\begin{theorem}\label{thm:diameter}
    Let $f:\mathbb{R}^n\to \mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. For any bounded $X\subset \mathbb{R}^n$, there exist $\bar{\alpha},\beta,\epsilon,\varsigma_1,\varsigma_2>0$ and $\theta\in (0,1)$ such that for any subgradient sequence $(x_k)_{k\in \mathbb{N}}$ with step sizes $0<\alpha_{K-1} \le \cdots \le \alpha_0 \le \bar{\alpha}$ and $x_{\llbracket 0,K \rrbracket} \subset X\cap [ |f| \le \epsilon]$ for some $K\in \mathbb{N}$, we have
        \begin{align*}
        \mathrm{diam}(x_{\llbracket 0,K \rrbracket}) \le~&\varsigma_1\left(\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\theta}-\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\theta}\right) +\cdots\\
        &+\varsigma_2\left(\alpha_0^\beta+ \sum_{k=0}^{K-1}\alpha_k^{1+\beta} +\left(\sum_{k=0}^{K-1}\alpha_k^{1+\beta}\right)^{1-\theta}+  \sum_{k=0}^{K-1} \alpha_k \left(\sum_{j=k}^{K-1} \alpha_j^{1+\beta}\right)^\theta\right).
    \end{align*}
\end{theorem}

Let $f:\mathbb{R}^n\to \mathbb{R}$ be locally Lipschitz. We study \emph{subgradient sequences} $(x_k)_{k\in \mathbb{N}}$ defined by
\begin{equation*}
    x_{k+1} \in x_k - \alpha_k \partial f(x_k)
\end{equation*}
for $k\in \mathbb{N}$, where $x_0\in \mathbb{R}^n$ is arbitrary, $(\alpha_k)_{k\in \mathbb{N}}$ is a sequence of positive scalars (called \emph{step sizes}) that is not summable, and $\partial f:\mathbb{R}^n\rightrightarrows \mathbb{R}^n$ denotes the Clarke subdifferential \cite{clarke1975,clarke1990} of $f$. Subgradient sequences are discretizations of continuous-time subgradient trajectories \cite{bolte2007lojasiewicz}, which are solutions to the differential inclusion $x'\in -\partial f(x)$. Subgradient trajectories can be viewed as generalizations of classical gradient trajectories of smooth functions \cite{absil2006stable,santambrogio2017euclidean}. In the optimization literature, subgradient sequences are realizations of the subgradient method \cite{shor1962application}, which generalizes Cauchy's steepest descent method \cite{cauchy1847methode} to minimize locally Lipschitz functions. The subgradient method and its variants garnered significant attention within the machine learning community recently, due to their success in solving large-scale optimization problems arising from deep learning and artificial intelligence \cite{sutskever2013importance,lecun2015deep,vaswani2017attention}.

Subgradient sequences can behave erratically, even for functions that are $C^\infty$ \cite{palis2012geometric,absil2005convergence,daniilidis2020pathological}. It is for this reason that we assume additional geometric structures of the objective function $f$, that is, the function is definable in o-minimal structures \cite{van1996geometric}. We defer the definition and discussion of o-minimal structures to Section \ref{sec:o-minimal}. At a high level, o-minimal structures generalize semialgebraic sets \cite{tarski1951decision}, and are families of ``tame'' subsets of $\mathbb{R}^n$ that possess certain finiteness properties. The study of (sub)gradient dynamics for definable functions was initiated in \L{}ojasiewicz's pioneer works \cite{lojasiewicz1963propriete,lojasiewicz1982trajectoires} on (real) analytic functions. It was shown that bounded (continuous-time) gradient trajectories of analytic functions have finite length, a consequence of the gradient inequality (known as the \L{}ojasiewicz gradient inequality \cite{lojasiewicz1958}) of analytic functions. This inequality can be extended to smooth functions definable in arbitrary o-minimal structures \cite{kurdyka1998gradients} and later to nonsmooth functions in \cite{bolte2007lojasiewicz}. Consequently, the convergence of subgradient trajectories in these two settings is also established \cite{kurdyka1998gradients,bolte2010characterizations}.

In contrast to their continuous-time counterparts, bounded subgradient sequences are known to converge only when $f$ is either 1)  definable and differentiable with a locally Lipschitz gradient \cite{absil2005convergence} or 2) convex \cite{alber1998projected}, given that the step sizes $(\alpha_k)_{k\in \mathbb{N}}$ are square summable. For nonconvex nonsmooth functions, recent works proposed to analyze subgradient sequences under the assumption that $f$ is ``path-differentiable'' \cite{davis2020stochastic,bolte2022long,bolte2025inexact}. Path-differentiable functions are functions that are almost everywhere differentiable when precomposed with any absolutely continuous arc \cite[Definition 5.1]{davis2020stochastic}\cite[Definition 3]{bolte2020conservative}. Locally Lipschitz functions definable in o-minimal structures are path-differentiable \cite[Theorem 5.8]{davis2020stochastic}, as their graphs can be stratified into smooth manifolds. If in addition $f$ satisfies the weak Sard property, i.e., $f$ is constant on connect components of its critical set\footnote{$x\in \mathbb{R}^n$ is a critical point of $f$ if $0\in \partial f(x)$. The collection of all critical points is the critical set.}, then the limit points of any bounded subgradient sequence $(x_k)_{k\in \mathbb{N}}$ are critical points, and the function values $(f(x_k))_{k\in \mathbb{N}}$ converges \cite[Theorem 3.2]{davis2020stochastic}\cite[Theorem 5]{bolte2022long}. It is worth noticing that their approach regards the subgradient sequence as an approximation of subgradient trajectories, inspired by previous works in stochastic approximation \cite{ljung1977analysis,kushner1977general,benaim2005stochastic,duchi2018stochastic}. Drawing tools from the theory of closed measures, one can further study the oscillation of subgradient sequences \cite{bolte2022long}.

It is natural to wonder whether and when the subgradient sequences with vanishing step sizes will converge. By an example of Ríos-Zertuche \cite[Section 2]{rios2022examples}, subgradient sequences of path-differentiable functions can indeed oscillate. In fact, the constructed ``pathological'' function is Whitney $C^\infty$ stratifiable and satisfies the nonsmooth Łojasiewicz gradient inequality \cite[Proposition 6]{rios2022examples}. It is noteworthy that the subgradient trajectories of the same function converge, due to the nonsmooth Łojasiewicz gradient inequality \cite{bolte2010characterizations}. This highlights the distinct dynamics of subgradient sequences compared to their continuous-time counterparts, emphasizing the need for additional geometric structures to guarantee their convergence.

In this work, we aim to conduct a refined analysis on subgradient sequences of locally Lipschitz functions definable in o-minimal structures. We seek to identify conditions under which the sequence will converge if bounded. Recall that the \emph{diameter} of a set $A\subset \mathbb{R}^n$ is given by $\mathrm{diam}(A):= \sup\{|a-b|:a,b\in A\}$. In our main result (Theorem \ref{thm:diameter}), we estimate the diameter of subgradient sequences when they stay close to a level set of $f$. We show that the diameter is controlled by a difference in function values, up to high-order accumulations of the step sizes. Let $a,b\in \mathbb{N}$ such that $a \le b$, we denote by $\llbracket a,b \rrbracket:= \{a,\ldots,b\}$. Given a sequence $(x_k)_{k\in \mathbb{N}}$, we denote by $x_{\llbracket a,b \rrbracket}:= \{x_a,\ldots,x_b\}$. For a function $g:\mathbb{R}^n\rightarrow \mathbb{R}$ and $v\in \mathbb{R}^n$, we denote by $[g \le v]:=\{x\in \mathbb{R}^n:g(x) \le v\}$ the sublevel set of $g$ with respect to the value $v$. We also define the sign function $\mathrm{sgn}:\mathbb{R}\to \{-1,0,1\}$ that returns $-1$ for negative values, $1$ for positive values, and $0$ for zero. We are now ready to present the main result of this paper.

Subgradient sequences can behave erratically, even for functions that are $C^\infty$ \cite{palis2012geometric,absil2005convergence,daniilidis2020pathological}. It is for this reason that we assume additional geometric structures of the objective function $f$, that is, the function is definable in o-minimal structures \cite{van1996geometric}. We defer the definition and discussion of o-minimal structures to Section \ref{sec:o-minimal}. At a high level, o-minimal structures generalize semialgebraic sets \cite{tarski1951decision}, and are families of ``tame'' subsets of $\mathbb{R}^n$ that possess certain finiteness properties. The study of (sub)gradient dynamics for definable functions was initiated in \L{}ojasiewicz's pioneer works \cite{lojasiewicz1963propriete,lojasiewicz1982trajectoires} on (real) analytic functions. It was shown that bounded (continuous-time) gradient trajectories of analytic functions have finite length, a consequence of the gradient inequality (known as the \L{}ojasiewicz gradient inequality \cite{lojasiewicz1958}) of analytic functions. This inequality can be extended to smooth functions definable in arbitrary o-minimal structures \cite{kurdyka1998gradients} and later to nonsmooth functions in \cite{bolte2007lojasiewicz}. Consequently, the convergence of subgradient trajectories in these two settings is also established \cite{kurdyka1998gradients,bolte2010characterizations}.

In contrast to their continuous-time counterparts, bounded subgradient sequences are known to converge only when $f$ is either 1) definable and differentiable with a locally Lipschitz gradient \cite{absil2005convergence} or 2) convex \cite{alber1998projected}, given that the step sizes $(\alpha_k)_{k\in \mathbb{N}}$ are square summable. For nonconvex nonsmooth functions, recent works proposed to analyze subgradient sequences under the assumption that $f$ is ``path-differentiable'' \cite{davis2020stochastic,bolte2022long,bolte2025inexact}. Path-differentiable functions are functions that are almost everywhere differentiable when precomposed with any absolutely continuous arc \cite[Definition 5.1]{davis2020stochastic}\cite[Definition 3]{bolte2020conservative}. Locally Lipschitz functions definable in o-minimal structures are path-differentiable \cite[Theorem 5.8]{davis2020stochastic}, as their graphs can be stratified into smooth manifolds. If in addition $f$ satisfies the weak Sard property, i.e., $f$ is constant on connect components of its critical set\footnote{$x\in \mathbb{R}^n$ is a critical point of $f$ if $0\in \partial f(x)$. The collection of all critical points is the critical set.}, then the limit points of any bounded subgradient sequence $(x_k)_{k\in \mathbb{N}}$ are critical points, and the function values $(f(x_k))_{k\in \mathbb{N}}$ converges \cite[Theorem 3.2]{davis2020stochastic}\cite[Theorem 5]{bolte2022long}. It is worth noticing that their approach regards the subgradient sequence as an approximation of subgradient trajectories, inspired by previous works in stochastic approximation \cite{ljung1977analysis,kushner1977general,benaim2005stochastic,duchi2018stochastic}. Drawing tools from the theory of closed measures, one can further study the oscillation of subgradient sequences \cite{bolte2022long}.

It is natural to wonder whether and when the subgradient sequences with vanishing step sizes will converge. By an example of Ríos-Zertuche \cite[Section 2]{rios2022examples}, subgradient sequences of path-differentiable functions can indeed oscillate. In fact, the constructed ``pathological'' function is Whitney $C^\infty$ stratifiable and satisfies the nonsmooth Łojasiewicz gradient inequality \cite[Proposition 6]{rios2022examples}. It is noteworthy that the subgradient trajectories of the same function converge, due to the nonsmooth Łojasiewicz gradient inequality \cite{bolte2010characterizations}. This highlights the distinct dynamics of subgradient sequences compared to their continuous-time counterparts, emphasizing the need for additional geometric structures to guarantee their convergence.

In this work, we aim to conduct a refined analysis on subgradient sequences of locally Lipschitz functions definable in o-minimal structures. We seek to identify conditions under which the sequence will converge if bounded. Recall that the \emph{diameter} of a set $A\subset \mathbb{R}^n$ is given by $\mathrm{diam}(A):= \sup\{|a-b|:a,b\in A\}$. In our main result (Theorem \ref{thm:diameter}), we estimate the diameter of subgradient sequences when they stay close to a level set of $f$. We show that the diameter is controlled by a difference in function values, up to high-order accumulations of the step sizes. Let $a,b\in \mathbb{N}$ such that $a \le b$, we denote by $\llbracket a,b \rrbracket:= \{a,\ldots,b\}$. Given a sequence $(x_k)_{k\in \mathbb{N}}$, we denote by $x_{\llbracket a,b \rrbracket}:= \{x_a,\ldots,x_b\}$. For a function $g:\mathbb{R}^n\rightarrow \mathbb{R}$ and $v\in \mathbb{R}^n$, we denote by $[g \le v]:=\{x\in \mathbb{R}^n:g(x) \le v\}$ the sublevel set of $g$ with respect to the value $v$. We also define the sign function $\mathrm{sgn}:\mathbb{R}\to \{-1,0,1\}$ that returns $-1$ for negative values, $1$ for positive values, and $0$ for zero. We are now ready to present the main result of this paper.

Combining with the existing guarantees for path-differentiable functions, Theorem \ref{thm:diameter} implies that bounded subgradient sequences converge to critical points of $f$, given that certain summations of step sizes are finite. This is the case when the step sizes $(\alpha_k)_{k\in \mathbb{N}}$ decrease and are of order $1/k$, which is widely used in the subgradient method for convex functions \cite{shor1985book,beck2017first}. Given two sequences of positive scalars $(a_k)_{k\in \mathbb{N}}$ and $(b_k)_{k\in \mathbb{N}}$, we write $(a_k)_{k\in \mathbb{N}} \sim (b_k)_{k\in \mathbb{N}}$ if there exist $c,C>0$ such that $c \le a_k/b_k \le C$ for all $k\in \mathbb{N}$.
\begin{corollary}
 \label{cor:1/k}
 Let $f:\mathbb{R}^n\to \mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. Any bounded subgradient sequence $(x_k)_{k\in \mathbb{N}}$ with decreasing step sizes $(\alpha_k)_{k\in \mathbb{N}} \sim (1/(k+1))_{k\in \mathbb{N}}$ converges to a critical point of $f$.
\end{corollary}
With Theorem \ref{thm:diameter}, the proof of Corollary \ref{cor:1/k} is quite straightforward: Given a bounded subgradient sequence $(x_k)_{k\in \mathbb{N}}$, by \cite[Theorem 3.2]{davis2020stochastic}, its limit points are critical and $(f(x_k))_{k\in \mathbb{N}}$ converges to $f^*\in \mathbb{R}^n$ as $(\alpha_k)_{k\in \mathbb{N}}$ is not summable. We apply Theorem \ref{thm:diameter} to the sequence with $f$ replaced by $f - f^*$, which yields an upper bound on $\mathrm{diam}(x_{\llbracket k_1,k_2 \rrbracket})$ for arbitrary $k_1,k_2\in \mathbb{N}$ that are sufficiently large. A direct calculation shows that this upper bound diminishes as $k_1\to \infty$, which implies that the sequence $(x_k)_{k\in \mathbb{N}}$ is Cauchy and thus convergent. This completes the proof of Corollary \ref{cor:1/k}.

In the following lemma, we lower bound the variation on function value (at projected sequence) from $k = l_m$ to $k = H(l_m)$. As it turns out, the lower bound depends on the number of distinct non-open strata that the sequence crosses. Intuitively, the more strata that the iterates cross, the less the function values will decrease. Formally, we consider the function $U:\mathcal{L}:\rightarrow \mathbb{N}$ defined by
\[
U(l):=|\{G(k):k\in (q(l),H(l))\cap I_C\}|+1
\]
for any $l\in \mathcal{L}$. This quantity is upper bounded by $T$, and the proof of Lemma \ref{lemma:l->H} is based on an induction on it.
\begin{lemma}\label{lemma:l->H}
Let $\mu>0$. After possibly reducing $\bar{\alpha}$, for any $l_m\in \mathcal{L}$ such that $H(l_m)\in \mathcal{L}$, we have
 \begin{subequations}\label{eq:inductive}
 \begin{align}
 \psi\left(z^{G(l_m)}_{l_m}\right)-\psi\left(z^{G(H(l_m))}_{H(l_m)}\right) \ge~& \frac{1-U(l_m)\mu}{c} \mathrm{diam}(x_{\llbracket l_m,H(l_m)\rrbracket}) + \cdots\\
 &-2\overline{C}U(l_m)^2\alpha_{l_m}^{\underline{\gamma}(1-\theta)}\sum_{j=0}^{\lfloor\log_2 \alpha_{l_m}/\alpha_{_{H(l_m)}}\rfloor-1}2^{-j\underline{\gamma}(1-\theta)}
 +\cdots\\
 &+ \overline{C} \left(\alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)} -\alpha_{H(l_m)}^{\gamma_{G(H(l_m))}(1-\theta)}\right)-\sum_{k = l_m}^{H(l_m)-1}(\alpha_k^{1+\beta}+\alpha_k g_k^\theta)+\cdots\\
 &-2\sum_{l_m \le l< H(l_m)<s(l)}\overline{C} \alpha_{l}^{\gamma_{G(l)}(1-\theta)}, \label{eq:extra}
 \end{align}
\end{subequations}
where $\overline{C}>0$ is the same constant that appears in Lemma \ref{lemma:lm->lm+1} and $\underline{\gamma}:= \min_{i\in \llbracket 1,T\rrbracket}\{\gamma_i\}$.
\end{lemma}
\begin{proof}
Let $l_m\in \mathcal{L}$ such that $H(l_m)\in \mathcal{L}$. Let $\mu>0$. We will prove the desired inequality by an induction on $U(l_m)$. We start with the base case where $U(l_m) = 1$, which means that $(q(l_m),H(l_m))\cap I_C = \emptyset$. In this case, $x_{\llbracket q(l_m)+1,H(l_m) \rrbracket} \subset M_i$ for some open stratum $M_i$ and $H(l_m) = l_{m+1}$. Thus, the summation \eqref{eq:extra} on the right hand side of the desired inequality is equal to zero.

Since $x_{s(l_m)+1}\notin \mathcal{N}(G(l_m),\alpha_{s(l_m)+1})$, we have either $d_{{s(l_m)+1}}^{G(l_m)}\ge c_{G(l_m)}\alpha_{s(l_m)+1}^{\beta_{G(l_m)}}$ or $d_{s(l_m)+1}^j<c_j\alpha_{s(l_m)+1}^{\gamma_j}$ for some $j$ such that $M_j\subset \partial M_{G(l_m)}$. In the former case, we have
\begin{equation}\label{eq:rea_start}
 \begin{aligned}
 |x_{s(l_m)}-x_{l_m}|&\ge |x_{s(l_m)+1}-x_{l_m}| - |x_{s(l_m)+1}-x_{s(l_m)}| \\
 &\ge c_{G(l_m)}\alpha_{s(l_m)+1}^{\beta_{G(l_m)}}-2c_{G(l_m)}\alpha_{l_m}^{\gamma_{G(l_m)}}-L\alpha_{s(l_m)}\\
 &\ge c_{G(l_m)}\alpha_{H(l_m)}^{\beta_{G(l_m)}}-2c_{G(l_m)}\alpha_{l_m}^{\gamma_{G(l_m)}}-L\alpha_{l_m}
 \end{aligned}
\end{equation}
as the step sizes $(\alpha_k)_{k\in \mathbb{N}}$ are decreasing. Thus, either $\alpha_{H(l_m)} \le \alpha_{l_m}/2$ or
\begin{equation}
 \frac{\mu}{c} |x_{s(l_m)}-x_{l_m}| \ge \frac{\mu }{c}(c_{G(l_m)}\alpha_{H(l_m)}^{\beta_{G(l_m)}}-2c_{G(l_m)}\alpha_{l_m}^{\gamma_{G(l_m)}}-L\alpha_{l_m} )\ge 2\overline{C}\alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)}.
\end{equation}
Indeed, this holds after possibly reducing $\bar{\alpha}$ as $\beta_{G(l_m)}< \gamma_{G(l_m)}(1-\theta)$. In the latter case,
\begin{equation}\label{V(n)2}
 \begin{aligned}
 |x_{s(l_m)}-x_{l_m}|&\ge |x_{s(l_m)+1}-x_{l_m}| - |x_{s(l_m)+1}-x_{s(l_m)}| \\
 &\ge 2c_{j}\alpha_{l_m}^{\gamma_{j}}-c_{j}\alpha_{s(l_m)+1}^{\gamma_{j}}-L\alpha_{s(l_m)}\\
 &\ge c_{j}\alpha_{l_m}^{\gamma_{j}} -L\alpha_{l_m}.
 \end{aligned}
\end{equation}
Thus,
\begin{equation*}
 \frac{\mu}{c} |x_{s(l_m)}-x_{l_m}| \ge \frac{\mu}{c} (c_{j}\alpha_{l_m}^{\gamma_{j}} -L\alpha_{l_m}) \ge 2\overline{C}\alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)}.
\end{equation*}
Indeed, this holds after possibly reducing $\bar{\alpha}$ as $\gamma_j< \gamma_{G(l_m)}(1-\theta)$ for all $M_j \subset \partial M_{G(l_m)}$. Thus, in both cases, we have either $\alpha_{H(l_m)} \le \alpha_{l_m}/2$ or
\begin{equation}\label{eq:rea_end}
 \frac{\mu}{c} \mathrm{diam}(x_{\llbracket l_m,H(l_m)\rrbracket}) \ge \frac{\mu}{c} |x_{s(l_m)}-x_{l_m}| \ge 2\overline{C}\alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)}.
\end{equation}
By Lemma \ref{lemma:lm->lm+1}, it holds that
 \begin{align*}
 &\psi\left(z^{G(l_m)}_{l_m}\right)-\psi\left(z^{G(H(l_m))}_{H(l_m)}\right)\\
 \ge & \frac{1}{c} \mathrm{diam}(x_{\llbracket l_m,H(l_m)\rrbracket}) -\sum_{k = l_m}^{H(l_m)-1}(\alpha_k^{1+\beta}+\alpha_k g_k^\theta)- \overline{C} \left(\alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)} +\alpha_{H(l_m)}^{\gamma_{G(H(l_m))}(1-\theta)}\right)\\
 =& \frac{1-\mu}{c} \mathrm{diam}(x_{\llbracket l_m,H(l_m)\rrbracket}) +\frac{\mu}{c} \mathrm{diam}(x_{\llbracket l_m,H(l_m)\rrbracket})- 2\overline{C} \alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)}+\cdots\\
 &+ \overline{C} \left(\alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)} -\alpha_{H(l_m)}^{\gamma_{G(H(l_m))}(1-\theta)}\right)-\sum_{k = l_m}^{H(l_m)-1}(\alpha_k^{1+\beta}+\alpha_k g_k^\theta)\\
 \ge& \frac{1-\mu}{c} \mathrm{diam}(x_{\llbracket l_m,H(l_m)\rrbracket}) -2\overline{C}\alpha_{l_m}^{\underline{\gamma}(1-\theta)}\sum_{j=0}^{\lfloor\log_2 \alpha_{l_m}/\alpha_{_{H(l_m)}}\rfloor-1}2^{-j\underline{\gamma}(1-\theta)}+\cdots\\
 &+ \overline{C} \left(\alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)} -\alpha_{H(l_m)}^{\gamma_{G(H(l_m))}(1-\theta)}\right)- \sum_{k = l_m}^{H(l_m)-1}(\alpha_k^{1+\beta}+\alpha_k g_k^\theta),
 \end{align*}
where the last inequality is due to \eqref{eq:rea_end} and the fact that
\[
- 2\overline{C} \alpha_{l_m}^{\gamma_{G(l_m)}(1-\theta)} \ge -2\overline{C}\alpha_{l_m}^{\underline{\gamma}(1-\theta)}\ge -2\overline{C}\alpha_{l_m}^{\underline{\gamma}(1-\theta)}\sum_{j=0}^{\lfloor\log_2 \alpha_{l_m}/\alpha_{_{H(l_m)}}\rfloor-1}2^{-j\underline{\gamma}(1-\theta)}
\]
when $\alpha_{H(l_m)} \le \alpha_{l_m}/2$. The desired inequality \eqref{eq:inductive} then follows as $U(l_m) = 1$.

Theorem \ref{thm:diameter} requires the objective function to be definable in a polynomially bounded o-minimal structure, which we recall at the beginning of Section \ref{sec:o-minimal}. We also need the step sizes $(\alpha_k)_{k\in \mathbb{N}}$ associated with the sequence to be small and decreasing, which coincides with the classical choices of step sizes in nonsmooth optimization \cite{polyak1967general,polyak1978subgradient,alber1998projected}. The constants that appear in the theorem depend only on the local geometry of $f$. In particular, $\theta$ is an exponent that appears at the \L{}ojasiewicz gradient inequality of $f$ when restricted on certain smooth manifolds.

Combining with the existing guarantees for path-differentiable functions, Theorem \ref{thm:diameter} implies that bounded subgradient sequences converge to critical points of $f$, given that certain summations of step sizes are finite. This is the case when the step sizes $(\alpha_k)_{k\in \mathbb{N}}$ decrease and are of order $1/k$, which is widely used in the subgradient method for convex functions \cite{shor1985book,beck2017first}. Given two sequences of positive scalars $(a_k)_{k\in \mathbb{N}}$ and $(b_k)_{k\in \mathbb{N}}$, we write $(a_k)_{k\in \mathbb{N}} \sim (b_k)_{k\in \mathbb{N}}$ if there exist $c,C>0$ such that $c \le a_k/b_k \le C$ for all $k\in \mathbb{N}$.
\begin{corollary}
    \label{cor:1/k}
    Let $f:\mathbb{R}^n\to \mathbb{R}$ be locally Lipschitz and definable in a polynomially bounded o-minimal structure. Any bounded subgradient sequence $(x_k)_{k\in \mathbb{N}}$ with decreasing step sizes $(\alpha_k)_{k\in \mathbb{N}} \sim (1/(k+1))_{k\in \mathbb{N}}$ converges to a critical point of $f$.
\end{corollary}
With Theorem \ref{thm:diameter}, the proof of Corollary \ref{cor:1/k} is quite straightforward: Given a bounded subgradient sequence $(x_k)_{k\in \mathbb{N}}$, by \cite[Theorem 3.2]{davis2020stochastic}, its limit points are critical and $(f(x_k))_{k\in \mathbb{N}}$ converges to $f^*\in \mathbb{R}^n$ as $(\alpha_k)_{k\in \mathbb{N}}$ is not summable. We apply Theorem \ref{thm:diameter} to the sequence with $f$ replaced by $f - f^*$, which yields an upper bound on $\mathrm{diam}(x_{\llbracket k_1,k_2 \rrbracket})$ for arbitrary $k_1,k_2\in \mathbb{N}$ that are sufficiently large. A direct calculation shows that this upper bound diminishes as $k_1\to \infty$, which implies that the sequence $(x_k)_{k\in \mathbb{N}}$ is Cauchy and thus convergent. This completes the proof of Corollary \ref{cor:1/k}.

From the previous discussions, it is evident that the proof of Theorem \ref{thm:diameter} must leverage the unique geometric properties of o-minimal structures. Our approach diverges from the literature \cite{davis2020stochastic,bolte2022long,bolte2025inexact}, which relies on the continuous-time limit of subgradient sequences. Instead, the proof hinges on the stratifications of definable sets with strong metric properties, which is discussed in Section \ref{sec:o-minimal}. Building on these stratifications, we decompose functions into smooth pieces with locally Lipschitz Riemannian gradients, which relate to their subdifferentials. This is the object of Section \ref{sec:decomposition}. Finally, we prove Theorem \ref{thm:diameter} in Section \ref{sec:main_proof}, which requires analyzing subgradient sequences as they alternate between the components proposed in Section \ref{sec:decomposition}.

The post-theorem introduction indicates that the proof of Theorem~\ref{thm:diameter} "must leverage the unique geometric properties of o-minimal structures" and that the authors' approach "diverges from the literature ... which relies on the continuous-time limit of subgradient sequences." Instead, "the proof hinges on the stratifications of definable sets with strong metric properties" (Section~\ref{sec:o-minimal}).

Building on these stratifications, they "decompose functions into smooth pieces with locally Lipschitz Riemannian gradients, which relate to their subdifferentials" (Section~\ref{sec:decomposition}). Finally, they "prove Theorem~\ref{thm:diameter} in Section~\ref{sec:main_proof}, which requires analyzing subgradient sequences as they alternate between the components proposed in Section~\ref{sec:decomposition}."

The post-theorem introduction indicates that, to prove the main theorem, the proof “must leverage the unique geometric properties of o-minimal structures” and that the authors’ approach “diverges from the literature ... which relies on the continuous-time limit of subgradient sequences.” Instead, “the proof hinges on the stratifications of definable sets with strong metric properties” (Section~\ref{sec:o-minimal}).

Building on these stratifications, they “decompose functions into smooth pieces with locally Lipschitz Riemannian gradients, which relate to their subdifferentials” (Section~\ref{sec:decomposition}). Finally, they prove the main theorem later, which requires analyzing subgradient sequences as they alternate between the components proposed earlier.

Existential–Universal
Inequality or Bound

Question: Let $f:\mathbb{R}^n\to\mathbb{R}$ be a locally Lipschitz function definable in a polynomially bounded o-minimal structure, and let $X\subset \mathbb{R}^n$ be bounded. Write $\mathrm{diam}(A):=\sup\{|a-b|:a,b\in A\}$ for the diameter of a set $A\subset\mathbb{R}^n$, $x_{\llbracket 0,K\rrbracket}:=\{x_0,\dots,x_K\}$, and $[|f|\le \epsilon]:=\{x\in\mathbb{R}^n:|f(x)|\le \epsilon\}$. Which statement holds about the existence of uniform constants controlling the diameter of every subgradient sequence $(x_k)_{k\in\mathbb{N}}$ with nonincreasing step sizes $0<\alpha_{K-1}\le \cdots \le \alpha_0$ whose iterates satisfy $x_{\llbracket 0,K\rrbracket}\subset X\cap[|f|\le \epsilon]$ for some $K\in\mathbb{N}$?

There exist constants $\bar{\alpha},\beta,\epsilon,\varsigma_1,\varsigma_2>0$ and $\theta\in(0,1)$ such that, whenever $0<\alpha_{K-1}\le\cdots\le\alpha_0\le\bar{\alpha}$ and $x_{\llbracket 0,K\rrbracket}\subset X\cap[|f|\le\epsilon]$, one has
\[
\mathrm{diam}(x_{\llbracket 0,K\rrbracket}) \le \varsigma_1\Big(\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\theta}-\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\theta}\Big)
\]
\[
\qquad\qquad +\;\varsigma_2\Bigg(\alpha_0^{\beta}+\sum_{k=0}^{K-1}\alpha_k^{1+\beta}+\Big(\sum_{k=0}^{K-1}\alpha_k^{1+\beta}\Big)^{1-\theta}+\sum_{k=0}^{K-1}\alpha_k\Big(\sum_{j=k}^{K-1}\alpha_j^{1+\beta}\Big)^\theta\Bigg).
\]

Choices:
B.
There exist constants $\bar{\alpha},\beta,\epsilon,\varsigma_1,\varsigma_2>0$ and $\theta\in(0,1)$ such that, whenever $0<\alpha_{K-1}\le\cdots\le\alpha_0\le\bar{\alpha}$ and $x_{\llbracket 0,K\rrbracket}\subset X\cap[|f|\le\epsilon]$, one has
\[
\mathrm{diam}(x_{\llbracket 0,K\rrbracket}) \le \varsigma_1\Big(|f(x_0)|^{1-\theta}-|f(x_K)|^{1-\theta}\Big)
\]
\[
\qquad\qquad +\;\varsigma_2\Bigg(\alpha_0^{\beta}+\sum_{k=0}^{K-1}\alpha_k^{1+\beta}+\Big(\sum_{k=0}^{K-1}\alpha_k^{1+\beta}\Big)^{1-\theta}+\sum_{k=0}^{K-1}\alpha_k\Big(\sum_{j=k}^{K-1}\alpha_j^{1+\beta}\Big)^\theta\Bigg).
\]
C.
There exist constants $\bar{\alpha},\beta,\epsilon,\varsigma_1,\varsigma_2>0$ and $\theta\in(0,1)$ such that, whenever $0<\alpha_{K-1}\le\cdots\le\alpha_0\le\bar{\alpha}$ and $x_{\llbracket 0,K\rrbracket}\subset X\cap[|f|\le\epsilon]$, one has
\[
\mathrm{diam}(x_{\llbracket 0,K\rrbracket}) \le \varsigma_1\Big(\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\theta}-\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\theta}\Big)
\]
\[
\qquad\qquad +\;\varsigma_2\Bigg(\alpha_0^{\beta}+\sum_{k=0}^{K-1}\alpha_k^{1+\beta}+\Big(\sum_{k=0}^{K-1}\alpha_k^{1+\beta}\Big)^{1-\theta}\Bigg).
\]
D.
There exist constants $\bar{\alpha},\beta,\epsilon,\varsigma_1,\varsigma_2>0$ and $\theta\in(0,1)$ such that, for every $K\in\mathbb{N}$ and every subgradient sequence $(x_k)_{k\in\mathbb{N}}$ with $0<\alpha_k\le\bar{\alpha}$ for all $0\le k\le K-1$ and $x_{\llbracket 0,K\rrbracket}\subset X\cap[|f|\le\epsilon]$, one has
\[
\mathrm{diam}(x_{\llbracket 0,K\rrbracket}) \le \varsigma_1\Big(\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\theta}-\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\theta}\Big)
\]
\[
\qquad\qquad +\;\varsigma_2\Bigg(\alpha_0^{\beta}+\sum_{k=0}^{K-1}\alpha_k^{1+\beta}+\Big(\sum_{k=0}^{K-1}\alpha_k^{1+\beta}\Big)^{1-\theta}+\sum_{k=0}^{K-1}\alpha_k\Big(\sum_{j=k}^{K-1}\alpha_j^{1+\beta}\Big)^\theta\Bigg).
\]
E.
There exist constants $\bar{\alpha},\beta,\epsilon,\varsigma_1,\varsigma_2>0$ and $\theta\in(0,1)$ such that, whenever $0<\alpha_{K-1}\le\cdots\le\alpha_0\le\bar{\alpha}$ and $x_{\llbracket 0,K\rrbracket}\subset X\cap[|f|\le\epsilon]$, one has
\[
\mathrm{diam}(x_{\llbracket 0,K\rrbracket}) \le \varsigma_1\Big(\mathrm{sgn}(f(x_0))|f(x_0)|^{1-\theta}-\mathrm{sgn}(f(x_K))|f(x_K)|^{1-\theta}\Big)
\]
\[
\qquad\qquad +\;\varsigma_2\Bigg(\sum_{k=0}^{K-1}\alpha_k^{1+\beta}+\Big(\sum_{k=0}^{K-1}\alpha_k^{1+\beta}\Big)^{1-\theta}+\sum_{k=0}^{K-1}\alpha_k\Big(\sum_{j=k}^{K-1}\alpha_j^{1+\beta}\Big)^\theta\Bigg),
\]
with no separate $\alpha_0^{\beta}$ term required.

The stem does not reveal the correct option explicitly or by a trivial cue. It only sets notation and asks which uniform-diameter statement is valid.

This is essentially a theorem-statement recognition item: the correct choice is the exact technical conclusion of the result, with nearby variants as distractors. It tests recall of the theorem more than application.

There is some reasoning pressure because the options differ in subtle but meaningful ways (signed potential, monotone step-size hypothesis, mixed accumulation term, initial-scale term). Still, the task is mainly to identify the exact theorem rather than generate or derive a conclusion from mathematical data.

The distractors are strong: they are plausible theorem variants, mathematically distinct, and reflect realistic failure modes such as dropping a needed hypothesis or omitting a necessary term.

A technically well-crafted but theorem-recall-heavy MCQ: little answer leakage and strong distractors, but low tautology avoidance and only moderate generative reasoning demand.


The present work characterizes, under the model-theoretic hypothesis of finite-dimensionality,
irreducible bi-modules, i.e., abelian groups together with two commuting subrings of endomorphisms, and which are "irreducible" for the bi-action.\\
This result is well-known for groups of finite Morley rank \cite{zilber1984some}. It is also known for $o$-minimal theories (see, for example \cite{peterzil2000simple}). The main problem is that the two proofs use techniques proper to the two families of theories involved (in the finite Morley rank case, the indecomposability theorem is necessary). A natural question arises: to which generality can we extend the linearization?\\
An interesting family of theories is finite-dimensional theories, in the sense of \cite{wagner2020dimensional}.
\begin{defn}
    A theory $T$ is said to be \emph{finite-dimensional} if there exists a function $\operatorname{dim}$ from the class of all interpretable subset in any model $\mathcal{M}$ of $T$ into $\omega\cup\{-\infty\}$ such that for any $\phi(x,y)$ formula, $X,Y$ interpretable sets in $T$ and $f$ interpretable function from $X$ to $Y$,
\begin{itemize}
    \item If $a,a'$ have same type over $\emptyset$, $\operatorname{dim}(\phi(x,a))=\operatorname{dim}(\phi(x,a'))$
    \item $\operatorname{dim}(\emptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
    \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
    \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
    \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
    \item $\Gamma,\Delta$ sharply commute;
    \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
    \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
    There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
    Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
    \begin{itemize}
        \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
        \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
        \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
        \item $A_0=\Kat(\Gamma,\Delta)$.
    \end{itemize}
    Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

mptyset)=-\infty$ and $\operatorname{dim}(X)=0$ if and only if $X$ is finite;
 \item $\operatorname{dim}(X\cup Y)=\max\{\operatorname{dim}(X),\operatorname{dim}(Y)\}$;
 \item If $\operatorname{dim}(f^{-1}(y))\geq k$ if for any $y\in Y$, then $\operatorname{dim}(X)\geq \operatorname{dim}(Y)+k$;
 \item If $\operatorname{dim}(f^{-1}(y))\leq k$ for any $y\in Y$, then $\operatorname{dim}(X)\leq \operatorname{dim}(Y)+k$.
\end{itemize}
\end{defn}
Examples of finite-dimensional theories are $o$-minimal theories, supersimple theories of finite Lascar rank, and supperosy theories of finite $U^{\text{\thorn}}$-rank.\\
 A first proof of Zilber's Field Theorem in this context has been given by Deloro in \cite{deloro2024zilber}. An important hypothesis of the theorem is that the module on which the ring of endomorphisms acts is connected. This clearly holds in the finite Morley rank and $o$-minimal context, but in general is not true: it is sufficient to take a supersimple theory of finite Lascar rank. Therefore, it remains unclear what happens for non virtually connected modules. The aim of this article is exactly to analyse this case.\\
 In reality, we will work with two more general notions of endomorphisms: endogenies and quasi-endomorphisms. An endogeny $\phi$ of an abelian group $A$ is a subgroup of $A\times A$ such that $\pi_1(\phi)=A$ and $\{a\in A:\ (0,a)\in \phi\}$ is finite. The latter subgroup is called the \emph{katakernel} of $\phi$. A \emph{quasi-endomorphism} $\phi$ is a subgroup of $A$ such that $\pi_1(\phi)$, that is called the domain of $\phi$, is of finite index in $A$ and the katakernel is finite. Therefore, an endogeny is a quasi-endomorphism that is total.\\
Endogenies and quasi-endomorphisms arise naturally in the finite-dimensional context. Indeed, let $H\geq H_1$ be definable subgroups of the definable abelian group $G$ and $\Gamma\leq \operatorname{End}(G)$. Assume that $H$ and $H_1$ are \emph{$\Gamma$-almost invariant} \hbox{i.e.} $\gamma(H)\cap H$ is of finite index in $\gamma(H)$ for any $\gamma\in \Gamma $. Then, $\Gamma$ acts on $G/H$ by endogenies (with katakernel of $\gamma$ equal to $\gamma(H)+H/H$) and $\Gamma$ acts by quasi-endomorphisms on $H/H_1$ with domain $\{h\in H:\ \gamma(h)\in H\}$. In the finite Morley rank context and in the $o$-minimal one, we can avoid working with endogenies or quasi-endomorphisms since, if $H$ is $\Gamma$-almost invariant, the connected component $H^0$ is $\Gamma$-invariant and the action of $\Gamma$ on $G/H^0$ is again by endomorphisms.\\
The article follows the ideas of \cite{WagnerDeloro}, which proves the linearization theorem for endogenies in the connected case.\\
In the first section, we introduce all the definitions that we need in the article, and we verify some easy lemmas.\\
Then, we proceed to prove the following result.
\begin{thmA}
In a finite-dimensional theory, let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that:
\begin{itemize}
 \item $\Gamma,\Delta$ sharply commute;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa.
\end{itemize}
 There is a finite $\Gamma$ and $\Delta$-invariant subgroup $F$ such that $\Gamma/{\sim}$ and $\Delta/{\sim}$ act by endomorphisms on $A/F$.
\end{thmA}
Moreover, $F$ will be given explicitly.\\
In the second section, we verify the easy case in which both $\Gamma$ and $\Delta$ act by endogenies with finite kernel or finite image.\\
The third section analyzes the case in which the unbounded pre-ring of endogenies has an endogeny with infinite kernel and infinite image. Finally, we complete the proof in the fourth section.\\
The main result of the fifth section is the following linearization theorem.
\begin{thmB}
 Let $A$ be an abelian definable group and $\Gamma,\Delta$ two invariant pre-rings of definable endogenies such that
 \begin{itemize}
 \item $C^{\#}(\Gamma)=\Delta$ and $C^{\#}(\Delta)=\Gamma$;
 \item $A$ is absolutely $(\Gamma,\Delta)$-minimal;
 \item $\Gamma$ is essentially unbounded and $\Delta$ is essentially infinite or vice-versa;
 \item $A_0=\Kat(\Gamma,\Delta)$.
 \end{itemize}
 Then, there exists $K$ a definable infinite field such that $A/A_0$ is a finite-dimensional $K$-vector space contained into $\big(\Gamma/{\sim}\big)\cap \big(\Delta/{\sim}\big)$ and $\Gamma/{\sim}$ and $\Delta/{\sim}$ act $K$-linearly on $A/A_0$.
\end{thmB}
\textbf{Remark}:\\
These two theorems are not direct extensions of the Theorems in \cite{WagnerDeloro}. Indeed, we do not assume $A$ minimal but \emph{absolutely minimal} \hbox{i.e.} there exists no almost invariant infinite definable subgroups of infinite index. In reality, if we assume connectivity, our proofs can also be applied in the minimal case and so, in this sense, the results are an extension of the connected case.\\
In the sixth section, we introduce some basic results for quasi-endomorphisms. Then, in sections $7,8,9$, we prove an extension of Theorem $A$ in the case one of the two pre-rings $\Gamma,\Delta$ is of quasi-endomorphisms and the other of endogenies.\\
In section $10$, we prove a version of Theorem $A$ and $B$ in the case that $A$ has finite $n$-torsion for every $n<\omega$. Finally, in section $11$, we prove a generalization of Zilber's Field Theorem in finite-dimensional theories.
\subsection{Finite-dimensional groups}
In this subsection, we recall the main results on finite-dimensional groups, in particular Lemma \ref{boundedind}, which will be fundamental in the proof of Theorem A and Theorem B.

\begin{lemma}\label{boundedind}
    Let $G$ be a definable group of finite dimension and $\{H_i\}_{i\in I}$ a family of uniformly definable subgroups. Then, there exists $n,d<\omega$ such that there is no $J=\{j_1,...,j_n\}\subseteq I$ of cardinality $n$ such that $|\bigcap_{i=1}^kH_{j_i}:\bigcap_{i=1}^{k+1} H_{j_i}|$ has index greater than $d$ for any $k\leq n-1$.
\end{lemma}

\def\notind#1#2{#1\setbox0=\hbox{$#1x$}\kern\wd0
\hbox to 0pt{\mathchardef\nn=12854\hss$#1\nn$\kern1.4\wd0\hss}
\hbox to 0pt{\hss$#1\mid$\hss}\lower.9\ht0 \hbox to 0pt{\hss$#1\smile$\hss}\kern\wd0}

\end{theorem}
\begin{proof}
 We prove the base case. Given $\Gamma$ by hypothesis essentialy unbounded then it not admits an ascending chain of finite $\Gamma$-invariant subgroups. Therefore let $A_0$ this subgroup, it contains the katakernel of $\Delta$, that therefore is finite.\\
 Being the katakernel finite, there exists only boundedly many elements for $\Delta$ or $\Delta$ is unbounded (and then the proof follows using the previous idea).\\
 Then we can take a representative with maximal image and we have to prove that
\end{proof}
\begin{theorem}

\begin{proof}
Let $n=\dim(G)$. We will work in a sufficiently saturated structure $\mathfrak{M}$ in which $G$ is definable and assume that $\phi(x,y)$ is the formula defining the family. Assume, by contrary, that the conclusion is false then for $N=n+2$ and for any $k\in \omega$, there exists $g_1,...,g_N$ such that $|\bigcap_{j\leq k-1} \phi(\mathfrak{M},g_j)/\bigcap_{j\leq k} \phi(\mathfrak{M},g_j)|\geq k$. Therefore, the partial type given by the formulas
$$\exists a^1_1,...,a^1_k,...,a^N_1,...,a_k^N: a^i_j\in \bigcap_{j=1}^i \phi(\mathfrak{M},x_j)\wedge a^i_j{a^i_k}^{-1}\not\in \phi(\mathfrak{M},x_j)\wedge \forall_{i\leq N}\ \phi(\mathfrak{M},x_i)\leq G(\mathfrak{M})$$
for all $k<\omega$, is finitely satisfable. By compactness and saturation, there exist subgroups $\{H_i\}_{i\leq N}$ such that $|\bigcap_{i<j}H_i/\bigcap_{i\leq j}H_i|$ is infinite for every $i$. This would imply that the dimension of $G$ is strictly greater than $n$, a contradiction.
\end{proof}
If we apply these results to the centralizers of elements in $G$, we obtain that a finite-dimensional definable group has the hereditary $\widetilde{\mathfrak{M}}_c$-property.
\begin{defn}
    A group $G$ is \emph{hereditarily-$\widetilde{\mathfrak{M}}_c$} if for any definable subgroups $H,N$, such that $N$ is normalised by $H$, there exist natural numbers $n_{HN}$ and $d_{HN}$ such that any sequence of centralisers
    $$C_H(a_0/N)\geq C_H(a_0,a_1/N)\geq...\geq C_H(a_0,a_1,...,a_n/N)\geq...$$
    with each centraliser of index at least $d_{HN}$ in the previous one, has length at most $n_{HN}$.
\end{defn}
The notions of almost containment and commensurability are fundamental in the study of groups definable in finite-dimensional theories.
\begin{defn}
\begin{itemize}
    \item  Let $G$ be a group and $A, B\leq G$ subgroups. $A$ \emph{almost contains} $B$, denoted $A\apprge B$, if $|B:B\cap A|$ is finite.
    \item  If $A\apprle B$ and $B\apprle A$, $A$ and $B$ are \emph{commensurable}.
    \item Given a family of subgroups $\{G_i\}_{i\in I}$ in $G$, the family $\{G_i\}_{i\in I}$ is \emph{uniformly commensurable} if there exists $n<\omega$ such that $|G_i/G_i\cap G_j|\leq n$ for every $i,j\in I$.
\end{itemize}
  \end{defn}

Assume towards a contradiction that the conclusion of Theorem~\ref{boundedind} fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. Hence the desired bounds $n,d<\omega$ exist.

Assume towards a contradiction that the conclusion of the main theorem fails. Let $n=\dim(G)$ and set $N=n+2$. Working in a sufficiently saturated structure $\mathfrak{M}$ where the uniformly definable family is given by $\phi(x,y)$, the failure yields that for every $k\in\omega$ there exist parameters $g_1,\dots,g_N$ such that
\[
\left|\bigcap_{j\leq k-1}\phi(\mathfrak{M},g_j)\,\Big/\,\bigcap_{j\leq k}\phi(\mathfrak{M},g_j)\right|\ge k.
\]
From this, one writes a partial type asserting (via witnesses $a^i_j$) that each successive intersection properly decreases in a way producing many distinct cosets, and that each $\phi(\mathfrak{M},x_i)$ is a subgroup of $G(\mathfrak{M})$. This partial type is "finitely satisfiable". By compactness and saturation, obtain subgroups $\{H_i\}_{i\le N}$ such that for every $i$,
\[
\left|\bigcap_{i<j}H_i\,\Big/\,\bigcap_{i\le j}H_i\right|\text{ is infinite}.
\]
Such an infinite drop at each step would force $\dim(G)$ to be strictly greater than $n$, contradicting $n=\dim(G)$. This completes the proof of the main theorem.

Question: Let $G$ be a definable group of finite dimension, and let $\{H_i\}_{i\in I}$ be a uniformly definable family of subgroups of $G$ (that is, all the subgroups $H_i$ are given by instances of a single definable formula). Which statement holds about finite chains of successive intersections of members of this family?

There exist $n,d<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.\] Equivalently, there do not exist $n$ distinct indices $j_1,\dots,j_n\in I$ such that each successive index \(\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\) is greater than $d$ for every $k\le n-1$.

Choices:
B. There exists $d<\omega$ such that for every $n<\omega$ and every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.\] Equivalently, one may choose a single bound $d$ working uniformly for chains of all finite lengths.
C. There exists $n<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|<\infty.\]
D. There exist $n,d<\omega$ such that for every sequence of distinct indices $j_1,\dots,j_n\in I$ and every $k\le n-1$, one has \[\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.\] In other words, all successive indices along every chain of length $n$ are uniformly bounded by the same finite constant.
E. There exists $n<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$, there is some $k\le n-1$ for which \[\bigcap_{i=1}^k H_{j_i}=\bigcap_{i=1}^{k+1} H_{j_i}.\] Equivalently, in every chain of $n$ successive intersections, one inclusion must already be equality.

dropped the uniform finite bound d, retaining only finiteness of one successive index

bounded index confused with equality of successive intersections

The stem does not reveal the correct conclusion or encode its key quantifier structure; it only specifies the setting and asks which statement is true.

The item is close to theorem-statement recognition: the correct option is essentially the precise theorem formulation, though the presence of nearby variants prevents it from being a pure verbatim restatement.

Selecting the right answer requires some reasoning about quantifiers, uniform bounds, and the difference between existential and universal conditions, but it mainly tests recognition of the exact statement rather than deeper derivation.

The distractors are mathematically plausible and distinct: one is stronger by over-uniformizing d, one is weaker by dropping the bound, one incorrectly strengthens 'some k' to 'every k', and one confuses bounded index with equality.

A solid theorem-recognition MCQ with strong distractors and no answer leakage, but it leans more toward exact-statement recall than genuinely generative mathematical reasoning.


For every pair of fixed integers $r,\ell\geqslant 3$, consider the random greedy high girth $r$-clique
removal process on vertex set $[n]$. Let $M$  be the number of edges in the
generated linear $r$-graph with girth larger than $\ell$ when the process terminates.
With high probability, there exists some positive constant $\lambda=\lambda(\ell)$  such that
\begin{align*}
M\geqslant n^{1+ \frac{1}{\ell-1}-\lambda \frac{\log\log n}{\log n}}.
\end{align*}

For an integer $r\geqslant 3$,
 a hypergraph $H=(V_H,E_H)$ on vertex set $[n]$ is an
 \textit{$r$-uniform hypergraph} (\textit{$r$-graph} for short) if each edge
  is an $r$-element subset (\textit{$r$-set} for short) of $[n]$,
 and is said to be \textit{linear} if any two distinct edges share at most
 one vertex. Write ${v}_H$ and ${e}_H$ to denote the number of vertices and
 edges of $H$. An $r$-graph $H$ is called a partial Steiner $(n,r,s)$-system
 for an integer $s$ with $2\leqslant s\leqslant r-1$,
 if every $s$-set lies in at most one
 edge of it. In particular,  partial Steiner $(n,r,2)$-systems are
 linear $r$-graphs, partial Steiner $(n,3,2)$-systems are partial
 Steiner triple systems. Given a family $\mathcal{H}$ of linear $r$-graphs, define the
\textit{linear Turán number} of $\mathcal{H}$, denoted by
 $\textmd{ex}_r^L(n,\mathcal{H})$, is the maximum
number of edges among  linear $r$-graphs on vertex set $[n]$ that
 contain no $r$-graphs of $\mathcal{H}$ as subgraphs ($\mathcal{H}$-\textit{free} for short).
 Recently, the research on the Turán problem in linear hypergraphs has attracted
 considerable attention. Let
  $\textmd{Forb}_r^L(n,\mathcal{H})$ be the set
 of all  linear $\mathcal{H}$-free $r$-graphs on
vertex set $[n]$. When $\mathcal{H}$ consists of a single linear
 $r$-graph $H$, we simply write them
as $\textmd{ex}_r^L(n,{H})$  and
$\textmd{Forb}_r^L(n,{H})$ instead. The Tur\'{a}n problem and the enumeration problem
for a family $\mathcal{H}$ of $r$-graphs are closely related. Every subgraph of a $\mathcal{H}$-free graph
is also $\mathcal{H}$-free, then a trivial bound on the size of  $\textmd{Forb}_r^L(n,\mathcal{H})$
is given in~\cite{balgoh17} as follows
\begin{align*}
2^{\textmd{ex}_r^L(n,\mathcal{H})}\leqslant |\textmd{Forb}_r^L(n,\mathcal{H})|\leqslant
\sum_{i\leqslant \textmd{ex}_r^L(n,\mathcal{H})}\binom{\binom{n}{r}}{i}\leqslant 2n^{r\cdot \textmd{ex}_r^L(n,\mathcal{H})}.
\end{align*}

 There are several different notions of hypergraph cycles.
  For an integer $\ell\geqslant 3$, a Berge cycle $B_\ell^r$
  of length $\ell$ is an alternating sequence of distinct vertices and distinct
  edges of the form ${v}_1,\boldsymbol{e}_1,{v}_2,\boldsymbol{e}_2,\ldots,{v}_\ell,\boldsymbol{e}_\ell$,
  where ${v}_i,{v}_{i+1}\in \boldsymbol{e}_i$ for each $i\in[\ell-1]$ and ${v}_\ell,{v}_1\in \boldsymbol{e}_\ell$.
 This definition
  of a hypergraph cycle is the classical definition due to Berge.
    Note that forbidding a Berge cycle $B_\ell^r$ actually forbids a family of hypergraphs.
    Another notion of hypergraph cycles that has been actively investigated   is that of a linear cycle, which is unique up to isomorphism. An $r$-uniform linear cycle  $C_\ell^r$ of length $\ell$
 is an alternating sequence ${v}_1,\boldsymbol{e}_1,{v}_2,\boldsymbol{e}_2,\ldots,{v}_\ell,\boldsymbol{e}_\ell$ of distinct vertices and distinct
  edges such that $\boldsymbol{e}_i\cap \boldsymbol{e}_{i+1}=\{{v}_{i+1}\}$ for each $i\in[\ell-1]$,
   $\boldsymbol{e}_1\cap \boldsymbol{e}_\ell=\{{v}_1\}$ and $\boldsymbol{e}_i\cap \boldsymbol{e}_j=\emptyset$ if $1<|j-i|<\ell-1$.
The \textit{girth} of a linear $r$-graph is to be the smallest integer $\ell$
such that it contains a $C_\ell^r$. If no such $\ell$ exists,
the girth of the linear $r$-graph is defined to be infinite.

Determining $\textmd{ex}_3^L(n,C_3^3)$ is equivalent to the
famous $(6,3)$-problem, which aims at determining the maximum number of edges
of triple systems not carrying three edges on six vertices. The $(6,3)$-problem is studied
by Brown, Erd\H{o}s and S\'{o}s~\cite{brown73}, and it is known that for some constant $c>0$,
\begin{align*}
n^{2- {c}{\sqrt{\log n}}}<\textmd{ex}_3^L(n,C_3^3)=o(n^2),
\end{align*}
where the lower bound is given by Behrend~\cite{behrend46} and the upper bound is given by
Rusza and Szemer\'{e}di~\cite{ruz78}.
Erd\H{o}s, Frankl and R\"{o}dl~\cite{erd86}
showed that for every $r\geqslant 3$, $\textmd{ex}_r^L(n,C_3^r)=o(n^2)$ and
$\textmd{ex}_r^L(n,C_3^r)>n^{2-o(1)}$.
Using the so-called $2$-fold Sidon sets,
Lazebnik and  Verstra\"{e}te~\cite{laz03} constructed linear $3$-graphs with girth $5$
and showed that
\begin{align*}
\textmd{ex}_3^L\bigl(n,\{B_3^3, B_4^3\}\bigr)= \frac{1}{6}n^{  \frac{3}{2}}+O(n).
\end{align*}
Kostochka, Mubayi and Verstra\"{e}te asked if for integers $r\geqslant 3$ and $\ell\geqslant 4$,
\begin{align*}
\textmd{ex}_r^L\bigl(n,C_\ell^r\bigr)=\Theta\bigl(n^{1+ \frac{1}{\lfloor \ell/2\rfloor}}\bigr).
\end{align*} Later, Collier-Cartaino, Graber and Jiang~\cite{coll14}
proved that $\textmd{ex}_r^L(n,C_\ell^r)=O(n^{1+ \frac{1}{\lfloor \ell/2\rfloor}})$.
 In detail,
they obtained there exist positive constants $c(r, k)$ and $d(r, k)$
 such that
 \begin{align*}
 \textmd{ex}_r^L(n,C_{2k}^r)\leqslant c(r,k)n^{1+ \frac{1}{k}}\ \text{and }
 \textmd{ex}_r^L(n,C_{2k+1}^r)\leqslant d(r,k)n^{1+ \frac{1}{k}},
 \end{align*}
where the integer $k\geqslant 2$, the constants $c(r, k)$ and $d(r, k)$ are
exponential in $k$ for fixed $r$.
As an immediate corollary of the main results in~\cite{jiang30}, Jiang, Ma and Yepremyan
further improved the coefficient $c(r, k)$ to be linear in $k$.
Ergemlidze et al.~\cite{ergem2019} strengthened some results in $3$-graphs to be
$\textmd{ex}_3^L(n, B_4^3)= \frac{1}{6}n^{  3/2}+O(n)$, $\textmd{ex}_3^L(n, B_5^3)= \frac{1}{3\sqrt{3}}n^{  3/2}+O(n)$,
and for $k=2,3,4$ and $6$,
\begin{align*}
\textmd{ex}_3^L\bigl(n,\{C_3^3,C_5^3,\ldots,C_{2k+1}^3\}\bigr)= \Theta\bigl(n^{1+ \frac{1}{k}}\bigr).
\end{align*}
In general,  for $k\geqslant 2$, they also showed that \begin{align*}
\textmd{ex}_3^L\bigl(n,\{C_3^3,C_5^3,\ldots,C_{2k+1}^3\}\bigr)= \Omega\bigl(n^{1+ \frac{2}{3k-4+\epsilon}}\bigr),
\end{align*}
where $\epsilon=0$ if $k$ is odd and $\epsilon=1$ if $k$ is even.
As a special case of the linear hypergraph extension of K\H{o}v\'{a}ri–S\'{o}s–Tur\'{a}n's theorem in~\cite{gao2021},
Gao and Chang showed that $\textmd{ex}_3^L(n, C_4^3)= \frac{1}{6}n^{  3/2}+O(n)$,
and Gao et al. in~\cite{gao2023} further proved that $\textmd{ex}_3^L(n,C_5^3)= \frac{1}{3\sqrt{3}}n^{  3/2}+O(n)$.
For the lower bound of $\textmd{ex}_r^L(n,C_\ell^r)$, the best known lower bound  for $r=3$ is \begin{align*}
\textmd{ex}_3^L\bigl(n,C_\ell^3\bigr)=\Omega(n^{1+ \frac{1}{\ell-1}})
\end{align*}
that was shown in~\cite{coll14,ergem2019} due to Verstra\"{e}te, by taking a random subgraph of a Steiner
triple system.
Using generalized Sidon sets~\cite{ergem2019}, it is conceivable that one can obtain a similar (or
better) constructive lower bound of $\textmd{ex}_3^L(n,C_\ell^3)$
(and maybe also for all $r\geqslant 3$), which is an area worth some exploration.
The lower bound on the linear Tur\'{a}n number of linear cycles is still
far from what is conjectured, especially in higher uniformities $r\geqslant 4$.
Sometimes to explore the lower bound of the corresponding
Tur\'{a}n problem is more difficult than the upper one.

Following the same logic with the Tur\'{a}n problems of cycles,
for integers $r\geqslant 3$ and $\ell\geqslant 4$, Balogh and Li~\cite{balgoh17} conjectured that
$ |\textmd{Forb}_r^L(n,C_\ell^r)|=2^{\Theta(n^{1+1/\lfloor \ell/2\rfloor})}$.
They confirmed the conjecture in the case of $\ell=4$.
For $\ell>4$, they provided a result on the girth version, that is,
there exists a constant $c=c(r,\ell)$ such that
the number of linear $r$-graphs with girth larger than $\ell$ is at most
$2^{c\cdot n^{1+1/\lfloor \ell/2\rfloor}}$.
Define $\textmd{ex}_L(n,r,\ell)$ and $\textmd{Forb}_L(n,r,\ell)$
as  $\textmd{ex}_r^L(n,\mathcal{H})$ and $\textmd{Forb}_r^L(n,\mathcal{H})$ of
$\mathcal{H}=\{C_i^r:\, 3\leqslant i\leqslant \ell\}$, respectively.
Hence, it implies that $|\textmd{Forb}_L(n,r,\ell)|=2^{O(n^{1+1/\lfloor \ell/2\rfloor})}$, while its lower bound remains open.
The idea of the proof in~\cite{balgoh17} is to reduce the
 hypergraph enumeration problems to some graph enumeration problems, and
then the graph container method is implemented in it.
Balogh, Narayanan and Skokan~\cite{bal18} indeed made use of
the hypergraph container method and provided a balanced supersaturation theorem
for linear cycles to  resolve a conjecture on the
enumeration of $r$-uniform hypergraphs with a forbidden $C_\ell^r$.

In order to solve a conjecture of Erd\H{o}s about the existence
of sparse Steiner triple systems,
Glock, K\"{u}hn, Lo and Osthus~\cite{glock20}, and
Bohman and Warnke~\cite{boh19}, studied a constrained triple process along
similar lines, which can be
seen as a generalization of the \textit{random greedy triangle removal process}~\cite{bohman15},
to iteratively build a sparse partial Steiner triple system,
that is,
adding the sparseness constraint does not affect
the evolution of the process significantly.
To be precise, a Steiner triple system is called $k$-sparse if it does not
contain any $(i+2,i)$-configurations for integers $2\leqslant i\leqslant k$,
where a $(i+2,i)$-configuration
is a set of $i$ triples  which span at most $i+2$ vertices.
Erd\H{o}s made his conjecture
in Steiner triple systems.
Some further problems to
higher uniformities have been proposed
and considered in~\cite{fure13,glock20}.

The
 \textit{random greedy $r$-clique removal process} starts with a complete graph
 on vertex set $[n]$, denoted as $\mathbb{G}(0)$,
 and $\mathbb{G}(i+1)$ is the remaining graph from $\mathbb{G}(i)$ by
 selecting one $r$-clique uniformly at random out of all $r$-cliques in $\mathbb{G}(i)$
 and deleting all its edges from the edge set ${E}(i)$ of $\mathbb{G}(i)$.
 The process terminates once the remaining graph contains no $r$-cliques,
 and along the way produces a linear $r$-graph, where it starts with the
 empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each
 chosen $r$-clique as a new edge in the hypergraph.
 Let $M=\min\{i\,:\mathbb{G}(i)\ {\mbox{is\ } r{\mbox {-clique free}}}\}$,
 and  ${E}(M)$ be the set of edges left unsaturated
 by the produced $r$-cliques.  In particular, this problem attracted much  attention when $r=3$ and it is also called the random greedy triangle removal process.
Fewer results are known on $r\geqslant 4$.
Bennett and Bohman~\cite{bennett15} considered the random greedy
hypergraph matching process, which can be seen as a generalization
of the $r$-clique removal process.
Tian et al.~\cite{tian23} directly discussed the structure
of the random greedy $r$-clique removal process to a natural barrier,
 using a heuristic assumption to find the trajectories of an ensemble
 of random variables when the process evolves.
Inspired by the above works, for integers $r\geqslant 3$ and $\ell\geqslant 3$,
we consider the constrained {random greedy $r$-clique removal process}
so that it
does not just produce a linear $r$-graph but with girth larger than $\ell$,
 and the corresponding process is called  the \textit{random greedy high girth $r$-clique removal process}
 or \textit{high girth linear $r$-uniform hypergraph process}.
Compared with their analysis in~\cite{glock20,boh19} discussing sparse Steiner triple systems,
the challenge
arising in the analysis of random greedy high girth $r$-clique removal process is that each individual
forbidden configuration $C_i^r$ with $3\leqslant i\leqslant \ell$ should be considered.
It is more sensitive and stricter to establish
 the trajectories and error functions of an ensemble
 of random variables which we track in this constrained process with a higher uniformity,
such that the one-step
 changes, trend hypotheses and boundedness hypotheses of these tracked variables can  support the proof about the concentration of all these variables to the end.
 Finally, we show the following theorem.

Following the same logic with the Tur\'{a}n problems of cycles,
for integers $r\geqslant 3$ and $\ell\geqslant 4$, Balogh and Li~\cite{balgoh17} conjectured that
$ |\textmd{Forb}_r^L(n,C_\ell^r)|=2^{\Theta(n^{1+1/\lfloor \ell/2\rfloor})}$.
They confirmed the conjecture in the case of $\ell=4$.
For $\ell>4$, they provided a result on the girth version, that is,
there exists a constant $c=c(r,\ell)$ such that
the number of linear $r$-graphs with girth larger than $\ell$ is at most
$2^{c\cdot n^{1+1/\lfloor \ell/2\rfloor}}$.
Define $\textmd{ex}_L(n,r,\ell)$ and $\textmd{Forb}_L(n,r,\ell)$
as $\textmd{ex}_r^L(n,\mathcal{H})$ and $\textmd{Forb}_r^L(n,\mathcal{H})$ of
$\mathcal{H}=\{C_i^r:\, 3\leqslant i\leqslant \ell\}$, respectively.
Hence, it implies that $|\textmd{Forb}_L(n,r,\ell)|=2^{O(n^{1+1/\lfloor \ell/2\rfloor})}$, while its lower bound remains open.
The idea of the proof in~\cite{balgoh17} is to reduce the
 hypergraph enumeration problems to some graph enumeration problems, and
then the graph container method is implemented in it.
Balogh, Narayanan and Skokan~\cite{bal18} indeed made use of
the hypergraph container method and provided a balanced supersaturation theorem
for linear cycles to resolve a conjecture on the
enumeration of $r$-uniform hypergraphs with a forbidden $C_\ell^r$.

In order to solve a conjecture of Erd\H{o}s about the existence
of sparse Steiner triple systems,
Glock, K\"{u}hn, Lo and Osthus~\cite{glock20}, and
Bohman and Warnke~\cite{boh19}, studied a constrained triple process along
similar lines, which can be
seen as a generalization of the \textit{random greedy triangle removal process}~\cite{bohman15},
to iteratively build a sparse partial Steiner triple system,
that is,
adding the sparseness constraint does not affect
the evolution of the process significantly.
To be precise, a Steiner triple system is called $k$-sparse if it does not
contain any $(i+2,i)$-configurations for integers $2\leqslant i\leqslant k$,
where a $(i+2,i)$-configuration
is a set of $i$ triples which span at most $i+2$ vertices.
Erd\H{o}s made his conjecture
in Steiner triple systems.
Some further problems to
higher uniformities have been proposed
and considered in~\cite{fure13,glock20}.

The
 \textit{random greedy $r$-clique removal process} starts with a complete graph
 on vertex set $[n]$, denoted as $\mathbb{G}(0)$,
 and $\mathbb{G}(i+1)$ is the remaining graph from $\mathbb{G}(i)$ by
 selecting one $r$-clique uniformly at random out of all $r$-cliques in $\mathbb{G}(i)$
 and deleting all its edges from the edge set ${E}(i)$ of $\mathbb{G}(i)$.
 The process terminates once the remaining graph contains no $r$-cliques,
 and along the way produces a linear $r$-graph, where it starts with the
 empty $r$-graph on vertex set $[n]$, iteratively adding the vertex set of each
 chosen $r$-clique as a new edge in the hypergraph.
 Let $M=\min\{i\,:\mathbb{G}(i)\ {\mbox{is\ } r{\mbox {-clique free}}}\}$,
 and ${E}(M)$ be the set of edges left unsaturated
 by the produced $r$-cliques. In particular, this problem attracted much attention when $r=3$ and it is also called the random greedy triangle removal process.
Fewer results are known on $r\geqslant 4$.
Bennett and Bohman~\cite{bennett15} considered the random greedy
hypergraph matching process, which can be seen as a generalization
of the $r$-clique removal process.
Tian et al.~\cite{tian23} directly discussed the structure
of the random greedy $r$-clique removal process to a natural barrier,
 using a heuristic assumption to find the trajectories of an ensemble
 of random variables when the process evolves.
Inspired by the above works, for integers $r\geqslant 3$ and $\ell\geqslant 3$,
we consider the constrained {random greedy $r$-clique removal process}
so that it
does not just produce a linear $r$-graph but with girth larger than $\ell$,
 and the corresponding process is called the \textit{random greedy high girth $r$-clique removal process}
 or \textit{high girth linear $r$-uniform hypergraph process}.
Compared with their analysis in~\cite{glock20,boh19} discussing sparse Steiner triple systems,
the challenge
arising in the analysis of random greedy high girth $r$-clique removal process is that each individual
forbidden configuration $C_i^r$ with $3\leqslant i\leqslant \ell$ should be considered.
It is more sensitive and stricter to establish
 the trajectories and error functions of an ensemble
 of random variables which we track in this constrained process with a higher uniformity,
such that the one-step
 changes, trend hypotheses and boundedness hypotheses of these tracked variables can support the proof about the concentration of all these variables to the end.
 Finally, we show the following theorem.

In order to solve a conjecture of Erd\H{o}s about the existence
of sparse Steiner triple systems,
Glock, K\"{u}hn, Lo and Osthus~\cite{glock20}, and
Bohman and Warnke~\cite{boh19}, studied a constrained triple process along
similar lines, which can be
seen as a generalization of the \textit{random greedy triangle removal process}~\cite{bohman15},
to iteratively build a sparse partial Steiner triple system,
that is,
adding the sparseness constraint does not affect
the evolution of the process significantly.
To be precise, a Steiner triple system is called $k$-sparse if it does not
contain any $(i+2,i)$-configurations for integers $2\leqslant i\leqslant k$,
where a $(i+2,i)$-configuration
is a set of $i$ triples which span at most $i+2$ vertices.
Erd\H{o}s made his conjecture
in Steiner triple systems.
Some further problems to
higher uniformities have been proposed
and considered in~\cite{fure13,glock20}.

The remainder of the paper is structured as follows.
Notation and auxiliary
results used throughout the paper are presented in Section 2.
We define the random greedy high girth $r$-clique
removal process in Section 3, introducing some key random variables of the process that we wish to track,
estimating the corresponding expected trajectory and
 choosing error function for each tracked variable.
 We formally prove the concentration of all these variables in Section 4,
 where the required one-step changes, trend hypotheses and boundedness hypotheses for
these tracked variables are analyzed in higher uniformities.
 Two important claims to bound the overcounting and boundedness parameters
 are discussed in detail in the final section.

The \textit{random greedy high girth $r$-clique removal process} starts with a complete graph
 on vertex set $[n]$, denoted by $\mathbb{G}(0)$,
 and $\mathbb{G}(i+1)$ is the remaining graph from $\mathbb{G}(i)$ by
 selecting one $r$-clique uniformly at random out of all \textit{available} $r$-cliques in $\mathbb{G}(i)$
 and deleting all its edges from the edge set ${E}(i)$ of $\mathbb{G}(i)$,
 where an $r$-clique is called \textit{available} in $\mathbb{G}(i)$ means that adding
 the vertex set of this $r$-clique as an edge
 does not produce an $r$-graph in $\mathcal{L}$ with the edges generated by
 the vertices of previously chosen available $r$-cliques.
 Let ${M}=\min\{i:\mathbb{G}(i) {\text{ is} \textit{ available } r{\text{-}\textit{clique} \text{ free}}}\}$.
 This process is equivalently viewed as creating
 a random linear $r$-graph with girth greater than $\ell$, that is, the
 \textit{random greedy high girth linear $r$-uniform hypergraph process}.
 Beginning with the empty $r$-graph $\mathbb{H}(0)$ on the same vertex set $[n]$ we
sequentially set $\mathbb{H}({i+1})=\mathbb{H}(i)+\boldsymbol{e}_{i+1}$, where the added $r$-set $\boldsymbol{e}_{i+1}$
is the vertex set of the chosen available $r$-clique in $\mathbb{G}(i)$, and
$\boldsymbol{e}_{i+1}$ is also called to be an available $r$-set or available edge for $\mathbb{H}(i)$.
 The process terminates at a maximal linear $r$-graph $\mathbb{H}({M})$ with girth larger than $\ell$
 and no available $r$-sets or available edges. At time $i$, graphs are defined with
 respect to edge sets on the vertex set $[n]$, where
 both ${\mathbb{G}}(i)$ and $\mathbb{H}(i)$ depend on the outcomes of the process up
 to this point, and ${\mathbb{G}}(i)$ is the graph given by the pairs of vertices
 that do not appear in any edge of $\mathbb{H}(i)$.
It will be convenient to study the relation
 between $\mathbb{G}(i)$ and $\mathbb{H}(i)$ via
\begin{align}
{E}(i)=\binom{[n]}{2}\Bigl\backslash \bigcup_{\boldsymbol{f}\in \mathbb{H}(i)}
\biggl\{\bigl\{x,y\bigr\}\in \binom{\boldsymbol{f}}{2}\biggr\}.
\end{align}

\begin{theorem}
Given any $\mu>0$ and fixed integers $r,\ell\geqslant 3$, there exist $\lambda>0$ and $\alpha\in (0,1)$
such that, for any ${\boldsymbol{f}}_m\in {K}_m(i)$ with $2\leqslant m\leqslant r-1$,
${\boldsymbol{f}}\in {Q}(i)$, $L\in \mathcal{L}$ and $0\leqslant k\leqslant e_L-2$, with probability at least $1-n^{-\mu}$,
\begin{align}
|{Q}(i)|&={q}(t)\pm\epsilon_{q}(t),\\
|{Y}_{{\boldsymbol{f}}_m}(i)|&={y}_m(t)\pm \epsilon_{{y}_m}(t),\\
|{W}_{{\boldsymbol{f}},L,k}(i)|&={w}_{L,k}(t)\pm \epsilon_{w_{L,k}}(t),
\end{align}
holding when $i$ satisfying $0\leqslant i\leqslant {M}$
with ${M}$ defined as
\begin{align}{M}=n^{1+ \frac{1}{\ell-1}-\lambda \frac{\log\log n}{\log n}}.
\end{align}
Here, for $t\in [0,t_M]$,
\begin{align}{q}(t)&= \frac{n^r}{r!}p_i^{\binom{r}{2}}\xi_i,\\
{y}_m(t)&= \frac{n^{r-m}}{(r-m)!}p_i^{ \binom{r}{2}- \binom{m}{2}}\xi_i,\\
{w}_{L,k}(t)&= \frac{r!e_L}{|\texttt{Aut}(L)|}\binom{e_L-1}{k}(r!t)^{k}(p_i^{\binom{r}{2}}\xi_i)^{e_L-1-k}n^{(r-1)(e_L-k)+k-r},
\end{align}
and \begin{align}
\epsilon_{q}(t)&=\sigma_i n^{\alpha+r-1},\\\epsilon_{y_m}(t)&=\sigma_i n^{\alpha+r-m-1},\\\epsilon_{w_{L,k}}(t)&=\beta_k\sigma_i^{k+2} n^{\alpha+(r-1)(e_L-k)+k-r-1}t_{\textmd{M}}^k,
\end{align}in which \begin{align}
\sigma_i=\sigma(t)=\log(n^\alpha+n^2t),
\end{align} $\beta_k$ is a constant depending on $k$ and
\begin{align}
t_{{M}}= \frac{{M}}{n^2}=n^{-1+ \frac{1}{\ell-1}-\lambda \frac{\log\log n}{\log n}}.
\end{align}
\end{theorem}

Theorem~3.4 is proved in Section 4 and we
always tacitly assume $0\leqslant i\leqslant {M}$ with ${M}$ defined in~(3.18).
 It implies that the process produces
a linear $r$-graph of size at least ${M}$ with its girth greater than $\ell$.
Theorem~3.4 verifies Theorem~1.2
with room to spare in the power of the logarithmic factor.
We make no attempt to optimize the constants $\lambda$, $\alpha$ and
the coefficients $\beta_k$ for any $L\in \mathcal{L}$
 and $0\leqslant k\leqslant e_L-2$.
There are many choices of
them that can be balanced to satisfy certain inequalities.
For example, for given positive integers $r$, $\ell$ and $k$, we
choose these constants to satisfy the equations \begin{align}
\lambda> \frac{\ell}{\ell-1},\quad\alpha_0<\alpha<1 \quad\text{and}\quad\beta_k= \Bigl(\frac{3\ell r!}{p_{{M}}^{ \binom{r}{2}}\xi_{{M}}}\Bigr)^{k},
\end{align}
where $\alpha_0= \frac{\ell-2}{\ell-1}$, $p_{{M}}$ and $\xi_{{M}}$ are constants
when $i=M$ defined in~(3.5) and~(3.9). We will see these choices are sufficient for our proof of Theorem 3.4 in next section.
 We do not replace them with their actual values, which is for the interest of
understanding the role of these constants played in the calculations.

\noindent  In the following proof of Theorem~1.1, we will show the term $O({\log\log n}/{\log n})$
 is necessary in the exponent of
 the lower bound of
 $M$. Furthermore, it is impossible to apply this
 approach to obtain a better lower bound than $\Omega(n^{1+ \frac{1}{\ell-1}})$.
 As a corollary of Theorem 1.1, it is natural to  obtain lower bounds
of $\textmd{ex}_L(n,r,\ell)$ and $|\textmd{Forb}_L(n,r,\ell)|$
when  $\ell\geqslant 3$.
\begin{corollary} For every pair of fixed integers $r,\ell\geqslant 3$,
there exists $\lambda=\lambda(\ell)>0$ such that as $n\rightarrow\infty$,  \begin{align*}
\textmd{ex}_L(n,r,\ell)> n^{1+ \frac{1}{\ell-1}- \frac{\lambda\log\log n}{\log n}}\quad \text{and}\quad
|\textmd{Forb}_L(n,r,\ell)|> 2^{n^{1+ \frac{1}{\ell-1}- \frac{\lambda\log\log n}{\log n}}}.
\end{align*}
\end{corollary}
\noindent

The remainder of the paper is structured as follows.
Notation and  auxiliary
results used throughout the paper are presented in Section 2.
We define the random greedy high girth $r$-clique
removal process in Section 3, introducing some key random variables of the process that we wish to track,
estimating the corresponding expected trajectory and
 choosing error function for each tracked variable.
 We formally prove the concentration of all these variables in Section 4,
 where the required one-step changes, trend hypotheses and boundedness hypotheses for
these tracked variables are analyzed in higher uniformities.
    Two important claims to bound the overcounting and boundedness parameters
    are discussed in detail in  the final section.

We define the random greedy high girth $r$-clique removal process (Section 3) and introduce “some key random variables of the process that we wish to track,” then estimate “the corresponding expected trajectory” and “choos[e] error function for each tracked variable.” Next, we “formally prove the concentration of all these variables” (Section 4) by analyzing “the required one-step changes, trend hypotheses and boundedness hypotheses for these tracked variables … in higher uniformities.” Finally, “two important claims to bound the overcounting and boundedness parameters are discussed in detail in the final section.” The proof also notes that the exponent’s term $O({\log\log n}/{\log n})$ is “necessary,” and that “it is impossible to apply this approach to obtain a better lower bound than $\Omega(n^{1+ \frac{1}{\ell-1}})$.”

Question: For fixed integers $r,\ell\ge 3$, consider the random greedy high girth $r$-clique removal process on vertex set $[n]$: starting from the complete graph on $[n]$, one repeatedly chooses an $r$-clique uniformly at random among the currently available choices subject to the condition that the $r$-sets chosen so far form a linear $r$-uniform hypergraph of girth greater than $\ell$, adds the chosen $r$-set as a hyperedge, and deletes all graph edges of that clique; the process stops when no further such $r$-clique can be chosen. Let $M$ denote the number of hyperedges in the final generated linear $r$-graph. Which statement is guaranteed with high probability (that is, with probability tending to $1$ as $n\to\infty$)?

There exists a positive constant $\lambda=\lambda(\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}-\lambda\frac{\log\log n}{\log n}}$.

Choices:
B. There exists a positive constant $\lambda=\lambda(\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}+\lambda\frac{\log\log n}{\log n}}$.
C. There exists a positive constant $\lambda=\lambda(\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}-\lambda}$.
D. There exists a positive constant $\lambda=\lambda(r,\ell)$ such that $M\ge n^{1+\frac{1}{\ell-1}-\lambda\frac{\log\log n}{\log n}}$.
E. With high probability, for every positive constant $\lambda=\lambda(\ell)$ one has $M\ge n^{1+\frac{1}{\ell-1}-\lambda\frac{\log\log n}{\log n}}$.

necessary negative logarithmic correction

replaced the sharper $\frac{\log\log n}{\log n}$ loss by a fixed constant loss in the exponent

dependence of the constant on parameters; theorem states $\lambda=\lambda(\ell)$, not on $r$

existential choice of $\lambda$ replaced by universal quantification over all positive constants

The stem does not reveal the correct option explicitly or through obvious wording cues; the answer must be identified from the fine details of the asymptotic statement.

The item is very close to theorem recall: the correct choice is essentially the precise theorem statement, though the alternatives introduce nearby variants in exponent loss, parameter dependence, and quantification.

Selecting A requires some comparison of subtle asymptotic and quantifier differences, especially against the weaker true statement C and the near-miss options B, D, and E, but it mainly tests precise recall rather than substantial derivation.

The distractors are mathematically plausible and target realistic failure modes: wrong sign in the logarithmic correction, weakening the exponent, incorrect parameter dependence, and existential-versus-universal quantifier confusion.

A solid but theorem-recall-heavy MCQ: no answer leakage and strong distractors, but only moderate resistance to tautology and limited generative reasoning.