\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.


\begin{figure}[h]
    \centering
    \includegraphics[width=0.55\linewidth]{continuity.pdf}
    \caption{Discontinuity of $f$ at $x$.}
    \label{fig:continuity}
\end{figure}

\subsection{Van Kampen--Flores theorem} \label{subsec:vk-f}
A simplicial complex of dimension $d$ can always be embedded (even affinely) in $\R^{2d+1}$, for example, by placing the vertices of the complex along the moment curve $\R \to \R^{2d+1},~t \mapsto (t, t^2, \dots, t^{2d+1})$. The result of van Kampen \cite{van1933komplexe} and Flores \cite{flores1933n} states that the $d$-skeleton of the $(2d+2)$-dimensional simplex does not embed in $\R^{2d}$, showcasing that the above general embedding result is the best possible. In fact, they prove a stronger statement. For a simplicial complex $K$, we say that a function $f \colon K \to \R^n$ is \emph{almost injective} if $f(x_1) = f(x_2)$ implies that $x_1$ and $x_2$ belong to a common face of~$K$. This describes a wider class of functions than just injective functions. A function that is almost injective and is continuous will be called an \emph{almost embedding}.

\begin{theorem}[van Kampen--Flores \cite{flores1933n,van1933komplexe}]
    \label{thm:vk-f}
    Let $d \ge 1$ be an integer. Then, there does not exists an almost embedding $\sk_d(\Delta_{2d+2}) \to \R^{2d}$.
\end{theorem}

As before, a natural question arises: \emph{Since every almost injective function $\sk_d(\Delta_{2d+2}) \to \R^{2d}$ is discontinuous, can we quantify how discontinuous it needs to be}?
To answer this question, we will need to adjust the notion of the scale invariant modulus of discontinuity $\alpha$. Namely, for a simplicial complex $K$, let
\begin{equation*}
    \Conf_2^{\Delta}(K) \coloneqq \{(x,y) \in K \times K \colon~ x,y~\text{belong to disjoint faces}\}
\end{equation*}
denote the appropriate configuration space, in the literature \cite{matousek2003using} also known as the \emph{2-fold deleted product of $K$}. We again equip it with a $\Z/2$-action that flips $x$ and $y$. Then, an almost embedding $f \colon~ K \to \R^d$ induces a $\Z/2$-map
\begin{equation} \label{eq:Pf-vk-f}
    P_f \colon \Conf_2^{\Delta}(K) \longrightarrow S^{2d-1},~(x,y) \longmapsto \frac{f(x)-f(y)}{\|f(x)-f(y)\|},
\end{equation}
which is the restriction of the map \eqref{eq:test-inj}. We now define the \emph{scale invariant modulus of discontinuity of an almost injective function} $f$ as $\alpha^{(2)}(f) \coloneqq \delta(P_f)$. The reason we have the number two in the superscript of $\alpha$ is that in Section \ref{sec:almost-r-emb} we will study the more general almost $r$-injective functions and define the appropriate modulus of discontinuity $\alpha^{(r)}$.

With the new notion at hand, we can prove the following generalization of Theorem \ref{thm:vk-f}.

\begin{theorem}[Quantified van Kampen--Flores]
    \label{thm:quant-vk-f}
    Let $d \ge 1$ be an integer. Then, every almost injective function $\sk_d(\Delta_{2d+2}) \to \R^{2d}$ satisfies $\alpha^{(2)}(f) \ge r_{2d-1} = \arccos{\left(\frac{-1}{2d}\right)}$.
\end{theorem}
\begin{proof}
    Let $f \colon~ \sk_d(\Delta_{2d+2}) \to \R^{2d}$ be almost injective. Analogously as in the proof of Theorem \ref{thm:quant-general}, the function $P_f$ from \eqref{eq:Pf-vk-f} induces a $\Z/2$-map
    \begin{equation} \label{eq:del-prod-to-sphere}
        \Conf_2^{\Delta}(\sk_d(\Delta_{2d+2})) \longrightarrow \vrm{S^{2d-1}}{\alpha^{(2)}(f) + \varepsilon}.
    \end{equation}
    Let us show that this can only happen if $\alpha^{(2)}(f) + \varepsilon \ge r_{2d-1}$.
    Indeed, if $\alpha^{(2)}(f) + \varepsilon < r_{2d-1}$, the homotopy equivalence \eqref{eq:P(Sn)-homotopy} (and the text thereafter) would yield a $\Z/2$-map
    \[
        \vrm{S^{2d-1}}{\alpha^{(2)}(f) + \varepsilon} \longrightarrow S^{2d-1}
    \] which when precomposed with \eqref{eq:del-prod-to-sphere} would yield a $\Z/2$-map
    \begin{equation*}
        F \colon \Conf_2^{\Delta}(\sk_d(\Delta_{2d+2})) \longrightarrow S^{2d-1}.
    \end{equation*}
    Let us introduce the \emph{2-fold deleted join} (see \cite[Section~5.5]{matousek2003using} for more detail) as a simplicial complex
    \begin{equation*}
        (\sk_d(\Delta_{2d+2}))^{*2}_{\Delta} \coloneqq \{tx + (1-t)y \in (\sk_d(\Delta_{2d+2}))^{*2} \colon~ x,y~\text{belong to disjoint faces}\}.
    \end{equation*}
    The deleted product can be seen as a subspace of the deleted join via the inclusion
    \begin{equation*}
        \Conf_2^{\Delta}(\sk_d(\Delta_{2d+2})) \longrightarrow (\sk_d(\Delta_{2d+2}))^{*2}_{\Delta},~(x,y) \longmapsto \tfrac{1}{2}x + \tfrac{1}{2}y.
    \end{equation*}
    The map $F$ can be extended to a $\Z/2$-map
    \begin{equation*}
        (\sk_d(\Delta_{2d+2}))^*_{\Delta} \longrightarrow S^{2d},\quad tx + (1-t)y \longmapsto \frac{(1-2t,t(1-t)F(x,y))}{\|(1-2t,t(1-t)F(x,y))\|}.
    \end{equation*}
    However, this contradicts \cite[Section~5.6, pg.~117]{matousek2003using} which says that $(\sk_d(\Delta_{2d+2}))^*_{\Delta}$ is a $(2d+1)$-sphere.
\end{proof}

We remark that we cannot improve the lower bound on the scale parameter for the non-existence of a $\Z/2$-map \eqref{eq:del-prod-to-sphere} since a $\Z/2$-map
    \begin{equation*}
        \Conf_2^{\Delta}(\sk_d(\Delta_{2d+2})) \longrightarrow \vrm{S^{2d-1}}{r_{2d-1}}
    \end{equation*} exists. Namely, the domain is a $(2d)$-dimensional cellular complex with a free $\Z/2$-action and by \eqref{eq:P(Sn)-homotopy} the codomain is $(2d-1)$-connected. Hence, all obstructions to defining such a $\Z/2$-map vanish (see also \cite[Lemma~6.2.2]{matousek2003using}).

\section{Almost \texorpdfstring{$r$}{r}-embeddings}
\label{sec:almost-r-emb}

In this section we will prove a quantified topological Tverberg's theorem \cite{BSSz1981,ozaydin1987}. We will first introduce the appropriate language in order to state it as a nonembeddability-type result \cite{avvakumov2021eliminating}.

\begin{definition}
    Let $r \ge 2$, $d \ge 0$ be integers and $K$ a simplicial complex. A function $f\colon K \to \R^d$ is \emph{almost $r$-injective} if any $r$ pairwise disjoint faces $\sigma_1,\dots ,\sigma_r \subseteq \Delta_N$ satisfy $f(\sigma_1) \cap \dots \cap f(\sigma_r) = \emptyset$.
\end{definition}

In another words, an almost $r$-injective function $f$ never identifies $r$ points from pairwise disjoint faces. This represents the $r$-fold generalization of the notion of the almost injectivity ($r=2$) from the previous section.
Tverberg \cite{tverberg1966} proved that for $r \ge 2$ and $d \ge 0$ any \emph{affine} function $f\colon \Delta_N \to \R^d$ is not almost $r$-injective if the dimension of the simplex is large enough, namely if $N \ge (r-1)(d+1)$.
B\'ar\'any, Shlosman and Sz\H ucs \cite{BSSz1981} and \"Ozaydin \cite{ozaydin1987} showed that no such \emph{continuous} almost $r$-injective function exists if $r$ is a prime power.

\begin{theorem}[Topological Tverberg, prime power case {\cite{BSSz1981, ozaydin1987}}] \label{theorem: topological tverberg}
    Let $d \ge 1$ be an integer, let $r \ge 2$ be a prime power, and let $N = (r-1)(d+1)$. Let $f\colon \Delta_N \to \R^d$ be any function. Then, if $f$ is a $r$-injective, it is not continuous.
\end{theorem}

The prime power condition is really necessary: Frick \cite{frick2015counterexamples} and Blagojevi\'c, Frick and Ziegler~\cite{blagojevic2015barycenters}, building on work of Mabillard and Wagner \cite{mabillard2014eliminating}, showed that if $r$ is not a prime power and $d \ge 3r$, then a continuous almost $r$-embedding $f \colon~ \Delta_N \to \R^d$ exists. The lower bound on $d$ was improved to $d \ge 2r+1$ by Avvakumov, Mabillard, Skopenkov and Wagner \cite{avvakumov2021eliminating}. In particular, when $r$ is a prime power, a natural question of quantifying discontinuity of almost $r$-injective functions emerges.

In order to answer it, in Definition \ref{def:alpha^(r)} below, we introduce a notion $\alpha^{(r)}$ that captures discontinuity of such maps. It is an $r$-fold analogue of scale-invariant moduli of discontinuity $\alpha$ and $\alpha^{(2)}$ introduced in Section \ref{sec:injectivity}. The following is the main theorem of the section.

\begin{theorem} [Quantified topological Tverberg] \label{theorem: discontinuous tverberg}
    Let $d \ge 1$ be an integer, $r$ a prime power, and $f\colon \Delta_{(r-1)(d+1)} \to \R^d$ any function. Then, if $f$ is almost $r$-injective it must satisfy
    \[
    	\alpha^{(r)}(f) ~\ge~  \arccos(\tfrac{-1}{d(r-1)}).
    \]
\end{theorem}

The constant $\arccos(\tfrac{-1}{d(r-1)})$ denotes the diameter of the regular $d(r-1)$-dimensional simplex inscribed in the $(d(r-1)-1)$-sphere equipped with the geodesic metric. As explained in Remark \ref{remark: R_delta and angles}, the fact that $\alpha^{(r)}(f) > 0$ implies that $f$ is not continuous. Therefore, Theorem \ref{theorem: discontinuous tverberg} presents a generalization of the topological Tverberg theorem.

\begin{definition} \label{def:r-del-prod}
	For an integers $r \ge 2$ and $N \ge r-1$ we define a cellular complex
	\[
        \Conf_r^{\Delta}(\Delta_N) := \{(x_1,...,x_r) \in  (\Delta_N)^{\times r}: x_i\text{'s belong to pairwise disjoint faces}\},
	\]
    called the \textit{$r$-fold deleted product} of $\Delta_N$.
\end{definition}

The symmetric group $\mathfrak{S}_r$ acts freely on $\Conf_r^{\Delta}(\Delta_N)$ by permuting the coordinates. Let us denote by
\(
	W_r^{\oplus d} := \{(z_1,...,z_r) \in (\R^d)^{\oplus r}: z_1 + \dots + z_r = 0\}
\)
a $d(r-1)$-dimensional $\mathfrak{S}_r$-representation, where the action is also given by permutation of coordinates. Next, we define an $\mathfrak{S}_r$-map
\begin{equation*} \label{eq:map Pf}
	\Conf_r^{\Delta}(f) \colon \Conf_r^{\Delta}(\Delta_N) \longrightarrow W_r^{\oplus d}
\end{equation*}
which sends a tuple $(x_1,...,x_r) \in \Conf_r^{\Delta}(\Delta_N)$ to a tuple obtained from
\(
	(f(x_1), \dots , f(x_r)) \in (\R^d)^{\oplus r}
\)
by subtracting the average $\frac{1}{r}(f(x_1) + \dots + f(x_r))$ from each of the $r$ coordinates $f(x_i)$. We note that $f$ is almost $r$-injective if and only if the image of $\Conf_r^{\Delta}(f)$ does not contain the origin $0 \in W_r^{\oplus d}$, which enables the following definition.

\begin{definition} \label{def:alpha^(r)}
	Let $f\colon \Delta_N \to \R^d$ be an almost $r$-embedding. Then we define
	\[
		\alpha^{(r)}(f) := \delta\Big(\Conf_r^{\Delta}(\Delta_N) ~ \xrightarrow{~\Conf_r^{\Delta}(f)~}~ W_r^{\oplus d} \setminus \{0\} ~\xrightarrow{~\nu~} ~S(W_r^{\oplus d})\Big),
	\]
	where $\nu\colon v \mapsto v/\|v\|$ is a deformation retraction and the unit sphere $S(W_r^{\oplus d}) \subseteq W_r^{\oplus d}$ is assumed to be endowed with the geodesic metric.
\end{definition}

\begin{remark} \label{remark: R_delta and angles}
    Let $f\colon \Delta_N \to \R^d$ be an almost $r$-embedding.  The value $\alpha^{(r)}(f)$ can be geometrically interpreted as follows.
    We have
\[
	\Conf_r^{\Delta}(f)(x_1,...,x_r) = \left(f(x_1) - \tfrac{1}{r}\bigl(f(x_1) + \dots + f(x_r)\bigr), \dots , f(x_r) - \tfrac{1}{r}\bigl(f(x_1) + \dots + f(x_r)\bigr)\right),
\]
as depicted in Figure \ref{fig:r-tuple}.
Then, there are arbitrarily close $r$-tuples $x, y \in \Conf_r^{\Delta}(\Delta_N)$ with the property that the angle between the vectors $\Conf_r^{\Delta}(f)(x),~ \Conf_r^{\Delta}(f)(y) \in W_r^{\oplus d}$
is at least $\alpha^{(r)}(f)$.
In other words, the notion $\alpha^{(r)}(f)$ captures how large angles between values of $\Conf_r^{\Delta}(f)$ at two close points in the $r$-fold deleted product can be. If $f$ is moreover continuous, then so is $\Conf_r^{\Delta}(f)$, and therefore $\alpha^{(r)}(f) = 0$. So, in particular, having $\alpha^{(r)}(f)>0$ ensures that $f$ is discontinuous.

\begin{figure}[h]
    \centering
    \includegraphics[width=0.52\linewidth]{r-tuple.pdf}
    \caption{The three vectors in the triple $\Conf_3^{\Delta}(f)(x_1,x_2,x_3)$ for $r=3$ and $d=2$.}
    \label{fig:r-tuple}
\end{figure}

\begin{theorem} \label{thm:RP}
    Any injective function $f \colon \RP^{2^k} \to \R^{2^{k+1}-1}$ satisfies $\alpha(f) \ge r_{2^{k+1}-2} = \arccos{\left(\frac{-1}{2^{k+1}-1}\right)}$.
\end{theorem}

\begin{theorem}
\label{thm:alpha-coind}
Let $X$ be a topological space, let $k = \operatorname{coind}_{\Z/2}(\Conf_2(X))$, and assume $k \ge d - 1$.
Then any injective function $f \colon X \to \R^d$ satisfies $\alpha(f) \geq c_{d-1,k}$.
\end{theorem}

\label{sec:intro}

The ``embedding problem'' is a classical problem of topology, which for a given space~$X$ and positive integer~$d$ asks whether $X$ embeds into~$\R^d$. Seminal results include constructions of Flores~\cite{flores1933n} and of van Kampen \cite{van1933komplexe} in the 1930s of $d$-dimensional complexes that do not embed into~$\R^{2d}$ (on the negative side), and Whitney's embedding theorem~\cite{whitney44} from the 1940s which asserts that any smooth $d$-manifold embeds into~$\R^{2d}$ (on the positive side). If $X$ indeed embeds into~$\R^d$ we may ask how rich the space of embeddings $X \hookrightarrow \R^d$ is---for example, we can investigate whether all such embeddings are the same up to isotopy (the ``unknotting problem''). If $X$ is endowed with a metric, quantitative bounds for embeddings, such as for their Lipschitz constants, have been studied \cite{aharoni1974every, kalton2008best}.

Here we complement this quantitative investigation on the negative side, that is, for nonembedd\-ability results. We augment classical results asserting that $X$ does not embed into~$\R^d$ to quantified bounds for the discontinuity of injective functions $X \to \R^d$. For compact $X$ a nonembeddability result simply asserts that there is no continuous injection of $X$ into~$\R^d$. We more generally establish lower bounds for a measure of discontinuity of injective functions $X \to \R^d$. This measure of discontinuity needs to be scale-invariant, since for a given injective function $f\colon X\to\R^d$, a suitable rescaled function has its image contained in some small $\varepsilon$-ball. In particular, the modulus of discontinuity (also called oscillation) of injective functions $X\to \R^d$ is arbitrarily small.

Instead, a suitable measure of discontinuity will only make reference to angles and not to distances in~$\R^d$. Let $f\colon X\to \R^d$ be continuous. For $u,v \in \R^d$ denote the line segment connecting $u$ to~$v$ by~$\overline{uv}$. For any given angle $\alpha >0$ and any point $(x,y) \in X\times X$ with $x \ne y$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \in U$ the line segments $\overline{f(x)f(y)}$ and $\overline{f(x')f(y')}$ make an angle of at most~$\alpha$. For discontinuous~$f$, even if $x$ is close to~$x'$ and $y$ is close to~$y'$, the corresponding line segments connecting their images in~$\R^d$ could make a large angle. We will refer to the infimal~$\alpha$ such that for any point $(x,y) \in X\times X$, there is a neighborhood $U$ of~$(x,y)$ such that for all $(x',y') \in U$ the line segments $\overline{f(x)f(y)}$ and $\overline{f(x')f(y')}$ make an angle of at most~$\alpha$ as the \emph{scale-invariant modulus of discontinuity}~$\alpha(f)$. In particular, continuous functions $f$ necessarily have $\alpha(f) = 0$. Moreover, we show in Section \ref{sec:suitability-alpha} that under mild assumptions $\alpha$ completely captures continuity in the sense that $\alpha(f) = 0$ happens if and only if $f$ is continuous. Therefore, bounding $\alpha(f)$ away from zero indeed represents a scale-invariant quantification of discontinuity of~$f$.

Here we list some consequences of our results for classical nonembeddability theorems, starting with the quantified version of the nonembeddability of $\RP^d$ into $\R^{2d-1}$ when $d$ is a power of two~\cite{milnor1974characteristic}. We will denote by $r_n$ the diameter of the vertices of a regular $(n+1)$-simplex inscribed in $S^n$ (see Section~\ref{sec:background}).

Here we list some consequences of our results for classical nonembeddability theorems, starting with the quantified version of the nonembeddability of $\RP^d$ into $\R^{2d-1}$ when $d$ is a power of two~\cite{milnor1974characteristic}. We will denote by $r_n$ the diameter of the vertices of a regular $(n+1)$-simplex inscribed in $S^n$ (see Section~\ref{sec:background}).

This is a consequence of the more general Theorem \ref{thm:quant-general} and Corollary \ref{cor:quant-hw} for manifolds and simplicial complexes. This approach relies on the seminal work of Haefliger~\cite{Haefliger} and Weber~\cite{Weber} relating the embedding problem to the existence of $\Z/2$-equivariant maps $\Conf_2(X) \to S^{d-1}$ (see Theorem \ref{thm:Haefliger}). For spaces with well-enough understood configuration space one can go beyond the Haefliger--Weber metastable range to obtain an improved lower bound:

\begin{theorem} \label{thm:vKF-intro}
 Every injective function $\sk_d(\Delta_{2d+2}) \to \R^{2d}$ satisfies $\alpha(f) \ge r_{2d-1} = \arccos{\left(\frac{-1}{2d}\right)}$.
\end{theorem}

\begin{theorem} [Quantified topological Tverberg] \label{thm:tv-intro}
 Let $d \ge 1$ be an integer and $r \ge 2$ a prime power. Then, any almost $r$-injective function $f\colon \Delta_{(r-1)(d+1)} \to \R^d$ satisfies $\alpha^{(r)}(f) \ge \arccos(\tfrac{-1}{d(r-1)})$.
\end{theorem}

\begin{theorem} \label{thm:quant-general}
 Let $X$ compact topological space. Assume that there does not exist a $\Z/2$-equivariant map $\Conf_2(X) \to S^{d-1}$. Then, every injective function $f \colon X \to \R^d$ satisfies $\alpha(f) \ge r_{d-1} = \arccos{(-1/d)}$.
\end{theorem}
\begin{proof}
 Let $f \colon~ X \to \R^{d}$ be an injective function. A straightforward generalization of \cite[Lemma~7.4]{GH-BU-VR} to metric thickenings implies that for any $\varepsilon > 0$ there exists a parameter $\rho = \rho(\varepsilon)$ such that the function
 \begin{equation*}
 \vrm{\Conf_2(X)}{\rho} \longrightarrow \vrm{S^{d-1}}{\delta(\Phi_f) + \varepsilon}
 \end{equation*}
 induced from $\Phi_f$ is continuous and $\Z/2$-equivariant. As noted in Section \ref{sec:background}, there is a canonical $\Z/2$-inclusion $\Conf_2(X) \hookrightarrow \vrm{\Conf_2(X)}{\rho}$, which then implies the existence of a $\Z/2$-map
 \begin{equation}\label{eq:conf-to-P}
 \Conf_2(X) \longrightarrow \vrm{S^{d-1}}{\delta(\Phi_f) + \varepsilon}.
 \end{equation}
 If $\delta(\Phi_f) + \varepsilon < r_{d-1}$, there would exist a $\Z/2$-map $\vrm{S^{d-1}}{\delta(\Phi_f) + \varepsilon} \to S^{d-1}$ from \eqref{eq:P(Sn)-homotopy}, which precomposed with \eqref{eq:conf-to-P} would yield a $\Z/2$-map
 \begin{equation*}
 \Conf_2(X) \longrightarrow S^{d-1}.
 \end{equation*}
 This is a contradiction with our assumption, hence $\delta(\Phi_f) + \varepsilon \ge r_{d-1}$. Letting $\varepsilon \to 0$ we obtain the claim $\alpha(f) = \delta(\Phi_f) \ge r_{d-1}$.
\end{proof}

\begin{corollary}\label{cor:quant-hw}
 Let $X$ be a smooth, closed manifold (resp.\ simplicial complex) of dimension $n$. Assume that there does not exists a differentiable (resp.\ linear) embedding $f \colon X \to \R^d$ for $d > \frac32(n + 1)$. Then, every injective function $f \colon X \to \R^d$ satisfies $\alpha(f) \ge r_{d-1} = \arccos{(-1/d)}$.
\end{corollary}

If the sequence of points $f(x_n)$ has a bounded subsequence, then it has a subsequence (also denoted by $f(x_n)$) that satisfies $f(x_n) \to z$, for some $z \in \ell$. Let
\[
 v \coloneqq \frac{z-f(y)}{\|z-f(y)\|} = \lim_{n \to \infty} \frac{f(x_n)-f(y)}{\|f(x_n)-f(y)\|} \in S^{d-1} .
\]
From $\|f(x)-f(x_n)\| \ge \varepsilon_1$ we obtain $\|f(x)-z\| \ge \varepsilon_1$, which implies $v \neq u$. Therefore there exists $\varepsilon > 0$ such that
\begin{equation} \label{eq:angle}
 \left\langle \frac{f(x)-f(y)}{\| f(x) - f(y) \|}, \frac{f(x_{n})-f(y)}{\| f(x_{n}) - f(y) \|} \right\rangle < 1 - \varepsilon
\end{equation}
for sufficiently large $n$, and $\alpha(f) > \arccos(1 - \varepsilon) > 0$, as desired.

\begin{theorem}[Quantified van Kampen--Flores]
 \label{thm:quant-vk-f}
 Let $d \ge 1$ be an integer. Then, every almost injective function $\sk_d(\Delta_{2d+2}) \to \R^{2d}$ satisfies $\alpha^{(2)}(f) \ge r_{2d-1} = \arccos{\left(\frac{-1}{2d}\right)}$.
\end{theorem}
\begin{proof}
 Let $f \colon~ \sk_d(\Delta_{2d+2}) \to \R^{2d}$ be almost injective. Analogously as in the proof of Theorem \ref{thm:quant-general}, the function $P_f$ from \eqref{eq:Pf-vk-f} induces a $\Z/2$-map
 \begin{equation} \label{eq:del-prod-to-sphere}
 \Conf_2^{\Delta}(\sk_d(\Delta_{2d+2})) \longrightarrow \vrm{S^{2d-1}}{\alpha^{(2)}(f) + \varepsilon}.
 \end{equation}
 Let us show that this can only happen if $\alpha^{(2)}(f) + \varepsilon \ge r_{2d-1}$.
 Indeed, if $\alpha^{(2)}(f) + \varepsilon < r_{2d-1}$, the homotopy equivalence \eqref{eq:P(Sn)-homotopy} (and the text thereafter) would yield a $\Z/2$-map
 \[
 \vrm{S^{2d-1}}{\alpha^{(2)}(f) + \varepsilon} \longrightarrow S^{2d-1}
 \] which when precomposed with \eqref{eq:del-prod-to-sphere} would yield a $\Z/2$-map
 \begin{equation*}
 F \colon \Conf_2^{\Delta}(\sk_d(\Delta_{2d+2})) \longrightarrow S^{2d-1}.
 \end{equation*}
 Let us introduce the \emph{2-fold deleted join} (see \cite[Section~5.5]{matousek2003using} for more detail) as a simplicial complex
 \begin{equation*}
 (\sk_d(\Delta_{2d+2}))^{*2}_{\Delta} \coloneqq \{tx + (1-t)y \in (\sk_d(\Delta_{2d+2}))^{*2} \colon~ x,y~\text{belong to disjoint faces}\}.
 \end{equation*}
 The deleted product can be seen as a subspace of the deleted join via the inclusion
 \begin{equation*}
 \Conf_2^{\Delta}(\sk_d(\Delta_{2d+2})) \longrightarrow (\sk_d(\Delta_{2d+2}))^{*2}_{\Delta},~(x,y) \longmapsto \tfrac{1}{2}x + \tfrac{1}{2}y.
 \end{equation*}
 The map $F$ can be extended to a $\Z/2$-map
 \begin{equation*}
 (\sk_d(\Delta_{2d+2}))^*_{\Delta} \longrightarrow S^{2d},\quad tx + (1-t)y \longmapsto \frac{(1-2t,t(1-t)F(x,y))}{\|(1-2t,t(1-t)F(x,y))\|}.
 \end{equation*}
 However, this contradicts \cite[Section~5.6, pg.~117]{matousek2003using} which says that $(\sk_d(\Delta_{2d+2}))^*_{\Delta}$ is a $(2d+1)$-sphere.
\end{proof}

\begin{corollary}\label{cor:quant-hw}
    Let $X$ be a smooth, closed manifold (resp.\ simplicial complex) of dimension $n$.  Assume that there does not exists a differentiable (resp.\ linear) embedding $f \colon X \to \R^d$ for $d > \frac32(n + 1)$. Then, every injective function $f \colon X \to \R^d$ satisfies $\alpha(f) \ge r_{d-1} = \arccos{(-1/d)}$.
\end{corollary}

\begin{theorem}[Haefliger~\cite{Haefliger}, Weber~\cite{Weber}]
\label{thm:Haefliger}
Let $X$ be a smooth, closed manifold (resp.\ simplicial complex) of dimension $n$.  If $d > \frac32(n + 1)$, then there exists a differentiable (resp.\ linear) embedding $f \colon X \to \R^d$ if and only if there exists a $\Z/2$-equivariant map $\Phi \colon \Conf_2(X) \to S^{d-1}$.
\end{theorem}

\begin{theorem} \label{thm:quant-general}
    Let $X$ compact topological space.  Assume that there does not exist a $\Z/2$-equivariant map $\Conf_2(X) \to S^{d-1}$. Then, every injective function $f \colon X \to \R^d$ satisfies $\alpha(f) \ge r_{d-1} = \arccos{(-1/d)}$.
\end{theorem}

This is a consequence of the more general Theorem \ref{thm:quant-general} and Corollary \ref{cor:quant-hw} for manifolds and simplicial complexes. This approach relies on the seminal work of Haefliger~\cite{Haefliger} and Weber~\cite{Weber} relating the embedding problem to the existence of $\Z/2$-equivariant maps $\Conf_2(X) \to S^{d-1}$  (see Theorem \ref{thm:Haefliger}). For spaces with well-enough understood configuration space one can go beyond the Haefliger--Weber metastable range to obtain an improved lower bound:

\begin{theorem} \label{thm:StoR}
    For $k \ge d-1$, any injective function $f \colon S^{k} \to \R^d$ satisfies $\alpha(f) \ge c_{d - 1, k}$.
\end{theorem}

This is a special case of a more general Theorem~\ref{thm:alpha-coind}; the constants~$c_{d - 1, k}$ are defined in Section~\ref{sec:background} and record certain homotopy information of the Vietoris--Rips complexes of spheres.

One of the earliest nonembeddability results asserts the non-planarity of the complete graph on five vertices~$K_5$. Our results more generally show that for any injective function $f\colon K_5\to \R^2$ there are tuples of arbitrarily close points $(x,y)$ and $(x',y')$ in~$K_5$ such that the line segments $\overline{f(x)f(y)}$ and $\overline{f(x')f(y')}$ make an angle of at least~$\arccos(-1/2) = 2\pi/3$. Namely, we obtain the following quantified version of the nonembeddability result of van Kampen and Flores:

\begin{theorem} \label{thm:vKF-intro}
    Every injective function $\sk_d(\Delta_{2d+2}) \to \R^{2d}$ satisfies $\alpha(f) \ge r_{2d-1} = \arccos{\left(\frac{-1}{2d}\right)}$.
\end{theorem}

In Section~\ref{subsec:vk-f} we study a wider notion of an \emph{almost injective} function from a simplicial complex $K$ into~$\R^d$, where the function does not identify two points from disjoint faces of~$K$. In Theorem~\ref{thm:quant-vk-f} we prove a quantified extension of the classical van Kampen--Flores theorem which says that there is no almost injective function $\sk_d(\Delta_{2d+2}) \to \R^{2d}$. The formulation of this result uses a slight variation $\alpha^{(2)}$ of the scale invariant modulus of discontinuity $\alpha$ adapted to almost embeddings.

Furthermore, Tverberg-type theory studies \emph{almost $r$-injective} functions~$f \colon~K \to \R^d$ for an integer $r \ge 2$, where $f$ does not identify $r$ points of $K$ lying in pairwise disjoint faces. A continuous almost $r$-injective function is in the literature also called an \emph{almost $r$-embedding} \cite{mabillard2014eliminating}. The topological Tverberg conjecture states that there is no almost $r$-embedding from the $(r-1)(d+1)$-dimensional simplex $\Delta_{(r-1)(d+1)}$ into $\R^d$. B\'ar\'any, Shlosman and Sz\H ucs~\cite{BSSz1981} and \"Ozaydin~\cite{ozaydin1987} proved the conjecture for $r$ a prime power, while Frick~\cite{frick2015counterexamples} (see also Blagojevi\'c, Frick, and Ziegler~\cite{blagojevic2015barycenters}) established counterexamples for composite $r \le d/3$ using the $r$-fold Whitney trick of Mabillard and Wagner~\cite{mabillard2014eliminating}. This was later improved upon by Avvakumov, Mabillard, Skopenkov, and Wagner~\cite{avvakumov2021eliminating} who obtained counterexamples for $r \le (d-1)/2$. To this end, in Section \ref{sec:almost-r-emb} we introduce the $r$-fold version $\alpha^{(r)}$ of the scale invariant modulus of discontinuity to study and quantify the failure of the existence of almost $r$-injective functions from the simplex. Namely, we establish the following:

\begin{theorem} [Quantified topological Tverberg] \label{thm:tv-intro}
    Let $d \ge 1$ be an integer and $r \ge 2$ a prime power. Then, any almost $r$-injective function $f\colon \Delta_{(r-1)(d+1)} \to \R^d$ satisfies $\alpha^{(r)}(f) \ge  \arccos(\tfrac{-1}{d(r-1)})$.
\end{theorem}

As before, this theorem represents a quantified extension of the topological Tverberg theorem, as continuous almost $r$-injective functions (i.e., almost $r$-embeddings) necessarily have $\alpha^{(r)}=0$.\\

The rest of the document is organized as follows. Section~\ref{sec:background} contains the notation and background. In Section~\ref{sec:injectivity} we prove quantified nonembeddability results, which represent generalizations of Theorems~\ref{thm:RP} and~\ref{thm:StoR}, as well as the proof of Theorem~\ref{thm:vKF-intro}. Finally, we develop quantified Tverberg-type theory in Section~\ref{sec:almost-r-emb}, where we prove Theorem~\ref{thm:tv-intro}.

The introduction does not give a step-by-step proof sketch for Theorem~\ref{thm:RP}. It only indicates that it is "a consequence of the more general Theorem \ref{thm:quant-general} and Corollary \ref{cor:quant-hw}" and that the overall approach "relies on" the Haefliger--Weber framework "relating the embedding problem to the existence of $\Z/2$-equivariant maps $\Conf_2(X) \to S^{d-1}$ (see Theorem \ref{thm:Haefliger})."

The introduction does not give a step-by-step proof sketch for the main theorem. It only indicates that it is a consequence of the more general result

\begin{theorem} \label{thm:quant-general}
    Let $X$ compact topological space.  Assume that there does not exist a $\Z/2$-equivariant map $\Conf_2(X) \to S^{d-1}$. Then, every injective function $f \colon X \to \R^d$ satisfies $\alpha(f) \ge r_{d-1} = \arccos{(-1/d)}$.
\end{theorem}

and

\begin{corollary}\label{cor:quant-hw}
    Let $X$ be a smooth, closed manifold (resp.\ simplicial complex) of dimension $n$.  Assume that there does not exists a differentiable (resp.\ linear) embedding $f \colon X \to \R^d$ for $d > \frac32(n + 1)$. Then, every injective function $f \colon X \to \R^d$ satisfies $\alpha(f) \ge r_{d-1} = \arccos{(-1/d)}$.
\end{corollary}

and that the overall approach relies on the Haefliger--Weber framework relating the embedding problem to the existence of $\Z/2$-equivariant maps $\Conf_2(X) \to S^{d-1}$. We first recall the following theorem.

\begin{theorem}[Haefliger~\cite{Haefliger}, Weber~\cite{Weber}]
\label{thm:Haefliger}
Let $X$ be a smooth, closed manifold (resp.\ simplicial complex) of dimension $n$.  If $d > \frac32(n + 1)$, then there exists a differentiable (resp.\ linear) embedding $f \colon X \to \R^d$ if and only if there exists a $\Z/2$-equivariant map $\Phi \colon \Conf_2(X) \to S^{d-1}$.
\end{theorem}

Question: Let k be a nonnegative integer, and let f\colon \mathbb{RP}^{2^k} \to \mathbb{R}^{2^{k+1}-1} be injective. For an injective map g\colon X\to \mathbb{R}^m, define its scale-invariant modulus of discontinuity \(\alpha(g)\) to be the infimum of all angles \(\alpha>0\) such that for every \((x,y)\in X\times X\) with \(x\neq y\), there is a neighborhood \(U\) of \((x,y)\) for which, for all \((x',y')\in U\), the line segments \(\overline{g(x)g(y)}\) and \(\overline{g(x')g(y')}\) make an angle at most \(\alpha\). Which statement holds for f?

One necessarily has \(\alpha(f)\ge r_{2^{k+1}-2}=\arccos\!\left(-\frac{1}{2^{k+1}-1}\right)\).

Choices:
B. One necessarily has \(\alpha(f)\ge r_{2^{k+1}-1}=\arccos\!\left(-\frac{1}{2^{k+1}}\right)\).
C. One necessarily has \(\alpha(f)>0\).
D. There exists a constant \(\delta>0\), independent of \(k\), such that every injective \(f\colon \mathbb{RP}^{2^k}\to\mathbb{R}^{2^{k+1}-1}\) satisfies \(\alpha(f)\ge \delta\).
E. One necessarily has \(\alpha(f)\ge r_{2^{k+1}-1}=\arccos\!\left(-\frac{1}{2^{k+1}-1}\right)\).

explicit sharp lower bound \(r_{2^{k+1}-2}\) replaced by mere positivity

dependence of the lower bound on target dimension replaced by a uniform constant

mismatch between the index in \(r_n\) and the cosine formula denominator

The stem defines the setup and the modulus \alpha(g) but does not state or strongly hint at the sharp lower bound r_{2^{k+1}-2}. The correct answer is not leaked by wording in the prompt.

The item appears to ask almost exactly for the theorem's conclusion for the stated map f\colon \mathbb{RP}^{2^k}\to\mathbb{R}^{2^{k+1}-1}. If a student knows the result, the correct option is essentially a direct restatement rather than a derived consequence.

There is some pressure to distinguish the sharp bound from nearby false variants and a weaker true statement, but the task is driven more by theorem recall and careful matching of indices than by substantial generative reasoning from first principles.

The distractors are plausible and mathematically targeted: one weaker true statement, one overly strong uniform claim, one index-shift error, and one notation/formula mismatch. These reflect realistic failure modes and are clearly distinct.

A technically well-constructed theorem-recall MCQ with strong distractors and no answer leakage, but it is largely tautological and only moderately tests reasoning.