}
    \end{equation}
\end{theorem}

Along the way, we also obtain a stronger Bogomolov--Gieseker type inequality for quintic surfaces.
\begin{theorem}[See \Cref{ChInequalityOnSurface} and \Cref{CorollaryOnSheaves}]
    Let $S_5\subset \mathbb{P}^3_{\mathbb{C}}$ be a smooth quintic surface and $H\coloneqq c_1(\mathcal{O}_X(1))$. Let $F$ be a torsion-free $H$-slope-semistable sheaf in $\Coh(X)$ with
    \begin{equation*}
        \mu_H(F)\in (0,1).
    \end{equation*}
    Then the following piecewise inequality holds:
    \begin{equation}\notag
        \xi_H(F)\leq
    \begin{cases}
    \frac{17}{26}\mu_H(F) - \frac{2}{13} & \text{when } 0 < \mu_H(F) \leq \frac{7}{46}; \\[0ex]
    -\frac{5}{14}\mu_H(F)  & \text{when } \frac{7}{46} \leq  \mu_H(F)\leq \frac{7}{20}; \\[0ex]
    \frac{1}{2}\mu_H(F)-\frac{3}{10} & \text{when } \frac{7}{20}\leq \mu_H(F) \leq \frac{13}{20}; \\[0ex]
    \frac{19}{14}\mu_H(F) - \frac{6}{7} & \text{when } \frac{13}{20}\leq \mu_H(F)< \frac{39}{46}; \\[0ex]
    \frac{9}{26}\mu_H(F) & \text{when } \frac{39}{46}\leq \mu_H(F)< 1.
\end{cases}
    \end{equation}
\end{theorem}

Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann–-Roch theorem combined with Clifford-type inequalities for plane quintic curves.

This approach parallels the methods of \cite{li_stability_2019,koseki_stability_2022,liu_stability_2022}, which established analogous inequalities for some complete intersection type Calabi--Yau threefolds. The main improvement here lies in a refined restriction theorem (\Cref{RestrictionTheorem}), which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$. As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates. The method employed here extends to weighted complete intersection hypersurfaces as well, though the computations in each case tend to be considerably more demanding.

Stronger Bogomolov--Gieseker type inequalities on Calabi--Yau threefolds are known to play a crucial role in constructing Bridgeland stability conditions, as demonstrated in \cite{li_stability_2018,li_stability_2019,koseki_stability_2022,liu_stability_2022} and the recent preprint \cite{feyzbakhsh2025stabilityconditionscalabiyauthreefolds}. In fact, Toda's conjecture (\Cref{TodaConj0}) was originally motivated by the search for a Gepner-type stability condition on the quintic threefold. However, as far as the author is aware, the existing inequalities are not yet sufficient for this purpose due to certain technical obstacles.

The construction of such a Gepner-type stability condition on the quintic remains an intriguing direction for future work.

Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann--Roch theorem combined with Clifford-type inequalities for plane quintic curves. The main improvement here lies in a refined restriction theorem (\Cref{RestrictionTheorem}), which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$. As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates.

Our proof relies on the notion of tilt-stability. The strategy is to employ a generalized restriction theorem that allows us to reduce to the case of quintic surfaces, where the desired bounds follow from the Riemann--Roch theorem combined with Clifford-type inequalities for plane quintic curves. The main improvement here lies in the following refined restriction theorem, which weakens the hypothesis and allows restriction to $|H|$ instead of $|2H|$.

\begin{lemma}[\textbf{Restriction theorem}]\label{RestrictionTheorem}
    Let $(X,H)$ be a polarized smooth projective variety with dimension $n \geq 2$. Let $E\in \Coh(X)$. Suppose there exists $\alpha > 0$ and $m\in\mathbb{Z}_{> 0 }$ such that both $E$ and  $E(-mH)[1]$ are in $\Coh^{0,H}(X)$ and $\nu_{\alpha,0,H}$-tilt-semistable.

    Then for a smooth irreducible subvariety $Y\in |mH|$, the restricted sheaf $E|_Y$ has $\rk(E) = \rk(E|_{Y})$, $H_Y^{n-2}\ch{1}(E|_Y) = mH^{n-1}\ch{1}(E)$ and when $n \geq 3$, $\ch{2}(E|_Y) = mH\ch{2}(E)$. Moreover, we have
    \begin{align*}
    [\mu_{H_Y}^-(E|_Y),\mu^+_{H_Y}(E|_Y)]\subseteq \llbracket \tfrac{m}{2}+\nu_{\alpha,0,H}(E), \tfrac{m}{2}+\nu_{\alpha,0,H}(E(-mH)[1]) \rrbracket.
    \end{align*} In particular, if $\nu_{\alpha,0,H}(E) = \nu_{\alpha,0,H}(E(-mH)[1])$, then $E|_Y$ is $H_Y$-slope-semistable.
\end{lemma}

As a consequence, we may directly use Clifford-type bounds for plane curves, leading to sharper estimates.

[Bogomolov--Gieseker Inequality \cite{bogomolov_holomorphic_1979,gieseker_theorem_1979}]
Let $X$ be a smooth projective complex variety and $H$ an ample divisor on $X$. For any torsion-free $H$-slope-semistable sheaf $E$ on $X$, we have
\begin{equation*}
    \Delta(E) \cdot H^{\dim(X) - 2} \geq 0,
\end{equation*}
where the discriminant is defined by
\begin{equation*}
    \Delta(E) \coloneqq \ch{1}(E)^2 - 2\ch{0}(E)\ch{2}(E).
\end{equation*}.

Question:
Let $X$ be a smooth projective complex variety of dimension $\dim(X)$, and let $H$ be an ample divisor on $X$. For a torsion-free sheaf $E$ on $X$ that is slope-semistable with respect to $H$, define its discriminant by
\[
\Delta(E)\coloneqq \ch_1(E)^2-2\ch_0(E)\ch_2(E).
\]
Which quantitative inequality holds for such a sheaf?

For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\ge 0.
\]

Choices:
B.
For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-1}\ge 0.
\]
C.
For every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\in \mathbb{R}.
\]
D.
There exists a constant $c_X>0$, depending only on $X$, such that for every torsion-free $H$-slope-semistable sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\ge c_X.
\]
E.
For every torsion-free sheaf $E$ on $X$, one has
\[
\Delta(E)\cdot H^{\dim(X)-2}\ge 0.
\]

intersection power matches codimension-two class

nonnegativity conclusion dropped, retaining only well-definedness of the intersection number

uniform lower bound strengthened from 0 to a positive constant depending only on X

The stem does not explicitly reveal the conclusion; it only sets up the hypotheses and asks for the relevant inequality. There is no direct answer leakage.

The correct choice is essentially the standard Bogomolov inequality stated under its usual hypotheses, so the item is very close to a direct theorem restatement.

Some discrimination is required because the distractors vary by codimension, strength, and hypotheses, but solving the item is still mostly a matter of theorem recall rather than genuine derivation or generative reasoning.

The distractors are mathematically meaningful and target common failure modes: wrong intersection power, a weaker true statement, an unjustified stronger bound, and omission of the semistability hypothesis.

A solid recall-based MCQ with strong distractors and no answer leakage, but it is largely tautological since it directly restates a standard theorem rather than testing deeper reasoning.


The fourth stable homotopy group of spheres is trivial, meaning that $\pi_{n+4}(S^n)=0$ for all $n>5$. In contrast to this, it is a theorem that $S^m$ has no trivial higher homotopy groups when $m\in\{2,3,4,5\}$, as we will briefly explain.

Curtis proved that $\pi_n(S^4)\neq 0$ for all $n\geq 4$ \cite{Cur69}. Curtis also proved that $\pi_n(S^2)\neq 0$ for all $n\not\equiv 1\mod 8$. These same results were obtained (via different methods) by Mimura, Mori, and Oda \cite{MMO75}. The proof that $\pi_n(S^5)\neq 0$ for all $n\geq 5$ was given by Mori \cite{Mor75} and Mahowald \cite{Mah75,Mah82}.

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
    Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

In fact, it appears that every positive integer has at least two Toda primes.

\begin{conj}\label{conj:at least two}
    If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

Since $\pi_n(S^2)\cong\pi_n(S^3)$ for all $n\geq 3$, the remaining case was $\pi_n(S^3)$ with $n\equiv 1\mod 8$. This last case was proved by Gray \cite{Gra84}, and later by Ivanov, Mikhailov, and Wu \cite{IMW16} using different methods. In \emph{op.~cit.}, the authors note that \cite[Theorem 5.2(ii)]{Tod66} implies that
\[\mathbb{Z}/p\subseteq\pi_{2(p-1)k+1}(S^3)\]
whenever $\gcd(p,k)=1$ \cite[p.~342, Equation (B)]{IMW16}. It follows that if every positive integer $n$ admits an odd prime $p$ and an integer $k$ such that $\gcd(p,k)=1$ and $4n=(p-1)k$, then $\pi_n(S^3)\neq 0$ for all $n\equiv 1\mod 8$. This leads one to the following definition.

\begin{defn}
 Let $n$ be an integer. A \emph{Toda prime} of $n$ is an odd prime $p$ such that $p-1\mid 4n$ and $\gcd(p,\frac{4n}{p-1})=1$. Denote the set of Toda primes of $n$ by $T(n)$ (see \cref{table:toda sets}), and let $t(n):=|T(n)|$ (see \cref{fig:tn}).
\end{defn}

If every positive integer has a Toda prime, then one can greatly simplify the proof that $\pi_n(S^3)\neq 0$ for $n\geq 3$. This was asked on MathOverflow \cite{MO} (and attributed to Roman Mikhailov) several years ago.

\begin{conj}\label{conj:at least two}
 If $n$ is a positive integer, then $t(n)\geq 2$.
\end{conj}

We tried to answer \cref{ques:at least one} in the affirmative, but our approach hits a snag. To turn our failed attempt into a theorem, we adopt the time-tested tradition of stating our snag as a conjecture (\cref{conj:main}). We will give some heuristic evidence for this conjecture in \cref{sec:heuristic}.

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{figure}[p]
    \includegraphics[width=.75\linewidth]{tn.png}
    \caption{$t(n)$ for $n\leq 100000$}\label{fig:tn}
\end{figure}

\begin{ques}[Mikhailov]\label{ques:at least one}
    Does every positive integer have a Toda prime?
\end{ques}

\begin{table}[t]
\centering
\caption{Sets of Toda primes}\label{table:toda sets}
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
1 & $3,5$\\
2 & $3,5$\\
3 & $5,7,13$\\
4 & $3,5,17$\\
5 & $3,11$\\
6 & $5,7,13$\\
7 & $3,5,29$\\
8 & $3,5,17$\\
9 & $5,7,13,19,37$\\
10 & $3,11,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
11 & $3,5,23$\\
12 & $5,7,13,17$\\
13 & $3,5,53$\\
14 & $3,5,29$\\
15 & $7,11,13,31,61$\\
16 & $3,5,17$\\
17 & $3,5$\\
18 & $5,7,13,19,37,73$\\
19 & $3,5$\\
20 & $3,11,17,41$\\
\hline
\end{tabular}
\quad
\begin{tabular}{|r|l|}
\hline
$n$ & $T(n)$\\
\hline
21 & $5,13,29,43$\\
22 & $3,5,23,89$\\
23 & $3,5,47$\\
24 & $5,7,13,17,97$\\
25 & $3,11,101$\\
26 & $3,5,53$\\
27 & $5,7,13,19,37,109$\\
28 & $3,5,17,29,113$\\
29 & $3,5,59$\\
30 & $7,11,13,31,41,61$\\
\hline
\end{tabular}
\end{table}

\begin{conj}\label{conj:main}
 Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\begin{lem}\label{lem:not 5 or not 3 and 5}
 Let $p\in\{3,5\}$. If $p\nmid n$, then $p\in T(n)$. In particular:
 \begin{itemize}
 \item If $5\nmid n$, then $t(n)\geq 1$.
 \item If $3,5\nmid n$, then $t(n)\geq 2$.
 \end{itemize}
\end{lem}
\begin{proof}
 Let $p\in\{3,5\}$. Then $p-1\mid 4n$, and $\gcd(4n,p)=1$ by assumption. Thus $\gcd(\frac{4n}{p-1},p)=1$, so $p\in T(n)$.
\end{proof}

\begin{lem}\label{lem:t(n)=3}
 Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}
\begin{proof}
 We will induct on $\omega(n)$. Our base cases will consist of $\omega(n)\leq 4$. Note that if $p\nmid n$ for each $p\in\{5,7,13\}$, then $T(n)\supseteq\{5,7,13\}$. In particular, we may restrict our attention to multiples of these three primes. Moreover, $t(ap)\geq t(a)$ for any prime $p\not\in T(a)$ by \cref{cor:multiply by prime}, so we may assume that every prime factor $p\mid n$ is a Toda prime of some divisor of $n$.
 \begin{itemize}
 \item The case of $\omega(n)=1$ is just the calculation $T(3)=\{5,7,13\}$.
 \item For $\omega(n)=2$, we just need to compute $t(15)=t(39)=5$ and $t(21)=4$.
 \item For $\omega(n)=3$, we first compute $t(3\cdot 5\cdot 7)=9$ and $t(3\cdot 5\cdot 13)=t(3\cdot 7\cdot 13)=8$. It remains to compute, for each $p\in\{5,7,13\}$, the Toda primes of $3pq$ for each $q\in T(3p)$. Using the code provided in \cref{sec:code}, we find that $t(3pq)\geq 4$ for all such $p,q$.
 \item For $\omega(n)=4$, we first compute $t(3\cdot 5\cdot 7\cdot 13)=16$. For the remaining computations in this case, we use the code in \cref{sec:code}.
 \begin{itemize}
 \item If $\{p,q\}\subseteq\{5,7,13\}$ and $r\in T(3pq)$, then $t(3pqr)\geq 9$.
 \item If $p\in\{5,7,13\}$ and $\{q,r\}\subseteq T(3p)$, then $t(3pqr)\geq 7$.
 \item If $p\in\{5,7,13\}$, $q\in T(3p)$, and $r\in T(3pq)$, then $t(3pqr)\geq 5$.
 \end{itemize}
 \end{itemize}

\begin{cor}\label{prop:divisible by 3}
 Assume \cref{conj:main}. If $3\mid n$, then $t(n)\geq 3$.
\end{cor}
\begin{proof}
 By \cref{cor:multiply by divisor}, we may assume that $n$ is square-free. By \cref{cor:odd}, we may further assume that $n$ is odd. The result now follows from \cref{lem:t(n)=3}.
\end{proof}

Firstly, if there exists $q'\in T(3p)-\{5,7,13,q\}$ such that $q'\nmid n$, then $\{5,7,13,q,q'\}-\{p\}\subseteq T(n)$, and we are done. In fact, the Toda primes of $n$ are precisely those primes among $\{2d+1:d\mid 2n\}$ that are not factors of $n$. Thus if
\begin{equation}\label{eq:heuristic}
\{2d+1\text{ prime}:d\mid 2n\}-(\Omega(n)\cup\{5,7,13,q\})
\end{equation}
is non-empty, then $t(n)\geq 4$. Our heuristic for \cref{conj:main} is that the set $\{2d+1:d\mid 2n\}$ consists of $2^{\omega(n)+1}$ elements, while $\Omega(n)\cup\{5,7,13,q\}$ consists of $\omega(n)+3$ elements.

\begin{prop}\label{prop:t(p)}
Assume $p\geq 7$ is a prime. Let $\vphi$ denote the totient function. If $\vphi(x)=4p$ for some integer $x$, then $T(p)=\{3,5,2p+1\}$ or $\{3,5,4p+1\}$. Otherwise, $T(p)=\{3,5\}$.
\end{prop}
\begin{proof}
 One can directly check that $3,5\in T(p)$ for all primes greater than 5. Now by Euler's product formula, we have $\vphi(x)=p_1^{e_1-1}(p_1-1)\cdots p_m^{e_m-1}(p_m-1)$, where $x=\prod_{i=1}^m p_i^{e_i}$ is the prime factorization of $x$. It follows that there exists $x$ such that $\vphi(x)=4p$ if and only if one of the following cases holds:
 \begin{enumerate}[(i)]
 \item $x=2^2\cdot q$, where $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot 3\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=2p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=2$.
 \item $x=2^r\cdot q$, where $r\in\{0,1\}$ and $q$ is an odd prime such that $q-1=4p$. In this case, $q$ is a Toda prime of $p$ with $\frac{4p}{q-1}=1$.
 \item $x=2^r\cdot 5^2$, where $r\in\{0,1\}$ (in which case $p=5$). This case is not relevant for this lemma, as we have assumed $p\geq 7$.
 \end{enumerate}
 It remains to show that no other primes can be the Toda prime of $p$. To this end, let $q>5$ be a Toda prime of $p$. Then $q-1\mid 4p$, so we either have $q-1=4p$ or $q-1=2p$ (as $q-1$ is even and $p$ is odd). The existence of such a $q$ gives us a solution to $\vphi(x)=4p$ as outlined in cases (i), (ii), and (iii).

\begin{lem}\label{lem:strategy for denoms}
 Let $d$ be a Bernoulli denominator with $F(d)=4a$ for some integer $a$. If $\{2pi+1:i\mid 2a\}$ contains a prime number for each $p\in T(a)$, then \cref{conj:general bernoulli} \eqref{conj:toda for bernoulli} holds for this Bernoulli denominator.
\end{lem}
\begin{proof}
 We know that $p-1\mid 4a$ with $\gcd(\frac{4a}{p-1},p)=1$ for all $p\in T(a)$. Thus $p-1\mid 4am$, and we have $\gcd(\frac{4am}{p-1},p)=1$ if and only if $p\nmid m$. It therefore suffices to show that if $p\mid m$ for some $p\in T(a)$, then $D_{4am}>D_{4a}$.

\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}

\begin{lem}\label{lem:t(n)=3}
    Assume \cref{conj:main}. Let $n$ be an odd, square-free multiple of 3. Then $t(n)\geq 3$. Moreover, $t(n)=3$ if and only if $T(n)=\{5,7,13\}$.
\end{lem}

\begin{thm}\label{thm:main}
    Assume \cref{conj:main}. If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.
\end{thm}

In \cref{sec:lemmas}, we state and prove a few simple lemmas. We prove \cref{thm:main} in \cref{sec:proof} by inducting on the number of odd prime factors of $n$. Essentially all of the real work happens in \cref{lem:t(n)=3}. We conclude with \cref{sec:observations}, where we pose a couple questions that arose while working on this project.

\subsection*{Acknowledgements}
We thank Nick Andersen, Pace Nielsen, and Kyle Pratt for helpful conversations. The author was partially supported by the NSF (DMS-2502365) and the Simons Foundation.

\label{thm:main}
Assume
\begin{conj}\label{conj:main}
    Let $n$ be an odd, square-free multiple of 3. Assume that there exists $p\in\{5,7,13\}$ such that $p\mid n$, and that $r\nmid n$ for $r\in\{5,7,13\}-\{p\}$. Finally, assume there exists $q\in T(3p)-\{5,7,13\}$ such that $q\nmid n$. Then $t(n)\geq 4$.
\end{conj}
If $n$ is a positive integer, then $t(n)\geq 1$. If $5\nmid n$, then $t(n)\geq 2$. If $3\mid n$, then $t(n)\geq 3$.,

Question: For a positive integer n, call an odd prime p a Toda prime of n if p-1 divides 4n and \(\gcd\!\left(p,\frac{4n}{p-1}\right)=1\). Let \(T(n)\) be the set of Toda primes of n, and let \(t(n)=|T(n)|\). Assume the following hypothesis holds universally: whenever m is an odd, square-free multiple of 3, there exists \(p\in\{5,7,13\}\) such that \(p\mid m\) and \(r\nmid m\) for every \(r\in\{5,7,13\}\setminus\{p\}\), and there exists \(q\in T(3p)\setminus\{5,7,13\}\) with \(q\nmid m\), then \(t(m)\ge 4\). Under this assumption, which quantitative estimate holds for Toda primes of a positive integer n?

For every positive integer n, \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

Choices:
B. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(3\nmid n\), then \(t(n)\ge 2\); and if \(5\mid n\), then \(t(n)\ge 3\).
C. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\).
D. For every positive integer \(n\), \(t(n)\ge 2\); moreover, if \(5\nmid n\), then \(t(n)\ge 3\); and if \(3\mid n\), then \(t(n)\ge 4\).
E. For every positive integer \(n\), \(t(n)\ge 1\); moreover, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(n\) is an odd, square-free multiple of \(3\), then \(t(n)\ge 4\).

special-role of the primes 3 and 5 in the baseline bounds

dropped the final clause requiring \(3\mid n\Rightarrow t(n)\ge 3\)

numerical lower bounds 1/2/3 replaced by stronger 2/3/4

scope of the \(t(m)\ge 4\) hypothesis enlarged by dropping the auxiliary existence conditions on \(p\) and \(q\)

The stem does not explicitly state the target estimate, and there is no direct textual giveaway of choice A. The hypothesis is technical, but it does not itself reveal the final 1/2/3 lower-bound pattern.

The item appears to ask for the theorem-level conclusion under a stated hypothesis, so it is close to theorem recall rather than a fully independent problem. Still, the answer choices differ by quantifiers, exceptional primes, and bound strength, so it is not a pure verbatim restatement.

There is some reasoning pressure in comparing nearby quantitative claims and tracking how the special roles of 3 and 5 affect the bounds. However, the stem is too specialized to support much derivation from first principles, so the task leans more toward recognizing the intended conclusion than generating it.

The distractors are plausible and well targeted: one swaps the roles of 3 and 5, one gives a weaker true-looking statement, one overstates the bounds, and one improperly broadens the hypothesis. These reflect realistic mathematical failure modes involving quantifiers and strength of conclusions.

A reasonably strong MCQ with little answer leakage and high-quality distractors, but it is somewhat theorem-recall driven and only moderately tests genuine generative reasoning.