\begin{theorem}  \label{thm: factor hazard}
Let the hazard function, $h_0:(0,\infty)\to(0,\infty)$, admit the factorization:
\[
h_0(x)=C\,x^{p-1}\,g(x),\qquad C>0,\; p\ge 0,
\]
where $g:(0,\infty)\to(0,\infty)$ is non-decreasing. For $\tau>0$ define:
\[
r(x,\tau)\;=\;\tau\,\frac{h_0(\tau x)}{h_0(x)}.
\]
Then, for every $x>0$,
\[
0\le \tau\le 1
\quad\Longleftrightarrow\quad
0\le r(x,\tau)\le 1.
\]
\end{theorem}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0(\tau x)}{h_0(x)}, \qquad x>0.
\]
Then:
\[
0 \le r(x,\tau) \le 1 \quad \forall x>0
\quad \Longleftrightarrow \quad
0 \le \tau \le 1.
\]
\end{theorem}

\label{app: thms}

\begin{theorem} \label{thm: ifr hazard}
Let $h_0:(0,\infty)\to(0,\infty)$ be a non-decreasing (i.e., increasing failure rate) hazard function. Define:
\[
r(x,\tau) = \tau\,\frac{h_0

\begin{theorem} \label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. Then:
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]
\end{theorem}

As per \citet{arnold2020bivariate}, the general accelerated failure conditionals model is of the form:
\begin{align}
 \bar{F}_X(x) = P(X>x) = \bar{F}_0(x),\quad x>0, \label{eq: general marginal}
\end{align}
for some survival function $\bar{F}_0(x) \in [0, 1] \ \forall x> 0$, and for each $x>0$:
\begin{equation}
 P(Y>y\mid X>x) = \bar{F}_1\left(\beta(x)y\right), \quad x, y >0,\label{eq: general cond}
\end{equation}
for some survival function $\bar{F}_1(\beta(x)y) \in [0, 1] \ \forall x, y >0$ and a suitable acceleration function $\beta(x)$. As per \citet{arnold2020bivariate}, in the analysis of dependent lifetimes of components in a system, it is more appropriate to consider the conditional density of $Y$ given that the first component, with lifetime $X$, remains operational at time $x$. Hence why, the conditioning event is taken to be $\{X > x\}$, rather than conditioning on the exact value $X = x$. The joint survival function is then:
\begin{align}
 P(X>x, Y>y) = \bar{F}_0(x) \bar{F}_1\left(\beta(x)y\right), \quad x, y>0. \label{eq: general joint}
\end{align}
Assuming differentiability and $X,Y$ are continuous, we obtain the marginal densities:
\[
f_X(x) = -\frac{d}{dx}P(X>x) = -\frac{d}{dx}\bar{F}_0(x) = f_0(x),
\]
and since $\lim_{x\to 0^+}\bar{F}_0(x)=1$, where $\beta(0):=\lim_{x\to 0^+}\beta(x)$:
\[
P(Y>y) = \lim_{x\to 0^+}P(Y>y,X>x) = \bar{F}_1\bigl(\beta(0)y\bigr).
\]
Now, the density of $Y$ is:
\[
f_Y(y) = -\frac{d}{dy}P(Y>y)
= -\frac{d}{dy}\bar{F}_1(\beta(0)y)
= \beta(0)\,f_1(\beta(0)y),
\]
where $f_1(t)=-\bar{F}_1'(t)$ denotes the density associated with $\bar{F}_1$. Hence the marginal distribution of $Y$ belongs to the same family as $\bar{F}_1$, but with its argument scaled by $\beta(0)$. For \eqref{eq: general joint} to be a valid survival function, it must have a non-negative
mixed partial derivative. Denoting $f_0$ and $f_1$ as the densities corresponding to the survival function $\bar{F}_0$ and $\bar{F}_1$, and differentiating, we obtain:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) &=
f_0(x) f_1(\beta(x) y)\beta(x)
- \bar{F}_0(x) f_1'(\beta(x) y) \beta(x) \beta'(x) y
- \bar{F}_0(x) f_1(\beta(x) y) \beta'(x) \nonumber \\
&= f_0(x) f_1(\beta(x) y)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(\beta(x) y) + f_1'(\beta(x) y) \beta(x) y\right]. \nonumber
\end{align}
Denoting $t = \beta(x)y$, we have:
\begin{align}
 \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y)
&= f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ f_1(t) + f_1'(t)t\right] \nonumber \\
& = f_0(x) f_1(t)\beta(x)
- \beta'(x) \bar{F}_0(x) \left[ t f_1(t) \right]' .\nonumber
\end{align}
Thus a necessary and sufficient condition for $ \frac{\partial}{\partial x} \frac{\partial}{\partial y} P(X>x, Y>y) \geq 0$ is:
\[
\beta'(x)\;\begin{cases}
\le\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'>0,\\[10pt]
\ge\; \dfrac{f_0(x)}{\bar F_0(x)}\;\dfrac{f_1(t)}{[t f_1(t)]'}\;\beta(x), & \text{if } [t f_1(t)]'<0, \\
\text{unconstrained}, & \text{if } [t f_1(t)]'=0.
\end{cases}
\]
Denoting the hazard function $h_0(x) = \frac{f_0(x)}{\bar{F}_0(x)}$ and $S(t) = \frac{f_1(t)}{[t f_1(t)]'}$, we may write the bounds as such:
\begin{align}
\sup_{t: [t f_1(t)]'<0} h_0(x) S(t)\beta(x)
\;\leq\; \beta'(x)
\;\leq\;
\inf_{t: [t f_1(t)]'>0} h_0(x) S(t)\beta(x). \label{eq: B'(x) bounds}
\end{align}
A special case of the upper bound for $B'(x)$ arises for distribution families satisfying $f_1'(t) \leq0$ - that is, for non-increasing densities. In this case, we note that $S(t) = \frac{f_1(t)}{\left[tf_1(t) \right]'} = \frac{f_1(t)}{f_1(t) + tf_1'(t) } \geq1$ for $t \geq 0$, hence $\beta'(x)
\;\leq\;
 h_0(x)\beta(x)$.\\

It should be noted that multiple functional forms for $\beta(x)$ are possible. The question of selecting a form of $\beta(x)$ that provides the best model fit has already been investigated by \citet{arnold2020bivariate}. Accordingly, the present study restricts attention to those choices of $\beta(x)$ that arise from the extreme case in which $\beta'(x)$ attains its theoretical upper bound, that is, given $\inf_{t: [t f_1(t)]'>0} S(t) = c^*$:
\begin{align}
 \beta'(x) &= c^*h_0(x) \beta(x), \nonumber \\
 \therefore \int \frac{1}{\beta(x)} d\beta(x) &= \int c^* h_0(x) dx, \nonumber\\
 \therefore \log\left( \beta(x) \right) &= -c^* \log\left( \bar{F}_0 (x)\right) + K. \nonumber \\
 \therefore \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(x)\right]^{c^*}}, \label{eq: beta(x) general}
 \end{align}
for $\gamma = e^K>0$. This suggests $\beta(x)$ is non-decreasing for $c^*>0$. Furthermore, $\beta(0): = \lim_{x\to 0^+} \beta(x) = \gamma$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$. We also note that since $\bar{F}_0(x)$ approaches zero only as $x\rightarrow \infty$, the expression in \eqref{eq: beta(x) general} remains finite for all finite $x$.\\

The role of the acceleration function $\beta(x)$ in determining the sign of the correlation between $X$ and $Y$ is established in Theorem~\ref{thm: cov(x, y)}. In particular, since the present study considers only non-decreasing specifications of $\beta(x)$, the resulting framework is restricted to modelling data in which $X$ and $Y$ exhibit negative correlation.

\subsection{Moments}
Noting $X \sim Weibull(\alpha, \lambda)$ and $Y \sim Weibull(\gamma, \nu)$, we have:
\begin{align}
 E(X) &= \frac{1}{\alpha} \Gamma\left(1+ \frac{1}{\lambda}\right), \label{eq: Weibull ex} \\
 E(Y) &= \frac{1}{\gamma} \Gamma\left(1+ \frac{1}{\nu}\right), \label{eq: Weibull ey} \\
 {Var}(X) &= \frac{1}{\alpha^2}\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right),\label{eq: Weibull varx} \\
 {Var}(Y) &= \frac{1}{\gamma^2}\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right).\label{eq: Weibull vary}
\end{align}
Accordingly:
\begin{align}
\operatorname{Cov}(X, Y)
&= \int_0^\infty \int_0^\infty
 \big( P(Y>y , X> x) - P(X> x)P(Y> y) \big) \,dy\,dx \nonumber \\
&= \int_0^\infty \int_0^\infty
 \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr) \right) \,dy\,dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \int_0^\infty \left( e^{-(\beta(x)y )^\nu}- e^{-(\beta(0)y )^\nu} \right) dy \right) dx \nonumber \\
&= \int_0^\infty e^{-(\alpha x)^\lambda}
 \left( \frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(x)} -\frac{\Gamma\!\left(1+\frac{1}{\nu}\right)}{\beta(0)} \right) dx \nonumber \\
&= \frac{1}{\gamma } \Gamma\!\left( 1+ \frac{1}{\nu}\right)
 \left[\int_0^\infty e^{-(\alpha x)^\lambda - \frac{1}{\nu}(\alpha \tau x)^\lambda} \, dx
 - \int_0^\infty e^{-(\alpha x)^\lambda} \, dx \right] \nonumber \\
&= \frac{1}{\alpha \gamma} \Gamma\!\left(1+ \frac{1}{\nu} \right)\Gamma\!\left(1+ \frac{1}{\lambda} \right)
 \left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right].
\label{eq: Weibull cov}
\end{align}
Now, using \eqref{eq: Weibull varx}, \eqref{eq: Weibull vary} and \eqref{eq: Weibull cov}, the correlation function is:
\begin{align}
 \rho(X,Y)
 &= \frac{\operatorname{Cov}(X,Y)}{\sqrt{\operatorname{Var}(X)\operatorname{Var}(Y)}} \nonumber\\
 &= \frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + \tau^\lambda} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}}.
 \label{eq: Weibull rho}
\end{align}
Clearly, \eqref{eq: Weibull rho} is strictly decreasing in $\tau \in [0, 1]$. Hence:
\begin{align}
 \rho_{\text{min}}(\lambda, \nu):=\frac{\Gamma\left(1+ \frac{1}{\nu} \right)\Gamma\left(1+ \frac{1}{\lambda} \right)\left[ \left(\frac{\nu}{\nu + 1} \right)^{\frac{1}{\lambda}} - 1 \right]}{\sqrt{\left( \Gamma\left( 1+ \frac{2}{\lambda}\right) - \left(\Gamma\left(1+\frac{1}{\lambda}\right)\right)^2 \right)\left( \Gamma\left( 1+ \frac{2}{\nu}\right) - \left(\Gamma\left(1+\frac{1}{\nu}\right)\right)^2 \right)}} \leq \rho(X, Y) \leq 0.
\end{align}
Now, $\inf_{\lambda, \nu > 0} \rho_{\text{min}}(\lambda, \nu) = \lim_{\nu \to \infty}\rho_{\text{min}}(\lambda = 1, \nu) = -\frac{\sqrt{6}}{\pi}$. Hence, this Weibull model with acceleration function $\beta(x) = \gamma e^{\frac{1}{\nu}(\alpha \tau x)^\lambda}$ will be able to accommodate correlations in the range $\rho(X,Y)\in\big[-\tfrac{\sqrt{6}}{\pi},\,0\big]$.

\appendix
\section*{Appendices}
\section{Joint Densities and Log-likelihoods} \label{app: joint lik}
\subsection{Exponential}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: exp joint survival}:
\[
f_{X,Y}(x,y)
= \alpha \gamma \,\bigl(\gamma \tau y e^{\alpha \tau x} - \tau + 1 \bigr)
 e^{\!\left(\alpha x(\tau - 1) - \gamma y e^{\alpha \tau x}\right)},
 \qquad x,y>0.
\]
where $\alpha, \gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\gamma,\tau)
&= n \log \alpha + n \log \gamma
+ \sum_{i=1}^n \log\!\Big( \gamma \tau y_i e^{\alpha \tau x_i} - \tau + 1 \Big) \nonumber \\
&\quad + \alpha (\tau - 1)\sum_{i=1}^n x_i
- \gamma \sum_{i=1}^n y_i e^{\alpha \tau x_i}. \nonumber
\end{align}
\subsection{Lomax}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Lomax joint survival}:
\begin{align*}
f_{X,Y}(x,y)
&= \nu \,(1+\alpha x)^{-3\lambda - 1}\,
 \bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{-3\nu - 4} \\[6pt]
&\quad \times \Biggl[
 \frac{\alpha \gamma^{2} \lambda \tau y (\nu+1)\,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{2\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+2}}
 {1+\alpha \tau x} \\[6pt]
&\qquad - \frac{\alpha \gamma \lambda \tau \,(1+\alpha x)^{2\lambda+1}(1+\alpha \tau x)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}}
 {1+\alpha \tau x} \\[6pt]
&\qquad + \alpha \gamma \lambda \,\bigl((1+\alpha x)^{2}(1+\alpha \tau x)\bigr)^{\lambda}
 \,\bigl(1+\gamma y (1+\alpha \tau x)^{\lambda}\bigr)^{2\nu+3}
 \Biggr], \quad x,y >0,
\end{align*}
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \log \nu
 - (3\lambda+1)\sum_{i=1}^n \log(1+\alpha x_i) \nonumber \\
&\quad - (3\nu+4)\sum_{i=1}^n \log\!\bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr) \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Biggl\{
 \frac{\alpha \gamma^{2} \lambda \tau y_i (\nu+1)\,(1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{2\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+2} \nonumber \\
&\qquad\qquad
 - \frac{\alpha \gamma \lambda \tau (1+\alpha x_i)^{2\lambda+1}(1+\alpha \tau x_i)^{\lambda}}
 {1+\alpha \tau x_i}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3} \nonumber \\
&\qquad\qquad
 + \alpha \gamma \lambda \,\bigl((1+\alpha x_i)^{2}(1+\alpha \tau x_i)\bigr)^{\lambda}
 \bigl(1+\gamma y_i (1+\alpha \tau x_i)^{\lambda}\bigr)^{2\nu+3}
 \Biggr\}. \nonumber
\end{align}
\subsection{Weibull}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Weibull joint survival}:
\[
f_{X, Y}(x,y) = \alpha^{\lambda} \gamma^{\nu} \lambda \nu x^{\lambda - 1} y^{\nu - 1}
\left( \tau^{\lambda} (\gamma y)^{\nu} e^{(\alpha \tau x)^\lambda}
- \tau^{\lambda} + 1 \right)
e^{(\alpha x)^\lambda(\tau^\lambda - 1)
- (\gamma y)^{\nu} e^{(\alpha\tau x)^{\lambda}}}, \quad x, y>0,
\]
where $\alpha, \lambda, \nu,\gamma > 0 $ and $\tau \in [0, 1]$. The log-likelihood is thus:
\begin{align}
\ell(\alpha,\lambda,\nu,\gamma,\tau)
&= n \lambda \log \alpha + n \nu \log \gamma
 + n \log \lambda + n \log \nu \nonumber\\
&\quad + (\lambda - 1) \sum_{i=1}^n \log x_i
 + (\nu - 1) \sum_{i=1}^n \log y_i \nonumber \\
&\quad + \sum_{i=1}^n
 \log\!\Bigl(\tau^{\lambda} (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}
 - \tau^{\lambda} + 1 \Bigr) \nonumber \\
&\quad + (\tau^{\lambda} - 1)\sum_{i=1}^n (\alpha x_i)^\lambda
 - \sum_{i=1}^n (\gamma y_i)^{\nu} e^{(\alpha \tau x_i)^\lambda}. \nonumber
\end{align}
\subsection{Log-logistic}
We obtain the joint density function by differentiating the joint survival function \eqref{eq: Log joint survival}:
\[
\begin{aligned}
f_{X,Y}(x,y)
&= \frac{\lambda \nu\,(\gamma y)^{\nu}}
{x\,y\,\bigl((\alpha x)^{\lambda}+1\bigr)^{2}
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)^{3}}
\\[4pt]
&\quad\times
\Biggl\{
(\alpha x)^{\lambda}\bigl((\alpha \tau x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
+ (\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Bigl( (\gamma y)^{\nu}\bigl((\alpha \tau x)^{\lambda}+1\bigr)+1 \Bigr)
\\
&\qquad\qquad
- 2\,(\alpha \tau x)^{\lambda}\bigl((\alpha x)^{\lambda}+1\bigr)
\Biggr\},
\qquad x,y>0,
\end{aligned}
\]

\begin{proof}
Since $P(Y>y , X> x) = \bar{F}_0(x)\bar{F}_1\bigl(\beta(x)y\bigr)$ then $P(Y>y) = \lim_{x\to 0^+}P(Y>y , X> x) = \bar{F}_0(0)\bar{F}_1\bigl(\beta(0)y\bigr) = \bar{F}_1\bigl(\beta(0)y\bigr)$ since $\lim_{x\to 0^+}\bar{F}_0(x) = 1$ and $\beta(0): = \lim_{x\to 0^+} \beta(x)$. We note from Hoeffding's covariance identity:
\begin{align}
    Cov(X, Y) & = \int_0^\infty \int_0^\infty\left( P(Y>y , X> x) - P(X> x)P(Y> y)   \right) \ dx dy \nonumber \\
    & = \int_0^\infty \int_0^\infty  \bar{F}_0(x) \left( \bar{F}_1\bigl(\beta(x)y\bigr) - \bar{F}_1\bigl(\beta(0)y\bigr)  \right) \ dx dy.\nonumber
\end{align}
Now since $\bar{F}_1'(\cdot)\le 0$ (as $\bar{F}_1$ is a survival function), if $\beta'(x) \geq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \leq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \leq 0$ for $x, y \geq 0$. Conversely, if $\beta'(x) \leq 0$ then $\bar{F}_1\bigl(\beta(x)y\bigr) \geq \bar{F}_1\bigl(\beta(0)y\bigr)$ and $Cov(X, Y) \geq 0$ for $x, y \geq 0$. Finally, if $\beta(x)$ is constant, say $\beta(x)\equiv c$, then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)\,P(Y>y)$, so $X$ and $Y$ are independent and ${Cov}(X,Y)=0$.
\end{proof}
Furthermore, as done in \citet{arnold2020bivariate}, we introduce a dependence parameter $\tau$ which controls the strength of the dependence between $X$ and $Y$. Extending from \eqref{eq: beta(x) general}, we define the acceleration function as such:
\begin{align}
    \beta(x) &= \gamma \frac{1}{\left[\bar{F}_0(\tau x)\right]^{c^*}}. \label{eq: beta(x) general 2}
\end{align}
Given the bounds established in \eqref{eq: B'(x) bounds}, and upon selecting a non-decreasing specification for $\beta(x)$ as defined in \eqref{eq: beta(x) general 2}, and again assuming $\inf_{t: [t f_1(t)]'>0} S(t) = c^* > 0$, we obtain bounds for $\tau$ as such:
\begin{align}
    0 &\leq \beta'(x) \leq c^* h_0(x)\, \beta(x), \nonumber\\
    \therefore
    0 &\leq
    \frac{c^* \tau \gamma f_0(\tau x)}{\big[\bar{F}_0(\tau x)\big]^{c^* + 1}}
    \leq
    c^* \gamma \,
    \frac{f_0(x)}{\big[\bar{F}_0(x)\big]^{c^*}}
    \frac{1}{\big[\bar{F}_0(\tau x)\big]^{c^*}}, \nonumber\\
    \therefore
    0 &\leq \tau \leq \frac{h_0(x)}{h_0(\tau x)}. \nonumber\end{align}
We re-write this inequality as such:
\begin{align}
    0 \le r(x, \tau) \le 1,
    \label{eq: tau bounds}
\end{align}
where $r(x, \tau) = \tau \frac{h_0(\tau x)}{h_0(x)}$, where the endpoint $\tau=0$ is included by continuity, with $r(x,0):=\lim_{\tau\to 0^+}r(x,\tau)=0$. Theorem \ref{thm: ABC} and Theorems \ref{thm: ifr hazard} and \ref{thm: factor hazard} in Appendix \ref{app: thms}, serve as a basis for establishing bounds on the dependence parameter $\tau$ that satisfy \eqref{eq: tau bounds}.

\begin{theorem}
\label{thm: ABC}
Let $h_0:(0,\infty)\to(0,\infty)$ be a hazard function. For $x>0$ and $\tau>0$ define:
\[
r(x,\tau):=\tau\,\frac{h_0(\tau x)}{h_0(x)},\qquad
\phi(x):=x\,h_0(x).
\]
Let
\[
A:\ \ 0\le r(x,\tau)\le 1\ \ \text{for all }x>0,\qquad
B:\ \ \tau\in(0,1],\qquad
C:\ \ \phi \text{ is non-decreasing on }(0,\infty).
\]
Then:
\begin{enumerate}
\item[(1)] $(A \wedge B)\ \Rightarrow\ C$.
\item[(2)] $(C \wedge B)\ \Rightarrow\ A$.
\item[(3)] $(C \wedge A)\ \Rightarrow\ B$, \emph{unless} $\phi$ is identically constant on $(0,\infty)$, in which case $A$ holds for all $\tau>0$ and $B$ cannot be inferred.
\end{enumerate}

\end{theorem}

\begin{proof}
\textbf{(1) $(A \wedge B)\Rightarrow C$.}
Fix $0<y<x$ and set $\tau:=y/x\in(0,1]$ so that $B$ holds. By $A$,
\[
1 \ \ge\ r(x,\tau)=\tau\,\frac{h_0(\tau x)}{h_0(x)}=\frac{y\,h_0(y)}{x\,h_0(x)}.
\]
Hence $y\,h_0(y)\le x\,h_0(x)$ for all $0<y<x$, i.e.\ $\phi$ is non-decreasing. Thus $C$ holds.

\medskip
\textbf{(2) $(C \wedge B)\Rightarrow A$.}
Assume $C$ and $\tau\in(0,1]$. Then $\tau x\le x$ and, by the non-decreasing nature of $\phi$,
\[
\phi(\tau x)\le \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x)\le x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau)\le 1.
\]
Since $h_0>0$, we also have $r(x,\tau)\ge 0$. Hence $A$ holds.

\medskip
\textbf{(3) $(C \wedge A)\Rightarrow B$, unless $\phi$ is constant.}
Assume $C$ and $A$ hold. Because $h_0(x)>0$, the lower bound $r(x,\tau)\ge 0$ implies $\tau \ge 0$. Now suppose, toward a contradiction, that $\tau>1$, then $\tau x > x$ and, by assuming $\phi$ is now strictly increasing:
\[
\phi(\tau x) > \phi(x)\ \Longleftrightarrow\ \tau x\,h_0(\tau x) > x\,h_0(x)
\ \Longleftrightarrow\ r(x,\tau) > 1.
\]
contradicting the upper bound. Hence $\tau \le 1$. If instead $\phi$ is identically constant,
say $\phi(x)\equiv K>0$, then $h_0(x)=K/x$ and
\[
r(x,\tau)=\tau\,\frac{K/(\tau x)}{K/x}=1 \quad (\forall x>0,\ \forall \tau>0),
\]
so $A$ holds for all $\tau>0$ and no restriction $B$ follows.
\end{proof}
\begin{remark} When convenient, extend $B$ to $\tau\in[0,1]$ by defining $r(x,0):=\lim_{\tau\to 0^+} r(x,\tau)=0$ (whenever the limit exists). All implications above then carry over with this convention.
\end{remark}
\begin{remark}
If $h_0$ is non-decreasing, then $\phi(x) =  x h_0(x)$ is non-decreasing (product of increasing, positive functions), so Theorem~\ref{thm: ABC} recovers Theorem \ref{thm: ifr hazard}.
If $h_0(x)=C\,x^{p-1}g(x)$ with $C>0$, $p\ge 0$, and $g(x)$ non-decreasing, then $\phi(x) = C x^pg(x)$ is non-decreasing, so Theorem \ref{thm: ABC} recovers Theorem \ref{thm: factor hazard}.
\end{remark}

The following sections present the range of models employed in this study. In each case, the survival function forms are specified according to a single particular rate-shape distributional family such that $X \sim Family (\alpha, \lambda)$ and $Y \sim Family (\beta(0), \nu)$ where $\alpha$ and $\beta(0)$ denote the rate parameters and $\lambda$ and $\nu$ denote the shape parameters (shape parameters are not applicable for exponential and half-Cauchy models, however). Closed-form expressions of the models' moments are derived, and additionally, theoretical correlation bounds are obtained (with the exception of the half-Cauchy and gamma models).

Using $P(Y>y,X>x)=\bar F_0(x)\,\bar F_1(\beta(x)y)$, compute the marginal tail of $Y$ by letting $x\to 0^+$:
\[
P(Y>y)=\lim_{x\to 0^+}P(Y>y,X>x)=\bar F_1(\beta(0)y),\qquad \beta(0):=\lim_{x\to 0^+}\beta(x),
\]
(since $\lim_{x\to 0^+}\bar F_0(x)=1$). Then apply Hoeffding's covariance identity:
\[
\operatorname{Cov}(X,Y)=\int_0^\infty\!\int_0^\infty \Big(P(Y>y,X>x)-P(X>x)P(Y>y)\Big)\,dx\,dy
=\int_0^\infty\!\int_0^\infty \bar F_0(x)\Big(\bar F_1(\beta(x)y)-\bar F_1(\beta(0)y)\Big)\,dx\,dy.
\]
Since $\bar F_1'(\cdot)\le 0$, if $\beta'(x)\ge 0$ then $\bar F_1(\beta(x)y)\le \bar F_1(\beta(0)y)$, making the integrand non-positive for $x,y\ge 0$, hence $\operatorname{Cov}(X,Y)\le 0$; conversely, if $\beta'(x)\le 0$ then $\bar F_1(\beta(x)y)\ge \bar F_1(\beta(0)y)$, so $\operatorname{Cov}(X,Y)\ge 0$. Finally, if $\beta(x)\equiv c$ is constant then $P(X>x,Y>y)=\bar F_0(x)\,\bar F_1(cy)=P(X>x)P(Y>y)$, so $X$ and $Y$ are independent and $\operatorname{Cov}(X,Y)=0$.

\label{thm: cov(x, y)}
Consider an accelerated conditional model of the form:
\[
P(X>x) = \bar{F}_0(x)
\quad \text{and} \quad
P(Y>y \mid X>x) = \bar{F}_1\bigl(\beta(x)y\bigr),
\]
where $\bar{F}_0$ and $\bar{F}_1$ are survival functions. To prove the main theorem, it suffices to show that
\[
\beta'(x) \ge 0 \quad \Longrightarrow \quad {Cov}(X,Y) \le 0,
\qquad
\beta'(x) \le 0 \quad \Longrightarrow \quad {Cov}(X,Y) \ge 0.
\]

Question: Let \(X\) and \(Y\) be positive random variables in an accelerated conditional model with marginal survival function \(P(X>x)=\bar F_0(x)\) and conditional survival function \(P(Y>y\mid X>x)=\bar F_1(\beta(x)y)\) for \(x,y>0\), where \(\bar F_0\) and \(\bar F_1\) are survival functions and \(\beta\) is the acceleration function. Which sign relationship between the monotonicity of \(\beta\) and the covariance \(\operatorname{Cov}(X,Y)\) holds?

If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\le 0\); if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\).

Choices:
B. If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\); if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\le 0\).
C. If \(\beta(x)\) is constant for all relevant \(x\), then \(\operatorname{Cov}(X,Y)=0\).
D. If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)< 0\); if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)> 0\).
E. If \(\beta'(x)\ge 0\) for all relevant \(x\), then \(X\) and \(Y\) are independent; if \(\beta'(x)\le 0\) for all relevant \(x\), then \(\operatorname{Cov}(X,Y)\ge 0\).

monotonicity of survival function flips covariance sign

dropped the full monotonicity-to-sign conclusion, retaining only the constant-\(\beta\) independence case

non-strict inequalities from pointwise sign of Hoeffding integrand

constant-\(\beta\) is required for factorization, not merely monotone increasing \(\beta\)

The stem defines the accelerated conditional model and asks for the covariance-sign conclusion, but it does not explicitly reveal or strongly hint at the correct sign pattern.

This is essentially a direct recall of the theorem linking monotonicity of the acceleration function to the sign of covariance, with minimal reformulation.

Some reasoning is needed to distinguish the correct weak sign conclusion from reversed-sign, strict-sign, and independence distractors, but the item mainly tests theorem recall rather than deeper derivation.

The distractors are plausible and well targeted: sign reversal, strict-vs-nonstrict strengthening, a weaker true special case, and an independence confusion all reflect realistic mathematical mistakes.

A solid MCQ with no answer leakage and strong distractors, but it is largely theorem-recall and therefore weak on tautology avoidance and deeper generative reasoning.


\label{main-2}  Let $R$ be a finite family of slopes containing $0,1,\infty$ of some cardinality $k+3$, and let $s$ be a slope not lying in $R$.
\begin{itemize}
    \item[(i)] (Three slopes) If $k=0$, then
\begin{equation}\label{2ab} 2 - \frac{c_2}{D(R;s)} \leq \SD\left(R; s\right) \leq 2 - \frac{c_1}{D(R;s)}
\end{equation}
    for some absolute constants $c_2 > c_1 > 0$.
    \item[(ii)]  (Many slopes) In general, one has
    \begin{equation}\label{sam} 2 - \frac{C_k \log(2+D(R;s))}{D(R;s)} \leq \SD(R; s) \leq 2 - \frac{c_k}{D(R;s)^{k+1}}
    \end{equation}
    for some absolute constants $c_k, C_k > 0$.
\end{itemize}

\begin{equation}\label{dxy}
d[X;Y], d[Y;Y] \ll \log K.
\end{equation}

For $a \in \Q$, define $g(a) \coloneqq \H[Y-aY'] - \H[Y]$, where $Y'$ is an independent copy of $Y$. From \eqref{hx}, \eqref{ruzsa}, \eqref{ruzsa-diff} we have $g(r) \ll \log K$ for $r \in R \backslash \{\infty\}$.  By Proposition \ref{dilate}(iv), we conclude that $g(s) \ll D^{k+1} \log K$, thus $d[Y; sY] \ll D^{k+1} \log K$.  Combining with \eqref{dxy} and \eqref{ruzsa}, \eqref{ruzsa-diff}, we obtain \eqref{dxsy} as required.

\section{A continuous limit}\label{cts-sec}

In this section we establish \Cref{cts-limit}.  Let $\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\infty$ be as in that theorem.  We allow implied constants in the asymptotic notation to depend on $\alpha, f$. We will establish the lower bound
\begin{equation}\label{sd-targ}
 \SD\left(\{0,1,\infty\},\frac{a}{b}\right) \geq 2 - \frac{h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))+o(1)}{\log b}.
 \end{equation}

\subsection{The arithmetic Kakeya conjecture}

Define a \emph{slope} to be an element $r$ of the projective rational\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic.  But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\Q \cup \{\infty\}$.  We then define the projection operators $\pi_r \colon \Q \times \Q \to \Q$ by setting
$$ \pi_r(x,y) \coloneqq x + ry$$
for $r \neq \{\infty\}$, and
$$ \pi_\infty(x,y) \coloneqq y.$$
Given a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \emph{sum-difference constant} $\SD(R;s)$ to be the least exponent such that the bound
\begin{equation}\label{sdef}
\H[ \pi_s(X,Y) ] \leq \SD(R;s) \max_{r \in R} \H[\pi_r(X,Y)]
\end{equation}
holds for $\Q$-valued random variables\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where
$$ \H[X] \coloneqq \sum_{x} \h(\P(X=x))$$
is the Shannon entropy of a random variable $X$, with $\h(t) \coloneqq t \log \frac{1}{t}$ using the convention $\h(0) \coloneqq 0$.  Thus for instance
$$\H[ X-Y ] \leq \SD(\{0,1,\infty\};-1) \max\left( \H[X], \H[X+Y], \H[Y] \right)$$
whenever $X,Y$ are (possibly dependent) $\Q$-valued random variables.

The quantity $\SD(R;s)$ can equivalently be defined as the least exponent such that the bound
\begin{equation}\label{pise}
|\pi_{s}(E)| \leq (\max_{r \in R} |\pi_s(E)|)^{\SD(R;s)}
\end{equation}
for all finite non-empty $E \subset \Q$, where $|A|$ denotes the cardinality of a finite set $A$; see \cite{green}.  However, it will be convenient in this paper to work with the entropy formulation, in order to take advantage of the ``entropic Pl\"unnecke--Ruzsa calculus'' that are founded on the Shannon entropy inequalities.

It is easy to see that one has the projective invariance
$$ \SD(\phi(R); \phi(s)) = \SD(R; s)$$
for any projective transformation $\phi \colon \Q \cup \{\infty\} \to \Q \cup \{\infty\}$, that is to say a map of the form $\phi(r) \coloneqq \frac{ar+b}{cr+d}$ for some $a,b,c,d \in \Q$ with $ad-bc \neq 0$, with the usual conventions when $r$ is infinite or $cr+d$ vanishes.  For instance, by using a dilation transformation, we have
$$ \SD(\{0,1,\infty\}; s) = \SD(\{0,-1/s,\infty\}; -1)$$
for any slope $s$ other than $0,1,\infty$.
In the literature it is conventional to use this $3$-transitive projective symmetry to normalize $s = -1$ and $0, \infty \in R$ (assuming that $|R|\geq 2$ of course), though in this paper it will be more convenient to adopt the normalization $0,1,\infty \in R$ (assuming $|R| \geq 3$).

It is easy to see that $\SD(R;s)=\infty$ when $|R|<2$.
From the entropy inequality $\H[X-Y] \leq \H[X,Y] \leq \H[X]+\H[Y]$ we see that $\SD(R;-1) \leq 2$ when $0,\infty \in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \{0,\infty\}$.  By projective invariance, this implies that $\SD(R;s)=2$ whenever $|R|=2$.  As these quantities are clearly non-decreasing in $R$, we then have the trivial bound
\begin{equation}\label{sdf-upper}
\SD(R;s) \leq 2
\end{equation}
for $|R| \geq 2$.

Improvements upon \eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions.  Indeed, in \cite{bourgain} it was observed that a bound of the form $\SD(R;-1) \leq \alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\'edi's theorem by Leng, Sawhney, and Sah \cite{lss}, that one also obtains this bound for the Hausdorff dimension as well.  For the corresponding results for packing dimension, see \cite{cowen}.} at most $\frac{d-1}{\alpha}+1$.  In particular, if one can establish the \emph{arithmetic Kakeya conjecture}
\begin{equation}\label{infr}
\inf_R \SD(R;-1) = 1
\end{equation}
then this would imply that Kakeya and Nikodym sets in $\R^d$ have full Minkowski and Hausdorff dimension for all $d$.  This is currently only known for $d \leq 3$ \cite{wang}. We refer the reader to \cite{green}, \cite{cowen}, \cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.

Nontrivial progress towards the arithmetic Kakeya conjecture was first obtained in  \cite{bourgain}, who in our notation showed that
$$ \SD(\{0,1,\infty\};-1)  \leq 2 - \frac{1}{13} = 1.923\dots,$$
and used this to obtain new bounds on the Kakeya conjecture in high dimensions.
Further improvements were then obtained in \cite{katz-tao}, \cite{katz-tao-new}. For instance, it is known that
$$ 1.77898 \leq \SD(\{0,1,\infty\};-1) \leq 2 - \frac{1}{6} = 1.833\dots$$
and
$$ 1.668 \leq \SD(\{0,1,2,\infty\};-1) \leq 2 - \frac{1}{4} = 1.75,$$
with the upper bounds established in \cite{katz-tao}, and the lower bounds in \cite{lemm}, \cite{gdm} respectively.
At present, the best upper bound known towards \eqref{infr} is
$$\inf_R \SD(R;-1) \leq 1.67513\dots;$$
see \cite{katz-tao-new}.

\subsection{Asymptotic behavior}

Informally, the arithmetic Kakeya conjecture asserts that in the asymptotic regime where the number of slopes $R$ is large, the constants $\SD(R;s)$ converge to $1$.  Here we consider a complementary regime, in which the number of slopes $R$ is fixed, but we instead let the elements of $R$ (or $s$) vary.  Our main results assert, roughly speaking, that the behavior of these constants is determined by the \emph{rational complexity} of $s$ relative to $R$, with the constants approaching $2$ as it becomes harder to express $s$ in terms of a rational expression of the $R$.  We give a (slightly artificial) definition of this quantity, restricting attention to the normalized setting $\{0,1,\infty\} \subset R$ for simplicity.

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

Informally, if the complexity of $s$ with respect to $R$ is equal to $D$, then $s$ can be expressed in terms of the slopes in $R$ by a rational expression whose length (when expressed as string of characters) is comparable to $D$.  The rational complexity is reminiscent of the \emph{arithmetic circuit complexity} of $s$ in terms of $R$, but with the key difference that the circuit must take the specific rational form \eqref{rat}.

\begin{example}  $D(\{0,1,\infty\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$.  In particular, if $a,b$ are coprime then
$$D \left( \{0,1,\infty\};\frac{a}{b} \right) \asymp \log(2+|a|+|b|).$$
\end{example}

Our main result, proven in Section \ref{main-sec}, is then as follows.

Define a \emph{slope} to be an element $r$ of the projective rational\footnote{One could also formulate the arithmetic Kakeya conjecture in other fields than the rationals, and in fact most of the results here apply to arbitrary infinite fields, with minor modifications in the case where the field has positive characteristic. But in this paper we shall restrict attention for simplicity to the rational case, which is the case of interest for applications to the Kakeya problem.} line $\Q \cup \{\infty\}$. We then define the projection operators $\pi_r \colon \Q \times \Q \to \Q$ by setting
$$ \pi_r(x,y) \coloneqq x + ry$$
for $r \neq \{\infty\}$, and
$$ \pi_\infty(x,y) \coloneqq y.$$
Given a finite set $R$ of slopes and a further slope $s$ not in $R$, we define the \emph{sum-difference constant} $\SD(R;s)$ to be the least exponent such that the bound
\begin{equation}\label{sdef}
\H[ \pi_s(X,Y) ] \leq \SD(R;s) \max_{r \in R} \H[\pi_r(X,Y)]
\end{equation}
holds for $\Q$-valued random variables\footnote{In this paper, all random variables are understood to be discrete, and in fact take only finitely many values.} $X,Y$ (not necessarily independent), where
$$ \H[X] \coloneqq \sum_{x} \h(\P(X=x))$$
is the Shannon entropy of a random variable $X$, with $\h(t) \coloneqq t \log \frac{1}{t}$ using the convention $\h(0) \coloneqq 0$. Thus for instance
$$\H[ X-Y ] \leq \SD(\{0,1,\infty\};-1) \max\left( \H[X], \H[X+Y], \H[Y] \right)$$
whenever $X,Y$ are (possibly dependent) $\Q$-valued random variables.

It is easy to see that $\SD(R;s)=\infty$ when $|R|<2$.
From the entropy inequality $\H[X-Y] \leq \H[X,Y] \leq \H[X]+\H[Y]$ we see that $\SD(R;-1) \leq 2$ when $0,\infty \in S$, and it is easy to see (using the uniform distribution on a long arithmetic progression) that we have equality when $R = \{0,\infty\}$. By projective invariance, this implies that $\SD(R;s)=2$ whenever $|R|=2$. As these quantities are clearly non-decreasing in $R$, we then have the trivial bound
\begin{equation}\label{sdf-upper}
\SD(R;s) \leq 2
\end{equation}
for $|R| \geq 2$.

Improvements upon \eqref{sdf-upper} directly lead to improved upper bounds on the dimension of Kakeya and Nikodym sets in high dimensions. Indeed, in \cite{bourgain} it was observed that a bound of the form $\SD(R;-1) \leq \alpha$ implies that Kakeya and Nikodym sets in dimension $d$ have (upper) Minkowski dimension\footnote{It was recently observed by Thomas Bloom (private communication) that, by combining Bourgain's arguments with the recent quantitative progress on Szemer\'edi's theorem by Leng, Sawhney, and Sah \cite{lss}, that one also obtains this bound for the Hausdorff dimension as well. For the corresponding results for packing dimension, see \cite{cowen}.} at most $\frac{d-1}{\alpha}+1$. In particular, if one can establish the \emph{arithmetic Kakeya conjecture}
\begin{equation}\label{infr}
\inf_R \SD(R;-1) = 1
\end{equation}
then this would imply that Kakeya and Nikodym sets in $\R^d$ have full Minkowski and Hausdorff dimension for all $d$. This is currently only known for $d \leq 3$ \cite{wang}. We refer the reader to \cite{green}, \cite{cowen}, \cite{pohoata} for a discussion of this conjecture (and other equivalent forms of it), and its connection with other variants of the Kakeya conjecture.

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

\begin{example} $D(\{0,1,\infty\};s)$ is the least natural number for which one can express $s$ as $a/b$ where $a,b$ are integers of magnitude at most $2^D$. In particular, if $a,b$ are coprime then
$$D \left( \{0,1,\infty\};\frac{a}{b} \right) \asymp \log(2+|a|+|b|).$$
\end{example}

Our main result, proven in Section \ref{main-sec}, is then as follows.

\label{main-sec}

We now prove Theorem \ref{main-2}.  We begin with part (i).  Write $s=a/b$, where $b \geq 1$ and $a \neq 0,b$ is coprime to $b$, then $D(R;s) \asymp \log(2+|a|+|b|)$.  Let $N$ be the

\begin{definition}[Rational complexity] Given a family of slopes $R = \{0,1,\infty,r_1,\dots,r_k\}$ containing $0,1,\infty$ and a further slope $s$ not in $R$, we define the \emph{rational complexity} $D = D(R;s)$ of $s$ relative to $R$ to be the least natural number $D$ for which one has a representation of the form
\begin{equation}\label{rat}
 s = \frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
 \end{equation}
where $P,Q$ are polynomials of degree at most $D$ with integer coefficients of magnitude at most $2^D$, with $Q(r_1,\dots,r_k)$ non-zero; this complexity is finite since $s$ is rational.
\end{definition}

Our main result, proven in Section \ref{main-sec}, is then as follows.

\begin{figure}
 \centering
\centerline{\includegraphics[width=\linewidth]{best.png}}
 \caption{Lower bounds on $\SD(\{0,1,\infty\}; s)$ obtained by \texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \log(|a|+b|)$. The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}
 \label{fig:kak}
\end{figure}

\begin{theorem}[Continuous limit]\label{cts-limit} Let $\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \to \infty$ and $a/b \to \alpha$. Then
\begin{equation}\label{up}
 \SD\left(\{0,1,\infty\};\frac{a}{b}\right) \geq 2 - \frac{c_\alpha+o(1)}{\log b}
 \end{equation}
 as $b\to \infty$ and $a/b \to \alpha$, where $0 < c_\alpha < \infty$ is the infimum of all quantities
\begin{equation}\label{hf}
 h_2(f_{(\alpha)}) - \max(h(f_0), h(f_\infty), h(f_\alpha))
 \end{equation}
where $f \colon \R^2 \to \R^+$ ranges over smooth compactly supported functions with total mass $\int_{\R^2} f(x,y)\ dx dy$ equal to $1$, $f_{(\alpha)} \colon \R \times [0,1) \to \R^+$ is a ``folded'' version
\begin{equation}\label{falpha-def}
 f_{(\alpha)}(x,y) \coloneqq \sum_{j \in \Z} f(x-\alpha j, y+j)
 \end{equation}
of $f$, $f_0, f_\infty, f_1 \colon \R \to \R^+$ are the projections
\begin{align}
f_0(x) &\coloneqq \int_\R f(x,y)\ dy\label{f0-def}\\
f_\infty(y) &\coloneqq \int_\R f(x,y)\ dx\label{finfty-def}\\
f_1(z) &\coloneqq \int_\R f(x,z- x)\ dx,\label{f1-def}
\end{align}
the two-dimensional differential entropy $h_2(f_{(\alpha)})$ of $f_{(\alpha)}$ is defined as
$$h_2(f_{(\alpha)}) \coloneqq \int_0^1 \int_\R \h(f_{(\alpha)}(x,y))\ dx dy,$$
and the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\infty$ is defined as
$$ h(f_r) \coloneqq \int_\R \h(f_r(x))\ dx.$$
\end{theorem}

\begin{proposition}\label{dilate} Let $X$ be an $\Q$-valued random variable, and for any $a \in \Q$, let $g(a)$ denote the quantity $g(a) \coloneqq \H[X-aX'] - \H[X]$, where $X'$ is an independent copy of $X$.
\begin{itemize}
 \item[(i)] $g(0) = 0$ and $g(1)=d[X;X]$.
 \item[(ii)] For any $a \in \Q$, we have $g(-a) \leq 3g(a)$, and if $a$ is non-zero, $g(a^{-1}) = g(a)$.
 \item[(iii)] For any $k \geq 1$ and $a_1,\dots,a_k \in \Q$, we have $g(a_1 \dots a_k) \leq g(a_1)+ \dots + g(a_k)$ and $g(a_1 + \dots + a_k) \leq g(a_1)+\dots+g(a_k)+(k-1) g(1)$.
 \item[(iv)] If $a$ is a non-zero integer, then $g(a) \leq (4 + 10 \lfloor \log_2 |a| \rfloor) g(1)$.
 \item[(v)] If $R$ is a finite set of slopes containing $0,1,\infty$ of cardinality $k+3$, then
 $$ g(a) \ll D(R;a)^{k+1} \max_{r \in R \backslash \{\infty\}} g(r).$$
\end{itemize}
\end{proposition}

In this section we establish \Cref{cts-limit}. Let $\alpha$, $a$, $b$, $f$, $f_0, f_1, f_\infty$ be as in that theorem. We allow implied constants in the asymptotic notation to depend on $\alpha, f$. We will establish the lower bound
\begin{equation}\label{sd-targ}
 \SD\left(\{0,1,\infty\},\frac{a}{b}\right) \geq 2 - \frac{h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))+o(1)}{\log b}.
 \end{equation}
Comparing this with Theorem \ref{main-2}, we conclude that the expression $h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))$ is bounded from below, hence $c_\alpha > 0$; by taking an arbitrary test function for $f$ we also see that $c_\alpha < \infty$. Taking $h_2(f_\alpha) - \max(h(f_0), h(f_\infty), h(f_1))$ arbitrarily close to $c_\alpha$, we obtain the desired claim \eqref{up}.

It remains to establish \eqref{sd-targ}.
By the Poisson summation formula, the rapid decrease of the Fourier transform of the smooth compactly supported $f$, and the mass one hypothesis, we have
\begin{equation}\label{bib}
 \frac{1}{b^2} \sum_{n,m \in \Z} f\left(\frac{n}{b}, \frac{m}{b}\right) = 1 + \eps
 \end{equation}
for some $\eps = O(1/b)$; in fact one can get much better decay than this, but for our purposes any decay faster than $1/\log b$ will suffice. We then take $X,Y$ to be supported on the grid $\Z^2$ with probability distribution
$$ \P((X,Y) = (n,m)) \coloneqq \frac{1}{(1+\eps) b^2} \left(\frac{n}{b}, \frac{m}{b}\right)$$
thus $(X,Y)$ takes values in a ball of radius $O(b)$ and
\begin{equation}\label{pnx}
\P((X,Y) = (n,m)) = \frac{1}{b^2} f\left(\frac{n}{b}, \frac{m}{b}\right) + O\left( \frac{1}{b^3} \right).
\end{equation}
By Bezout's theorem, any integer can be uniquely written in the form $bn+am$ for some $0 \leq m < b$, and any other representation of the form $bn'+am'$ takes the form $b(n-aj) + a(m+bj)$. Thus
$$ \P(bX+aY = bn+am) = \sum_{j \in\Z} \P( (X,Y) = (n-aj, m+bj) ).$$
There are only $O(1)$ values of $j$ for which this sum is non-zero, so from \eqref{pnx} one has
$$ \P(bX+aY = bn+m) = \frac{1}{b^2} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) + O\left( \frac{1}{b^3} \right).$$
where $f_{(a,b)}: \R \times [0,1) \to \R^+$ is the function
$$f_{(a/b)}(x,y) \coloneqq \sum_{j \in \Z} f\left(x-\frac{a}{b} j, y+j\right).$$
Applying the entropy function $\h$, we conclude that
$$ \h(\P(bX+aY = bn+am)) = \frac{2 \log b}{b^2} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) +
\frac{1}{b^2} \h\left(f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right)\right) + O\left( \frac{\log b}{b^3} \right).$$
From \eqref{bib} one has
$$ \frac{1}{b^2} \sum_{n \in\Z; 0 \leq m < b} f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right) = 1 + \eps = 1 + O\left(\frac{1}{b} \right)$$
and from (uniform) Riemann integrability of the $f_{(a/b)}$ one has
$$ \frac{1}{b^2} \sum_{n \in\Z; 0 \leq m < b} \h\left(f_{(a/b)}\left(\frac{n}{b}, \frac{m}{b}\right)\right) = h_2(f_{(a/b)}) + o(1).$$
Finally, from dominated convergence one has
$$ h_2(f_{(a/b)}) = h_2(f_{(\alpha)})+o(1)$$
so we conclude that
$$ \H\left[\pi_{a/b}(X,Y)\right] = \H[bX+aY] = 2 \log b + h_2(f_{(\alpha)}) + o(1).$$
In a similar vein, from another application of Poisson summation and \eqref{f0-def} we see that
$$ \P(X=n) = \frac{1}{b} f_0\left(\frac{n}{b}\right) + O\left(\frac{1}{b^2}\right)$$
for any integer $n$, hence
$$ \h(\P(X=n)) = \frac{\log b}{b} f_0\left(\frac{n}{b}\right) + \frac{1}{b} \h\left(f_0 \left(\frac{n}{b}\right)\right) + O\left(\frac{\log b}{b^2}\right).$$
Since $X = O(b)$, one can sum using Riemann integrability and \eqref{bib} to conclude that
$$ \H[\pi_0(X,Y)] = \H[X]= \log b + h(f_0) + o(1).$$
Similar arguments give
$$ \H[\pi_\infty(X,Y)] = \H[Y] = \log b + h(f_\infty) + o(1)$$
and
$$ \H[\pi_1(X,Y)] = \H[X+Y] = \log b + h(f_1) + o(1).$$
Comparing this with \eqref{sdef}, we obtain \eqref{sd-targ}.

\begin{figure}
    \centering
\centerline{\includegraphics[width=\linewidth]{best.png}}
    \caption{Lower bounds on $\SD(\{0,1,\infty\}; s)$ obtained by \texttt{AlphaEvolve} for various $s=a/b$, plotted against $2 - 0.16 / \log(|a|+b|)$.  The horizontal axis is $|a|+|b|$ (so in some cases, multiple fractions $a/b$ are plotted on a single vertical line).}
    \label{fig:kak}
\end{figure}

The logarithmic convergence in \eqref{2ab} was suggested to us by experiments in \cite{gdm} using \texttt{AlphaEvolve} to obtain lower bounds for $\SD(\{0,1,\infty\}; s)$ for various slopes $s$.  This data was of low accuracy, as \texttt{AlphaEvolve} could only provide lower bounds and not upper bounds for these quantities; nevertheless, a logarithmic decay was numerically evident (see Figure \ref{fig:kak}), and furthermore the approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$ obtained by this tool (see \cite[Figure 18]{gdm}) suggested an approach to make the lower bound in \eqref{2ab} rigorous.  Once this was accomplished, the author was also able to obtain the matching upper bound in \eqref{2ab} using the entropic Pl\"unnecke--Ruzsa calculus.   A modification of these arguments, with some inefficiencies, then gives \eqref{sam}. We tentatively conjecture that the bounds in \eqref{sam} can be improved to be of the form \eqref{2ab} for all $k$, not just $k=0$ (possibly after some slight adjustments to the definition of rational complexity).

In the limit as $a/b$ converges to some real number $\alpha$ and $b \to \infty$, we can obtain a more precise lower bound (also suggested by the aforementioned \texttt{AlphaEvolve} numerics) as follows.

\begin{theorem}[Continuous limit]\label{cts-limit}  Let $\alpha$ be a real number, and let $a,b$ be coprime integer parameters with $b \to \infty$ and $a/b \to \alpha$.  Then
\begin{equation}\label{up}
 \SD\left(\{0,1,\infty\};\frac{a}{b}\right) \geq 2 - \frac{c_\alpha+o(1)}{\log b}
 \end{equation}
 as $b\to \infty$ and $a/b \to \alpha$, where $0 < c_\alpha < \infty$ is the infimum of all quantities
\begin{equation}\label{hf}
 h_2(f_{(\alpha)}) - \max(h(f_0), h(f_\infty), h(f_\alpha))
 \end{equation}
where $f \colon \R^2 \to \R^+$ ranges over smooth compactly supported functions with total mass $\int_{\R^2} f(x,y)\ dx dy$ equal to $1$, $f_{(\alpha)} \colon \R \times [0,1) \to \R^+$ is a ``folded'' version
\begin{equation}\label{falpha-def}
 f_{(\alpha)}(x,y) \coloneqq \sum_{j \in \Z} f(x-\alpha j, y+j)
 \end{equation}
of $f$, $f_0, f_\infty, f_1 \colon \R \to \R^+$ are the projections
\begin{align}
f_0(x) &\coloneqq \int_\R f(x,y)\ dy\label{f0-def}\\
f_\infty(y) &\coloneqq \int_\R f(x,y)\ dx\label{finfty-def}\\
f_1(z) &\coloneqq \int_\R f(x,z- x)\ dx,\label{f1-def}
\end{align}
the two-dimensional differential entropy $h_2(f_{(\alpha)})$  of $f_{(\alpha)}$ is defined as
$$h_2(f_{(\alpha)}) \coloneqq \int_0^1 \int_\R \h(f_{(\alpha)}(x,y))\ dx dy,$$
and the one-dimensional differential entropies $h(f_r)$ for $r=0,1,\infty$ is defined as
$$ h(f_r) \coloneqq \int_\R \h(f_r(x))\ dx.$$
\end{theorem}

We establish this result in \Cref{cts-sec}.
We tentatively conjecture that the upper bound in \eqref{up} is in fact an asymptotic equality, so that the asymptotic behavior of
$ \SD\left(\{0,1,\infty\}, \frac{a}{b}\right)$ is controlled not only by the rational complexity (as represented by the $\log b$ denominator), but also by the variational quantity $c_\alpha$ appearing in the numerator.  It is not clear whether this quantity $c_\alpha$ can be computed exactly; numerically, two-dimensional gaussians are reasonably good candidates for $f$, but in practice they do not extremize the functional \eqref{hf} precisely.

\subsection{Notation}

We use the asymptotic notation $X = O(Y)$, $X \ll Y$, or $Y \gg X$ to denote the assertion that $|X| \leq CY$ for some absolute constant $C$; if we need this implied constant $C$ to depend on some fixed quantities (such as the number $k$ of slopes), we will indicate this in the text.

When there is no possibility of ambiguity, we omit parentheses from pairs $(X,Y)$ of random variables, for instance abbreviating $\H[(X,Y)]$ as $\H[X,Y]$.  Given a random variable $Y$ taking values in some set $S$ and some function $f \colon S \to \R$, we define the expectation
$$ \E_{Y=y} f(y) \coloneqq \sum_y \P(Y=y) f(y)$$
where $y$ ranges over the essential range of $Y$.  While this expression could also be abbreviated as $\E f(Y)$, it will be notationally useful to distinguish between the random variable $Y$ and the possible values $y$ that this variable could take.  For instance, with this notation, the \emph{conditional entropy} $\H[X|Y]$ of one random variable $X$ with respect to another $Y$ can now be defined by the formula
$$ \H[X|Y] \coloneqq \E_{Y=y} \H[ X | Y = y ]$$
where $(X|Y=y)$ is $X$ conditioned to the event $Y=y$ (again, we omit parentheses when there is no possibility of ambiguity).  The chain rule asserts that $\H[X|Y]$ can also be expressed by the formula
$$ \H[X|Y] = \H[X,Y] - \H[Y].$$
The \emph{mutual information} $\I(X:Y)$ between two random variables is given by the formula
$$ \I(X:Y) = \H[X] - \H[X|Y] = \H[Y] - \H(Y|X) = \H[X] + \H[Y] - \H[X,Y].$$
As is well known, $\I(X:Y)$ is non-negative, and vanishes precisely when $X,Y$ are independent.  Equivalently, one has the subadditivity property
$$ \H[X,Y] \leq \H[X] + \H[Y]$$
with equality precisely when $X,Y$ are independent.

We also define the conditional mutual information
$$ \I[X:Y|Z] \coloneqq \E_{Z=z} \I[(X|Z=z):(Y|Z=z)].$$
Clearly, $\I[X:Y|Z]$ is non-negative, and vanishes precisely when $X,Y$ are independent conditionally on $Z$.
From the chain rule we have
\begin{equation}\label{i-split}
\begin{split}
\I[X:Y|Z] &= \H[X|Z] - \H[X|Y,Z] \\
&= \H[Y|Z] - \H[Y|X,Z] \\
&= \H[X|Z] + \H[Y|Z] - \H[X,Y|Z].
\end{split}
\end{equation}

\subsection{Acknowledgments}

The author was supported by the James and Carol Collins Chair, the Mathematical Analysis \& Application Research Fund, and by NSF grants DMS-2347850, and is particularly grateful to recent donors to the Research Fund.  He particularly thanks his coauthors Bogdan Georgiev, Javier G\'omez-Serrano, and Adam Zsolt Wagner for the highly productive and enjoyable collaboration \cite{gdm}, and for generously sharing the outputs of that collaboration for the purposes of writing the current paper.

While some of the results proven here were suggested by the outcome of AI-assisted experiments, the arguments in this paper are completely human-generated.

The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in Theorem~\ref{main-2}. For the three-slope case \eqref{2ab}, the “logarithmic convergence” was “suggested… by experiments… using \texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in \eqref{2ab} rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in \eqref{2ab} using the entropic Pl\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives \eqref{sam}.”

The post-theorem introduction does not give a step-by-step proof, but it does outline the methods used to establish the bounds in the main theorem. For the three-slope case
\begin{equation}\label{2ab} 2 - \frac{c_2}{D(R;s)} \leq \SD\left(R; s\right) \leq 2 - \frac{c_1}{D(R;s)}
\end{equation}
, the “logarithmic convergence” was “suggested… by experiments… using \texttt{AlphaEvolve} to obtain lower bounds,” and the “approximate discrete gaussian shape of the joint distributions of the random variables $X,Y$… suggested an approach to make the lower bound in the equation above rigorous.” “Once this was accomplished, the author was also able to obtain the matching upper bound in the equation above using the entropic Pl\"unnecke--Ruzsa calculus.” Finally, “a modification of these arguments, with some inefficiencies, then gives
\begin{equation}\label{sam} 2 - \frac{C_k \log(2+D(R;s))}{D(R;s)} \leq \SD(R; s) \leq 2 - \frac{c_k}{D(R;s)^{k+1}}
    \end{equation}
.”

Question:
Let a slope mean an element of \(\mathbb{Q}\cup\{\infty\}\). For each slope \(r\), define \(\pi_r:\mathbb{Q}^2\to\mathbb{Q}\) by \(\pi_r(x,y)=x+ry\) if \(r\neq\infty\), and \(\pi_\infty(x,y)=y\). For a finite set of slopes \(R\) and a slope \(s\notin R\), define \(\mathrm{SD}(R;s)\) to be the least real number such that
\[
H[\pi_s(X,Y)]\le \mathrm{SD}(R;s)\max_{r\in R} H[\pi_r(X,Y)]
\]
for all finitely supported discrete \(\mathbb{Q}\)-valued random variables \(X,Y\) (not necessarily independent), where \(H[Z]=\sum_z \mathbb{P}(Z=z)\log\frac1{\mathbb{P}(Z=z)}\) is Shannon entropy.

Assume \(R=\{0,1,\infty,r_1,\dots,r_k\}\) is a finite family of slopes of cardinality \(k+3\), and \(s\notin R\). Define the rational complexity \(D=D(R;s)\) to be the least natural number such that
\[
s=\frac{P(r_1,\dots,r_k)}{Q(r_1,\dots,r_k)}
\]
for some polynomials \(P,Q\) of degree at most \(D\) with integer coefficients of magnitude at most \(2^D\), with \(Q(r_1,\dots,r_k)\neq 0\).

Which quantitative estimate holds for \(\mathrm{SD}(R;s)\)?

If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)}.
\]
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_k}{D(R;s)^{k+1}}.
\]

Choices:
B.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1\log(2+D(R;s))}{D(R;s)}.
\]
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_k}{D(R;s)^{k+1}}.
\]
C.
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2.
\]
D.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\) such that
\[
2-\frac{c_2}{D(R;s)^2}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)^2}.
\]
In general, there exist positive constants \(c_k,C_k\) such that
\[
2-\frac{C_k}{D(R;s)^{k+1}}\le \mathrm{SD}(R;s)\le 2-\frac{c_k\log(2+D(R;s))}{D(R;s)}.
\]
E.
If \(k=0\) (so \(R=\{0,1,\infty\}\)), then there exist absolute constants \(c_2>c_1>0\), independent of \(s\), such that
\[
2-\frac{c_2}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c_1}{D(R;s)}.
\]
Moreover, there exist absolute constants \(c,C>0\), independent of \(k\), such that for every finite \(R\) as above,
\[
2-\frac{C\log(2+D(R;s))}{D(R;s)}\le \mathrm{SD}(R;s)\le 2-\frac{c}{D(R;s)^{k+1}}.
\]

three-slope logarithmic-free rate vs many-slope logarithmic-loss rate

dropped the nontrivial upper improvement below 2

power of D and side of the estimate carrying the logarithmic loss

dependence of constants on k in the many-slopes bounds

The stem defines the objects carefully but does not reveal the quantitative estimate itself or uniquely signal the correct asymptotic form. There is no explicit answer leakage.

The item is essentially a direct theorem-recall question: it asks which estimate holds for the defined quantity, and the correct option states the theorem almost verbatim rather than requiring transfer to a new setting.

Some discrimination is needed among subtle asymptotic variants involving logarithmic losses, powers of D, and dependence on k, but the task mainly tests recognition/memorization of the exact result rather than generative mathematical reasoning.

The distractors are mathematically plausible and distinct: they alter log factors, swap exponents, weaken the upper bound, or mishandle dependence on k. These reflect realistic failure modes in recalling or parsing such estimates.

Strong on avoiding leakage and using plausible distractors, but weak as a reasoning assessment because it is largely a direct restatement/recall of a specialized theorem.


[Rank-decomposition and rankwidth \cite{Oum_Seymour_2006, Voigt}] Given a graph $G=(V, E)$, a \emph{rank-decomposition} of $G$ is a pair $(T,\gamma)$ where $T$ is a subcubic tree (i.e., every internal node has degree at most $3$) and $\gamma\colon V(G)\to\operatorname{Leaf}(T)$ is a bijection. Observe that
deleting any edge $e\in E(T)$ splits the set of leaves into parts $(X_e,Y_e)$, thereby inducing a cut in $G$.
Then, the rankwidth of $G$ is given by
\begin{align}\mathrm{rankwidth}(G) \coloneqq \min_{(T,\gamma)} \max_{e\in E(T)}\operatorname{cutrank}_G(X_e,Y_e).\end{align}

The relationship between combinatorial width parameters and connectivity properties is a cornerstone of structural and algorithmic graph theory \cite{BODLAENDER19981, SEYMOUR199322, ROBERTSON1994323, harvey2016parameterstiedtreewidth, BOTTCHER20101217}.
For tree-width, the celebrated Grid-Minor Theorem of Chekuri and Chuzhoy \cite{10.1145/2820609} and its recent developments \cite{doi:10.1137/1.9781611977912.48, Geelen_2023} characterizes large width via the presence of an $r\times r$ grid minor, while branch-width \cite{fomin2021fastfptapproximationbranchwidth,https://doi.org/10.1002/net.20050} admits a well-linked-set characterization due to Oum and Seymour \cite{Oum_Seymour_2006}. While such relationships are well-understood for treewidth and branchwidth, a corresponding characterization for rankwidth and its sibling, clique-width, remains a gap in the literature: indeed, no well-linkedness or expander-style certificate has been pinned down. Here, we provide a proof of existence of such a certificate by establishing a linear dependence between rankwidth and another parameter, the minimum balanced cutrank. We begin by defining the minimum balanced cutrank. \\

A \emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\subseteq V(G)$ such that $X\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \emph{cutrank} of the separation, $\operatorname{cutrank}(X,Y)$, is the $\mathbb{F}_2$-rank of its $|X|\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \emph{balanced} if $\tfrac13|V(G)|\le|X|$ and $|Y|\le\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \emph{minimum balanced cutrank}, $\operatorname{min-bal-cutrank}(G)$, by
\begin{align}\operatorname{min-bal-cutrank}(G) = \min\mathopen{}\mathclose\bgroup\originalleft\{\operatorname{cutrank}(X,Y)\text{ }\Big| \text{ }X\sqcup Y = V(G)\text{ with }|X|, |Y| \leq \frac{2n}{3}\aftergroup\egroup\originalright\}. \end{align}

Informally, a graph whose minimum balanced cutrank is large should have no bisection with low rank, and therefore behaves like an \emph{expander graph} when measured in $\mathbb{F}_2$ rank. A vertex set is said to be an expander if it has many external neighbors. We call a graph bounded if every sufficiently large subgraph contains a subset which is not expanding. There is a long line of literature on expander graphs for their ubiquitous use in pseudorandomness \cite{Alon1986Eigenvalues,anand2025pseudorandomnessstickyrandomwalk,golowich_et_al:LIPIcs.CCC.2022.27,anand2025pseudorandomnessexpanderrandomwalks}, unique games \cite{bafna2021highdimensionalexpanderseigenstripping}, log-space computation \cite{Reingold2008}, the PCP theorem \cite{10.1145/1236457.1236459}, fault-tolerant computation \cite{AjtaiAlonBruckEtAl1992,chaudhari2025peer}, and a number of other applications. We refer the interested reader to the (incredible) surveys of Lubotzky \cite{lubotzky2011expandergraphspureapplied} and Hoory, Linial, and Wigderson \cite{Hoory2006}. However, the combinatorial notion of expansion is only meaningful for sparse graphs, and it is desirable in a number of applications to have an analog of expansion for dense graphs \cite{alon2003densegraphsantimagic, alon1992almost, doi:10.1137/18M122039X, 10.1016/0012-365X(95)00242-O, Alon_Nachmias_Shalev_2022, alon1998packing, srivastava2017alonboppanatypeboundweighted}. There are a number of concepts that have been studied in the literature previously, such as flip-width \cite{K_hn_2014,toruńczyk2024flipwidthcopsrobberdense} and twin-width \cite{bonnet2022}, but prior to now there was no agreed-upon notion.\\

For instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion. While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \emph{rank-expansion} is based on the $\mathbb{F}_2$-rank of the adjacency matrix.
This offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \ref{thm:main} proposes that \emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

A \emph{separation} of a graph $G=(V, E)$ is a pair $X,Y\subseteq V(G)$ such that $X\sqcup Y = V(G)$. Given a separation $(X,Y)$ of $V(G)$, the \emph{cutrank} of the separation, $\operatorname{cutrank}(X,Y)$, is the $\mathbb{F}_2$-rank of its $|X|\times|Y|$ biadjacency matrix. We say that a separation $(X,Y)$ is \emph{balanced} if $\tfrac13|V(G)|\le|X|$ and $|Y|\le\tfrac23|V(G)|$. Next, for any graph $G=(V,E)$, define the \emph{minimum balanced cutrank}, $\operatorname{min-bal-cutrank}(G)$, by
\begin{align}\operatorname{min-bal-cutrank}(G) = \min\left\{\operatorname{cutrank}(X,Y)\text{ }\Big| \text{ }X\sqcup Y = V(G)\text{ with }|X|, |Y| \leq \frac{2n}{3}\right\}. \end{align}

For instance, dense-graph expansion could be defined by the existence of a graphon sequence of finite cut-norm with a growing number of expander subgraphs of degree $O(1)$; however, this notion is maximized by a clique, for which we instead desire a small value. Information-theoretically, it is helpful for a graph and its complement to have similar dense-graph measures of expansion. While twin-width is defined via contraction sequences and flip-width via a game-theoretic process, \emph{rank-expansion} is based on the $\mathbb{F}_2$-rank of the adjacency matrix.
This offers a direct linear-algebraic perspective that is intuitively similar to the spectral definitions of expansion for sparse graphs. Moreover, a good parameter for expansion in dense graphs should be small on structurally-simple dense graphs (like cliques) and large on complex, random-looking dense graphs. This is satisfied by rankwidth, since the rankwidth of a clique is $1$ while the rankwidth of a random graph $G\sim G(n,p)$ is linear in $n$ for constant $p$, with high probability. With this perspective, our main result in Theorem \ref{thm:main} proposes that \emph{rank-expansion is a suitable candidate for the analog of expansion for dense graphs}. Moreover, when working over $\mathbb{F}_2$, taking the complement of the graph is equivalent to adding the all-ones matrix, which only changes the cutrank by at most $1$.

Motivated by this definition of rankwidth, we now provide a formal definition of \emph{rank-expansion}.

\begin{definition}[Rank-Expansion]\label{def: rank-expansion}
For a graph $G=(V,E)$ where $|V|=n$, the \emph{rank-expansion} is given by
 \begin{align}\operatorname{rank-expansion}(G) \coloneqq \max_{S\subseteq V(G): |S|\leq n/2} \frac{\operatorname{cutrank}(S, V(G)\setminus S)}{|S|}.\end{align}
\end{definition}

\noindent Theorem~\ref{thm:main} parallels the grid-minor theorem of Chekuri and Chuzhoy \cite{doi:10.1137/1.9781611977912.48, Geelen_2023, 10.1145/2820609}: delete vertices instead of contracting edges, and replace edge cuts by $\mathbb{F}_2$-rank cuts. For any graph $G$, this implies that
\begin{align}
\tfrac1{72}\,\mathrm{rankwidth}(G)\;\le\;
\max_{H\subseteq G}\operatorname{min-bal-cutrank(H)}
\;\le\;
\mathrm{rankwidth}(G).
\end{align}

\noindent Here, the right inequality holds because for any graph $H$ and any rank-decomposition of $H$ of width $k$, there exists a balanced cut $(A, B)$ with $|A|,|B|\geq |V(H)|/3$ such that $\operatorname{cutrank}_H(A,B) \leq k$. Therefore, for every induced $H\subseteq G$, the minimum cutrank (across balanced sets $A$ and $B$) is $\operatorname{cutrank}_H(A,B)\le \mathrm{rankwidth}(H)\le \mathrm{rankwidth}(G)$: by properties of tree decompositions, any rank-decomposition of width $k$ contains a balanced cut of rank at most $k$. The left inequality follows from Theorem \ref{thm:main}, since if $\mathrm{rankwidth}(G)\ge 72r$, then
there is an induced $H$ satisfying $\min_{(A,B)}\operatorname{cutrank}_H(A,B)\ge r$, for some balanced partition $(A,B)$ of $V(H)$.\\

First, in Claim \ref{claim:rank-tangle existence}, we adapt the tangle-and-congestion machinery of Dvořák and Norin \cite{Dvorak_Norin_2018} to extract the more strongly-connected side $X$ of a low-rank separation $(X,Y)$ that is maximally large yet \emph{well-behaved}. This well-behaved property formalizes the idea that $X$ is robustly connected to $Y$, meaning no small component of $X$ can be separated from the rest of the graph with a low-rank cut. Next, we observe that the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$ is bounded. This allows us to find a small subset of representative vertices in $X$ that captures all of these connections, effectively compressing $X$ without losing essential rank information. Specifically, we compress $X$ by taking one representative per $Y$-neighborhood class (Claim \ref{claim:findingH}), which preserves $\operatorname{cutrank}(X,Y)$; then, in Lemma \ref{lemma: splitting Y reduction}, we form an induced subgraph $H'$ so that any balanced cut of $H'$ of rank at most $r$ splits $Y$ and yields two lifted separations of $G$ whose cutranks are at most $72r$; combined with the tangle axiom this yields the existence of a strongly-connected side in $V(G)$ that is strictly larger than $X$, thereby contradicting the maximality assumption.\\

\begin{definition}
 Suppose $(X, Y)$ is a partition of $V(G)$. For each $u,v\in Y$, we define equivalence classes, where each class has the relation $u\sim v$ iff $N_G(u)\cap X = N_G(v) \cap X$. The family $\mathcal F=\{\,N_G(y)\cap X : y\in Y\}$ of subsets of $X$ has \emph{VC-dimension} $\operatorname{VC}(\mathcal F)$, the largest $S\subseteq X$ shattered by $\mathcal F$.
Then, Sauer–Shelah's lemma \cite{SAUER1972145,pjm1102968432} bounds $|\mathcal F|$ (the number of $X$-equivalence classes) by $\sum_{i=0}^{d}\binom{|X|}{i}=O(|X|^{d})$ where $d=\operatorname{VC}(\mathcal F)$.
\end{definition}
Let the number of equivalence classes be denoted by $N_Y$. Then, we have that $\log(N_Y) \leq \operatorname{cutrank}(X,Y) \leq {N_Y}$ since the rank cannot exceed the number of distinct column types (equivalence classes), and over $\mathbb{F}_2$, $k$ dimensions can encode at most $2^k$ distinct neighborhood patterns. Furthermore, if $N_Y \leq O(n^{1-\epsilon})$, then a number of separator theorems emerge, due to the works of Feige, Hajiaghayi, and Lee \cite{Feige_Hajiaghayi_Lee_2008}.

\begin{claim}
\label{claim:findingH}
    There exists an induced subgraph $H\subseteq G$ which is obtained from $G$ by deleting some vertices from $X$ such that for  all $v\in X$, there exists at least one vertex $v'\in X\cap V(H) \eqqcolon X^*$ such that $N(v)\cap Y = N(v')\cap Y$.
\end{claim}

\begin{claim}
\label{claim:rank-tangle existence}
    There exists a partition $(X,Y)$ of $V(G)$ such that $\operatorname{cutrank}(X,Y) < 70r$, the rank-tangle $\tau$ of order $72r$ orients this partition towards $X$ (i.e., $X \in \tau$), and $(X,Y)$ is $(3|Y|, 1/7)$-well-behaved. Moreover, $|X|$ is maximum among all separations oriented by $\tau$ with $\mathrm{cutrank}<70r$ (and, subject to this, $\mathrm{cutrank}(X,Y)$ is minimum).
\end{claim}

\begin{lemma}[Splitting $Y$]
    \label{lemma: splitting Y reduction}
    There exists an induced subgraph $H'\subseteq H$ and a balanced partition $(A', B')$ of $H'$ such that $A' \cap Y \neq \emptyset$ and $B'\cap Y\neq\emptyset$, $\operatorname{cutrank}_{H'}(A', B') < r$, and if $X^{**} \eqqcolon V(H') \cap X^*$, then \[\operatorname{cutrank}_{H'}(X^{**}, Y) \leq \operatorname{cutrank}_H(X^*, Y) = \operatorname{cutrank}_G(X,Y) < 70r.\]
\end{lemma}

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

Motivated by this definition of rankwidth, we now provide a formal definition of \emph{rank-expansion}.

\begin{definition}[Rank-Expansion]\label{def: rank-expansion}
For a graph $G=(V,E)$ where $|V|=n$, the \emph{rank-expansion} is given by
    \begin{align}\operatorname{rank-expansion}(G) \coloneqq \max_{S\subseteq V(G): |S|\leq n/2} \frac{\operatorname{cutrank}(S, V(G)\setminus S)}{|S|}.\end{align}
\end{definition}

We show that large rankwidths of a graph $G$ enforce the existence of a large, rank-robust subgraph $H$.

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

\noindent Theorem~\ref{thm:main} parallels the grid-minor theorem of   Chekuri and Chuzhoy \cite{doi:10.1137/1.9781611977912.48, Geelen_2023, 10.1145/2820609}: delete vertices instead of contracting edges, and replace edge cuts by $\mathbb{F}_2$-rank cuts. For any graph $G$, this implies that
\begin{align}
\tfrac1{72}\,\mathrm{rankwidth}(G)\;\le\;
\max_{H\subseteq G}\operatorname{min-bal-cutrank(H)}
\;\le\;
\mathrm{rankwidth}(G).
\end{align}

\noindent Here, the right inequality holds because for any graph $H$ and any rank-decomposition of $H$ of width $k$, there exists a balanced cut $(A, B)$ with $|A|,|B|\geq |V(H)|/3$ such that $\operatorname{cutrank}_H(A,B) \leq k$. Therefore, for every induced $H\subseteq G$, the minimum cutrank (across balanced sets $A$ and $B$) is $\operatorname{cutrank}_H(A,B)\le \mathrm{rankwidth}(H)\le \mathrm{rankwidth}(G)$: by properties of tree decompositions, any rank-decomposition of width $k$ contains a balanced cut of rank at most $k$. The left inequality follows from Theorem \ref{thm:main}, since if $\mathrm{rankwidth}(G)\ge 72r$, then
there is an induced $H$ satisfying $\min_{(A,B)}\operatorname{cutrank}_H(A,B)\ge r$, for some balanced partition $(A,B)$ of $V(H)$.\\

Oum’s vertex-minor theorem states that rankwidth~$k$ forces a $K_t$ minor with $t\approx2^k$ and provides a set of excluded vertex-minors for graphs of bounded rankwidth, which is analogous to the Robertson-Seymour theorem for graph minors \cite{Oum_Seymour_2006}.
In contrast, our result finds an {induced} subgraph such that  {every} balanced separation of the subgraph has rank $\Theta(k)$. Thus, graphs with large width must contain a highly connected induced subgraph with robust cutrank. Dvořák and Norin show that large tree-width forces large minimum balanced \emph{edge}-separators~\cite{Dvorak_Norin_2018}; in our work, we show how to transfer this to vertex-rank cuts. Finally, Oum and Seymour provide approximation algorithms for rankwidth \cite{Oum_Seymour_2006}; in this spirit, Theorem \ref{thm:main} supplies a proof of existence of an alternative certificate that may lead to new parameter-testing routines. \\

Through our result, we suggest rank-expansion as a candidate notion of expansion suitable for dense graphs. Indeed, our result can analogously be viewed as a Cheeger-type inequality for dense graphs \cite{Cheeger1970,AlonMilman1985}, which adds to the growing literature in the field \cite{khetan2018cheegerinequalitiesgraphlimits,jost2024cheegerinequalitiessimplicialcomplexes,10.1145/3564246.3585139,briani2022classcheegerinequalities, 10.1145/2582112.2582118, schild2018schurcomplementcheegerinequality, first2025cheegerinequalitycoboundaryexpansion}. For instance, Cheeger gives that $c_1(1 - \lambda_2) \leq \phi(G) \leq c_2\sqrt{1-\lambda_2}$, where $\phi(G)$ is the conductance of $G$, and $\lambda_2$ is the second largest (in absolute value) eigenvalue of $G$. Conversely, Theorem \ref{thm:main} gives $\frac{1}{72}\operatorname{rankwidth}(G) \leq \rho(G) \leq \operatorname{rankwidth}(G)$, where $\rho(G)$ is the minimum balanced cutrank of any subgraph of $G$, i.e. $\rho(G) = \max_{H\subseteq G} \operatorname{min-bal-cutrank}_H(A,B)$. In this vein, our work is most related to that of \cite{K_hn_2014} which shows a different result that dense regular graphs can be partitioned into robust expanders, which are subgraphs that still have good expansion properties after removing some vertices and edges. In turn, this can be applied to Hamiltonicity problems in dense regular graphs.

\paragraph{Proof overview.} Our proof proceeds by way of contradiction. If a graph has high rankwidth but low min-bal-cutrank, then every part of the graph can be cleanly separated with a low-rank cut. We introduce the tangle in definition \ref{def: rank-tangle of order k}, which points to the `more connected' component of a separation of a graph. We use it to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected. Subject to this, we assume $X$ is maximal and call it the {small side}, following the convention of \cite{Dvorak_Norin_2018}. We then \emph{compress} $X$ by selecting a subset of {representative} vertices, and show that finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality. Our proof continues via three stages.\\

First, in Claim \ref{claim:rank-tangle existence}, we adapt the tangle-and-congestion machinery of Dvořák and Norin \cite{Dvorak_Norin_2018} to extract the more strongly-connected side $X$ of a low-rank separation $(X,Y)$ that is maximally large yet \emph{well-behaved}. This well-behaved property formalizes the idea that $X$ is robustly connected to $Y$, meaning no small component of $X$ can be separated from the rest of the graph with a low-rank cut. Next, we observe that the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$ is bounded. This allows us to find a small subset of representative vertices in $X$ that captures all of these connections, effectively compressing $X$ without losing essential rank information. Specifically, we compress $X$ by taking one representative per $Y$-neighborhood class (Claim \ref{claim:findingH}), which preserves $\operatorname{cutrank}(X,Y)$; then, in Lemma \ref{lemma: splitting Y reduction}, we form an induced subgraph $H'$  so that any balanced cut of $H'$ of rank at most $r$ splits $Y$ and yields two lifted separations of $G$ whose cutranks are at most $72r$; combined with the tangle axiom this yields the existence of a strongly-connected side in $V(G)$ that is strictly larger than $X$, thereby contradicting the maximality assumption.\\

\paragraph{Proof overview.} The proof is "by way of contradiction": assuming "a graph has high rankwidth but low min-bal-cutrank," so that "every part of the graph can be cleanly separated with a low-rank cut." The argument introduces "the tangle in definition \\ref{def: rank-tangle of order k}, which points to the `more connected' component of a separation," and uses it "to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected." With $X$ chosen "maximal" (the "small side"), one then "compress[es] $X$ by selecting a subset of representative vertices," and shows that "finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality."

The proof proceeds in three stages: (1) In Claim~\\ref{claim:rank-tangle existence}, "adapt the tangle-and-congestion machinery of Dvořák and Norin" to extract a low-rank separation $(X,Y)$ where $X$ is "maximally large" and "well-behaved," meaning "$X$ is robustly connected to $Y$" ("no small component of $X$ can be separated from the rest of the graph with a low-rank cut"). (2) Bound "the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$" and thereby find "a small subset of representative vertices in $X$"; concretely, "compress $X$ by taking one representative per $Y$-neighborhood class (Claim~\\ref{claim:findingH}), which preserves $\\operatorname{cutrank}(X,Y)$." (3) In Lemma~\\ref{lemma: splitting Y reduction}, form an induced subgraph $H'$ so that "any balanced cut of $H'$ of rank at most $r$" forces a split of $Y$ and gives "two lifted separations of $G$ whose cutranks are at most $72r$"; "combined with the tangle axiom" this yields "a strongly-connected side" strictly larger than $X$, contradicting maximality, and thus establishing Theorem~\\ref{thm:main}.

\paragraph{Proof overview.} The proof is "by way of contradiction": assuming "a graph has high rankwidth but low min-bal-cutrank," so that "every part of the graph can be cleanly separated with a low-rank cut." The argument introduces the following definition.

\begin{definition}[Rank-tangle of order $k$] \label{def: rank-tangle of order k}
Following Robertson-Seymour \cite{ROBERTSON1991153}, a rank-tangle of order $k$ is a set $\mathcal{T}$ of subsets of $V(G)$ such that \begin{enumerate}\item For every partition $(X,Y)$ of $V(G)$ with $\operatorname{cutrank}_G(X,Y)<k$, exactly one of $X$ or $Y$ is in $\mathcal{T}$ (and is called the ``small side'')\item No three chosen small sides have union $V(G)$; i.e., for any sets $S_1, S_2, S_3 \in \mathcal{T}$, their union $S_1 \cup S_2 \cup S_3 \neq V(G)$.\end{enumerate}
The existence of a rank-tangle of order $k$ witnesses that $\mathrm{rankwidth}(G)\geq k$.
\end{definition}

This tangle points to the `more connected' component of a separation, and we use it to find a separation $(X,Y)$ such that the tangle-selected component is maximally connected. With $X$ chosen "maximal" (the "small side"), one then "compress[es] $X$ by selecting a subset of representative vertices," and shows that "finding a low-rank balanced cut in this compressed graph would allow us to extend $X$, which contradicts its maximality."

The proof proceeds in three stages: (1) We use the following claim to extract a low-rank separation $(X,Y)$ where $X$ is "maximally large" and "well-behaved," meaning "$X$ is robustly connected to $Y$" ("no small component of $X$ can be separated from the rest of the graph with a low-rank cut").

\begin{claim}
\label{claim:rank-tangle existence}
    There exists a partition $(X,Y)$ of $V(G)$ such that $\operatorname{cutrank}(X,Y) < 70r$, the rank-tangle $\tau$ of order $72r$ orients this partition towards $X$ (i.e., $X \in \tau$), and $(X,Y)$ is $(3|Y|, 1/7)$-well-behaved. Moreover, $|X|$ is maximum among all separations oriented by $\tau$ with $\mathrm{cutrank}<70r$ (and, subject to this, $\mathrm{cutrank}(X,Y)$ is minimum).
\end{claim}

(2) We bound "the number of distinct neighborhood patterns that vertices in $Y$ can form within $X$" and thereby find "a small subset of representative vertices in $X$"; concretely, we "compress $X$ by taking one representative per $Y$-neighborhood class" using the following claim, which preserves $\operatorname{cutrank}(X,Y)$.

\begin{claim}
\label{claim:findingH}
    There exists an induced subgraph $H\subseteq G$ which is obtained from $G$ by deleting some vertices from $X$ such that for  all $v\in X$, there exists at least one vertex $v'\in X\cap V(H) \eqqcolon X^*$ such that $N(v)\cap Y = N(v')\cap Y$.
\end{claim}

(3) We then use the following lemma to form an induced subgraph $H'$ so that "any balanced cut of $H'$ of rank at most $r$" forces a split of $Y$ and gives "two lifted separations of $G$ whose cutranks are at most $72r$"; combined with the tangle axiom, this yields a strongly-connected side strictly larger than $X$, contradicting maximality.

\begin{lemma}[Splitting $Y$]
    \label{lemma: splitting Y reduction}
    There exists an induced subgraph $H'\subseteq H$ and a balanced partition $(A', B')$ of $H'$ such that $A' \cap Y \neq \emptyset$ and $B'\cap Y\neq\emptyset$, $\operatorname{cutrank}_{H'}(A', B') < r$, and if $X^{**} \eqqcolon V(H') \cap X^*$, then \[\operatorname{cutrank}_{H'}(X^{**}, Y) \leq \operatorname{cutrank}_H(X^*, Y) = \operatorname{cutrank}_G(X,Y) < 70r.\]
\end{lemma}

This contradiction completes the proof of the following theorem.

\begin{restatable}{theorem}{MainThm}
\label{thm:main}
Let $r\ge1$ and $G=(V, E)$ be a graph. If $\mathrm{rankwidth}(G)\ge 72r$, then $G$ contains an induced subgraph $H$ with $\operatorname{min-bal-cutrank(H)}\ge r$.
\end{restatable}

Classification or Bijection
Inequality or Bound

Question: Let $G=(V,E)$ be a graph. A separation of $G$ is a pair $(X,Y)$ with $X\sqcup Y=V(G)$, and $\operatorname{cutrank}_G(X,Y)$ is the $\mathbb F_2$-rank of the $|X|\times |Y|$ biadjacency matrix between $X$ and $Y$. A rank-decomposition of $G$ is a pair $(T,\gamma)$ where $T$ is a subcubic tree (every internal node has degree at most $3$) and $\gamma:V(G)\to \operatorname{Leaf}(T)$ is a bijection. For an edge $e\in E(T)$, deleting $e$ partitions the leaves of $T$ into two parts, which via $\gamma^{-1}$ induce a partition $(X_e,Y_e)$ of $V(G)$. Which quantitative formula gives the rankwidth of $G$?

$\displaystyle \operatorname{rankwidth}(G)=\min_{(T,\gamma)}\max_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$, where the minimum is over all rank-decompositions $(T,\gamma)$ of $G$.

Choices:
B. $\displaystyle \operatorname{rankwidth}(G)=\max_{(T,\gamma)}\min_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$, where the maximum is over all rank-decompositions $(T,\gamma)$ of $G$.
C. $\displaystyle \operatorname{rankwidth}(G)\le \max_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$ for every rank-decomposition $(T,\gamma)$ of $G$.
D. $\displaystyle \operatorname{rankwidth}(G)=\min_{(T,\gamma)}\sum_{e\in E(T)} \operatorname{cutrank}_G(X_e,Y_e)$, where the minimum is over all rank-decompositions $(T,\gamma)$ of $G$.
E. $\displaystyle \operatorname{rankwidth}(G)=\min_{(T,\gamma)}\max_{e\in E(T)} |X_e|\,|Y_e|$, where the minimum is over all rank-decompositions $(T,\gamma)$ of $G$.

min-max order in the defining optimization

global minimum over decompositions dropped, keeping only the upper-bound inequality for each fixed decomposition

width measured by worst edge replaced by total aggregate over edges

cutrank replaced by cut size proxy depending only on part sizes

The stem does not explicitly state the defining min-max formula. It provides the setup and notation needed for rank-decompositions, but the crucial optimization structure is left to the choices.

This is essentially asking for the definition of rankwidth itself. It is a direct restatement/recognition question rather than a nontrivial consequence or application.

Some discrimination is required among min-max order, weaker inequalities, and incorrect width measures, but the task is primarily definition recall rather than genuine generative mathematical reasoning.

The distractors are plausible and well-targeted: B swaps min/max, C gives a weaker true statement, D replaces worst-edge width by a sum, and E confuses cutrank with a size-based proxy. These reflect realistic misconceptions.

A clear and technically sound MCQ with strong distractors, but it mainly tests recall of the definition of rankwidth, so it is low on tautology avoidance and only moderate on generative reasoning.