[Rigidity of Siegel polynomials of bounded type]
\label{Thm:A}
Let $f \colon \mathbb C \to \mathbb C$ be an atomic Siegel polynomial of bounded type and degree $d \ge 2$. Then:

\begin{enumerate}[\rm (A)]
\item
\label{it:main:1}
each point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.
\item
\label{it:main:2}
$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).
\item
\label{it:main:3}
If $\widetilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\widetilde f$ are affinely conjugate.
\end{enumerate}

\label{sec:intro}

A central theme in one-dimensional complex dynamics is the study of \emph{rigidity phenomena}, that is, understanding to what extent the combinatorial or topological data associated with a dynamical system determines its conformal or analytic structure. This perspective has driven substantial progress in the field, tracing back to the seminal work of Douady and Hubbard \cite{DoHu}, who introduced the notion of polynomial-like mappings and used them to establish deep structural results about the Mandelbrot set and quadratic polynomials. The landmark results of Yoccoz \cite{HY} later established rigidity for a broad class of quadratic polynomials, showing that combinatorial data (the landing pattern of external rays and normalization in the Fatou set) suffice to determine the conformal conjugacy class of the map. These ideas were subsequently extended to higher-degree polynomials, most notably through the development of enhanced puzzle techniques and renormalization theory in the works of Lyubich \cite{Ly}, Lyubich and Kahn \cite{KaLy1, KaLy2}, Avila, Kahn, Lyubich, and Shen \cite{AKLS}, Kozlovski, Shen, van Strien \cite{KSS}, Kozlovski and van Strien \cite{KvS}, and others.

However, rigidity results often rely on the absence of neutral behaviour in the dynamics. In cases where the dynamics has neutral periodic points, the situation becomes more delicate. For instance, in the parabolic setting, powerful tools like parabolic implosion and Lavaurs maps allow for a fine control of the local and global dynamics, and rigidity results can often be recovered. In contrast, in the presence of irrationally indifferent periodic points, the problem becomes substantially more subtle. One of the better-understood scenarios within neutral dynamics arises in the case of Siegel disks with bounded type rotation numbers.

Let $f \colon \mathbb C \to \mathbb C$ be a polynomial with a fixed point at $0$. We say that $0$ is \emph{irrationally indifferent} (with {\it rotation number} $\theta  \in (\mathbb R \setminus \mathbb Q) / \mathbb Z$) if $f'(0) = e^{2\pi i \theta}$. If additionally the dynamics of $f$ is linearizable locally near $0$, then $0$ is called a {\it Siegel point}. This means that in some neighbourhood of $0$ the map $f$ is conformally conjugate to the rigid rotation $z \mapsto e^{2\pi i \theta} \cdot z$. In this case, the maximal domain of definition for such a conjugacy is called the \emph{Siegel disk}, denoted $\Delta_f \ni 0$, and $f$ is referred to as a {\it Siegel polynomial}. Lastly, the rotation number, the Siegel point, the Siegel disk and the Siegel polynomial are said to be \emph{of bounded type} if all entries in the continued fraction expansion of $\theta$ are uniformly bounded.

The first seminal result on Siegel polynomials was due to Douady--Ghys--Herman--Shishikura (see \cite{Do, He2}). Using quasi-surgery, they constructed a Blaschke product model for the dynamics of $f$ in case $f$ is quadratic ($\deg(f) = 2$) and the rotation number is of bounded type. This allowed them to conclude that the Siegel boundary $\partial \Delta_f$ is a quasi-circle containing at least one critical point (which is unique in the quadratic case). This result was generalized to the cubic case $\deg(f) =3$ by Zakeri in \cite{Za}, then to any degree by Shishikura and Zhang \cite{Zh} (see Theorem~\ref{siegel bound quasi}). (See also \cite{WaYaZhZh}.)

The rigidity of Siegel polynomials of bounded type was settled in the quadratic case by Petersen in \cite{Pe} (see also the work of Yampolsky \cite{Ya1}). Petersen proved that the Julia set $J_f$ of a quadratic Siegel polynomial $f$ of bounded type is locally connected, thereby showing that the topology of the dynamics of $f$ is entirely determined by its combinatorial data. He then showed that $J_f$ has zero measure, which implies that it has a trivial conformal conjugacy class.

In Petersen's work, it is crucial that $\deg(f) = 2$, so that the unique critical point is forced to be contained in the Siegel boundary $\partial \Delta_f$. In the case $\deg(f) \geq 3$, it is possible that there exists an external, recurrent critical point whose orbit accumulates to $\partial \Delta_f$, but does not land exactly on it. In this highly non-linear situation, the geometry of $\partial \Delta_f$ could, a priori, become highly distorted, causing a failure of rigidity. This problem is addressed in \cite{Y}, where it is shown that even in the higher degree case, local connectivity still holds at $\partial \Delta_f$. This work is followed in \cite{YZh}, where it is shown that global rigidity holds for cubic Siegel polynomials of bounded type.

In this paper, we address the rigidity of Siegel polynomials of bounded type for arbitrary degrees. We focus on polynomials that are non-renormalizable in the sense of Douady and Hubbard. This means that the dynamics cannot be decomposed into simpler, more basic parts that can be analyzed separately. We say that such maps are \emph{atomic}.

The main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.

Let $f \colon \mathbb C \to \mathbb C$ be a polynomial with a fixed point at $0$. We say that $0$ is \emph{irrationally indifferent} (with {\it rotation number} $\theta \in (\mathbb R \setminus \mathbb Q) / \mathbb Z$) if $f'(0) = e^{2\pi i \theta}$. If additionally the dynamics of $f$ is linearizable locally near $0$, then $0$ is called a {\it Siegel point}. This means that in some neighbourhood of $0$ the map $f$ is conformally conjugate to the rigid rotation $z \mapsto e^{2\pi i \theta} \cdot z$. In this case, the maximal domain of definition for such a conjugacy is called the \emph{Siegel disk}, denoted $\Delta_f \ni 0$, and $f$ is referred to as a {\it Siegel polynomial}. Lastly, the rotation number, the Siegel point, the Siegel disk and the Siegel polynomial are said to be \emph{of bounded type} if all entries in the continued fraction expansion of $\theta$ are uniformly bounded.

The first seminal result on Siegel polynomials was due to Douady--Ghys--Herman--Shishikura (see \cite{Do, He2}). Using quasi-surgery, they constructed a Blaschke product model for the dynamics of $f$ in case $f$ is quadratic ($\deg(f) = 2$) and the rotation number is of bounded type. This allowed them to conclude that the Siegel boundary $\partial \Delta_f$ is a quasi-circle containing at least one critical point (which is unique in the quadratic case). This result was generalized to the cubic case $\deg(f) =3$ by Zakeri in \cite{Za}, then to any degree by Shishikura and Zhang \cite{Zh} (see Theorem~\ref{siegel bound quasi}). (See also \cite{WaYaZhZh}.)

The rigidity of Siegel polynomials of bounded type was settled in the quadratic case by Petersen in \cite{Pe} (see also the work of Yampolsky \cite{Ya1}). Petersen proved that the Julia set $J_f$ of a quadratic Siegel polynomial $f$ of bounded type is locally connected, thereby showing that the topology of the dynamics of $f$ is entirely determined by its combinatorial data. He then showed that $J_f$ has zero measure, which implies that it has a trivial conformal conjugacy class.

In Petersen's work, it is crucial that $\deg(f) = 2$, so that the unique critical point is forced to be contained in the Siegel boundary $\partial \Delta_f$. In the case $\deg(f) \geq 3$, it is possible that there exists an external, recurrent critical point whose orbit accumulates to $\partial \Delta_f$, but does not land exactly on it. In this highly non-linear situation, the geometry of $\partial \Delta_f$ could, a priori, become highly distorted, causing a failure of rigidity. This problem is addressed in \cite{Y}, where it is shown that even in the higher degree case, local connectivity still holds at $\partial \Delta_f$. This work is followed in \cite{YZh}, where it is shown that global rigidity holds for cubic Siegel polynomials of bounded type.

In this paper, we address the rigidity of Siegel polynomials of bounded type for arbitrary degrees. We focus on polynomials that are non-renormalizable in the sense of Douady and Hubbard. This means that the dynamics cannot be decomposed into simpler, more basic parts that can be analyzed separately. We say that such maps are \emph{atomic}.

The main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.

\begin{thm}[Shishikura, Zhang]\label{siegel bound quasi}
The Siegel boundary $\partial \Delta_f$ is a quasi-circle that contains at least one critical point $c_0$ of $f$. \qed
\end{thm}

\maketitle
\begin{abstract}
In this paper, we study rigidity of polynomials of arbitrary degree in the presence of neutral dynamics. Specifically, we focus on \emph{non-renormalizable} (in the sense of Douady and Hubbard) complex polynomials of degree $d \ge 2$ that possess a Siegel disk of bounded type rotation number. We refer to such maps as \emph{atomic Siegel polynomials of bounded type}. In this setting, our main results are:
\smallskip
\begin{enumerate}
\item[(A)] Atomic Siegel polynomials of bounded type have locally connected Julia sets;
\item[(B)] These Julia sets are quasiconformally rigid, i.e., they do not support invariant line fields;
\item[(C)] Any two combinatorially equivalent atomic Siegel polynomials of bounded type coincide up to an affine change of coordinates.
\end{enumerate}
\smallskip
In particular, item (C) verifies the notorious \emph{Combinatorial Rigidity Conjecture} for atomic Siegel polynomials of bounded type and arbitrary degree. By bringing neutral Siegel dynamics into the picture, we extend the celebrated higher-degree rigidity theorems of Avila--Kahn--Lyubich--Shen and Kozlovski--van Strien, which until now applied only in the Yoccoz setting, i.e., for finitely many times renormalizable polynomials without irrationally indifferent periodic points.
\end{abstract}

The main result of this paper is the following rigidity theorem for atomic Siegel polynomials of bounded type. This is the first time in the literature that the neutral irrational dynamics is incorporated into the rigidity picture for polynomial maps of arbitrary degree.

We will define all the concepts in the follow-up sections; in particular, combinatorial equivalence is defined in Section~\ref{sec:topmodel} (see Definition~\ref{Def:CombEq}). This result extends a recent result due to the second author and Zhang \cite{YZh} on rigidity of bounded-type Siegel polynomials of degree $3$ (one free critical point). Main Theorem~\eqref{it:main:3} confirms the \emph{Combinatorial Rigidity Conjecture} in the above-mentioned class of polynomials. This conjecture asserts that for polynomials under an \emph{appropriate} definition of the combinatorial equivalence, combinatorial rigidity holds. We remark that our definition includes the condition that $f$ and $\tilde f$ are \emph{Fatou normalized} and the positions of critical points on the boundary of the Siegel disk are combinatorially the same, see Section~\ref{sec:fatoutrees} for details. Since we assume that $f, \tilde f$ are non-renormalizable, their rational laminations, i.e., the combinatorial pattern of landing external rays, are empty. In our context, Fatou normalization means that each critical point $c$ in a preimage of a Siegel disk $\Delta_f$ lands at $0$.

\begin{thm}[Parameter rigidity]
\label{Thm:BoxParamRigidity}
Let $F\colon \U\to \V$ be a non-renormalizable dynamically natural complex box mapping. Suppose $\tilde F \colon \tilde \U \to \tilde \V$ is another dynamically natural complex box mapping for which there exists a quasiconformal homeomorphism $H \colon \V \to \tilde\V$ so that
\begin{enumerate}
\item
\label{part3c}
$\tilde F$ is combinatorially equivalent to $F$ w.r.t.\ $H$, and so in particular $H(\U) = \tilde \U$,
\item
\label{part3b}
$\tilde F \circ H = H \circ F$ on $\partial \U$, i.e.\ $H$ is a conjugacy on $\partial \U$.
\end{enumerate}
Then $F$ and $\tilde F$ are quasiconformally conjugate, and this conjugacy agrees with $H$ on $\V\setminus \U$. \qed
\end{thm}

\begin{thm}
\label{Thm:DynRigidity}
Let $f$ be an atomic Siegel polynomial of bounded type. Then
\begin{enumerate}
\item
all fibers of points in the Julia set $J(f)$ are trivial, and hence, $J(f)$ is locally connected;
\item
$J(f)$ carries no invariant line fields.
\end{enumerate}
\end{thm}

\begin{defn}[Combinatorial equivalence]
\label{Def:CombEq}
Two atomic Siegel polynomials $f, \tif : \bbC \to \bbC$ of bounded type are said to be {\it combinatorially equivalent} if
\begin{enumerate}
 \item
 $\deg(f) = \deg(\tif\,)$, $\rho(f)=\rho(\tif\,)$;
 \item
 $\adr_{\crit}(f)=\adr_{\crit}(\tif)$ (see Definition~\ref{Def:CritAdr});
 \item
 the orbits of each pair $c \in \Crit(f) \cap \inter(K_f)$, $\tilde c\in \Crit(\tif\,) \cap \inter(K_{\tif})$ of the corresponding critical points land in $\Delta_f$, $\Delta_{\tif}$ in the same conformal position with respect to the uniformizing coordinates.
\end{enumerate}
\end{defn}

\begin{thm}[Topological rigidity]\label{poly topo rigid}
Let $f, \tif : \bbC \to \bbC$ be atomic Siegel polynomials of bounded type. If $f$ and $\tif$ are combinatorially equivalent, then they are topologically conjugate, and, restricted to the Fatou sets, this conjugacy is conformal.\qed
\end{thm}

\begin{prop}
\label{Prop:CombEquivalent}
If $f$ and $\tf$ are two topologically conjugate, atomic Siegel polynomials of bounded type, then in Theorem~\ref{Thm:Extract} one can choose dynamically natural complex box mappings $F \colon \U \to \V$ and $\tilde F \colon \tilde \U \to \tilde \V$ to be combinatorially equivalent in the sense of Definition~\ref{DefA:CombEquivBoxMappings}, with respect to some quasiconformal homeomorphism $H \colon \V \to \tilde \V$ such that $H(\U) = \tilde \U$ and $\tilde F \circ H = H \circ F$ on $\partial \U$. \qed
\end{prop}

\begin{defn}[Combinatorial equivalence]
\label{Def:CombEq}
Two atomic Siegel polynomials $f, \tif : \bbC \to \bbC$ of bounded type are said to be {\it combinatorially equivalent} if
\begin{enumerate}
    \item
    $\deg(f) = \deg(\tif\,)$, $\rho(f)=\rho(\tif\,)$;
    \item
    $\adr_{\crit}(f)=\adr_{\crit}(\tif)$ (see Definition~\ref{Def:CritAdr});
    \item
    the orbits of each pair $c \in \Crit(f) \cap \inter(K_f)$, $\tilde c\in \Crit(\tif\,) \cap \inter(K_{\tif})$ of the corresponding critical points land in $\Delta_f$, $\Delta_{\tif}$ in the same conformal position with respect to the uniformizing coordinates.
\end{enumerate}
\end{defn}

\begin{enumerate}[\rm (A)]
\item
\label{it:main:1}
each point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.
\item
\label{it:main:2}
$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).
\item
\label{it:main:3}
If $\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\tilde f$ are affinely conjugate.
\end{enumerate}

We will define all the concepts in the follow-up sections; in particular, combinatorial equivalence is defined in Section~\ref{sec:topmodel} (see Definition~\ref{Def:CombEq}). This result extends a recent result due to the second author and Zhang \cite{YZh} on rigidity of bounded-type Siegel polynomials of degree $3$ (one free critical point). Main Theorem~\eqref{it:main:3} confirms the \emph{Combinatorial Rigidity Conjecture} in the above-mentioned class of polynomials. This conjecture asserts that for polynomials under an \emph{appropriate} definition of the combinatorial equivalence, combinatorial rigidity holds. We remark that our definition includes the condition that $f$ and $\widetilde f$ are \emph{Fatou normalized} and the positions of critical points on the boundary of the Siegel disk are combinatorially the same, see Section~\ref{sec:fatoutrees} for details. Since we assume that $f, \widetilde f$ are non-renormalizable, their rational laminations, i.e., the combinatorial pattern of landing external rays, are empty. In our context, Fatou normalization means that each critical point $c$ in a preimage of a Siegel disk $\Delta_f$ lands at $0$.

\begin{rem}
Our Main Theorem can be easily extended to the cases where one allows for finitely many orbits of periodic Siegel disks and finitely many renormalizations. In those cases, in the definition of combinatorial equivalence one needs to include in the combinatorial data also the landing patterns of external and internal rays. Furthermore, one can also include cycles of attracting and parabolic periodic points.

For clarity of theexposition, in this paper, we restrict our attention to atomic Siegel polynomials.
\end{rem}

\subsection{Outline of the proof}

The proof uses the following strategy, which was successful in a number of cases (see \cite[Section 2]{CDKvS}), including the one in the current paper:

\begin{enumerate}
\item
\emph{Markov partition}: In a given family $\mathcal F$ of mappings and a map $f \in \mathcal F$ in this family, construct a puzzle partition (i.e., a Yoccoz- or Markov-type partition) around points in the Julia set $J_f$ of $f$. In the context of the current paper, the partition is given in terms of Siegel bubble rays extended by external rays, see Section~\ref{sec:puzzle}.

\item
\emph{Rigidity on the Julia boundary of puzzle pieces}: Establish (dynamical) rigidity on the boundary of each puzzle piece, i.e., for a puzzle piece $P$, show that each point in $\partial P \cap J_f$ can be surrounded by a union of arbitrary small puzzle pieces. In our case, this result was proven in \cite{Y} (see Section~\ref{sec:puzzle}).

\item
\emph{Extract a dynamically natural complex box mapping}: Construct a dynamically natural box mapping (in the sense of Section~\ref{sec:box}) around the critical points of $f$ that do not land on the puzzle boundary. This mapping controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary. In our paper, the construction of such a mapping is carried out in Section~\ref{sec:DNBM}.

\item
\emph{Local connectivity and quasiconformal rigidity of the Julia set}: Use the rigidity result for box mappings (i.e., puzzle pieces shrink to points, no invariant line fields) to infer the corresponding results for $f \in \mathcal F$. In our case, this is done in Section~\ref{sec:proofAB}, where we prove the items \eqref{it:main:1} and \eqref{it:main:2} of the Main Theorem.

\item
\emph{Topological model of the Julia set}: Construct topological models of the dynamics for maps $f \in \mathcal F$ and prove that they are determined uniquely by combinatorial data. This is done in Section~\ref{sec:combtrees} and \ref{sec:fatoutrees}.

\item
\emph{Topological rigidity of combinatorially equivalent maps}: Prove that the filled Julia sets of maps in $\mathcal F$ are homeomorphic to the topological models constructed in the previous step. It follows that two maps within the same combinatorial class must be topologically conjugate. This is carried out in Section~\ref{sec:topmodel}.

\item
\emph{Conformal rigidity of topologically equivalent maps}: Use quasiconformal rigidity of the corresponding complex box mappings for the topologically conjugate $f, \widetilde f$ to conclude that $f, \widetilde f$ are quasiconformally conjugate, and hence, because of quasiconformal rigidity of $J_f, J_{\widetilde f}$ and Fatou normalization, $f, \widetilde f$ are conformally (affine) conjugate.

In our paper, this step is done in Section~\ref{sec:top2qc}. This concludes the proof of Main Theorem~\eqref{it:main:3}.
\end{enumerate}

The post-theorem introduction contains an explicit \subsection{Outline of the proof} giving the strategy (successful in other cases, cf. \cite[Section 2]{CDKvS}):
\begin{enumerate}
\item \emph{Markov partition}: for $f$ in the family $\mathcal F$, construct a puzzle (Yoccoz-/Markov-type) partition around points in $J_f$; here it is “given in terms of Siegel bubble rays extended by external rays” (Section~\ref{sec:puzzle}).
\item \emph{Rigidity on the Julia boundary of puzzle pieces}: prove dynamical rigidity on $\partial P$ for each puzzle piece $P$, i.e. each point of $\partial P\cap J_f$ can be surrounded by “a union of arbitrary small puzzle pieces”; in this setting it is cited as proven in \cite{Y} (Section~\ref{sec:puzzle}).
\item \emph{Extract a dynamically natural complex box mapping}: build a “dynamically natural box mapping” (Section~\ref{sec:box}) around critical points not landing on the puzzle boundary; it “controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary” (Section~\ref{sec:DNBM}).
\item \emph{Local connectivity and quasiconformal rigidity of the Julia set}: apply rigidity for box mappings (“puzzle pieces shrink to points, no invariant line fields”) to obtain the corresponding statements for $f$; this yields items \eqref{it:main:1} and \eqref{it:main:2} in Section~\ref{sec:proofAB}.
\item \emph{Topological model of the Julia set}: construct topological models for $f\in\mathcal F$ and show they are “determined uniquely by combinatorial data” (Sections~\ref{sec:combtrees} and \ref{sec:fatoutrees}).
\item \emph{Topological rigidity of combinatorially equivalent maps}: show filled Julia sets are homeomorphic to these models, so “two maps within the same combinatorial class must be topologically conjugate” (Section~\ref{sec:topmodel}).
\item \emph{Conformal rigidity of topologically equivalent maps}: use quasiconformal rigidity of the associated box mappings for topologically conjugate $f,\widetilde f$ to get a quasiconformal conjugacy; then, using “quasiconformal rigidity of $J_f, J_{\widetilde f}$ and Fatou normalization,” conclude $f,\widetilde f$ are “conformally (affine) conjugate” (Section~\ref{sec:top2qc}), which “concludes the proof of Main Theorem~\eqref{it:main:3}.”
\end{enumerate}

The post-theorem introduction contains an explicit \subsection{Outline of the proof} giving the strategy (successful in other cases, cf. \cite[Section 2]{CDKvS}):
\begin{enumerate}
\item \emph{Markov partition}: for $f$ in the family $\mathcal F$, construct a puzzle (Yoccoz-/Markov-type) partition around points in $J_f$; here it is “given in terms of Siegel bubble rays extended by external rays” (Section~\ref{sec:puzzle}).
\item \emph{Rigidity on the Julia boundary of puzzle pieces}: prove dynamical rigidity on $\partial P$ for each puzzle piece $P$, i.e. each point of $\partial P\cap J_f$ can be surrounded by “a union of arbitrary small puzzle pieces”; in this setting it is cited as proven in \cite{Y} (Section~\ref{sec:puzzle}).
\item \emph{Extract a dynamically natural complex box mapping}: build a “dynamically natural box mapping” (Section~\ref{sec:box}) around critical points not landing on the puzzle boundary; it “controls the dynamics of most critical points, except for those whose orbits accumulate only at the puzzle boundary” (Section~\ref{sec:DNBM}).
\item \emph{Local connectivity and quasiconformal rigidity of the Julia set}: apply rigidity for box mappings (“puzzle pieces shrink to points, no invariant line fields”) to obtain the corresponding statements for $f$; this yields the following two conclusions:
\begin{enumerate}[\rm (A)]
\item
\label{it:main:1}
each point in the Julia set $J_f$ is dynamically rigid (i.e., the orbits of any two orbits can be distinguished in terms of symbolic dynamics); in particular, $J_f$ is locally connected.
\item
\label{it:main:2}
$J_f$ is quasiconformally rigid (i.e., it does not admit non-trivial invariant line fields).
\item
\label{it:main:3}
If $\tilde f$ is another atomic Siegel polynomial of bounded-type that is combinatorially equivalent to $f$, then $f$ and $\tilde f$ are affinely conjugate.
\end{enumerate}
 in Section~\ref{sec:proofAB}.
\item \emph{Topological model of the Julia set}: construct topological models for $f\in\mathcal F$ and show they are “determined uniquely by combinatorial data” (Sections~\ref{sec:combtrees} and \ref{sec:fatoutrees}).
\item \emph{Topological rigidity of combinatorially equivalent maps}: show filled Julia sets are homeomorphic to these models, so “two maps within the same combinatorial class must be topologically conjugate” (Section~\ref{sec:topmodel}).
\item \emph{Conformal rigidity of topologically equivalent maps}: use quasiconformal rigidity of the associated box mappings for topologically conjugate $f,\widetilde f$ to get a quasiconformal conjugacy; then, using “quasiconformal rigidity of $J_f, J_{\widetilde f}$ and Fatou normalization,” conclude $f,\widetilde f$ are “conformally (affine) conjugate” (Section~\ref{sec:top2qc}), which “concludes the proof of Main Theorem~\eqref{it:main:3}.”
\end{enumerate}

Question: Let \(f:\mathbb C\to\mathbb C\) be an atomic Siegel polynomial of bounded type and degree \(d\ge 2\), meaning that \(f\) is a non-renormalizable complex polynomial of degree at least \(2\) with a Siegel disk whose irrational rotation number is of bounded type (equivalently, the continued-fraction coefficients of the rotation number are uniformly bounded). For another atomic Siegel polynomial of bounded type \(\widetilde f\), say that \(f\) and \(\widetilde f\) are combinatorially equivalent if they have the same degree and rotation number, the same critical-address data, and the orbits of corresponding critical points in the interiors of their filled Julia sets land in their Siegel disks in the same conformal position with respect to uniformizing coordinates. Under these assumptions, which statement about \(J_f\) and about such a map \(\widetilde f\) holds?

Each point of the Julia set \(J_f\) is dynamically rigid (that is, the orbits of any two orbits can be distinguished in terms of symbolic dynamics), and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then \(f\) and \(\widetilde f\) are affinely conjugate.

Choices:
B. Each point of the Julia set \(J_f\) is dynamically rigid, and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type with the same degree and rotation number as \(f\), then \(f\) and \(\widetilde f\) are affinely conjugate.
C. The Julia set \(J_f\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then \(f\) and \(\widetilde f\) are topologically conjugate.
D. Each point of the Julia set \(J_f\) is dynamically rigid, and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid, meaning that it admits no non-trivial invariant line fields; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then there exists a quasiconformal conjugacy between \(f\) and \(\widetilde f\).
E. Each point of the Julia set \(J_f\) that does not lie on the boundary of a puzzle piece is dynamically rigid, and in particular \(J_f\) is locally connected; moreover, \(J_f\) is quasiconformally rigid away from the Siegel boundary; and if \(\widetilde f\) is another atomic Siegel polynomial of bounded type that is combinatorially equivalent to \(f\), then \(f\) and \(\widetilde f\) are affinely conjugate whenever all critical orbits avoid accumulation on the puzzle boundary.

full combinatorial-equivalence data required for rigidity

affinely conjugate weakened to topologically conjugate; dynamic rigidity/local connectivity clause dropped

final affine conclusion weakened to quasiconformal-conjugacy endpoint

boundary-vs-nonboundary critical orbit split in box-mapping extraction

The stem does not explicitly reveal the correct option. It gives hypotheses and asks for the resulting conclusion, without naming the rigidity/local connectivity/affine-conjugacy conclusion outright.

The item is essentially a theorem-recall question: under the stated hypotheses, the correct choice reproduces the full theorem conclusion almost verbatim. It does not substantially reframe the result into a new problem.

There is some reasoning pressure because the options differ in subtle but important ways (same degree and rotation number vs full combinatorial equivalence, affine vs topological vs quasiconformal conjugacy, full vs partial rigidity). However, the task is still mainly recognition of the exact theorem statement rather than generating a conclusion from mathematical reasoning.

The distractors are plausible and mathematically meaningful. They reflect common weakening/strengthening mistakes: insufficient hypotheses, weaker conclusions, conflation of affine and quasiconformal rigidity, and technical boundary-case tampering.

A technically well-constructed theorem-recognition MCQ with strong distractors and little answer leakage, but it is largely tautological and only moderately tests genuine generative reasoning.


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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Question: For a positive integer m, let T(m) be the set of odd primes p such that p-1 divides 4m and \(\gcd\!\left(p,\frac{4m}{p-1}\right)=1\), and let \(t(m)=|T(m)|\). Assume the following statement is true: whenever m is an odd, square-free multiple of 3, there exists some \(p\in\{5,7,13\}\) with \(p\mid m\), no prime \(r\in\{5,7,13\}\setminus\{p\}\) divides m, and there exists \(q\in T(3p)\setminus\{5,7,13\}\) such that \(q\nmid m\), then \(t(m)\ge 4\). Under this assumption, which statement about a positive integer n holds?

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

Choices:
B. For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\); and if \(3\nmid n\), then \(t(n)\ge 3\).
C. For every positive integer n, \(t(n)\ge 1\). In addition, if \(5\nmid n\), then \(t(n)\ge 2\).
D. For every positive integer n, \(t(n)\ge 2\). In addition, if \(5\nmid n\), then \(t(n)\ge 3\); and if \(3\mid n\), then \(t(n)\ge 4\).
E. For every positive integer n, \(t(n)\ge 1\). In addition, if \(3\nmid n\), then \(t(n)\ge 2\); and if \(3\mid n\), then \(t(n)\ge 3\).

divisibility condition on 3 in the final clause

dropped the strongest conclusion \(3\mid n\Rightarrow t(n)\ge 3\)

replaced the specific hypothesis \(5\nmid n\) by the unrelated \(3\nmid n\)

The stem gives a technical assumed implication about special integers m, but it does not explicitly state the broader conclusion in choice A. There is no direct wording overlap that reveals the correct option.

The question is not a restatement of the assumption. It asks the test-taker to infer a global conclusion about t(n) from a conditional statement on a restricted class of m, so the correct answer is not tautologically built into the stem.

Selecting the correct option requires distinguishing between a strongest valid conclusion, a weaker true-looking alternative, and several subtle overgeneralizations or condition swaps. This creates real pressure to reason about how the assumption propagates to statements about arbitrary n.

The distractors are plausible and target common failure modes: reversing a divisibility condition, choosing a weaker-but-safe statement, and overstrengthening all bounds. They are distinct and mathematically aligned with likely errors.

A strong MCQ on these criteria: it avoids answer leakage, is clearly non-tautological, and uses high-quality distractors that force genuine comparative reasoning, though the stem is quite technically dense.


\begin{proposition}\label{prop:ball-meas-upper-bound}
For any compact set $K\subset \Omega_f$ there exists $r_K>0$ such that
\begin{align*}
    \Ha^n(B_{\Omega_f}(z,r))\le \omega_nd^{n/2}K_IK_O r^n,\quad z\in K,\ r<r_K.
\end{align*}
\end{proposition}

\begin{theorem}\label{thm:higher-integrability}
If $f:\Omega\to \R^n$ is a non-constant quasiregular map of degree $d$, then there exists $p>n$ such that $\minv f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_d(\Omega))$.
\end{theorem}

\begin{theorem}\label{thm:geom-QC}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$, and equip $\Omega_f=\minv f(f(\Omega))$ with the metric from $\mathcal A_d(\R^n)$ and the Hausdorff $n$-measure on $\Omega_f$.
\begin{itemize}
    \item[(1)] $\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\'e inequality;
    \item[(2)] $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal. More precisely,
\begin{align*}
    \frac 1{K_IK_O}\Mod_n\Gamma\le \Mod_n\minv f(\Gamma)\le K_IK_O\Mod_n\Gamma
\end{align*}
fir any path family $\Gamma $ in $f(\Omega)$.
\end{itemize}
\end{theorem}

\begin{theorem}\label{thm:QR-curve}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\minv{f}:f(\Omega)\to \mathcal A_d(\Omega)$ is an $\omega$-quasiregular curve, where $\omega=\operatorname{tr}(\operatorname{vol}_{\R^n})$ is given by \eqref{eq:natural-n-form}. More precisely,  $\minv f\in W_{loc}^{1,n}(f(\Omega),\mathcal A_d(\R^n))$, and
\[
\|\omega\|\circ \minv f\ |D\minv f|^n\le d^{n/2-1}K_I(f)\star \minv f^*\omega\quad \textrm{almost everywhere on }\Omega.
\]
\end{theorem}

\begin{theorem}\label{thm:wug}
For $\Mod_n$-a.e. curve $\gamma$ in $f(\Omega)$, $\minv f\circ\gamma$ is an absolutely continuous curve in $\mathcal{A}_d(\Omega)$, and satisfies
\begin{align}\label{eq:ug-ineq}
H(\gamma_t)|\gamma_t'|\le |(\minv f\circ\gamma)'_t|\le (K_IK_O)^{1/n} H(\gamma_t)|\gamma_t'|
\end{align}
a.e. $t$, where
\begin{align}\label{eq:H}
    H(y):=\big(f_\ast(\|Df\|^{-2})(y)\big)^{1/2},\quad y\in f(\Omega).
\end{align}
\end{theorem}

\begin{lemma}\label{lem:homeo}
The generalized inverse $\minv f:f(X)\to \mathcal A_d(X)$ is a homeomorphism onto its image.
\end{lemma}

\begin{lemma}\label{lem:barycenter}
The barycenter map \eqref{eq:barycenter} is $1/\sqrt d$-Lipschitz.
\end{lemma}

\begin{corollary}\label{cor:QR-curve}
Let $f:\Omega\to \R^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse
\[
g:f(\Omega)\to \R^n,\quad g(y):=\sum_{x\in f\inv(y)}\iota(f,x)x
\]
belongs to $W^{1,p}_{loc}(f(\Omega),\R^n)$.
\end{corollary}

\begin{align}\label{eq:inf-n-PI}
|Dg|_p=\Lip g\quad \mu\textrm{-a.e. on }X
\end{align}

\begin{theorem}\label{thm:sob-pull-back}
Let $n,m,Q\ge 1$ be natural numbers, $U\subset \R^m$ open, and $f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_Q(\R^n))$. If $\omega$ is a $S_Q$-invariant smooth $k$-form on $(\R^n)^Q$, and either $k+1\le p$ of $\ud \omega=0$ and $k\le p$, then $\omega$ has a well-defined pull-back $f^*\omega \in L^{p/k}_{loc}(U,\bigwedge^k\R^m)$ and moreover
\begin{align*}
    \ud(f^*\omega)=f^*(\ud\omega)
\end{align*}
weakly.
\end{theorem}

\subsection{Overview}
A continuous map $f\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\le k$ is a $K$-quasiregular $\omega$-curve, if
\begin{align}\label{eq:QR-curve}
\|\omega\|\circ f\|Df\|^n\le K\star f^*\omega\quad \textrm{ almost everywhere}.
\end{align}
Here, $\omega$ is a (closed) $n$-form on $N$, $\star f^*\omega$ is the Hodge-star dual of $f^*\omega$, and $\|\omega\|$ is the pointwise comass of $\omega$. When $k=n$ and $\omega$ is the volume form on $N$, \eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as
\begin{align}\label{eq:QR}
\|Df\|^n\le K\mathbf{J} f\quad\textrm{ almost everywhere},
\end{align}
where $\mathbf{J} f=\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \cite{vai66}. We call an open discrete map a branched cover.

Quasiregular curves (QR curves) in turn were introduced by Pankka \cite{pankka20}, and they generalize quasiregular maps by allowing the dimension of the target to be higher than that of the domain; for example pseudoholomorhic curves \cite{gro85} are QR curves. Much of the classical theory of QR maps extends to QR curves \cite{pankka20,pan-onn21}, with one notable exception: Reshetnyak's theorem. Namely, QR curves need not be branched covers. Indeed, if $k>n$, a QR curve cannot be open, while the more non-trivial failure of discreteness follows from an example in \cite{IVV02}.

Compared to QR maps, one of the challenges in QR curves is understanding how the choice of $\omega$ reflects the behaviour of $f$, see for example \cite{hei-pan-pry23,iko-pan24} for some rigidity results and connections with calibrations.

\bigskip\noindent In this paper, we identify a fruther connection between QR maps and QR curves: we construct a multi-valued inverse $\minv f$ for a QR map $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$, and show that $\minv f$ is naturally a QR curve (see Theorem \ref{thm:QR-curve}). In particular, QR maps have a local (multi-valued) inverse across the singular branch set (and not only outside it) with suitable Sobolev regularity. Earlier, related constructions include a push-forward operator \cite[p. 263]{HKM06}, \cite[Section 4]{teripekka} and a generalized local inverse \cite[Chapter II.5]{ric93}, \cite[Section 8]{onn-raj09}, both of which are important for obtaining modulus inequalities.

Our approach is based on Almgren's space of unordered tuples \cite{del11} (also known as the symmetric product, cf. \cite[Chapter 4.K]{hat02}), and brings a new perspective to quasiregular theory; together with a pull-back theory by multi-valued maps developed in this paper (see Theorems \ref{thm:sob-pull-back} and \ref{thm:lip-pullback-flat-form}) we obtain e.g. higher integrability properties of the generalized inverse \cite[Chapter II.5]{ric93} (see Corollary \ref{cor:QR-curve} and Theorem \ref{thm:higher-integrability}). We describe the construction and briefly discuss some its advantages over alternative approaches below.

\subsection{Multi-valued inverse}

Given a non-injective (proper) branched cover $f:\Omega\to \mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\Omega)\ni y\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\bar f\circ P_f$, where $P:\Omega\to \widetilde\Omega_f:=\Omega/\sim$ is the projection onto the quotient space given by the equivalence relation $x\sim y$ if $f(x)=f(y)$, and $\bar f:\widetilde \Omega_f\to f(\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\Omega)\to \mathcal{H}(\Omega)$ into the Hausdorff space of non-empty compact subsets of $\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\mathcal{H}(\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \cite[Theorem 2.1]{del11}).

Instead, we give a construction based on Almgren's space and local degree theory. Let $Q\ge 1$ be a natural number. The Almgren space $\mathcal A_Q(\mathbb{R}^n)$ of unordered $Q$-tuples in $\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \cite{alm2000}, and was further explored by De Lellis et. al. \cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as
\begin{align*}
\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q,\quad d_\mathcal A(\bb x,\bb y)=\inf_{\sigma\in S_Q}\Big(\sum_j^d|x_j-y_{\sigma(j)}|^2\Big)^{1/2},
\end{align*}
where $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\mathbb{R}^n)^Q$. (See Section \ref{sec:almgren-space} for more details.)

\bigskip\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\Omega\to \mathbb{R}^n$ of degree $d>0$ from an open set $\Omega\subset \mathbb{R}^n$, the local index $\iota(f,x)$ is defined for all $x\in\Omega$ and satisfies $\displaystyle \sum_{x\in f^{-1}(y)}\iota(f,x)=d$, $y\in f(\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\Omega$ (counted with multiplicities) for each $y\in f(\Omega)$. The multi-valued inverse $\minv f$ of the proper branched cover $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$ is the map
\begin{align}\label{eq:minv}
\minv f:f(\Omega)\to \mathcal A_d(\mathbb{R}^n),\quad \minv f(y):= \sum_{x\in f^{-1}(y)}\iota(f,x)\bb x.
\end{align}
We refer the reader to Section \ref{sec:branched-covers} (see in particular Definition \ref{def:multi-valued-inv}) for the notation and further details used here.

\subsection{Statement of results}

While it is not a manifold, $\mathcal A_Q(\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\mathbb{R}^n$; that is, the $n$-form
\begin{align}\label{eq:natural-n-form}
\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n}):=\sum_j^d P_j^*(\operatorname{vol}_{\mathbb{R}^n})
\end{align}
on $(\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\sigma^*\omega=\omega$ for all $\sigma\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \ref{sec:diff-forms} and \ref{sec:pull-back}), which is of independent interest.

\subsection{Overview}
A continuous map $f\in W^{1,n}_{loc}(M^n,N^k)$ between oriented Riemannian manifolds with $n\le k$ is a $K$-quasiregular $\omega$-curve, if
\begin{align}\label{eq:QR-curve}
\|\omega\|\circ f\|Df\|^n\le K\star f^*\omega\quad \textrm{ almost everywhere}.
\end{align}
Here, $\omega$ is a (closed) $n$-form on $N$, $\star f^*\omega$ is the Hodge-star dual of $f^*\omega$, and $\|\omega\|$ is the pointwise comass of $\omega$. When $k=n$ and $\omega$ is the volume form on $N$, \eqref{eq:QR-curve} recovers the notion of quasiregular (QR) maps, written more classically as
\begin{align}\label{eq:QR}
\|Df\|^n\le K\mathbf{J} f\quad\textrm{ almost everywhere},
\end{align}
where $\mathbf{J} f=\det(Df)$ and $Df$ is the weak differential of $f$. Quasiregular maps have a rich theory \cite{ric93,HKM06}, and enjoy many of the same analytic properties as quasiconformal (QC) maps. While they are not necessarily homeomorphisms, a seminal result of Reshetnyak \cite{res67} states that QR maps are open and discrete, and thus in particular locally injective outside a singular branch set of topological dimension at most $n-2$ \cite{vai66}. We call an open discrete map a branched cover.

Given a non-injective (proper) branched cover $f:\Omega\to \mathbb{R}^n$, there are several natural ways to interpret the set valued ``inverse'' map $f(\Omega)\ni y\mapsto f^{-1}(y)$. One is to consider the natural factorization $f=\bar f\circ P_f$, where $P:\Omega\to \widetilde\Omega_f:=\Omega/\sim$ is the projection onto the quotient space given by the equivalence relation $x\sim y$ if $f(x)=f(y)$, and $\bar f:\widetilde \Omega_f\to f(\Omega)$ the induced map, which in this case is a homeomorphism and thus invertible. Another approach is to view the set valued inverse as a map $f(\Omega)\to \mathcal{H}(\Omega)$ into the Hausdorff space of non-empty compact subsets of $\Omega$. A drawback of the first approach is that the target space of the inverse depends on $f$ and, more crucially, that $\bar f$ need not be geometrically quasiconformal unless $f$ is a BLD-map. The second approach, on the other hand, does not see the multiplicities in the preimage, and $\mathcal{H}(\Omega)$ does not carry any natural differential forms, nor any bi-Lipschitz embedding in Euclidean space (cf. \cite[Theorem 2.1]{del11}).

Instead, we give a construction based on Almgren's space and local degree theory. Let $Q\ge 1$ be a natural number. The Almgren space $\mathcal A_Q(\mathbb{R}^n)$ of unordered $Q$-tuples in $\mathbb{R}^n$ was used by Almgren in his seminal work on regularity of minimal currents \cite{alm2000}, and was further explored by De Lellis et. al. \cite{del11}. It is a $nQ$-dimensional smooth orbifold, but not a manifold (unless $Q=1$), defined as
\begin{align*}
\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q,\quad d_\mathcal A(\bb x,\bb y)=\inf_{\sigma\in S_Q}\Big(\sum_j^d|x_j-y_{\sigma(j)}|^2\Big)^{1/2},
\end{align*}
where $S_Q$ is the permutation group on $Q$ elements acting naturally on $(\mathbb{R}^n)^Q$. (See Section \ref{sec:almgren-space} for more details.)

\bigskip\noindent The link between the multi-valued inverse and Almgren space is the following simple observation based on local degree theory. Given a proper branched cover $f:\Omega\to \mathbb{R}^n$ of degree $d>0$ from an open set $\Omega\subset \mathbb{R}^n$, the local index $\iota(f,x)$ is defined for all $x\in\Omega$ and satisfies $\displaystyle \sum_{x\in f^{-1}(y)}\iota(f,x)=d$, $y\in f(\Omega)$. Thus, $f^{-1} (y)$ can be regarded as an unordered $d$-tuple in $\Omega$ (counted with multiplicities) for each $y\in f(\Omega)$. The multi-valued inverse $\minv f$ of the proper branched cover $f:\Omega\to \mathbb{R}^n$ of finite degree $d>0$ is the map
\begin{align}\label{eq:minv}
\minv f:f(\Omega)\to \mathcal A_d(\mathbb{R}^n),\quad \minv f(y):= \sum_{x\in f^{-1}(y)}\iota(f,x)\bb x.
\end{align}
We refer the reader to Section \ref{sec:branched-covers} (see in particular Definition \ref{def:multi-valued-inv}) for the notation and further details used here.

\subsection{Statement of results}

While it is not a manifold, $\mathcal A_Q(\mathbb{R}^n)$ nevertheless carries a natural $n$-form given by the trace of the volume form of $\mathbb{R}^n$; that is, the $n$-form
\begin{align}\label{eq:natural-n-form}
\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n}):=\sum_j^d P_j^*(\operatorname{vol}_{\mathbb{R}^n})
\end{align}
on $(\mathbb{R}^n)^Q$ (where $P_j$ is the projection to the $j$th factor) is $S_Q$-invariant: $\sigma^*\omega=\omega$ for all $\sigma\in S_Q$. Using $S_Q$-invariance, we develop a pull-back theory of multi-valued maps (see Sections \ref{sec:diff-forms} and \ref{sec:pull-back}), which is of independent interest.

\begin{definition}\label{def:multi-valued-inv}
Let $f:X\to Y$ be a proper branched cover of finite degree $d>0$ between topological $n$-manifolds. The multi-valued inverse $\minv f:f(X)\to \mathcal A_d(X)$ of $f$ is defined by
\begin{align*}
\minv f(y)=\sum_{x\in f\inv(y)}\iota(f,x)\bb x
\end{align*}
\end{definition}

\subsection{Statement of results}

In particular, if $f:U\to \mathcal A_Q(\R^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \ref{thm:lip-pullback-flat-form}).

An element $\omega\in \Omega^k(\mathcal A_d(V))$ is a smooth map $\omega:V^d\to \bigwedge^k(\R^n)^d$ with
\begin{align}\label{eq:S_d-invariant}
\omega_x(v_1,\ldots,v_k)=\omega_{\sigma(x)}(\sigma v_1,\ldots,\sigma v_k),\quad x\in V^d,\ v_1,\ldots, v_k\in (\R^n)^d.
\end{align}
In particular, for each $\bb{x}\in \mathcal A_d(V)$, the value $\omega_{\bb{x}}:=\omega_{\sigma(x)}\circ \sigma\in \bigwedge^k(\R^n)^d$
is well-defined and independent of $\sigma\in S_d$. We define the comass $\|\omega\|:\mathcal A_d(V)\to \R$ of a $k$-form $\omega\in \Omega^k(\mathcal A_d(V))$ by
\begin{align*}
\|\omega\|_{\bb x}=\|\omega_{x}\|,\quad \bb x\in \mathcal A_d(V).
\end{align*}
This is well-defined by \eqref{eq:S_d-invariant}.

\begin{lemma}\label{lem:pullback-est}
Suppose $U\subset \R^m$ and $V\subset \R^n$ are open sets, $\omega$ is a smooth $k$-form on $\mathcal A_d(V)$ and $h:U\to \mathcal A_d(V)$ a.e. approximately differentiable. Then $h^*\omega$ is a measurable $k$-form on $U$ and
\begin{align*}
\|h^*\omega\|_x\le \|D_xh\|_x^k\|\omega_{h(x)}\|\quad\textrm{a.e. }x\in U.
\end{align*}
\end{lemma}
\begin{remark}
In particular $h^*\omega\in L^{p/k}_{loc}(U)$ if $h\in W^{1,p}_{loc}(U,\mathcal A_d(V))$.
\end{remark}
\begin{proof}
The approximate differentials $Dh_1,\ldots,Dh_d$ are measurable, cf. ***. For all $x\in U$ where $h$ is approximately differentiable, we have
\begin{align*}
\|h^*\omega\|_x&=\max\{\omega_{h(x)}(Dh(v_1),\ldots,Dh (v_d)):\ |v_1|,\ldots,|v_d|\le 1\}\\
&\le \|D_xh\|^k\|\omega_{h(x)}\|,
\end{align*}
completing the proof.
\end{proof}

\begin{lemma}\label{lem:pull-back-of-decomp}
Suppose $f_i:U\to \mathcal A_{d_i}(V)$ is Lipschitz ($i=0,1$), $d=d_0+d_1$, and $\omega$ is a $S_{d_0}\times S_{d_1}$-invariant form on $V^d$. Then the pull-back $\bb{f_0,f_1}^*\omega$ is a well-defined $k$-form on $U$. Moreover,
\begin{align*}
\bb{f_0,f_1}^*(\omega_0\otimes\omega_1)=f_0^*\omega_0\wedge f_1^*\omega_1
\end{align*}
whenever $\omega_i\in \Omega^{k_i}(\mathcal A_{d_i}(V))$ ($i=0,1$) and $k=k_0+k_1$.
\end{lemma}
\begin{proof}
Let $x\in U$ be such that $f_0,f_1$ are differentiable at $x$, and let $f_i^1(x),\ldots, f_i^{d_i}(x)$ and $D_xf_i^1,\ldots,D_xf_i^{d_i}$ be as in Theorem \ref{thm:multi-valued-rademacher} (i) and (ii). (In particular, note that $f=\bb{f_0,f_1}$ is differentiable at $x$ and $Df=\bb{D_xf_0^1,\ldots,D_xf_0^{d_0},D_xf_1^1,\ldots,D_xf_1^{d_1}}$.) Given $(\sigma_0,\sigma_1)\in S_{d_0}\times S_{d_1}$, denote
\[
D_xf^{(\sigma_0,\sigma_1)}=(D_xf_0^{\sigma_0\inv(1)},\ldots D_xf_0^{\sigma_0\inv(d_0)},D_xf_1^{\sigma_1\inv(1)},\ldots,D_xf_1^{\sigma_1\inv(d_1)})
\]
and similarly
\[
(f_0^{\sigma_0}(x),f_1^{\sigma_1}(x))=(f_0^{\sigma_0\inv(1)}(x),\ldots,f_0^{\sigma_0\inv(d_0)},f_1^{\sigma_1\inv(1)},\ldots,f_1^{\sigma_1\inv(d_1)}).
\]
We define
\begin{align*}
 (f^*\omega)_x(v_1,\ldots,v_k)=\omega_{(f_0^{\sigma_0}(x),f_1^{\sigma_1}(x))}(D_xf^{(\sigma_0,\sigma_1)}v_1,\ldots,D_xf^{(\sigma_0,\sigma_1)}v_k)
\end{align*}
for $v_1,\ldots,v_k\in T_xU=\R^m$. By the $S_{d_0}\times S_{d_1}$-invariance of $\omega$, this expression is independent of $(\sigma_0,\sigma_1)$ and gives a well-defined element in $\bigwedge^k\R^m$, which we denote by $(f^*\omega)_x=\omega_{(f_0(x),f_1(x))}\circ(D_xf_0,D_xf_1)$.

\begin{proposition}\label{prop:flat-form}
Let $d_0,d_1\ge 1$ be natural numbers, $d:=d_0+d_1$, and let $U\subset \R^m$ be an open set. If $f_i:U\to \mathcal A_{d_i}(\R^n)$ are Lipschitz functions such that $\ud(f_i^*\omega)=f_i^*(\ud\omega)$ weakly for all $l$-forms $\omega\in \Omega^l(\mathcal A_{d_i}(\R^n))$, $l\le k$ ($i=0,1$), then $\ud(\bb{f_0,f_1}^*\omega)=\bb{f_0,f_1}^*(\ud\omega)$ weakly for all $k$-forms $\omega\in \Omega^k(\mathcal A_d(\R^n))$.
\end{proposition}
\begin{proof}
Denote $f:=\bb{f_0,f_1}$, and let $\omega\in \Omega^k(\mathcal A_d(\R^n))$. Let
\[
(\omega_j)\subset \bigcup_{r+l=k}\Omega^l((\R^n)^{d_0})\otimes \Omega^r((\R^n)^{d_1})
\]
be a sequence such that $(\omega_j)_x\to \omega_x$ and $(\ud\omega_j)_x\to (\ud\omega)_x$ for all $x\in (\R^n)^d$ and $\sup_j\max\{\|\omega_j\|_\infty,\|\ud\omega_j\|_\infty\}<\infty$. Then
\begin{align*}
\tilde\omega_j&:=P_{S_{d_0}\times S_{d_1}}\omega_j\to P_{S_{d_0}\times S_{d_1}}\omega=\omega,\\
\ud\tilde\omega_j&=P_{S_{d_0}\times S_{d_1}}(\ud\omega_j)\to P_{S_{d_0}\times S_{d_1}}(\ud\omega)=\ud\omega
\end{align*}
and $\sup_j\max\{\|\tilde \omega_j\|_\infty,\|\ud\tilde \omega_j\|_\infty\}<\infty$, cf. Lemma \ref{lem:proj-onto-invariant-forms}. By Lemma \ref{lem:pull-back-of-decomp} each $\tilde\omega_j$ is a sum of tensor products $\omega_0\otimes\omega_1$ where $\omega_i$ is $S_{d_i}$-invariant and $f^*(\omega_0\otimes\omega_1)=(f_0^*\omega_0)\wedge(f^*_1\omega_1)$. Together with the assumption that $f_i^*(\ud\omega_i)=\ud(f^*_i\omega)$ weakly for $S_{d_i}$-invariant forms $\omega$, this yields
\begin{align*}
f^*(\ud(\omega_0\otimes\omega_1))&=f^*(\ud\omega_0\otimes \omega_1+(-1)^{\deg\omega_0}\omega_0\otimes\ud\omega_1)\\
&=f^*_0(\ud\omega_0)\wedge(f^*_1\omega_1)+(-1)^{\deg\omega_0}(f_0^*\omega_0)\wedge (f_1^*(\ud\omega_1))\\
&=\ud(f_0^*\omega_0)\wedge(f_1^*\omega_1)+(-1)^{\deg\omega_0}(f_0^*\omega_0)\wedge\ud(f_1^*\omega_1)\\
&=\ud(f_0^*\omega_0\wedge f_1^*\omega_1)=\ud f^*(\omega_0\otimes\omega_1)
\end{align*}
weakly. Consequently $f^*(\ud\tilde\omega_j)=\ud(f^*\tilde\omega_j)$ weakly. Now the pointwise convergence $f^*\tilde\omega_j\to f^*\omega$, $f^*(\ud\tilde\omega_j)\to f^*(\ud\omega)$ together with the dominated convergence theorem yield
\begin{align*}
\int_U \alpha\wedge f^*(\ud\omega)&=\lim_{j\to\infty}\int_U \alpha\wedge f^*(\ud\tilde\omega_j)=(-1)^{k+1}\lim_{j\to\infty}\int_U\ud\alpha\wedge f^*\tilde\omega_j\\
&=(-1)^{k+1}\int_U\ud\alpha\wedge f^*\omega
\end{align*}
for all $\alpha\in \Omega_c^{m-k-1}(U)$. This proves the claim.
\end{proof}

\begin{proof}
Denote $H(y):=(f_\ast(|Df|^{-2}))^{1/2}$ and observe that
\[
\lim_{h\to 0}\frac{d_\mathcal A(\minv f(y+hv),\minv f(y))}{|h|}=\md_y\minv f(v) \quad\textrm{ a.e. }y\in f(\Omega)
\]
for each $v\in \mathbb S^{n-1}$. By considering the family $\Gamma_v=\{\gamma\subset f(\Omega): \gamma'=v\}$, it follows from a Fubini type argument and Theorem \ref{thm:wug} that
\begin{align}\label{eq:metric-diff-est}
H(y)\le \md_y\minv f(v)\le (K_IK_O)^{1/n}H(y)\quad \textrm{ for all } v\in \mathbb S^{n-1}.
\end{align}
This implies \eqref{eq:jacob-vs-H}. The remaining claim follows from \eqref{eq:metric-diff-est}. Let $\Gamma$ be a path family in $f(\Omega)$ and let $\rho$ be admissible for $\minv f(\Gamma)$. If $\Gamma_0$ is the exceptional family in Theorem \ref{thm:wug} then $\Mod_v\Gamma_0=0$ and, for any $\gamma\in \Gamma\setminus \Gamma_1$ we have
\begin{align*}
1\le \int_{\minv f\circ\gamma}\rho\ud s&=\int_0^1\rho(\minv f(\gamma_t))|(\minv f\circ\gamma)_t'|\ud t\le (K_IK_O)^{1/n}\int_0^1\rho\circ \minv f(\gamma_t)H(\gamma_t)|\gamma_t'|\ud t\\
&=(K_IK_O)^{1/n}\int_\gamma(\rho\circ \minv f)\cdot H\ud s.
\end{align*}
Thus $(K_IK_O)^{1/n}(\rho\circ \minv f)H$ is admissible for $\Gamma\setminus \Gamma_0$ implying
\begin{align*}
\Mod_n\Gamma&=\Mod_n\Gamma\setminus\Gamma_0\le K_IK_O\int_{f(\Omega)}\rho^n\circ \minv f H^n\ud y\\
&\le K_IK_O\int_{f(\Omega)}\rho^n\circ \minv f\J \minv f\ud y=K_IK_O\int_{\Omega_f}\rho^n\ud\Ha^n.
\end{align*}
Taking infimum over $\rho$ yields $\Mod_n\Gamma\le K_IK_O\Mod_n\minv f(\Gamma)$ and completes the proof.
\end{proof}

In particular, if $f:U\to \mathcal A_Q(\mathbb{R}^n)$ is a multi-valued (locally) Lipschitz map, the pull-back $f^*\omega$ of any $S_Q$-invariant smooth $k$-form is a (locally)flat $k$-form on $U$ in the sense of Whitney (cf. Theorem \ref{thm:lip-pullback-flat-form}).

Applying the definition of pull-back in the QR context we can show that, if $f:\Omega\to \mathbb{R}^n$ is a quasiregular map, the multi-valued inverse is a quasiregular curve.

\begin{theorem}\label{thm:QR-curve}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then the multivalued inverse $\accentset{\leftarrow}{f}:f(\Omega)\to \mathcal A_d(\Omega)$ is an $\omega$-quasiregular curve, where $\omega=\operatorname{tr}(\operatorname{vol}_{\mathbb{R}^n})$ is given by \eqref{eq:natural-n-form}. More precisely,  $\minv f\in W_{loc}^{1,n}(f(\Omega),\mathcal A_d(\mathbb{R}^n))$, and
\[
\|\omega\|\circ \minv f\ |D\minv f|^n\le d^{n/2-1}K_I(f)\star \minv f^*\omega\quad \textrm{almost everywhere on }\Omega.
\]
\end{theorem}

Theorems \ref{thm:sob-pull-back} and \ref{thm:QR-curve} together with standard arguments found e.g. in \cite{pan-onn21} yield in particular the higher integrability of the generalized inverse of \cite[Chapter II.5]{ric93}.
\begin{corollary}\label{cor:QR-curve}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$. Then there exists $p>n$ such that the generalized inverse
\[
g:f(\Omega)\to \mathbb{R}^n,\quad g(y):=\sum_{x\in f^{-1}(y)}\iota(f,x)x
\]
belongs to $W^{1,p}_{loc}(f(\Omega),\mathbb{R}^n)$.
\end{corollary}

The multi-valued inverse can also be seen as a homeomorphism $\minv f:f(\Omega)\to \Omega_f$ onto its image $\Omega_f:=\minv f(f(\Omega))\subset \mathcal A_d(\mathbb{R}^n)$ (Lemma \ref{lem:homeo}). Thus, although $\mathcal A_d(\mathbb{R}^n)$ is not a manifold, $\Omega_f$ is a metric $n$-manifold when equipped with the metric from $\mathcal A_d(\mathbb{R}^n)$. In the following theorem we obtain that $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal.

\begin{theorem}\label{thm:geom-QC}
Let $f:\Omega\to \mathbb{R}^n$ be a proper quasiregular map of finite degree $d$, and equip $\Omega_f=\minv f(f(\Omega))$ with the metric from $\mathcal A_d(\mathbb{R}^n)$ and the Hausdorff $n$-measure on $\Omega_f$.
\begin{itemize}
    \item[(1)] $\Omega_f$ is $n$-rectifiable, upper Ahlfors $n$-regular and satisfies the infinitesimal $n$-Poincar\'e inequality;
    \item[(2)] $\minv f:f(\Omega)\to \Omega_f$ is geometrically quasiconformal. More precisely,
\begin{align*}
    \frac 1{K_IK_O}\Mod_n\Gamma\le \Mod_n\minv f(\Gamma)\le K_IK_O\Mod_n\Gamma
\end{align*}
fir any path family $\Gamma $ in $f(\Omega)$.
\end{itemize}
\end{theorem}

We refer to \eqref{eq:inf-n-PI} in Section \ref{sec:Omega_f-prop} for the definition of infinitesimal Poincar\'e inequality and further commentary. We mention here that the infinitesimal Poincar\'e inequality is not a quantitative condition.

\subsection{Main ideas and further discussion}
Theorem \ref{thm:sob-pull-back} follows by an approximation argument from it Lipschitz counterpart, Theorem \ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. We outline the main idea.

In order to apply induction in the proof of Theorem \ref{thm:lip-pullback-flat-form}, we will need to approximate a given multi-valued Lipschitz map by maps which are locally decomposable. A given map $f:U\to \mathcal A_d(\mathbb{R}^n)$ fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$. The restriction of $f$ to $F$ agrees with the map $d\llbracket b(f)\rrbracket$, where $b:\mathcal A_d(\mathbb{R}^n)\to \mathbb{R}^n$ is the barycenter map
\begin{align}\label{eq:barycenter}
b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d},
\end{align}
which is $\frac 1{\sqrt d}$-Lipschitz, see Lemma \ref{lem:barycenter} below. The equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies that the differentials of the two functions agree almost everywhere. However, since $F$ is closed and not open, we need an approximation argument for the induction step to work. Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$.

The main step in the proof of Theorem \ref{thm:QR-curve} are the Sobolev estimates, which we obtain in Theorem \ref{thm:wug}. Our argument here uses Vitali's covering theorem, which is valid in high generality, and thus Theorem \ref{thm:wug} probably holds e.g. in the setting of \cite{onn-raj09}.

The proof of Theorem \ref{thm:geom-QC} employs a similar induction argument as in Theorem \ref{thm:lip-pullback-flat-form} together with some metric quasiconformal theory. We remark here that a large part of metric quasiconformal theory is based on the Ahlfors regularity of the spaces under consideration. Indeed, in our setting this property would imply the Loewner property of the image set $\Omega_f:=\minv f(f(\Omega)$ by a (deep) result of Semmes \cite{sem96}, which in turn implies the equivalence of the various notions of quasiconformality in the metric setting \cite{hei98}. In this paper, we establish the upper Ahlfors $n$-regularity of $\Omega_f$ (Proposition \ref{prop:ball-meas-upper-bound}), and an infinitesimal Poincar\'e inequality. It would be interesting to analyze the geometry of $\Omega_f$ further, in particular whether $\Omega_f$ admits local bi-Lipschitz parametrizations. By Almgren's bi-Lipschitz embedding $\mathcal A_d(\mathbb{R}^n)\hookrightarrow \mathbb{R}^N$, $\Omega_f$ can be regarded as a subset of Euclidean space, and the existence of bi-Lipschitz parametrizations is linked with the existence of Cartan--Whitney presentations in a suitable Sobolev class  \cite[[Theorem 1.2]{hei11}.

Theorem~\ref{thm:sob-pull-back} "follows by an approximation argument from its Lipschitz counterpart, Theorem~\ref{thm:lip-pullback-flat-form}, whose proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme." The key issue for the induction is that one must "approximate a given multi-valued Lipschitz map by maps which are locally decomposable." A Lipschitz map $f:U\to \mathcal A_d(\mathbb{R}^n)$ "fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$." On $F$, one has that $f$ "agrees with the map $d\llbracket b(f)\rrbracket$, where $b$ is the barycenter map" $b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d}$, and the equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies "the differentials of the two functions agree almost everywhere." Since $F$ is "closed and not open," an approximation is needed: "Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$." This Lipschitz-level result is then used, via approximation, to obtain the Sobolev pull-back statement of Theorem~\ref{thm:sob-pull-back}.

To prove the main theorem, we proceed by an approximation argument from its Lipschitz counterpart. We first prove the following theorem.

\begin{theorem}\label{thm:lip-pullback-flat-form}
Suppose $U\subset\R^m$ is an open set, $f:U\to\mathcal A_d(\R^n)$ is Lipschitz and $\omega\in \Omega^k(\mathcal A_d(\R^n))$. Then $\ud(f^*\omega)=f^*(\ud\omega)$ weakly. In particular, $f^*\omega$ is a flat form on $U$ in the sense of Whitney.
\end{theorem}

Its proof uses induction, decomposition of multi-valued maps (cf. \cite[Proposition 1.6]{del11}), and a somewhat delicate approximation scheme. The key issue for the induction is that one must "approximate a given multi-valued Lipschitz map by maps which are locally decomposable." A Lipschitz map $f:U\to \mathcal A_d(\mathbb{R}^n)$ "fails to be locally decomposable on the set $F:=f^{-1}(\operatorname{diag}\mathcal A_d(\mathbb{R}^n))$." On $F$, one has that $f$ "agrees with the map $d\llbracket b(f)\rrbracket$, where $b$ is the barycenter map" $b(\llbracket x_1,\ldots,x_d\rrbracket)=\frac{x_1+\cdots+x_d}{d}$, and the equality $f|_F=d\llbracket b(f)\rrbracket|_F$ implies "the differentials of the two functions agree almost everywhere." Since $F$ is "closed and not open," an approximation is needed: "Our approximation scheme consists of a careful interpolation between $f$ and $d\llbracket b(f)\rrbracket$ in a neighbourhood of $F$." This Lipschitz-level result is then used, via approximation, in establishing the main theorem.

\label{thm:sob-pull-back}
Let $n,m,Q\ge 1$ be natural numbers, $U\subset \mathbb{R}^m$ open, and $f\in W_{loc}^{1,p}(f(\Omega),\mathcal A_Q(\mathbb{R}^n))$. If $\omega$ is a $S_Q$-invariant smooth $k$-form on $(\mathbb{R}^n)^Q$, and either $k+1\le p$ of $\mathrm {d} \omega=0$ and $k\le p$, then $\omega$ has a well-defined pull-back $f^*\omega \in L^{p/k}_{loc}(U,\bigwedge^k\mathbb{R}^m)$ and moreover
\begin{align*}
    \mathrm {d}(f^*\omega)=f^*(\mathrm {d}\omega)
\end{align*}
weakly.,

Question: Let $n,m,Q\ge 1$ be natural numbers, let $U\subset \mathbb{R}^m$ be open, and let $f\in W^{1,p}_{\mathrm{loc}}(U,\mathcal A_Q(\mathbb{R}^n))$, where $\mathcal A_Q(\mathbb{R}^n)=(\mathbb{R}^n)^Q/S_Q$ is the Almgren space of unordered $Q$-tuples and $S_Q$ acts on $(\mathbb{R}^n)^Q$ by permuting the $Q$ factors. Let $\omega$ be a smooth $k$-form on $(\mathbb{R}^n)^Q$ that is $S_Q$-invariant, meaning $\sigma^*\omega=\omega$ for every permutation $\sigma\in S_Q$. Assume either (i) $k+1\le p$, or (ii) $\mathrm d\omega=0$ and $k\le p$. Under these hypotheses, which conclusion about the pull-back of $\omega$ by $f$ is valid?

The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; moreover its weak exterior derivative satisfies
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$.

Choices:
B.
The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$ whenever $k\le p$; moreover one always has
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$, without any additional assumption such as $k+1\le p$ or $\mathrm d\omega=0$.
C. The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$ under the stated hypotheses.
D.
The pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; moreover, if $\mathrm d\omega=0$ and $k\le p$, then
\[
\mathrm d(f^*\omega)=0
\]
weakly on $U$, whereas the identity \(\mathrm d(f^*\omega)=f^*(\mathrm d\omega)\) is guaranteed only in the case $k+1\le p$.
E.
For every smooth $k$-form $\omega$ on $(\mathbb{R}^n)^Q$, the pull-back $f^*\omega$ is well defined and belongs to $L^{p/k}_{\mathrm{loc}}(U,\bigwedge^k\mathbb{R}^m)$; if in addition $\omega$ is $S_Q$-invariant, then one has
\[
\mathrm d(f^*\omega)=f^*(\mathrm d\omega)
\]
weakly on $U$ under the stated hypotheses.

integrability threshold k+1<=p versus merely k<=p for non-closed forms

differential identity conclusion dropped

closed-form case still yields full chain-rule identity, not just vanishing derivative

S_Q-invariance is required for the pull-back itself to be well defined on Almgren space

The stem does not explicitly state the conclusion. It gives hypotheses and asks for the valid consequence, so the correct answer is not leaked directly.

The item is quite close to a theorem-recall question: the correct option essentially states the theorem under the given hypotheses. However, it is not purely tautological because the distractors alter thresholds, invariance requirements, and the derivative conclusion.

Some reasoning is needed to distinguish the exact chain-rule conclusion from weaker or overgeneralized variants, especially around the conditions k+1<=p versus dω=0 and k<=p. Still, the question mainly tests precise recall of a theorem rather than substantial derivation.

The distractors are mathematically plausible and target common mistakes: overextending regularity assumptions, omitting the differential identity, confusing the closed-form case, and ignoring the need for S_Q-invariance for well-definedness on Almgren space.

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


\label{thm:trans}
If $x\in \mathcal L_\varepsilon$ for some $\varepsilon>0$, then $x^x$ is transcendental.

A real number $x$ is a \emph{Liouville number} if for every $N\in\mathbb{N}$ there exist integers $p,q$ with $q\ge2$ such that
\[
\bigl|x-\tfrac{p}{q}\bigr|<q^{-N}.
\]
We denote by $\mathcal{L}$ the set of all Liouville numbers. It is classical that $\mathcal{L}$ is dense in $\mathbb{R}$, has Lebesgue measure zero, and is a dense $G_\delta$ subset of $\mathbb{R}$. In particular, $\mathcal{L}$ is uncountable and topologically large while being arithmetically very thin.

In 1962, \cite{erdos} proved that every real number can be expressed both as a sum and, if nonzero, as a product of two Liouville numbers. This also follows from the fact that the set $\mathcal{L}$ of Liouville numbers is a dense $G_{\delta}$ subset of $\mathbb{R}$, although Erd\H{o}s gave an explicit elementary proof. Equivalently, for any nonzero real number $r$, there exist $\sigma,\tau\in\mathcal{L}$ such that $r=f(\sigma,\tau)$ with $f(x,y)=x+y$ or $f(x,y)=xy$.

Later, \cite{burger} extended this phenomenon to a broad family of functions $f(x,y)$. In particular, for $f(x,y)=y^x$, he showed that every positive real number $r\ne1$ can be written as $r=\sigma^{\tau}$ for suitable $\sigma,\tau\in\mathcal{L}$. We refer to~\cite{alniacik1995, Morris2024, Kumar2014} for further topological studies of $\mathcal{L}$ and to~\cite{LG2024} for analogous results concerning $U$-numbers. Consequently, there is no analogue of the Gelfond--Schneider theorem within the class of Liouville numbers. Recently, \cite{ChalebgwaMorris2025} proved that if $S$ is any set of real numbers of full Lebesgue measure, then every positive real number $r$ equals $x^y$, for $x,y\in S$. So, for example, this applies to the case that $S$ is the set of all normal numbers, or the set of real numbers of irrationality exponent 2, or the Mahler set of S-numbers.

Despite this abundance of representations, it remains unknown whether the self-power $x^x$ of a Liouville number $x$ must always be transcendental.

In 2023, \cite{MarquesOliveira2023} introduced a dense $G_\delta$ set (and so with cardinality $\mathfrak{c}$) whose elements have this property. Such numbers are defined as follows:

\begin{definition}
Let $\varepsilon>0$. A real number $x$ is called an \emph{$\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that
\[
\Bigl|x-\frac{p_k}{q_k}\Bigr| < q_k^{-(\log q_k)^{1+\varepsilon}}, \quad \forall k\ge1.
\]
We denote the set of such numbers by $\mathcal L_\varepsilon$.
\end{definition}

\begin{proposition}
Every $\varepsilon$-strong Liouville number is a Liouville number.
\end{proposition}

\begin{proof}
For any $N>0$, take $k$ large enough that $(\log q_k)^{1+\varepsilon} > N$ (e.g., $k>3N^{1/(1+\varepsilon)}+2$). Then
\[
\left|x - \frac{p_k}{q_k}\right| < q_k^{-(\log q_k)^{1+\varepsilon}} < q_k^{-N},
\]
so $x$ is Liouville.
\end{proof}

A real number $x$ is a \emph{Liouville number} if for every $N\in\mathbb{N}$ there exist integers $p,q$ with $q\ge2$ such that
\[
\bigl|x-\tfrac{p}{q}\bigr|<q^{-N}.
\]
We denote by $\mathcal{L}$ the set of all Liouville numbers. It is classical that $\mathcal{L}$ is dense in $\mathbb{R}$, has Lebesgue measure zero, and is a dense $G_\delta$ subset of $\mathbb{R}$. In particular, $\mathcal{L}$ is uncountable and topologically large while being arithmetically very thin.

In 1962, \cite{erdos} proved that every real number can be expressed both as a sum and, if nonzero, as a product of two Liouville numbers. This also follows from the fact that the set $\mathcal{L}$ of Liouville numbers is a dense $G_{\delta}$ subset of $\mathbb{R}$, although Erd\H{o}s gave an explicit elementary proof. Equivalently, for any nonzero real number $r$, there exist $\sigma,\tau\in\mathcal{L}$ such that $r=f(\sigma,\tau)$ with $f(x,y)=x+y$ or $f(x,y)=xy$.

Later, \cite{burger} extended this phenomenon to a broad family of functions $f(x,y)$. In particular, for $f(x,y)=y^x$, he showed that every positive real number $r\ne1$ can be written as $r=\sigma^{\tau}$ for suitable $\sigma,\tau\in\mathcal{L}$. We refer to~\cite{alniacik1995, Morris2024, Kumar2014} for further topological studies of $\mathcal{L}$ and to~\cite{LG2024} for analogous results concerning $U$-numbers. Consequently, there is no analogue of the Gelfond--Schneider theorem within the class of Liouville numbers. Recently, \cite{ChalebgwaMorris2025} proved that if $S$ is any set of real numbers of full Lebesgue measure, then every positive real number $r$ equals $x^y$, for $x,y\in S$. So, for example, this applies to the case that $S$ is the set of all normal numbers, or the set of real numbers of irrationality exponent 2, or the Mahler set of S-numbers.

\begin{definition}
Let $\varepsilon>0$. A real number $x$ is called an \emph{$\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that
\[
\Bigl|x-\frac{p_k}{q_k}\Bigr| < q_k^{-(\log q_k)^{1+\varepsilon}}, \quad \forall k\ge1.
\]
We denote the set of such numbers by $\mathcal L_\varepsilon$.
\end{definition}

\begin{proposition}
Every $\varepsilon$-strong Liouville number is a Liouville number.
\end{proposition}

\begin{proof}
For any $N>0$, take $k$ large enough that $(\log q_k)^{1+\varepsilon} > N$ (e.g., $k>3N^{1/(1+\varepsilon)}+2$). Then
\[
\left|x - \frac{p_k}{q_k}\right| < q_k^{-(\log q_k)^{1+\varepsilon}} < q_k^{-N},
\]
so $x$ is Liouville.
\end{proof}

\begin{definition}
Let $\varepsilon>0$. A real number $x$ is called an \emph{$\varepsilon$-strong Liouville number} if there exists an infinite sequence of rationals $p_k/q_k$ with $q_k>1$ such that
\[
\Bigl|x-\frac{p_k}{q_k}\Bigr| < q_k^{-(\log q_k)^{1+\varepsilon}}, \quad \forall k\ge1.
\]
We denote the set of such numbers by $\mathcal L_\varepsilon$.
\end{definition}

\begin{proof}
For any $N>0$, take $k$ large enough that $(\log q_k)^{1+\varepsilon} > N$ (e.g., $k>3N^{1/(1+\varepsilon)}+2$). Then
\[
\left|x - \frac{p_k}{q_k}\right| < q_k^{-(\log q_k)^{1+\varepsilon}} < q_k^{-N},
\]
so $x$ is Liouville.
\end{proof}

\begin{proof}
The proof relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}. We refer the reader to \cite[Proposition 3.2]{MarquesOliveira2023} for details.
\end{proof}

\begin{theorem}
Every spiffy constant belongs to $\mathcal L_\varepsilon$ for all $\varepsilon>0$.
\end{theorem}

\begin{proof}
Let $x=x_a$ be a spiffy constant and define the rational truncations
\begin{equation}
\label{eq:trunc}
r_m = \sum_{n=1}^m \frac{a_n}{3^{e_n}} = \frac{p_m}{q_m}, \qquad q_m = 3^{e_m}.
\end{equation}
Then
\[
|x - r_m| \le q_m^{-3^{e_m-1}}.
\]
Since $\log q_m = e_m \log 3$, we have $(\log q_m)^{1+\varepsilon} = o(3^{e_m-1})$, and therefore, for sufficiently large $m$,
\[
|x - r_m| < q_m^{-(\log q_m)^{1+\varepsilon}}.
\]
Hence, $x \in \mathcal{L}_\varepsilon$ for every $\varepsilon>0$.
\end{proof}

\begin{theorem}\label{thm:strong-Liouville-selfpowers-nonLiouville}
For every $\varepsilon>0$ there exist continuum many $\varepsilon$-strong
Liouville numbers $x$ such that $x^x$ is transcendental and not a Liouville
number.

For any $\xi\in C\setminus\mathcal{B}$, we have $|\xi^\xi-a/b|\geq b^{-\tau}$ for all but finitely many rationals $a/b$, so $\xi^\xi$ is not Liouville. Moreover, since every $\xi\in C$ is $\varepsilon$-strong Liouville, Theorem 1.2 implies $\xi^\xi$ is transcendental. Thus $C\setminus\mathcal{B}$ consists of continuum many $\varepsilon$-strong Liouville numbers $\xi$ for which $\xi^\xi$ is transcendental and not Liouville.
\end{proof}

Having settled that:
\begin{enumerate}
\item for $x$ an $\varepsilon$-strong Liouville number, $x^x$ is a transcendental number;
\item the set of $\varepsilon$-strong Liouville numbers is a dense $G_\delta$-subset of $\mathbb{R}$;
\item for cardinality $\mathfrak{c}$ members of $\mathcal{L}_\varepsilon$, $x^x$ are Liouville numbers, and an equal number are not Liouville numbers.
\end{enumerate}
 We ask: what can be said about the set all Liouville numbers $x$ such that $x^x$ is transcendental, or indeed a Liouville number? The theorems below shed some light on this question.

\begin{proof}
The proof relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}. We refer the reader to \cite[Proposition 3.2]{MarquesOliveira2023} for details.
\end{proof}

In this article, we introduce the set of tuned spiffy constants, a proper  subset of cardinality $\mathfrak{c}$ of $\varepsilon$-strong Liouville numbers, with the property that if $x$ belongs to this set, then $x^x$ is a Liouville number.

\begin{remark}
Throughout this work, we employ standard asymptotic notation. For functions $f$ and $g$, we write $f(x) = O(g(x))$ or equivalently $f(x) \ll g(x)$ if $|f(x)| \le C|g(x)|$ for an absolute constant $C>0$ and all sufficiently large $x$. We denote a negligible error by $f(x) = o(g(x))$, which means $f(x)/g(x) \to 0$ as $x \to \infty$, and $f(x) \asymp g(x)$ signifies that $f\ll g$ and $g\ll f$. Unless stated otherwise, all implied constants are absolute.
\end{remark}

The proof of Theorem~\ref{thm:trans} \emph{relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}}. (Details are deferred to \cite[Proposition 3.2]{MarquesOliveira2023}.)

In establishing the main theorem, the proof \emph{relies on a linear forms in logarithms estimate due to Baker combined with the defining property of \emph{$\varepsilon$-strong Liouville numbers}}. (Details are deferred to \cite[Proposition 3.2]{MarquesOliveira2023}.)

Question:
Let $x$ be a real number such that for some $varepsilon>0$ there exists an infinite sequence of rational numbers $p_k/q_k$ with $q_k>1$ satisfying
\[
\left|x-\frac{p_k}{q_k}\right|<q_k^{-(\log q_k)^{1+\varepsilon}}\qquad \text{for all }k\ge 1.
\]
Such a number is called an $\varepsilon$-strong Liouville number, and the set of all such numbers is denoted $\mathcal L_\varepsilon$. Under these assumptions, which statement about $x^x$ holds?

$x^x$ is transcendental.

Choices:
B. $x^x$ is a Liouville number.
C. $x^x$ is irrational.
D.
There exists a constant $\tau>0$, depending only on $\varepsilon$, such that for every $x\in\mathcal L_\varepsilon$ and all but finitely many rationals $a/b$ one has
\[
\left|x^x-\frac ab\right|\ge b^{-\tau}.
\]
E. If $x\in\mathcal L_\varepsilon$ for every $\varepsilon>0$, then $x^x$ is transcendental; but from membership in a single class $\mathcal L_\varepsilon$ for some fixed $\varepsilon>0$ one cannot conclude transcendence of $x^x$.

The stem states only the hypothesis defining an varepsilon-strong Liouville number and asks for a consequence about x^x; it does not explicitly reveal or strongly hint that transcendence is the intended conclusion.

The item is essentially a direct theorem-recall question: given the exact hypothesis, the correct choice states the theorem's conclusion almost verbatim.

There is some need to compare transcendence with nearby alternatives such as 'Liouville' or merely 'irrational,' but the problem mainly tests recognition of the known result rather than substantial reasoning or derivation.

The distractors are mathematically meaningful and reflect common confusions: strengthening transcendence to Liouville, weakening it to irrationality, introducing an unjustified uniform approximation bound, and altering the quantifier on varepsilon.

A solid recall-style MCQ with strong distractors and no answer leakage, but it is largely tautological and offers only moderate pressure for genuine mathematical reasoning.