\label{introthm:transserial_char}
    The following are equivalent for an exponential o-minimal theory $T$:
    \begin{enumerate}[ref=(\arabic*)]
        \item\label{introthmenum:transserial} $T$ is transserial;
        \item\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\mathbb E, \mathcal{O}) \models T_\mathrm{convex}$, all $\mathcal{O} < b <\mathbb E^{>\mathcal{O}}$, for every $y \in \mathbb E \langle b \rangle$ there is a $\mathbb E$-definable gne $g$, such that $y \equiv_\mathbb E g(b)$;
        \item\label{introthmenum:gne_wim_1} all 1-$\dcl_T$-dimensional wim-constructible extension of models of $T_\mathrm{convex}$ have the gne-property;
        \item\label{introthmenum:gne_res_1} all 1-$\dcl_T$-dimensional res-constructible extensions of models of $T_\mathrm{convex}$ have the gne property;
        \item\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\mathrm{convex}$ have the gne property;
        \item\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\mathrm{convex}$ have the gne property.
    \end{enumerate}

\begin{proof}
        This is essentially already in Example~\ref{main:examples}(2). Let $\cO_x^+\coloneqq \cO_*^+\cap \bE\langle x \rangle$ and  Consider the valued differential field $(\bE \langle x \rangle, \cO_x^+, \partial_x)$.
        By Proposition~\ref{prop:der_order-convexity} and Corollary~\ref{cor:order-convex_implies_val-convex}(1), $\partial_x$ is weakly-$\cO_x$-Liouville-convex. But since it contains a $\cO$-wim element over $\bE$, by Corollary~\ref{cor:der-types_and_absorbed-elements} $\partial_x$ has type (W) and $\res_{\cO_*^+}(\bE)=\res_{\cO_*^+}(\bE\langle x \rangle)$.
    \end{proof}
\end{corollary}

\subsection{Tressl signature-alternative in simply exponential o-minimal fields}\label{ssec:first_application}
In this subsection we show how Proposition~\ref{prop:height_over_absorbed}, can be used to partially answer a problem posed in \cite{tressl2005model}. Our setting will be the one of an elementary extension $\bE \prec \bE_*\models T$ of models of an \emph{exponential} o-minimal theory $T$.

In \cite[Def.~3.16]{tressl2005model}, Tressl considers the following condition on a unary type $p=\tp(x/\bE)$ over $\bE$: $p$ satisfies the \emph{signature alternative} if it is either non-symmetric, or
\begin{enumerate}
    \item $\Br(x/\bE)(\bE)$ is cofinal in $\Br(x/\bE)(\bE\langle x\rangle)$, that is, $\bE\langle x \rangle$ does not realize the cut above $\Br(x/\bE)(\bE)$;
    \item if $y$ is the realization of such cut, i.e.\ if $y \in \Br(y/\bE)(\bE_*)$ and $y>\Br(x/\bE)(\bE)$, then $\Br(\log y/\bE)$ is cofinal in $\Br(\log y/\bE)(\bE \langle y \rangle)$.
\end{enumerate}

Note that if (1) above is satisfied by all types over $\bE$, then all types over $\bE$ satisfy (2) as well. We point out the following immediate Corollary of Proposition~\ref{prop:breadth-ortho} and

\begin{corollary}\label{cor:finite_hight_and_br_ortho}
    If $x \in \bE_* \setminus \bE$ is symmetric, and all $y \in \bE \langle x\rangle$ have finite height in $(\bE\langle x \rangle, \cO_*^-, \partial_x)$ with $\cO_*^-=\CH(\bZ)$, then $\Br(x/\bE)(\bE)$ is cofinal in $\Br(x/\bE)(\bE\langle x\rangle)$.
    \begin{proof}
        By Proposition~\ref{prop:breadth-ortho}(1) we have for all $y \in \bE \langle x \rangle$ that if $y \in \Br(x/\bE)(\bE\langle x\rangle)\setminus \CH(\Br(x/\bE)(\bE))$, then
        \[ \lder_x^k(y)\in \Br(x/\bE)(\bE\langle x\rangle)\setminus \CH(\Br(x/\bE)(\bE)) \quad \text{for all}\; k \in \bN.\]
        But then this would imply that $\val_{\cO_*^-}(\lder^k y) \notin \val_{\cO_*^-}(\bE)$ and $y$ would have infinite height.
    \end{proof}

\begin{thmA}\label{introthm:transserial_char}
    The following are equivalent for an exponential o-minimal theory $T$:
    \begin{enumerate}[ref=(\arabic*)]
        \item\label{introthmenum:transserial} $T$ is transserial;
        \item\label{introthmenum:exp-bdd_gne_at_special} $T$ is exponentially bounded and for all $(\bE, \cO) \models T_\convex$, all $\cO < b <\bE^{>\cO}$, for every $y \in \bE \langle b \rangle$ there is a $\bE$-definable gne $g$, such that $y \equiv_\bE g(b)$;
        \item\label{introthmenum:gne_wim_1} all 1-$\dcl_T$-dimensional wim-constructible extension of models of $T_\convex$ have the gne-property;
        \item\label{introthmenum:gne_res_1} all 1-$\dcl_T$-dimensional res-constructible extensions of models of $T_\convex$ have the gne property;
        \item\label{introthmenum:gne_wim} all wim-constructible extensions of models of $T_\convex$ have the gne property;
        \item\label{introthmenum:gne_res} all res-constructible extensions of models of $T_\convex$ have the gne property.
    \end{enumerate}
\end{thmA}

\begin{definition}\label{def:rosenlicht_levels}
    Let $T$ be an exponential o-minimal theory and $(\bE, \cO)\models T_\convex$. Recall that $x, y \in \bE^{> \cO}$ have the same \emph{Rosenlicht level} if there is a nautral number $m$ such that $\val(\log_m x)=\val(\log_m y)$ We write $x\asymp_Ly$ when $x$ and $y$ have the same Rosenlicht level. $\asymp_L$ is an equivalence relation with convex classes, so the set of Rosenlicht levels is naturally ordered.
\end{definition}

\begin{proof}
        Note that it is valuation convex for $\CH^{\le}(\bZ)$ by Remark~\ref{rmk:order-convex_to_val-convex}, then apply Lemma~\ref{lem:convex_der_val_relaxation} to conclude that $\partial$ is in fact $\cO_*$-valuation-convex.

        (2) Now suppose that $\partial$ is furthermore logarithmically order-convex. Note that for every order-convex subgroup $M<1$, we have $\CH^{\le}(\lder(1+M))=\CH(\partial M)$ as for all $m \in M$, $|\partial m/2| \le |\lder(1+m)| \le |2\partial m|$.
        So to conclude that it is weakly-$\cO_*$-Liouville convex, by Lemma~\ref{lem:Lv-convex_implies_log-convex} it suffices to show that $1+\co_*$ is $\lder$-convex for $\preceq_{\cO_*}$. Since by logarithmic order-convexity we have $\lder(1+\co_*)=\CH^{\le}(\lder(1+\co_*)) \cap \lder(\bE_*^{\neq0})$, it suffices to show that $\CH^{\le}(\lder(1+\co_*))=\CH^{\le}(\partial \co_*)$ is a $\cO_*$-module, but this follows from the first part of the statement.

        (3) By (1) and (2), we only need to show that $\partial$ satisfies (\ref{axiom:new-type-H-relative}).
        Let $\cO_\circ=\CH^{\le}_{\bE_*}(\bZ)$.
        Suppose that there are no constants $c$ with $x \preceq_{\cO_*} c \preceq_{\cO_*} y$, then a fortiori there are no constants $c$ with $x \preceq_{\cO_\circ} c \preceq_{\cO_\circ} y$, so by Lemma~\ref{lem:der_local_monotone}, we can conclude that if $\lder(x) \prec \lder(y)$, we have that for all $x\prec_{\cO_\circ} z \prec_{\cO_\circ} y$, and thus a fortiori for all $x \prec_{\cO_*} \prec z \prec_{\cO_*} y$, we have $\lder(z)\preceq_{\cO_\circ} \lder(y)$ and a fortiori $\lder(z) \preceq_{\cO_*}\lder(y)$.
    \end{proof}
\end{corollary}

\subsection{The case of few constants}\label{ssec:few-constants}
When the constants are included in the valuation ring, several notions of convexity collapse to notions already present in the literature and thoroughly studied in \cite{aschenbrenner2019asymptotic}.

\begin{remark}[Asymptotic and pre-d-valued fields]
    Recall the following notions from \cite[Sec.~9.1, 10.1]{aschenbrenner2019asymptotic}, that a valued differential field $(\bE_*, \cO_*, \partial)$ is
    \begin{enumerate}
        \item \emph{asymptotic} if for all $x,y\prec 1$ (or equivalently, by \cite[Prop.~9.1.3]{aschenbrenner2019asymptotic}, all $x,y \notin \cO_* \setminus \co_*)$, $x\prec y \Leftrightarrow \partial x \prec \partial y$;
        \item \emph{pre-d-valued} if for all $y\preceq 1$ and all $x \prec 1$ (or equivalently all $x \notin \cO_* \setminus \co_*$), $\lder(x) \succ \partial y$;
        \item \emph{asymptotic of type H}, if it is asymptotic and furthermore for all $x\prec y\prec 1$, $\lder(x) \succeq \lder(y)$.
    \end{enumerate}
    Also recall that every pre-d-valued field is asymptotic, \cite[Lem.~10.1.1]{aschenbrenner2019asymptotic}.
    The next Lemma shows that if the constants of $\partial$ are included in $\cO_*$ (i.e.\ there are \emph{few constants}, \cite[p.192]{aschenbrenner2019asymptotic}), then the notion of $\cO_*$-Liouville-convex specializes to the notion of a pre-d-valued field of type H and the notion of almost $\cO_*$-Liouville-convex specializes to the notion of pre-d-valued field.
\end{remark}

\begin{lemma}\label{lem:pre-d-valued}
    If $\partial$ is a derivation on $(\bE_*, \cO_*)$ with constants $\bE=\Kr(\partial) \subseteq \cO_*$, and $\val_{\cO_*}(\bE)$ divisible, then
    \begin{enumerate}
        \item if $\partial$ is $\cO_*$-valuation-convex, then $(\bE_*, \cO_*, \partial)$ is asymptotic;
        \item if $(\bE_*, \cO_*, \partial)$ is pre-d-valued if and only if $\cO_* \setminus \co_*$ is $\lder$-convex;
        \item if $(\bE_*, \cO_*, \partial)$ is pre-d-valued, then $\partial$ is $\cO_*$-valuation-convex, so $(\bE_*, \cO_*, \partial)$ is pre-d-valued if and only if it is almost Liouville-$\cO_*$-convex;
        \item if $(\bE_*, \cO_*, \partial)$ is pre-d-valued, then it is of type H if and only if it is logarithmically convex.
    \end{enumerate}
    \begin{proof}
        (1). Suppose that $\partial$ is $\cO_*$-valuation convex and $x,y\prec 1$. If $\partial x \in \cO_*\partial y \subseteq \cO_* \partial (y \cO_*)$, then by convexity $\partial x \in \partial (y \cO_*)$, so $x \in \bE+ y \cO_*$, but since $x, y \prec \bE^{\neq 0}$ by construction, this implies $x \in \cO_* y$. Similarly if $\partial x \in \co_* \partial y\subseteq \cO_* \partial (y \co_*)$ (here we are using Lemma~\ref{lem:module-derivatives}(4)),
        then $\partial x \in \partial (y \co_*)$, so by the same argument $x \prec y$.

        (2). $\cO_* \setminus \co_*$ is $\lder$-convex if and only if $\val(x) \notin \val(\bE)\Rightarrow \lder(x) \succ \partial \cO_*$. But since $\bE \subseteq \cO_*$, we have $\val(x) \notin \val(\bE) \Leftrightarrow x \not \asymp 1$ and thus this is equivalent to $\lder(x) \succ \partial \cO_*$ for all $x\not \asymp 1$ which is being pre-d-valued.

        (3)         We need to show to show that for all $y$, $\partial(y \cO_*)$ is $\preceq$-convex in $\partial \bE_*$ for all $y$, that is: $x \notin \bE+y\cO_* \Rightarrow \partial x \succ \partial (y \cO_*)$.

        Note that since $\bE \subseteq \cO_*$, we have $x \notin \bE+y \cO_* \Leftrightarrow x\succ y$.
        So if $y \not\asymp 1$, we get $\partial x \succ \partial y$. Furthermore we have $\lder(y)\succ \partial \cO_*$, so by Lemma~\ref{lem:module-derivatives}(1) $\cO_*\partial (y \cO_*)=\cO_*\partial y  + y \cO_*\partial \cO_*=\cO_* \partial y$ and we are done.

        For the case $y \in \cO_* \setminus \co_*$, note that by the hypothesis that $\val(\cO_*)$ is divisible we have $x\succ z$ for some $1\prec z \prec x$, so we can reduce to the previous case and deduce $x \succ \partial(z\cO_*)\supseteq \partial (\cO_*)$.

        (4) Note that since $\bE^{\neq0} \subseteq \cO_*\setminus \co_*$, we have that for $1\prec x\prec y$, $\not\exists c \in \bE,\, 1 \prec c \preceq y$, so (\ref{axiom:new-type-H}) is equivalent to $1 \prec x\prec y \Rightarrow \lder(x) \preceq \lder(y)$. On the other hand it is easy to see that if $1 \prec x\prec y \Rightarrow \lder(x) \preceq \lder(y)$, then (\ref{axiom:new-type-H-relative}) holds.
    \end{proof}

\begin{definition}\label{def:v-convex_derivation}
    Let $(\bE_*, \cO_*)$ be a valued field and $\partial: \bE_* \to \bE_*$ be a derivation.
    I will say that a derivation $\partial$ on $\bE_*$ is \begin{enumerate}
        \item \emph{$\cO_*$-valuation-convex} if all $\cO_*$-submodules are $\partial$-convex (w.r.t.\ $\preceq_{\cO_*}$);
        \item \emph{weakly logarithmically $\cO_*$-valuation-convex} if for all $\cO_*$-modules $M\subseteq \co_*$, $1+M$ is $\lder$-convex;
        \item \emph{almost logarithmically $\cO_*$-valuation-convex} if it is almost $\cO_*$-valuation-convex and $\cO_*\setminus \co_*$ is $\lder$-convex;
        \item \emph{logarithmically $\cO_*$-valuation convex} if it is weakly logarithmically $\cO_*$-valuation-convex and for all $x,y,z$
        \begin{equation}\tag{H1}\label{axiom:new-type-H}
            (x \preceq z \preceq y \;\& \; \lder(x) \prec \lder(y)\prec \lder(z)) \Rightarrow \exists c, \; (\partial c=0 \; \& \; x \prec c \preceq y).
        \end{equation}
        \item \emph{weakly $\cO_*$-Liouville-convex} if it is valuation-convex and $1+\co_*$ is $\lder$-convex;
        \item \emph{almost $\cO_*$-Liouville-convex} if it is valuation-convex and $\cO_*\setminus\co_*$ is $\lder$-convex;
        \item \emph{$\cO_*$-Liouville-convex} if it is valuation-convex and logarithmically valuation-convex.
    \end{enumerate}
    We will often drop the ``$\cO_*$-'' in front of the definition if it is clear from the context or specified otherwise (e.g.\ saying w.r.t.\ $\cO_*$).
\end{definition}

\begin{definition}\label{def:order-convex_der}
    Let $\bE_*$ be an ordered field. We will say that $\partial: \bE_* \to \bE_*$ is \emph{order-convex}, if for all $y \in \bE_*^{\ge0}$, $[-y,y]$ is $\partial$-convex, we will say that $\partial$ is \emph{logarithmically order-convex} if for all $y \in \bE_*^{\ge1}$, $[1/y,y]$ is $\lder$-convex. We will say that it is \emph{order-Liouville-convex} if it is both order-convex and logarithmically order-convex.
\end{definition}

Let $T$ be the theory of an o-minimal expansion of an ordered field and let $T_\mathrm{convex}$ be the (complete) theory of models of $T$ expanded by a predicate for a non-trivial $T$-convex valuation subring (cf \cite[Def.~2.7, Cor.~3.13]{dries1995t}). If $\mathcal{O}$ is a convex subring of a field $\mathbb E$ we denote by $\mathfrak{o}$ the corresponding maximal ideal. We will use the standard notations $\val_\mathcal{O}$, $\rv_\mathcal{O}$, and $\res_\mathcal{O}$, respectively for the valuation induced by $\mathcal{O}$, the rv-sort induced by $\mathcal{O}$, and the map $\res_\mathcal{O}: \mathbb E \to \mathcal{O}/\mathfrak{o}$ which is $0$ outside of $\mathcal{O}$ and is given by the quotient modulo $\mathfrak{o}$ on $\mathcal{O}$. We drop the subscript $\mathcal{O}$ if it is clear from the context.

In \cite{freni2024t}, the author introduced the $T$-$\lambda$-spherical completion of a model $(\mathbb E, \mathcal{O})\models T_\mathrm{convex}$ (where $\lambda$ is an uncountable cardinal): this is a prime elementary $\lambda$-spherically extension of $(\bE_\lambda, \cO_\lambda)\succ(\mathbb E, \mathcal{O})$, it is in general unique up to isomorphism and has the same residue field as $(\mathbb E,\mathcal{O})$, \cite[Thm.~B]{freni2024t}.

Toward the goal of better understanding these completions, it was shown in the same paper that if $T$ is simply exponential (that is, an expansion of a power-bounded o-minimal theory by a compatible exponential), then $(\bE_\lambda, \cO_\lambda)$ only realizes unary types $p$ over $\mathbb E$ which are images of some other $\mathcal{O}$-weakly immediate type $q$ over $\mathbb E$ by a $\mathbb E$-definable \emph{generalized nested exponential} (henceforth a \emph{gne}): a finite composition $g$ of translations by elements of $\mathbb E$, sign changes and exponentials \cite[Prop.~5.27 and Cor.~5.30]{freni2024t}.

In this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:
\begin{enumerate}[label=(TS), ref=(TS)]
 \item\label{axiom:transserial0} for all $n\in \mathbb N$ and for all $(n+1)$-ary $T$-definable functions $f$, said
 \[\dagger_n f(x_0, \ldots, x_{n-1}, t)=\frac{\partial f(x_0, \ldots, x_{n-1}, t)/\partial t}{f(x_0, \ldots, x_{n-1}, t)} \quad \text{if it is defined}\]
 and $\dagger_n f(\overline{x}, t)=\infty$ otherwise, and set $G_m^f(\overline{x},t)\coloneqq\min\{|(\lder^k_nf)(\overline{x},t)|: k \le m\}$ one has that for some $m \in \mathbb N$
 \[T_\mathrm{convex} \models \forall \overline{x},\; \exists r\in \mathcal{O}, \;\forall t \in \mathcal{O},\; \big(t>r \rightarrow G_m^f(\overline{x},t) < 1 \big).\]
\end{enumerate}

It turns out that for transserial theories, a similar property to the one outlined above for $T$-$\lambda$-spherical completions, is shared by other kind of extensions. So, to state the main result in a compact way, let us say that an elementary extension $(\mathbb E, \mathcal{O}) \prec (\bE_*, \cO_*)$ of models of $T_\mathrm{convex}$,
\begin{enumerate}[label=(E\arabic*)]
 \item\label{df:gne-ppty} has the \emph{gne-property} if for all elements $y\in \bE_*$, there is a $\mathbb E$-definable gne such that $\mathbf{v}(g^{-1}(y)-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$.
\end{enumerate}
and recall that an elementary extension $(\mathbb E, \mathcal{O}) \preceq (\bE_*, \cO_*)\models T_\mathrm{convex}$ is said to be
\begin{enumerate}[label=(E\arabic*), resume]
 \item\label{df:res-constructible} \emph{residually constructible} (or \emph{res-constructible}, (cf \cite[Def.~14]{freni2025residually}), if there is a $\dcl_T$-basis $\overline{r}$ of $\bE_*$ over $\mathbb E$, such that $\mathbf{r}(\overline{r})$ is $\dcl_T$-independent over $\mathbf{r}(\mathbb E)$;
 \item\label{df:wim-constructible} \emph{($\lambda$-bounded) wim-constructible} (cf \cite[Def.~3.15]{freni2024t}) if there is an ordinal indexed $\dcl_T$-basis $(x_i: i<\alpha)$ of $\bE_*$ over $\mathbb E$, such that for all $j<\alpha$, $x_j$ is a pseudolimit of a p.c.\ sequence in $\mathbb E \langle x_i: i<j\rangle$ without pseudolimits therein and of length $<\lambda$.
\end{enumerate}
Also recall that by \cite[Thm.~A]{freni2024t},
\begin{enumerate}[label=(\alph*)]
 \item if $(\mathbb E, \mathcal{O}) \prec (\bE_*, \cO_*)$ is wim-constructible then $\mathbf{r}(\mathbb E, \mathcal{O})=\mathbf{r}(\bE_*, \cO_*)$, hence if $y \in \bE_*\setminus \mathbb E$, $\mathbf{v}(y-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$ means that $y$ is a pseudolimit;
 \item if $(\mathbb E, \mathcal{O}) \prec (\bE_*, \cO_*)$ is residually constructible, then $\bE_*$ does not contain pseudolimits of limitless p.c. sequences in $\mathbb E$, hence if $y \in \bE_*\setminus \mathbb E$, $\mathbf{v}(y-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$ means that $y=c+dz$ for some $c,d \in \mathbb E$, and $z\in \bE_*\setminus \mathbb E$ with $\mathbf{r}(z)\notin \mathbf{r}(\mathbb E, \mathcal{O})$.
\end{enumerate}
Note that a $T$-$\lambda$-spherical completion is a wim-constructible extension (in fact a maximal $\lambda$-bounded one).
The main result is thus stated as follows:

In this paper I characterize those complete exponential o-minimal theories $T$ for which this fact holds true. These are called \emph{transserial} o-minimal theories and are those satisfying the following strengthening of exponential-boundedness:
\begin{enumerate}[label=(TS), ref=(TS)]
 \item\label{axiom:transserial0} for all $n\in \bN$ and for all $(n+1)$-ary $T$-definable functions $f$, said
 \[\dagger_n f(x_0, \ldots, x_{n-1}, t)=\frac{\partial f(x_0, \ldots, x_{n-1}, t)/\partial t}{f(x_0, \ldots, x_{n-1}, t)} \quad \text{if it is defined}\]
 and $\dagger_n f(\overline{x}, t)=\infty$ otherwise, and set $G_m^f(\overline{x},t)\coloneqq\min\{|(\lder^k_nf)(\overline{x},t)|: k \le m\}$ one has that for some $m \in \bN$
 \[T_\convex \models \forall \overline{x},\; \exists r\in \cO, \;\forall t \in \cO,\; \big(t>r \rightarrow G_m^f(\overline{x},t) < 1 \big).\]
\end{enumerate}

The name \emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\log(f(x)-x)=f(\log_2(x))$ are prohibited.

The way results are obtained is an analysis of the ordered differential valued fields $(\bE\langle x\rangle, \cO_x, \partial_x)$ where
\begin{enumerate}[label=(C\arabic*)]
 \item $\bE\langle x\rangle\coloneqq\dcl_T(\bE, x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\bE\models T$ and $\cO_x$ is a $T$-convex valuation subring;
 \item $\partial_x$ is the unique non-trivial $\bE$-linear $T$-derivation with $\partial_x (x)=1$ (cf \cite{fornasiero2021generic}, \cite{kaplan2023t});
 \item $\val_{\cO_x}(x-\bE)\subseteq \val_{\cO_x}(\bE)$.
\end{enumerate}

\begin{corollary}\label{cor:sOabs_to_sOabs_der}
 Let $x \in \bE_*\setminus \bE$ and $f,g,h: \bE_* \to \bE_*$ be $\bE$-definable functions. Then the following hold:
 \begin{enumerate}
 \item if $x \in \bE_* \setminus \bE$ is strongly $\cO$-absorbed and $h(x) \in (B_x^+:\cO)$, then $h'(x) \in \co_*^-$, furthermore if $\cO$ is $T$-convex, then $h'(x) \in \co_*^+$;
 \item if $x \in \bE_* \setminus \bE$ is $\cO$-wim and $f(x)\equiv_\bE g(x)$ are strongly $\cO$-absorbed, then $f(x), g(x)$ are $\cO$-wim and moreover $f'(x) \sim_{\cO_*^-} g'(x)$.
 \end{enumerate}
 \begin{proof}
 (1) is a direct application of Lemma~\ref{lem:sOabs_to_sOabs_der} with $f(x)=x$ and $g(x)=x+h(x)$. The ``furthermore'' follows from the fact that if $\cO$ is $T$-convex, and $x$ is strongly $\cO$-absorbed, then by \cite[Thm.~A]{freni2024t} $\co_*^-\cap \bE \langle x \rangle = \co_*^+\cap \bE \langle x \rangle$.
 (2) if $x$ is $\cO$-wim, then $B_x^+$ is a $\cO$-module. Furthermore since $f(x), g(x)$ are strongly $\cO$-absorbed, by Lemma~\ref{lem:wim-cofres_orto}, we must have that $f(x)$ and $g(x)$ are $\cO$-wim as well so $B_{f(x)}$ is a $\cO$-module as well, but then Lemma~\ref{lem:sOabs_to_sOabs_der}, implies $|f'(x)-g'(x)|<f'(x)/r$ for all $r \in \cO$, whence $f'(x) \sim_{\cO_*^-} g'(x)$.
 \end{proof}
\end{corollary}

\begin{proposition}\label{prop:der_order-convexity}
 Let $T$ be an exponential o-minimal theory, $\bE \prec \bE\langle x\rangle\models T$ a 1-$\dcl_T$-dimensional extension of models of $T$, and $\partial_x$ be the only $\bE$-linear $T$-derivation with $\partial_x x=1$, then for all $y \in \bE\langle x\rangle^{\neq 0}$, $[-y, y]$ is $\partial_x$-convex and $[1/y, y]$ is $\dagger_x$-convex.
 \begin{proof}
 We have to show that if $f, h$ are continuously differentiable $\bE$-definable unary functions, with $|f'(x)|\le |h'(x)|$ and $|h(x)|\le y$, then $f(x) \in \bE + [-y,y]$.
 Note that we can in fact assume that $0<|h(x)|\le y$ and that $h'(x) \neq 0$, otherwise $f'(x)=0$ and the thesis is trivial. Similarly we can assume $|f'(x)|<|h'(x)|$, because otherwise $f'=h'$ in a $\bE$-definable neighborhhod of $x$ and $f(x)-h(x)\in \bE$ and again the thesis is trivial.

\begin{corollary}\label{cor:main-cor}
 The following are equivalent:
 \begin{enumerate}
 \item every element of $\bE \langle x\rangle\setminus \bE$ has finite height over $\partial$;
 \item $(\bE\langle x\rangle, \cO_*)\succ (\bE, \cO_*\cap \bE)$ has the gne-property;
 \end{enumerate}
 If furthermore $\val_{\cO_*}(x-\bE)\subseteq \val_{\cO_*}(\bE)$, then they are equivalent to
 \begin{enumerate}[resume]
 \item for every $y \in \bE \langle x \rangle\setminus \bE$, there is $k\in\bN$ such that $\val(\lder^k(y)) \in \val(\bE)$.
 \end{enumerate}
 \begin{proof}
 Suppose (1), then by Lemma~\ref{lem:approximate_to_exact_log_scale}, we for all $y \in \bE\langle x \rangle$ find a $\bE$-definable gne $g$ and a $z \in \bE \langle z\rangle$ with $g(z)=y$ and $\val(z-\bE)\subseteq \val(\bE)$, thus we get (2).

\begin{lemma}\label{lem:unary_functions}
 Let $T$ be an exponential o-minimal theory.
 \begin{enumerate}
 \item $T$ is weakly transserial if and only if for all $(\bE, \cO) \models T_\convex$ and every unary $\bE$-definable function $f$, there is a natural number $m$ such that
 \begin{equation}\tag{$\mathrm{WTS}^{f}_m$}
 (\bE, \cO) \models \exists r \in \cO, \forall t\in \cO, \big(t>r \rightarrow G_m^f(t) \in \cO\big).
 \end{equation}
 \item $T$ is transserial if and only if for all $(\bE, \cO) \models T_\convex$ and every unary $\bE$-definable function $f$, there is a natural number $m$ such that
 \begin{equation}\tag{$\mathrm{TS}^{f}_{m,1}$}
 (\bE, \cO) \models \exists r \in \cO, \forall t\in \cO, \big(t>r \rightarrow G_m^f(t) \le 1\big).
 \end{equation}
 \end{enumerate}
 \begin{proof}
 We prove (2), the proof of (1) is similar.
 $\Rightarrow$ is obvious.
 The converse is a compactness argument. Suppose that for some $T$-definable $n$-ary $f: \bE^{n+1} \to \bE$ we have for all $m \in \bN$
 \[T_\convex \models \exists \overline{x},\, \forall r\in \cO, \exists t\in \cO, (t > r \land G_m^f(\overline{x}, t)>1)\]
 which by weak o-minimality of $T_\convex$ is equivalent to \begin{equation}\label{eqn:strong_negation}
 T_\convex \models \exists \overline{x},\, \exists r\in \cO, \forall t\in \cO,\, (t > r \rightarrow G_m^f(\overline{x}, t)>1).
 \end{equation}
 Notice that since
 \[G_{m+1}^f(\overline{x}, t)=\min\{G_{m}^f(\overline{x}, t), \lder_n (G_m^f)(\overline{x},t)\} \le G_{m}^f(\overline{x}, t),\]
 said
 \[\varphi_m(\overline{x}):= \big(\exists r\in \cO, \forall t\in \cO,\, (t > r \rightarrow G_m^f(\overline{x}, t)>1)\big),\]
 we have $\varphi_{m+1}(\overline{x}) \rightarrow \varphi_{m}(\overline{x})$ hence the validity of \ref{eqn:strong_negation} for all $m$ is equivalent to the consistency of the type $p(\overline{x})=\{\varphi_m(\overline{x}): m \in \bN\}$. If $(\bE, \cO)$ is a model of $T_\convex$ realizing such type and $\overline{c} \in \bE^n$ is a realization of $p$, we get that $g(t)=f(\overline{c}, t)$ is such that for all $m$ $(\bE, \cO)\models \exists r \in \cO,\, \forall t \in \cO,\, (t>r \rightarrow G_m^g(t)>1)$.
 \end{proof}
\end{lemma}

The name \emph{transserial}, comes from the field of logarithmic exponential transseries, where certain configurations such as formal asymptotic solutions of the equation $\log(f(x)-x)=f(\log_2(x))$ are prohibited.

I also show that if a transserial theory admits an Archimedean prime model, then Tressl's signature alternative (cf \cite[Def.~3.16]{tressl2005model} or see Subsection~\ref{ssec:first_application}) holds true for every cut over every model. Moreover all Hardy fields of a transserial theory have the the same Rosenlicht levels (as defined in \cite[Def.~5.2]{berarducci2015surreal} in the context of surreal numbers with the natural valuation, see Definition~\ref{def:rosenlicht_levels}).

\begin{thmA}\label{introthm:transserial_consequences}
    Let $T$ be a transserial theory, then:
    \begin{enumerate}
        \item wim-constructible and residually constructible extensions do not add new Rosenlicht levels, in particular for all $\mathbb K\prec \mathbb K'\models T$, the Hardy fields of $\mathbb K$ and $\mathbb K'$ have the same Rosenlicht levels;
        \item if $T$ has an Archimedean prime model, then for every $\mathbb E\models T$, every unary type $p$ over $\mathbb E$ satisfies Tressl's signature alternative.
    \end{enumerate}
\end{thmA}

Note that the condition (1) is a weakening of the notion of ($\mathbb Z$-)\emph{levelled} o-minimal theory of \cite{marker1997levelled}.

\smallskip

The way results are obtained is an analysis of the ordered differential valued fields $(\mathbb E\langle x\rangle, \cO_x, \partial_x)$ where
\begin{enumerate}[label=(C\arabic*)]
    \item $\mathbb E\langle x\rangle\coloneqq\dcl_T(\mathbb E, x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\mathbb E\models T$ and $\cO_x$ is a $T$-convex valuation subring;
    \item $\partial_x$ is the unique non-trivial $\mathbb E$-linear $T$-derivation with $\partial_x (x)=1$ (cf \cite{fornasiero2021generic}, \cite{kaplan2023t});
    \item $\val_{\cO_x}(x-\mathbb E)\subseteq \val_{\cO_x}(\mathbb E)$.
\end{enumerate}

Note that this situation comprises the cases in which $(\mathbb E \langle x\rangle, \cO_x)\succ (\mathbb E, \mathcal{O})$ is wim-constructible or res-constructible.

A key ingredient in Theroem~\ref{introthm:transserial_char}, is that in the setting above, if $T$ is exponential, an element $y \in \mathbb E \langle x \rangle$ is of the form $g(z)$ for a $\mathbb E$-definable gne and a $z$ such that $\mathbf{v}(z-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$, if and only if there is a natural number $m$ such that $\mathbf{v}(\lder_x^m \partial_x(y))\in \mathbf{v}(\mathbb E)$ (Proposition~\ref{prop:height_over_absorbed}).

\smallskip

The treatment points out some general properties of ordered differential valued fields the form $(\mathbb E \langle x\rangle, \cO_x, \partial_x)$ satisfying (C1) and (C2) above. When $x>\mathbb E$, and $\cO_x$ is the convex hull of $\mathbb E$, $(\mathbb E \langle x\rangle, \cO_x, \partial_x)$ is the Hardy field of the o-minimal structure $\mathbb E$ which is an example of an $H$-field in the sense of \cite{aschenbrenner2002fields}.

In a similar vein, even $\mathrm{tp}(x/\mathbb E)$ is not a definable type, many important properties are captured by the notions of an \emph{order-convex} (resp.\ \emph{logarithmically ---}) derivation and a \emph{valuation-convex} (resp.\ \emph{logarithmically ---}) derivation (Definitions~\ref{def:order-convex_der} and \ref{def:v-convex_derivation}).

I call a derivation $\partial$ on an ordered field $\bE_*$, \emph{order-convex} if it maps order-convex sets onto order-convex subsets of $\partial\bE_*$, similarly it is \emph{logarithmically order-convex} if the corresponding logarithmic derivative $\lder(t)\coloneqq (\partial t)/t$ sends order-convex sets to order-convex subsets of $\lder(\mathbb E^{\neq0}_*)$. These can be regarded as traces of the mean-value theorem for functions definable in o-minimal ordered fields.
The notion of valuation-convex and logarithmically valuation-convex arise naturally as traces of the corresponding notions of order-convexity with respect to the minimal order-convex valuation, specifically logarithmic convexity has several weakening that are sufficient for most purposes.

As expected, when $\cO_*$ contains the field of constants, this notions specialize to several notions already studied in the literature (\cite{rosenlicht1980differential},\cite{rosenlicht1983rank}, \cite{aschenbrenner2002fields}, and \cite{aschenbrenner2019asymptotic}) such as pre-H-field, asymptotic field, pre-d-valued field and pre-d-valued field of H-type (see Subsection~\ref{ssec:few-constants}).

Given suitable convexity assumptions on the derivative on a valued field $(\bE_*, \cO_*)$ with constants $\mathbb E\coloneqq\Kr(\partial)$, the condition $\mathbf{v}(x-\mathbb E)\subseteq \mathbf{v}(\mathbb E)$ is equivalent to the fact that the derivation ``at $x$'' $\partial_x=\partial/\partial x$ has the property that $y \succ \partial_x\cO_x\Rightarrow \lder_x(y)\prec y$. We will say that such derivations are \emph{absorbed} (Definition~\ref{def:absorbed-type-der}), a property that will be at the center of several arguments.

\subsection{Acknowledgments} I thank Vincenzo Mantova for several conversations on the topic and for encouraging me to try to convert some previous proofs of the main theorems into general arguments about valued differential fields.

The post-theorem introduction does not give a full step-by-step proof of Theorem~\ref{introthm:transserial_char}, but it does describe the main ingredients and framework used.

- The results (including Theorem~\ref{introthm:transserial_char}) are obtained by “an analysis of the ordered differential valued fields $(\mathbb E\langle x\rangle, \cO_x, \partial_x)$” satisfying:
  (C1) $\mathbb E\langle x\rangle:=\dcl_T(\mathbb E,x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\mathbb E\models T$ and $\cO_x$ is a $T$-convex valuation subring;
  (C2) $\partial_x$ is the “unique non-trivial $\mathbb E$-linear $T$-derivation with $\partial_x(x)=1$”; and
  (C3) $\val_{\cO_x}(x-\mathbb E)\subseteq \val_{\cO_x}(\mathbb E)$.
  This “comprises the cases in which $(\mathbb E\langle x\rangle,\cO_x)\succ (\mathbb E,\mathcal O)$ is wim-constructible or res-constructible.”

- A “key ingredient in Th[e]orem~\ref{introthm:transserial_char}” is the following criterion (stated as Proposition~\ref{prop:height_over_absorbed}): in the above setting, when $T$ is exponential, for $y\in\mathbb E\langle x\rangle$,
  \[
  y\text{ is of the form }g(z)\text{ for a }\mathbb E\text{-definable gne and }z\text{ with }\mathbf v(z-\mathbb E)\subseteq \mathbf v(\mathbb E)
  \]
  “if and only if there is a natural number $m$ such that $\mathbf v(\lder_x^m\partial_x(y))\in \mathbf v(\mathbb E)$.”

- The treatment isolates “general properties” of such $(\mathbb E\langle x\rangle,\cO_x,\partial_x)$ and introduces convexity conditions on derivations (order-convex / logarithmically order-convex; valuation-convex / logarithmically valuation-convex) as “traces of the mean-value theorem,” used to control behavior even when “$\mathrm{tp}(x/\mathbb E)$ is not a definable type.”

- Under “suitable convexity assumptions,” the condition $\mathbf v(x-\mathbb E)\subseteq \mathbf v(\mathbb E)$ is reformulated via an ‘absorbed’ condition on $\partial_x$ (Definition~\ref{def:absorbed-type-der}): “$y\succ \partial_x\cO_x\Rightarrow \lder_x(y)\prec y$,” and such absorbed derivations are said to be “at the center of several arguments.”

The post-theorem introduction does not give a full step-by-step proof of the main theorem, but it does describe the main ingredients and framework used.

- The results (including the main theorem) are obtained by “an analysis of the ordered differential valued fields $(\mathbb E\langle x\rangle, \cO_x, \partial_x)$” satisfying:
  (C1) $\mathbb E\langle x\rangle:=\dcl_T(\mathbb E,x)$ is a 1-$\dcl_T$-dimensional elementary extension of $\mathbb E\models T$ and $\cO_x$ is a $T$-convex valuation subring;
  (C2) $\partial_x$ is the “unique non-trivial $\mathbb E$-linear $T$-derivation with $\partial_x(x)=1$”; and
  (C3) $\val_{\cO_x}(x-\mathbb E)\subseteq \val_{\cO_x}(\mathbb E)$.
  This “comprises the cases in which $(\mathbb E\langle x\rangle,\cO_x)\succ (\mathbb E,\mathcal O)$ is wim-constructible or res-constructible.”

- A “key ingredient in the main theorem” is the following criterion.

\begin{proposition}\label{prop:height_over_absorbed}
    If $\partial\br(\partial) \prec 1$, then
    \[\height(x,\partial)=\inf\{k\in \omega: \val(\lder^k(\partial x))\in \val(\bE)\}\in \omega+1.\]
    \begin{proof}
        Set $a=\partial x$. Clearly the height of $x$ is $0$ if and only if $\val(a)\in \val(\bE)$.

        It suffices to show that for all $m\ge 0$, there is an essential approximate logarithmic scale $(y_k)_{k < m}$ with $a \sim \partial y_0$ (in the case $m>0$) if and only if $\val(\lder^k(a))\notin \val(\bE)$ for all $k<m$.

        If there is such a sequence, then $(\partial y_i)_{i<m}$ is a $\lder$-sequence by Lemma~\ref{lem:essential_log-scales_are_dagger_seq}, so $\lder^k(\partial x)\sim \lder(\partial y_0) \sim \partial y_k$ and we have $\val(\lder^k(a))\notin \val(\bE)$ for all $k<m$.

        Conversely if by $\val(\lder^k(a))\notin \val(\bE)$ for all $k<m$, then by Lemma~\ref{lem:essential_log-scales_are_dagger_seq}(2), there is an essential approximate $\partial$-logarithmic scale $(y_{k})_{k<m}$ with $\partial y_{0}\sim a=\partial x$.
    \end{proof}
\end{proposition}

  In the above setting, when $T$ is exponential, for $y\in\mathbb E\langle x\rangle$,
  \[
  y\text{ is of the form }g(z)\text{ for a }\mathbb E\text{-definable gne and }z\text{ with }\mathbf v(z-\mathbb E)\subseteq \mathbf v(\mathbb E)
  \]
  “if and only if there is a natural number $m$ such that $\mathbf v(\lder_x^m\partial_x(y))\in \mathbf v(\mathbb E)$.”

- The treatment isolates “general properties” of such $(\mathbb E\langle x\rangle,\cO_x,\partial_x)$ and introduces convexity conditions on derivations (order-convex / logarithmically order-convex; valuation-convex / logarithmically valuation-convex) as “traces of the mean-value theorem,” used to control behavior even when “$\mathrm{tp}(x/\mathbb E)$ is not a definable type.”

- Under “suitable convexity assumptions,” the condition $\mathbf v(x-\mathbb E)\subseteq \mathbf v(\mathbb E)$ is reformulated via an ‘absorbed’ condition on $\partial_x$.

\begin{definition}\label{def:absorbed-type-der}
    Let $(\bE_*, \cO_*)$ be a valued field, $\partial$ be an almost $\cO_*$-Lioville-convex derivation on $\bE_*$. Let $\bE\coloneqq\Kr(\partial)$.
    We will say that $\partial$ is \emph{weakly $\cO_*$-absorbed} if $\partial(\br_{\cO_*}(\partial))\prec 1$. We will say that it is \emph{$\cO_*$-absorbed} if furthermore there is $x\in \bE_* \setminus \bE$ such that $\val(x-\bE)\subseteq \val(\bE)$.
\end{definition}

  In particular, such absorbed derivations are said to be “at the center of several arguments.”

Biconditional or Equivalence
Universal–Existential

Question:
Let $T$ be an exponential o-minimal theory, and let $T_{\mathrm{convex}}$ be the complete theory of models $(\mathbb E,\mathcal O)$ of $T$ expanded by a predicate for a nontrivial $T$-convex valuation subring $\mathcal O$. For a $T$-definable function $f(\bar x,t)$, write
\[
\dagger_n f(\bar x,t)=\frac{\partial f(\bar x,t)/\partial t}{f(\bar x,t)}
\]
when defined and $\infty$ otherwise, and let
\[
G_m^f(\bar x,t):=\min\{\,|(\dagger_n^k f)(\bar x,t)|:k\le m\,\}.
\]
Call $T$ transserial if for every $n$ and every $(n+1)$-ary $T$-definable $f$ there exists $m\in\mathbb N$ such that
\[
T_{\mathrm{convex}}\models \forall \bar x\,\exists r\in\mathcal O\,\forall t\in\mathcal O\,(t>r\to G_m^f(\bar x,t)<1).
\]
A generalized nested exponential (gne) over $\mathbb E$ is a finite composition of translations by elements of $\mathbb E$, sign changes, and exponentials. An elementary extension $(\mathbb E,\mathcal O)\preceq(\mathbb E_*,\mathcal O_*)$ has the gne-property if for every $y\in\mathbb E_*$ there exist an $\mathbb E$-definable gne $g$ and some $z\in\mathbb E_*$ with $g(z)=y$ and $\mathbf v(z-\mathbb E)\subseteq \mathbf v(\mathbb E)$. A res-constructible extension is one admitting a $\operatorname{dcl}_T$-basis over $\mathbb E$ whose residues are $\operatorname{dcl}_T$-independent over the base residue field; a wim-constructible extension is one admitting an ordinal-indexed $\operatorname{dcl}_T$-basis such that each new element is a pseudolimit of a pseudo-Cauchy sequence over the previously generated structure with no pseudolimit there; and 1-$\operatorname{dcl}_T$-dimensional means $\operatorname{dcl}_T$-dimension $1$ over the base. If $b$ lies in an elementary extension and realizes the cut $\mathcal O<b<\mathbb E^{>\mathcal O}$, let $\mathbb E\langle b\rangle=\operatorname{dcl}_T(\mathbb E b)$, and write $y\equiv_{\mathbb E} g(b)$ to mean that $y$ and $g(b)$ have the same type over $\mathbb E$. Which statement is valid?

The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists an $\mathbb E$-definable gne $g$ such that $y\equiv_{\mathbb E} g(b)$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.

Choices:
B. The following are equivalent for $T$: (1) $T$ is transserial; (2) for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists an $\mathbb E$-definable gne $g$ such that $y\equiv_{\mathbb E} g(b)$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.
C. If $T$ is transserial, then every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property, and every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.
D. The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists a single $\mathbb E$-definable gne $g$ depending only on $b$ such that $y\equiv_{\mathbb E} g(b)$ for all such $y$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.
E. The following are equivalent for $T$: (1) $T$ is transserial; (2) $T$ is exponentially bounded and for every $(\mathbb E,\mathcal O)\models T_{\mathrm{convex}}$, every $b$ realizing $\mathcal O<b<\mathbb E^{>\mathcal O}$, and every $y\in\mathbb E\langle b\rangle$, there exists an $\mathbb E$-definable gne $g$ such that $y=g(b)$; (3) every 1-$\operatorname{dcl}_T$-dimensional wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (4) every 1-$\operatorname{dcl}_T$-dimensional res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; (5) every wim-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property; and (6) every res-constructible elementary extension of models of $T_{\mathrm{convex}}$ has the gne-property.

The stem gives definitions and setup but does not explicitly reveal the theorem or directly signal choice A. The correct answer is not stated or trivially inferable from the wording alone.

The item is not a direct restatement of a definition in the stem, but it is very close to asking for the exact formulation of a theorem. It mainly tests recognition of the right equivalence statement rather than deriving a new conclusion.

There is some reasoning pressure because the choices differ by subtle logical modifications: omitted hypotheses, weakened implication, altered quantifier dependence, and equality versus type-equivalence. However, the task is still largely theorem-recall/statement-discrimination rather than genuinely generative mathematical reasoning.

The distractors are strong and mathematically meaningful. Each reflects a plausible failure mode: dropping exponential boundedness, replacing equivalence by one implication, making the gne uniform in the wrong way, or strengthening type-equivalence to equality.

A solid theorem-discrimination MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a theorem statement than toward deep generative reasoning.


In this paper, all groups are finite. Take $G$ to be a group, $p$ to be a prime, and $\pi$ to be a set of primes. We write ${\rm Irr} (G)$ for the set of irreducible characters of $G$, ${\rm IBr} (G)$ for the set irreducible ($p$-)Brauer characters of $G$, and ${\rm I}_\pi (G)$ for the set of irreducible $\pi$-partial characters of $G$.

Gallagher's theorem \cite[Corollary 6.17]{Isaacs1976} can be stated as follows: let $N$ be a normal subgroup of $G$ and let $\chi\in {\rm Irr} (G)$ be such that $\chi_{N} = \theta \in {\rm Irr} (N)$, where $\chi_{N}$ is the restriction of $\chi$ to $N$. Then the characters $\beta\chi$ for $\beta\in {\rm Irr} (G/N)$ are irreducible and distinct for distinct $\beta$ and are all of the irreducible constituents of $\theta^{G}$.

The first goal in this paper is to prove the converse of this theorem:

\begin{abstract}
Let $G$ be a finite group. Suppose $N$ is a normal subgroup of $G$. Recall that Gallagher's theorem states that if $\chi \in {\rm Irr} (G)$ satisfies $\chi_N$ is irreducible, then $\chi \beta$ is irreducible and distinct for all $\beta \in {\rm Irr} (G/N)$. Furthermore, if $\theta = \chi_N$, then these are all of the irreducible constituents of $\theta^G$. We prove that the converse of this theorem holds. We also prove that a partial converse of the Brauer version of this theorem holds. Finally, we prove that an analog of Gallagher's theorem holds for Isaacs' $\pi$-partial characters and that a partial converse of that theorem is true.
\end{abstract}

The first goal in this paper is to prove the converse of this theorem:

We will see that we will obtain Theorem \ref{main1} as a corollary to the version for the Brauer character version of the theorem. The Brauer character version of Gallagher's theorem \cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\lhd G$ and let $\eta\in \IBr (G)$. If $\eta_{N} = \theta \in \IBr (N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$ and are all of the irreducible constituents of $\theta^{G}$. Now we consider a partial converse of the Brauer character version of Gallagher's theorem.

\begin{introthm} \label{main2}
Let $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime. Suppose $\eta\in {\rm IBr}(G)$ and ${\rm IBr}(G/N)=\{\beta_{1}, \cdots, \beta_{n}\}$, and suppose that $p\nmid |G/N|$ and the $\beta_{1}\eta, \cdots, \beta_{n}\eta$ are distinct and irreducible. Then $\eta_{N}$ is an irreducible Brauer character of $N$.
\end{introthm}

\begin{cor} \label{bpi Gall,2inpi,|G|odd}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose $2 \in \pi$ or $|G|$ is odd. Suppose there exists a character $\chi \in {\rm B}_{\pi} (G)$ so that $\theta = \chi_N \in \irr (N)$. Then the map $\beta \mapsto \beta\chi$ is a bijection from ${\rm B}_{\pi} (G/N)$ to ${\rm B}_\pi (G \mid \theta)$.
\end{cor}

In fact, the partial converse of Gallagher's theorem will apply for sets of lifts of the $\pi$-partial characters. To see this we make the following definition we again only need that we have subsets of $\irr (G)$ and $\irr (N)$ that are in bijection with ${\rm I}_\pi (G)$ and ${\rm I}_\pi (N)$, and that irreducible constituents lie in this set. Thus, we prove the result in this generality so that it could be applied to any set of lifts.
\begin{cor}\label{lifts conv}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$ so that $G/N$ is a $\pi$-group. Assume there exist subsets $X_\pi (G) \subseteq \irr (G)$ and $X_\pi (N) \subseteq \irr (N)$ so that the map $\theta \mapsto \theta^o$ is a bijection from $X_\pi (G)$ to ${\rm I}_\pi (G)$ and from $X_\pi (N)$ to ${\rm I}_\pi (N)$, and assume for $\chi \in X_\pi (G)$ that the constituents of $\chi_N$ lie in $X_\pi (N)$. If there exist $\chi \in X_\pi (G)$ and $\irr (G/N) = \{ \gamma_1 = 1, \dots, \gamma_n \}$ that satisfy $(\gamma_1 \chi)^o, \dots, (\gamma_n \chi)^o$ are irreducible and distinct, then $\chi_N \in X_\pi (N)$.
\end{cor}

\begin{thm}
Let $\pi$ be a set of primes and let $G$ be a $\pi$-separable group. Let $N$ a normal subgroup of $G$. Let $\eta \in {\rm I}_\pi (G)$ and ${\rm I}_{\pi} (G/N) = \{\beta_{1}, \cdots, \beta_{n}\}$. Suppose that $|G/N|$ is divisible only by primes in $\pi$ and that the $\{\beta_{1}\eta, \cdots, \beta_{n}\eta\}$ are distinct and irreducible. Then $\eta_{N}$ is an irreducible $\pi$-partial character of $N$.
\end{thm}

\begin{thm}\label{bpi conv}
 Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$. Suppose there exist characters $\chi \in {\rm B}_{\pi} (G)$ and ${\rm B}_{\pi} (G/N) = \{\beta_1, \dots, \beta_n \}$. Assume that $G/N$ is a $\pi$-group and the $\{\beta_1 \chi, \dots, \beta_n \chi\}$ are distinct and irreducible Then $\chi_N$ is irreducible.
\end{thm}

\begin{introthm} \label{main1}
Let $G$ be a group and let $N$ be a normal subgroup of $G$.  Suppose ${\rm Irr} (G/N)=\{\beta_{1}, \cdots, \beta_{n}\}$ and $\chi\in {\rm Irr} (G)$, and suppose that the $\beta_{1}\chi, \cdots, \beta_{n}\chi$ are distinct and irreducible.  Then $\chi_{N}$ is an irreducible character of $N$.
\end{introthm}

We will see that we will obtain Theorem \ref{main1} as a corollary to the version for the Brauer character version of the theorem.  The Brauer character version of Gallagher's theorem \cite[Corollary 8.20]{Navarro1998} is as follows: Let $N\lhd G$ and let $\eta\in \IBr (G)$. If $\eta_{N} = \theta \in \IBr (N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$ and are all of the irreducible constituents of $\theta^{G}$.  Now we consider a partial converse of the Brauer character version of Gallagher's theorem.

\begin{introthm} \label{main2}
Let $G$ be a group, let $N$ be a normal subgroup of $G$, and let $p$ be a prime.  Suppose $\eta\in {\rm IBr}(G)$ and ${\rm IBr}(G/N)=\{\beta_{1}, \cdots, \beta_{n}\}$, and suppose that $p\nmid |G/N|$ and the $\beta_{1}\eta, \cdots, \beta_{n}\eta$ are distinct and irreducible. Then $\eta_{N}$ is an irreducible Brauer character of $N$.
\end{introthm}

We next consider Isaacs' $\pi$-theory, where $\pi$ is a set of primes.  For a $\pi$-separable group $G$, Isaacs has defined an analog of Brauer characters that are defined on the $\pi$-elements of $G$ that are called the $\pi$-partial characters of $G$.  We will write ${\rm I}_\pi (G)$ for the set of irreducible $\pi$-partial characters of $G$.  We will review the points of $\pi$-theory we need in Sections \ref{secn: pi theory} and \ref{secn: nucl}.  Interestingly, while Isaacs proved analogs of many other results for $\pi$-partial characters, he does not seem to have proved an analog of Gallagher's theorem.  We do that next.

\begin{introthm}\label{ipi Gall,gen}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$.  Suppose there exists a character $\zeta \in {\rm I}_{\pi} (G)$ so that $\xi = \zeta_N \in {\rm I}_\pi (N)$.  Then the map $\kappa \mapsto \kappa\zeta$ is a bijection from ${\rm I}_{\pi} (G/N)$ to ${\rm I}_\pi (G \mid \xi)$.
\end{introthm}

Now, in \cite{Isaacs1976}, there is a generalization of Gallagher's theorem (Theorem 6.16 of \cite{Isaacs1976}) of which Gallaher's theorem is obtained as a corollary.  We have not been able to obtain an analog of this generalization in cases, but we can obtain an analog when we add the extra assumption that $2 \in \pi$ or $|G|$ is odd.

\begin{introthm} \label{ipi pre,2inpi,|G|odd}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$.  Suppose $2 \in \pi$ or  $|G|$ is odd.  Assume there exist partial characters $\eta, \xi \in {\rm I}_\pi (N)$ so that $\eta$ is $G$-invariant, $\xi$ extends to $\zeta \in {\rm I}_\pi (G)$ and $\eta \xi \in {\rm I}_{\pi} (N)$.  Then the map $\kappa \mapsto \kappa \zeta$ is a bijection from ${\rm I}_\pi (G \mid \eta) \rightarrow {\rm I}_\pi (G \mid \eta \xi)$.
\end{introthm}

We close with a partial converse for the analog of Gallagher's theorem for $\pi$-partial characters.  We will show that it is not possible to have a full converse in this case.

\begin{introthm}\label{ipi conv}
Let $\pi$ be a set of primes, let $G$ be a $\pi$-separable group, and let $N$ be a normal subgroup of $G$.
Suppose there exists partial characters $\zeta \in {\rm I}_{\pi} (G)$ and ${\rm I}_{\pi} (G/N) = \{\kappa_1, \dots, \kappa_n \}$.  Assume that $G/N$ is a $\pi$-group and the $\kappa_1 \zeta, \dots, \kappa_n \zeta $ are distinct and irreducible  Then $\zeta_N$ is irreducible.
\end{introthm}

We will see that we will obtain Theorem~\ref{main1} as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\eta\in \IBr(G)$ and $\eta_N=\theta\in\IBr(N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$, and are all of the irreducible constituents of $\theta^G$; then it says: "Now we consider a partial converse" (stated as Theorem~\ref{main2}).

We will see that we will obtain the main theorem as a corollary to the version for the Brauer character version of the theorem. In particular, the introduction cites the Brauer character version of Gallagher's theorem: if $\eta\in \IBr(G)$ and $\eta_N=\theta\in\IBr(N)$, then the characters $\beta\eta$ for $\beta\in \IBr(G/N)$ are irreducible, distinct for distinct $\beta$, and are all of the irreducible constituents of $\theta^G$; then it says: "Now we consider a partial converse" (stated as Theorem~\ref{main2}).,

Question: Let $G$ be a finite group and $N\trianglelefteq G$. Write ${\rm Irr}(H)$ for the set of irreducible complex characters of a finite group $H$. Suppose ${\rm Irr}(G/N)=\{\beta_1,\dots,\beta_n\}$, let $\chi\in {\rm Irr}(G)$, and assume that the characters $\beta_1\chi,\dots,\beta_n\chi$ on $G$ (where each $\beta_i$ is inflated from $G/N$ to $G$ and then multiplied pointwise with $\chi$) are pairwise distinct and irreducible. Which conclusion holds for the restriction $\chi_N$ of $\chi$ to $N$?

The restricted character $\chi_N$ is an irreducible character of $N$.

Choices:
B. The restricted character $\chi_N$ is a sum of pairwise distinct irreducible characters of $N$, each occurring with multiplicity $1$.
C. The restricted character $\chi_N$ has at least one irreducible constituent of $N$.
D. The induced character $(\chi_N)^G$ has irreducible constituents exactly $\beta_1\chi,\dots,\beta_n\chi$, each occurring with multiplicity $1$.
E. For every irreducible constituent $\theta$ of $\chi_N$, the characters $\beta_1\chi,\dots,\beta_n\chi$ are precisely the irreducible constituents of $\theta^G$.

irreducible-restriction conclusion weakened to multiplicity-free reducibility

dropped irreducibility of the whole restriction, keeping only existence of an irreducible constituent

converse conclusion replaced by full Gallagher constituent description for induction

hidden dependence on choosing the unique constituent of $\chi_N$; falsified by quantifying over every constituent

The stem does not explicitly reveal the correct conclusion about _N. It gives a nontrivial hypothesis about the twists _i and asks for a consequence, so there is no direct answer leakage.

The item is very close to a standard character-theoretic result: the hypothesis is essentially tailored to force the conclusion that _N is irreducible. So it is not a literal restatement, but it is only a mild reformulation of a known theorem.

Some reasoning is required to distinguish the exact conclusion from stronger or weaker nearby statements, especially against options D and E. However, for a student who recognizes the theorem, the answer is largely pattern-matching rather than deep generative reasoning.

The distractors are mathematically plausible and reflect common failure modes: weakening to a trivially true statement (C), confusing irreducibility with multiplicity-free reducibility (B), and overstrengthening via induced-character conclusions (D, E). They are distinct and well aligned with the topic.

A solid MCQ with good distractors and no answer leakage, but it is quite close to a standard theorem statement, so it only moderately tests genuine generative reasoning.


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 $H_i$ are given by instances of a single defining formula). Which statement holds about finite chains of successive intersections of members of this family?

There exist natural numbers $n,d<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|>d
\]
for every $k\le n-1$. Equivalently, for any $n$ distinct indices $j_1,\dots,j_n$, at least one successive index
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|
\]
is at most $d$.

Choices:
B.
There exists a natural number $d<\omega$ such that for every natural number $n<\omega$, no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|>d
\]
for every $k\le n-1$.
C.
There exists a natural number $n<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\text{ is infinite}
\]
for every $k\le n-1$.
D.
There exist natural numbers $n,d<\omega$ such that for every subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ and every $k\le n-1$,
\[
\left|\bigcap_{i=1}^k H_{j_i} : \bigcap_{i=1}^{k+1} H_{j_i}\right|\le d.
\]
E.
There exist natural numbers $n,d<\omega$ such that no subset $J=\{j_1,\dots,j_n\}\subseteq I$ of cardinality $n$ satisfies
\[
\left|H_{j_k}:H_{j_k}\cap H_{j_{k+1}}\right|>d
\]
for every $k\le n-1$.

simultaneous existence of both bounds n and d

replace uniform finite bound by mere finiteness of some drop

at least one bounded drop versus all drops bounded

successive intersections replaced by pairwise intersection indices

The stem does not state or strongly hint at the specific quantified conclusion. It only asks which structural property of successive intersections holds, without leaking the existence/placement of the bounds n and d.

This is close to a theorem-identification question: under standard hypotheses, one choice reproduces the precise chain-condition statement. However, it is not a pure restatement because the options differ in meaningful ways (quantifier strength, weakening, and altered intersection structure).

Selecting the correct option requires moderate reasoning about quantifiers and the difference between 'some successive drop is bounded' versus stronger or weaker variants. Still, it primarily tests recognition of the exact theorem statement rather than substantial derivation.

The distractors are strong: they reflect natural mathematical mistakes such as overstrengthening quantifiers, replacing uniform boundedness by mere infinitude/weakening, and confusing successive intersections with pairwise intersections.

A solid theorem-discrimination MCQ with little answer leakage and high-quality distractors, though it leans more toward precise recall of a known result than deep generative reasoning.