The interplay between multiplicative structures (such as geometric progressions)
and digital structures (such as numeration systems with local constraints) is a
fertile source of problems at the interface of number theory, combinatorics,
and symbolic dynamics. A classical example is Erd\H{o}s' ternary conjecture
\cite{Erdos1979}, asserting that for all $n\ge 9$ the base-$3$ expansion of
$2^n$ contains the digit $2$.  Interpreting the middle-third Cantor set as the
set of reals whose ternary expansions avoid the digit $1$, this conjecture can
be viewed as a finiteness statement for the intersection of a geometric
progression with a Cantor-type digit-restricted set.

For standard integer bases $b\ge 2$, Cui, Ma, and Jiang \cite{Cui2022}
developed an algorithmic framework to study intersections of geometric
progressions with Cantor-like subsets of the integers defined by forbidden
blocks in base-$b$ expansions.  Their approach encodes the carry propagation of
multiplication into a finite combinatorial object, leading to effective
decidability results.

In the present paper we pursue an analogous viewpoint in a non-standard
numeration system, namely the \textit{Zeckendorf representation} associated with the
Fibonacci numbers.  Let $(F_n)_{n\ge 0}$ be defined by $F_0=0$, $F_1=1$,
$F_{n+2}=F_{n+1}+F_n$.  By Zeckendorf's theorem, every integer $N\ge 1$ admits a
unique decomposition
\[
   N=\sum_{i=2}^k \varepsilon_i F_i,
\]
with $\varepsilon_i\in\{0,1\}$, $\varepsilon_k=1$, and
$\varepsilon_i\varepsilon_{i+1}=0$ for $2\le i<k$.  We write
$Z(N)=\varepsilon_k\varepsilon_{k-1}\cdots \varepsilon_2$ for this Zeckendorf
word.  Symbolically, the set of Zeckendorf words is the \emph{Golden Mean shift},
i.e., the language over $\{0,1\}$ with the block $11$ forbidden (together with
the usual ``no leading zeros'' convention).

\smallskip

\noindent\emph{Remark on digit order.}
The representation $Z(N)=\varepsilon_k \varepsilon_{k-1}\cdots \varepsilon_2$ lists digits from the most significant to the least significant.
For local arithmetic (carry propagation) it is more natural to work with the \emph{reversed} order
\[
\operatorname{rev}(Z(N))=\varepsilon_2\varepsilon_3\cdots \varepsilon_k,
\]
i.e., reading from the least significant digit upward.
All subsequent notation will refer to this LSD-first stream.

\smallskip

Fix a finite family $\mathcal F$ of non-empty binary words, thought of as
additional forbidden patterns.  This defines a Zeckendorf-restricted subset
\[
   \mathcal K_{\mathcal F}
   =\{N\ge 1:\ \mathrm{rev}(Z(N))\ \text{contains no factor from }\mathcal F\},
\]
where $\mathrm{rev}(Z(N))$ lists the Zeckendorf digits from the least significant digit
(the coefficient of $F_2$) upward.  Given integers $u\ge 1$ and $q\ge 2$, we
seek to understand the set of exponents
\[
   S_u=\{n\ge 0:\ uq^n\in \mathcal K_{\mathcal F}\}.
\]
If $\mathcal F=\{11\}$, then $\mathcal K_{\mathcal F}$ coincides with the
full Zeckendorf language and $S_u=\N_{\ge 0}$, so the interesting case is when
$\mathcal F$ imposes genuinely new constraints.

A direct analysis of $S_u$ requires inspecting the entire Zeckendorf expansion
of $uq^n$, whose length grows linearly with $n$.  To obtain an effective result
capturing the local arithmetic, we introduce a window-restricted variant.  Let
$L=\max\{|f|:f\in\mathcal F\}$ and fix $M\ge L$. Since we will take prefixes of least-significant-digit streams, we adopt a padding convention that avoids artificially creating forbidden patterns when the true Zeckendorf expansion is shorter than the window.  Let \(\#\notin\{0,1\}\) be a neutral padding symbol and define the right\-infinite word
\[
\widetilde{Z}(N):=\operatorname{rev}(Z(N))\,\#^{\omega}\in\{0,1,\#\}^{\mathbb{N}}.
\]
The symbol \(\#\) never equals \(0\) or \(1\); thus a forbidden binary pattern cannot be created by involving a \(\#\).
Hence the prefix of length \(M\) of \(\widetilde{Z}(N)\) consists only of genuine Zeckendorf digits whenever \(N\) is large enough, and otherwise contains \(\#\)'s that do not affect the avoidance condition.

We say that a finite word $v\in\{0,1,\#\}^*$ \emph{avoids} $\mathcal F$ if no
$f\in\mathcal F$ occurs as a factor of $v$ (equivalently, occurrences cannot use
the symbol $\#$).  We then define
\[
   \mathcal K_{\mathcal F}^{(M)}
   =\{N\ge 1:\ \pref_M(\widetilde{Z}(N))\ \text{avoids }\mathcal F\},
\]
where for a word \(w = w_0 w_1 w_2 \dots \in \{0,1,\#\}^{\mathbb N}\) and an integer \(M \ge 1\), \(\pref_M(w)\) denotes its prefix of length \(M\), i.e., \(w_0 w_1 \dots w_{M-1}\).
Thus constraints are enforced only within the first \(M\) least significant Zeckendorf digits. Clearly $\mathcal K_{\mathcal F}\subseteq \mathcal
K_{\mathcal F}^{(M)}$ for every $M$, and with the above padding convention one
has
\[
   \mathcal K_{\mathcal F}=\bigcap_{M\ge L}\mathcal K_{\mathcal F}^{(M)}.
\]
Accordingly, we set
\[
   S_u^{(M)}=\{n\ge 0:\ uq^n\in \mathcal K_{\mathcal F}^{(M)}\}.
\]

Our main result shows that, for each fixed window length $M$, the set
$S_u^{(M)}$ is always ultimately periodic, and the period and preperiod can be
computed effectively.

The interplay between multiplicative structures (such as geometric progressions)
and digital structures (such as numeration systems with local constraints) is a
fertile source of problems at the interface of number theory, combinatorics,
and symbolic dynamics. A classical example is Erd\H{o}s' ternary conjecture
\cite{Erdos1979}, asserting that for all $n\ge 9$ the base-$3$ expansion of
$2^n$ contains the digit $2$. Interpreting the middle-third Cantor set as the
set of reals whose ternary expansions avoid the digit $1$, this conjecture can
be viewed as a finiteness statement for the intersection of a geometric
progression with a Cantor-type digit-restricted set.

In the present paper we pursue an analogous viewpoint in a non-standard
numeration system, namely the \textit{Zeckendorf representation} associated with the
Fibonacci numbers. Let $(F_n)_{n\ge 0}$ be defined by $F_0=0$, $F_1=1$,
$F_{n+2}=F_{n+1}+F_n$. By Zeckendorf's theorem, every integer $N\ge 1$ admits a
unique decomposition
\[
 N=\sum_{i=2}^k \varepsilon_i F_i,
\]
with $\varepsilon_i\in\{0,1\}$, $\varepsilon_k=1$, and
$\varepsilon_i\varepsilon_{i+1}=0$ for $2\le i<k$. We write
$Z(N)=\varepsilon_k\varepsilon_{k-1}\cdots \varepsilon_2$ for this Zeckendorf
word. Symbolically, the set of Zeckendorf words is the \emph{Golden Mean shift},
i.e., the language over $\{0,1\}$ with the block $11$ forbidden (together with
the usual ``no leading zeros'' convention).

Fix a finite family $\mathcal F$ of non-empty binary words, thought of as
additional forbidden patterns. This defines a Zeckendorf-restricted subset
\[
 \mathcal K_{\mathcal F}
 =\{N\ge 1:\ \mathrm{rev}(Z(N))\ \text{contains no factor from }\mathcal F\},
\]
where $\mathrm{rev}(Z(N))$ lists the Zeckendorf digits from the least significant digit
(the coefficient of $F_2$) upward. Given integers $u\ge 1$ and $q\ge 2$, we
seek to understand the set of exponents
\[
 S_u=\{n\ge 0:\ uq^n\in \mathcal K_{\mathcal F}\}.
\]
If $\mathcal F=\{11\}$, then $\mathcal K_{\mathcal F}$ coincides with the
full Zeckendorf language and $S_u=\N_{\ge 0}$, so the interesting case is when
$\mathcal F$ imposes genuinely new constraints.

A direct analysis of $S_u$ requires inspecting the entire Zeckendorf expansion
of $uq^n$, whose length grows linearly with $n$. To obtain an effective result
capturing the local arithmetic, we introduce a window-restricted variant. Let
$L=\max\{|f|:f\in\mathcal F\}$ and fix $M\ge L$. Since we will take prefixes of least-significant-digit streams, we adopt a padding convention that avoids artificially creating forbidden patterns when the true Zeckendorf expansion is shorter than the window. Let \(\#\notin\{0,1\}\) be a neutral padding symbol and define the right\-infinite word
\[
\widetilde{Z}(N):=\operatorname{rev}(Z(N))\,\#^{\omega}\in\{0,1,\#\}^{\mathbb{N}}.
\]
The symbol \(\#\) never equals \(0\) or \(1\); thus a forbidden binary pattern cannot be created by involving a \(\#\).
Hence the prefix of length \(M\) of \(\widetilde{Z}(N)\) consists only of genuine Zeckendorf digits whenever \(N\) is large enough, and otherwise contains \(\#\)'s that do not affect the avoidance condition.

We say that a finite word $v\in\{0,1,\#\}^*$ \emph{avoids} $\mathcal F$ if no
$f\in\mathcal F$ occurs as a factor of $v$ (equivalently, occurrences cannot use
the symbol $\#$). We then define
\[
 \mathcal K_{\mathcal F}^{(M)}
 =\{N\ge 1:\ \pref_M(\widetilde{Z}(N))\ \text{avoids }\mathcal F\},
\]
where for a word \(w = w_0 w_1 w_2 \dots \in \{0,1,\#\}^{\mathbb N}\) and an integer \(M \ge 1\), \(\pref_M(w)\) denotes its prefix of length \(M\), i.e., \(w_0 w_1 \dots w_{M-1}\).
Thus constraints are enforced only within the first \(M\) least significant Zeckendorf digits. Clearly $\mathcal K_{\mathcal F}\subseteq \mathcal
K_{\mathcal F}^{(M)}$ for every $M$, and with the above padding convention one
has
\[
 \mathcal K_{\mathcal F}=\bigcap_{M\ge L}\mathcal K_{\mathcal F}^{(M)}.
\]
Accordingly, we set
\[
 S_u^{(M)}=\{n\ge 0:\ uq^n\in \mathcal K_{\mathcal F}^{(M)}\}.
\]

Our main result shows that, for each fixed window length $M$, the set
$S_u^{(M)}$ is always ultimately periodic, and the period and preperiod can be
computed effectively.

Fix a finite family $\mathcal F$ of non-empty binary words, thought of as
additional forbidden patterns. This defines a Zeckendorf-restricted subset
\[
 \mathcal K_{\mathcal F}
 =\{N\ge 1:\ \rev(Z(N))\ \text{contains no factor from }\mathcal F\},
\]
where $\rev(Z(N))$ lists the Zeckendorf digits from the least significant digit
(the coefficient of $F_2$) upward. Given integers $u\ge 1$ and $q\ge 2$, we
seek to understand the set of exponents
\[
 S_u=\{n\ge 0:\ uq^n\in \mathcal K_{\mathcal F}\}.
\]
If $\mathcal F=\{11\}$, then $\mathcal K_{\mathcal F}$ coincides with the
full Zeckendorf language and $S_u=\N_{\ge 0}$, so the interesting case is when
$\mathcal F$ imposes genuinely new constraints.

A direct analysis of $S_u$ requires inspecting the entire Zeckendorf expansion
of $uq^n$, whose length grows linearly with $n$. To obtain an effective result
capturing the local arithmetic, we introduce a window-restricted variant. Let
$L=\max\{|f|:f\in\mathcal F\}$ and fix $M\ge L$. Since we will take prefixes of least-significant-digit streams, we adopt a padding convention that avoids artificially creating forbidden patterns when the true Zeckendorf expansion is shorter than the window. Let \(\#\notin\{0,1\}\) be a neutral padding symbol and define the right\-infinite word
\[
\widetilde{Z}(N):=\operatorname{rev}(Z(N))\,\#^{\omega}\in\{0,1,\#\}^{\mathbb{N}}.
\]
The symbol \(\#\) never equals \(0\) or \(1\); thus a forbidden binary pattern cannot be created by involving a \(\#\).
Hence the prefix of length \(M\) of \(\widetilde{Z}(N)\) consists only of genuine Zeckendorf digits whenever \(N\) is large enough, and otherwise contains \(\#\)'s that do not affect the avoidance condition.

Our main result shows that, for each fixed window length $M$, the set
$S_u^{(M)}$ is always ultimately periodic, and the period and preperiod can be
computed effectively.

The proof is entirely finite-state. The crucial input is the fact that
multiplication by a fixed integer $q$ in the Fibonacci numeration system is
realizable by a finite sequential transducer \cite{Frougny1992}. Reading digits
from the least significant side, this transducer induces, for each $M$, a
deterministic update rule on the set of length-$M$ windows
$\{0,1,\#\}^M$. After composing with the finite automaton that checks avoidance
of $\mathcal F$ inside the window, we obtain a deterministic dynamical system on
a finite state space. The orbit associated with the progression $(uq^n)_{n\ge
0}$ is therefore eventually periodic, which yields ultimate periodicity of
$S_u^{(M)}$ as an immediate corollary. The same finite-state construction gives
an explicit algorithm to compute the eventual period.

\textbf{Ultimate periodicity in finite-state dynamics.} A fundamental observation, used repeatedly in our proof, is that any deterministic process on a finite state space must eventually become periodic. More precisely, if \(X\) is a finite set and \(f: X \to X\) is a function, then for any starting point \(x_0 \in X\) the sequence \(x_{n+1} = f(x_n)\) is \emph{ultimately periodic}: there exist integers \(n_0 \ge 0\) and \(p \ge 1\) such that \(x_{n+p} = x_n\) for all \(n \ge n_0\). This elementary combinatorial fact drives the periodicity conclusion in Theorem~\ref{thm:main_window}.

\begin{lemma}\label{prop:frougny}
Let $q\ge 2$ be an integer. There exists a finite-state synchronous
(letter-to-letter) transducer $\mathcal T_q$ over the alphabet $\{0,1,\#\}$,
reading the input from the least significant digit upward, with the following
property: for every $N\ge 1$,
\[
 \mathcal T_q\bigl(\widetilde Z(N)\bigr)=\widetilde Z(qN),
 \qquad \widetilde Z(N):=\rev(Z(N))\,\#^{\omega}.
\]
Moreover, such a transducer can be effectively constructed from $q$.
\end{lemma}

Fix $u\ge 1$, $q\ge 2$, a finite family $\mathcal F$ of non-empty binary words,
and an integer $M\ge 1$. Set $x_n=uq^n$ for $n\ge 0$ and define
\[
 w_n:=\pref_M(\widetilde{Z}(x_n))\in\{0,1,\#\}^M.
\]
By Lemma~\ref{lem:locality}, we have the recursion
\[
 w_{n+1}=\Theta_{q,M}(w_n)\qquad(n\ge 0).
\]
Thus $(w_n)_{n\ge 0}$ is the forward orbit of $w_0$ under a deterministic map on
the finite set $\{0,1,\#\}^M$. Consequently, the sequence $(w_n)$ is ultimately
periodic: there exist $n_0\ge 0$ and $p\ge 1$ such that
\[
 w_{n+p}=w_n\qquad(n\ge n_0).
\]

\medskip
\noindent\emph{Effectiveness.}
Lemma~\ref{prop:frougny} provides an effective construction of the
transducer $\mathcal T_q$, hence of the induced map $\Theta_{q,M}$ in
Lemma~\ref{lem:locality}. Starting from $w_0=\pref_M(\widetilde{Z}(u))$, we
iterate $w_{n+1}=\Theta_{q,M}(w_n)$ and record the first repeated state. The
first repetition yields explicit $(n_0,p)$, and the same computation decides
whether $S_u^{(M)}$ is finite or infinite. In the finite case, the algorithm
outputs all elements of $S_u^{(M)}$ by checking $P(w_n)$ along the finite orbit
segment before the cycle.
\qed

\begin{theorem}\label{thm:main_window}
Let $u\ge 1$, $q\ge 2$ be integers, and let $\mathcal F$ be a finite family of
non-empty binary words.  Let $L=\max\{|f|:f\in\mathcal F\}$ and fix $M\ge L$.
Then the set
\[
   S_u^{(M)}=\{n\ge 0:\ uq^n\in \mathcal K_{\mathcal F}^{(M)}\}
\]
is either finite or ultimately periodic.  That is, there exist integers $n_0\ge 0$ and $p\ge 1$
such that
\[
   n\in S_u^{(M)}\ \Longleftrightarrow\ n+p\in S_u^{(M)}\qquad(n\ge n_0).
\]
Moreover, one can effectively compute such a pair $(n_0,p)$ and decide whether
$S_u^{(M)}$ is finite or infinite.  In the finite case, the same procedure
outputs all elements of $S_u^{(M)}$.
\end{theorem}

The proof is entirely finite-state.  The crucial input is the fact that
multiplication by a fixed integer $q$ in the Fibonacci numeration system is
realizable by a finite sequential transducer \cite{Frougny1992}.  Reading digits
from the least significant side, this transducer induces, for each $M$, a
deterministic update rule on the set of length-$M$ windows
$\{0,1,\#\}^M$.  After composing with the finite automaton that checks avoidance
of $\mathcal F$ inside the window, we obtain a deterministic dynamical system on
a finite state space.  The orbit associated with the progression $(uq^n)_{n\ge
0}$ is therefore eventually periodic, which yields ultimate periodicity of
$S_u^{(M)}$ as an immediate corollary.  The same finite-state construction gives
an explicit algorithm to compute the eventual period.

Theorem~\ref{thm:main_window} gives a rigorous and completely effective solution
to the window-restricted problem.  While passing from $S_u^{(M)}$ to the global
set $S_u$ entails additional difficulties (one must control where forbidden
patterns may first appear as $n$ varies), the window-restricted variant already
captures the local dynamics of Zeckendorf multiplication and provides a
practical method to detect all exponents for which the low-order Zeckendorf
digits of $uq^n$ satisfy prescribed constraints.

The proof of Theorem~\ref{thm:main_window} is \u201centirely finite-state\u201d. The key input is that \u201cmultiplication by a fixed integer $q$ in the Fibonacci numeration system is realizable by a finite sequential transducer\u201d, and when \u201creading digits from the least significant side\u201d this yields (for each $M$) \u201ca deterministic update rule on the set of length-$M$ windows $\{0,1,\#\}^M$.\u201d Composing this with \u201cthe finite automaton that checks avoidance of $\mathcal F$ inside the window\u201d produces \u201ca deterministic dynamical system on a finite state space.\u201d The orbit corresponding to $(uq^n)_{n\ge 0}$ is thus \u201ceventually periodic\u201d, and \u201cwhich yields ultimate periodicity of $S_u^{(M)}$ as an immediate corollary.\u201d Finally, \u201cthe same finite-state construction gives an explicit algorithm to compute the eventual period\u201d (hence to compute $(n_0,p)$ and decide finiteness, and list elements in the finite case).

To prove the main theorem, the argument is “entirely finite-state”. The key input is that “multiplication by a fixed integer $q$ in the Fibonacci numeration system is realizable by a finite sequential transducer”, and when “reading digits from the least significant side” this yields (for each $M$) “a deterministic update rule on the set of length-$M$ windows $\{0,1,\#\}^M$.” Composing this with “the finite automaton that checks avoidance of $\mathcal F$ inside the window” produces “a deterministic dynamical system on a finite state space.” The orbit corresponding to $(uq^n)_{n\ge 0}$ is thus “eventually periodic”, and “which yields ultimate periodicity of $S_u^{(M)}$ as an immediate corollary.” Finally, “the same finite-state construction gives an explicit algorithm to compute the eventual period” (hence to compute $(n_0,p)$ and decide finiteness, and list elements in the finite case).

Existential–Universal
Algorithmic or Constructive

Question:
Let \((F_n)_{n\ge 0}\) be the Fibonacci sequence defined by \(F_0=0\), \(F_1=1\), and \(F_{n+2}=F_{n+1}+F_n\). For each integer \(N\ge 1\), let \(Z(N)=\varepsilon_k\varepsilon_{k-1}\cdots \varepsilon_2\) be its Zeckendorf representation, meaning \(N=\sum_{i=2}^k \varepsilon_i F_i\) with \(\varepsilon_i\in\{0,1\}\), \(\varepsilon_k=1\), and \(\varepsilon_i\varepsilon_{i+1}=0\) for all \(2\le i<k\). Let \(\operatorname{rev}(Z(N))\) denote the Zeckendorf digits written from least significant to most significant, and define \(\widetilde Z(N)=\operatorname{rev}(Z(N))\,\#^{\omega}\), where \(\#\notin\{0,1\}\) is a padding symbol. For a word \(w\in\{0,1,\#\}^{\mathbb N}\), let \(\operatorname{pref}_M(w)\) be its prefix of length \(M\). If \(\mathcal F\) is a finite family of non-empty binary words, say that a word in \(\{0,1,\#\}^*\) avoids \(\mathcal F\) if none of the words in \(\mathcal F\) occurs as a factor. Define
\[
\mathcal K_{\mathcal F}^{(M)}=\{N\ge 1:\operatorname{pref}_M(\widetilde Z(N))\text{ avoids }\mathcal F\}
\]
and, for integers \(u\ge 1\), \(q\ge 2\), define
\[
S_u^{(M)}=\{n\ge 0: uq^n\in \mathcal K_{\mathcal F}^{(M)}\}.
\]
Let \(L=\max\{|f|:f\in\mathcal F\}\) and fix \(M\ge L\). Which statement holds for \(S_u^{(M)}\)?

The set \(S_u^{(M)}\) is either finite or ultimately periodic: there exist integers \(n_0\ge 0\) and \(p\ge 1\) such that
\[
 n\in S_u^{(M)}\iff n+p\in S_u^{(M)}\qquad (n\ge n_0).
\]
Moreover, one can effectively compute such a pair \((n_0,p)\) and decide whether \(S_u^{(M)}\) is finite or infinite; in the finite case, the same procedure outputs all elements of \(S_u^{(M)}\).

Choices:
B.
The set \(S_u^{(M)}\) is ultimately periodic: there exist integers \(n_0\ge 0\) and \(p\ge 1\) such that
\[
 n\in S_u^{(M)}\iff n+p\in S_u^{(M)}\qquad (n\ge n_0).
\]
Moreover, there is a procedure depending only on \(q,\mathcal F\), and \(M\) which computes such a pair \((n_0,p)\) simultaneously for every \(u\ge 1\).
C.
The set \(S_u^{(M)}\) is ultimately periodic: there exist integers \(n_0\ge 0\) and \(p\ge 1\) such that
\[
 n\in S_u^{(M)}\iff n+p\in S_u^{(M)}\qquad (n\ge n_0).
\]
D.
The set \(S_u^{(M)}\) is either finite or ultimately periodic provided \(M>L\): there exist integers \(n_0\ge 0\) and \(p\ge 1\) such that
\[
 n\in S_u^{(M)}\iff n+p\in S_u^{(M)}\qquad (n\ge n_0).
\]
Moreover, one can effectively compute such a pair \((n_0,p)\) and decide whether \(S_u^{(M)}\) is finite or infinite. In the finite case, the same procedure outputs all elements of \(S_u^{(M)}\).
E.
The set \(S_u^{(M)}\) is either finite or eventually a finite union of arithmetic progressions: there exist integers \(n_0\ge 0\), \(r\ge 1\), and pairs \((a_i,p_i)\) with \(p_i\ge 1\) for \(1\le i\le r\) such that
\[
 n\in S_u^{(M)}\iff n\in \bigcup_{i=1}^r \{a_i+k p_i:k\ge 0\}\qquad (n\ge n_0).
\]
Moreover, one can effectively compute such data and decide whether \(S_u^{(M)}\) is finite or infinite; in the finite case, the same procedure outputs all elements of \(S_u^{(M)}\).

dependence of computed period/preperiod on the initial state \(u\)

dropped the effectiveness and finite-case output conclusions

boundary condition on the window length, replacing \(M\ge L\) by \(M>L\)

replacing a single eventual period from one deterministic orbit by a general semilinear eventual description

The stem does not explicitly reveal the correct option, nor does it contain wording that uniquely points to choice A. It sets up definitions and hypotheses without directly stating the final theorem.

The item is largely a theorem-recall/restatement question: after giving the full setup, it asks which conclusion holds. The alternatives introduce nearby variants, so it is not completely tautological, but it still mostly tests recognition of the theorem statement.

There is some reasoning pressure in comparing subtle variants (finite-or-periodic vs. always periodic, effective vs. noneffective, dependence on u, restricted vs. unrestricted set). However, it mainly rewards recall/discrimination rather than substantial generative mathematical reasoning.

Most distractors are mathematically plausible and target real failure modes, especially B, D, and E. However, choice C is a weaker true statement implied by A, so as written it is not a clean distractor and creates ambiguity unless the prompt explicitly asks for the strongest valid statement.

Technically sophisticated item with mostly strong distractors, but it is weakened by theorem-restatement structure and, more importantly, ambiguity from including a weaker true option alongside the intended correct answer.


\begin{theorem}\label{thm:planar}
For every real number $q\notin\{0,1,2\}$,
the evaluation spectrum $|\mathsf{S}_n^{\pl}(q)|$ grows exponentially in $n$.

Moreover, we can show the following uniform lower bounds:
\[
|\mathsf{S}^{\pl}_n(q)| \;\ge\;
\begin{cases}
2^{(n-2)/2}, & q<0,\\[4pt]
2^{n/8 - o(n)}, & \hspace{-4mm} 0 < q < 2,\\[4pt]
\sqrt{F_{n-2}}= \Theta(\varphi^{(n-2)/2}), & q>2,\
\end{cases}
\]
where $(F_n)_n$ is the Fibonacci sequence and $\varphi = (1+\sqrt{5})/2$ is the golden ratio.
\end{theorem}

\begin{remark}
    \label{rmk:2-connec}
    Note that the operations defined in Definition \ref{def:ops-neg} preserve graph planarity and connectivity, and increase the number of vertices of the graph by one. In fact, if we apply a sequence of these operations starting with $(K_3, e)$ or $(K_4,e)$, $e$ being an arbitrary edge of the respective graph, then it is not hard to inductively show that the resulting graph is $2$--connected. Indeed, this follows from Whitney's theorem \cite{WhitneyEar}, since the graph will always have an ear decomposition: after applying $S$, replace $e$ by $a-w-b$ within the ears containing $e$; after applying $B$, we simply gain a new ear.

    In particular, replacing the starting graph in Lemma~\ref{lem:pingpong-neg} by $(K_3, e)$, Lemma~\ref{lem:pingpong-neg}, and the subsequent Lemma ~\ref{lem:ping_pong_02} and Lemma~\ref{lem:pingpong-pos}, all yield $2$--connected, planar witnesses, so after applying Lemma \ref{lem:sqrt-neg} and Lemma \ref{lem:sqrt-pos} respectively, we get the same bound asymptotically on the evaluation spectrum even if we require $G\in \mG_n^{\pl}$ to be connected.
\end{remark}

\label{sec:intro}

A \emph{proper $q$--coloring} of a finite undirected graph $G=(V,E)$ is an assignment
$\varphi:V\to[q]$ such that adjacent vertices receive different colors.
When $q\in\mathbb{Z}_{\ge 1}$, the number of proper $q$--colorings is denoted $P_G(q)$, and it is a classical theorem of Birkhoff and Whitney that there is a unique polynomial $P_G(x)\in\mathbb{Z}[x]$ which encodes the number of proper $q$-colorings of $G$ when evaluated at $x=q$. This is the so-called {\it{chromatic polynomial}} of graph $G$, a polynomial of degree $|V|$ and with leading coefficient equal to $1$. This object was first introduced by Birkhoff~\cite{Birkhoff1912} in 1912 as part of an algebraic approach for the Four Color Problem. Birkhoff’s original motivation was to study colorings of planar maps by encoding them in a polynomial and analyzing its behavior at small integer values, particularly at $x=4$. Although this approach did not yield a proof of the Four Color Theorem, it initiated a rich line of research and established the chromatic polynomial as a fundamental graph invariant. Whitney~\cite{Whitney1932} subsequently extended Birkhoff’s ideas to general graphs and introduced many of the structural properties of chromatic polynomials that are now standard. Among these is the deletion--contraction identity: for any edge $e$ of $G$,
\begin{equation}
    P_G(x)=P_{G-e}(x)-P_{G/e}(x), \label{eq:dc-intro}
\end{equation}
where $G-e$ denotes deletion and $G/e$ denotes contraction.
This identity places chromatic polynomials within the broader framework of the Tutte polynomial~\cite{Tutte1954}.

The chromatic polynomial encodes substantial information about $G$.
For instance, the chromatic number satisfies
\[
\chi(G)=\min\{q\in\mathbb{Z}_{\ge 1}: P_G(q)>0\}.
\]
Evaluations at negative integers also have a rich combinatorial meaning:
Stanley~\cite{Stanley1973} showed that
\[
|P_G(-1)|=\#\{\text{acyclic orientations of }G\},
\]
and more generally his chromatic reciprocity theorem interprets $(-1)^{|V|}P_G(-m)$, for $m\in\mathbb{Z}_{\ge 1}$,
as a weighted count of acyclic orientations together with order-preserving maps to $[m]$.

\medskip

\noindent {\bf{Chromatic polynomial spectrum.}} In \cite{AKTutte}, Agol and Krushkal proved that the number of chromatic polynomials of planar triangulations on $n$ vertices grows exponentially with $n$. In a few words, their argument works with duals of planar cubic graphs and considers the number of distinct values taken by the \emph{flow polynomial} at the special value
$Q=(3-\sqrt5)/2$ (for a planar graph $G$, its flow polynomial is precisely $P_{G^{*}}(x)$, where $G^{*}$ is the dual of $G$). At this particular parameter $Q$, the chromatic algebra (a certain algebra over $\mathbb{C}[Q]$ formed from linear combinations of isotopy classes of planar graphs in a rectangle with a fixed number of univalent points on top and bottom) satisfies an additional local relation (coming from Temperley--Lieb / Jones--Wenzl theory),
which collapses a natural boundary module from dimension three to dimension two.
Two explicit local operations on planar graphs $G$ then act on $P_{G^{*}}(Q)$ by $2\times2$ matrices whose squares generate a free semigroup,
yielding exponentially many distinct evaluations and hence exponentially many distinct chromatic polynomials.
See~\cite{AKFlow,AKTutte} for more details and further references.

In this paper we study the asymptotics of the \emph{evaluation spectra} of the chromatic polynomial. First, let $\mG_n$ be the set of all simple graphs on $n$ vertices. For each fixed real number $q$ and integer $n\ge 1$, we can now define
\[
\mathsf{S}_n(q)\;\coloneqq \;\{\,P_G(q)\,:\,G \in \mG_n\,\}.
\]
In words, $\mathsf{S}_n(q)$ represents the set of distinct values attained by $P_G(q)$ as $G$ ranges over all simple graphs on $n$ vertices. Furthermore, let
\begin{equation*}
    \mathsf{S}_n^{\pl}(q) \; \coloneqq \; \{\, P_G(q) \, : \, G \in \mG_n^{\pl} \, \}
\end{equation*}
where $\mG_n^{\pl}$ is the set of all $n$--vertex, simple, planar graphs. Trivially, $\abs{\mathsf{S}_n(q)} \geq \abs{\smash{\mathsf{S}_n^{\pl}}(q)}$.

\medskip
\noindent\textbf{Our results.}
We establish lower bounds for $\mathsf{S}_n(q)$ and $\mathsf{S}_n^{\pl}(q)$ for arbitrary fixed $q$. For integer $q\ge 3$ the value $P_G(q)$ counts proper $q$--colorings, and Agol conjectured that the spectrum $\mathsf{S}_n^{\pl}(3)$ should also grows exponentially over $\mG_n^{\pl}$; see~\cite{TreumannMO} for more context, in particular a motivation coming from the {\sf NP}-completeness of $3$--colorability of planar graphs.
Our main theorem confirms this conjecture and, more generally, shows that the evaluation spectrum is exponential in $n$ for every fixed
$q\notin\{0,1,2\}$.

\label{sec:intro}

A \emph{proper $q$--coloring} of a finite undirected graph $G=(V,E)$ is an assignment
$\varphi:V\to[q]$ such that adjacent vertices receive different colors.
When $q\in\mathbb{Z}_{\ge 1}$, the number of proper $q$--colorings is denoted $P_G(q)$, and it is a classical theorem of Birkhoff and Whitney that there is a unique polynomial $P_G(x)\in\mathbb{Z}[x]$ which encodes the number of proper $q$-colorings of $G$ when evaluated at $x=q$. This is the so-called {\it{chromatic polynomial}} of graph $G$, a polynomial of degree $|V|$ and with leading coefficient equal to $1$. This object was first introduced by Birkhoff~\cite{Birkhoff1912} in 1912 as part of an algebraic approach for the Four Color Problem. Birkhoff’s original motivation was to study colorings of planar maps by encoding them in a polynomial and analyzing its behavior at small integer values, particularly at $x=4$. Although this approach did not yield a proof of the Four Color Theorem, it initiated a rich line of research and established the chromatic polynomial as a fundamental graph invariant. Whitney~\cite{Whitney1932} subsequently extended Birkhoff’s ideas to general graphs and introduced many of the structural properties of chromatic polynomials that are now standard. Among these is the deletion--contraction identity: for any edge $e$ of $G$,
\begin{equation}
 P_G(x)=P_{G-e}(x)-P_{G/e}(x), \label{eq:dc-intro}
\end{equation}
where $G-e$ denotes deletion and $G/e$ denotes contraction.
This identity places chromatic polynomials within the broader framework of the Tutte polynomial~\cite{Tutte1954}.

The chromatic polynomial encodes substantial information about $G$.
For instance, the chromatic number satisfies
\[
\chi(G)=\min\{q\in\mathbb{Z}_{\ge 1}: P_G(q)>0\}.
\]
Evaluations at negative integers also have a rich combinatorial meaning:
Stanley~\cite{Stanley1973} showed that
\[
|P_G(-1)|=\#\{\text{acyclic orientations of }G\},
\]
and more generally his chromatic reciprocity theorem interprets $(-1)^{|V|}P_G(-m)$, for $m\in\mathbb{Z}_{\ge 1}$,
as a weighted count of acyclic orientations together with order-preserving maps to $[m]$.

\noindent {\bf{Chromatic polynomial spectrum.}} In \cite{AKTutte}, Agol and Krushkal proved that the number of chromatic polynomials of planar triangulations on $n$ vertices grows exponentially with $n$. In a few words, their argument works with duals of planar cubic graphs and considers the number of distinct values taken by the \emph{flow polynomial} at the special value
$Q=(3-\sqrt5)/2$ (for a planar graph $G$, its flow polynomial is precisely $P_{G^{*}}(x)$, where $G^{*}$ is the dual of $G$). At this particular parameter $Q$, the chromatic algebra (a certain algebra over $\mathbb{C}[Q]$ formed from linear combinations of isotopy classes of planar graphs in a rectangle with a fixed number of univalent points on top and bottom) satisfies an additional local relation (coming from Temperley--Lieb / Jones--Wenzl theory),
which collapses a natural boundary module from dimension three to dimension two.
Two explicit local operations on planar graphs $G$ then act on $P_{G^{*}}(Q)$ by $2\times2$ matrices whose squares generate a free semigroup,
yielding exponentially many distinct evaluations and hence exponentially many distinct chromatic polynomials.
See~\cite{AKFlow,AKTutte} for more details and further references.

In this paper we study the asymptotics of the \emph{evaluation spectra} of the chromatic polynomial. First, let $\mG_n$ be the set of all simple graphs on $n$ vertices. For each fixed real number $q$ and integer $n\ge 1$, we can now define
\[
\mathsf{S}_n(q)\;\coloneqq \;\{\,P_G(q)\,:\,G \in \mG_n\,\}.
\]
In words, $\mathsf{S}_n(q)$ represents the set of distinct values attained by $P_G(q)$ as $G$ ranges over all simple graphs on $n$ vertices. Furthermore, let
\begin{equation*}
 \mathsf{S}_n^{\pl}(q) \; \coloneqq \; \{\, P_G(q) \, : \, G \in \mG_n^{\pl} \, \}
\end{equation*}
where $\mG_n^{\pl}$ is the set of all $n$--vertex, simple, planar graphs. Trivially, $\abs{\mathsf{S}_n(q)} \geq \abs{\smash{\mathsf{S}_n^{\pl}}(q)}$.

\medskip
\noindent\textbf{Our results.}
We establish lower bounds for $\mathsf{S}_n(q)$ and $\mathsf{S}_n^{\pl}(q)$ for arbitrary fixed $q$. For integer $q\ge 3$ the value $P_G(q)$ counts proper $q$--colorings, and Agol conjectured that the spectrum $\mathsf{S}_n^{\pl}(3)$ should also grows exponentially over $\mG_n^{\pl}$; see~\cite{TreumannMO} for more context, in particular a motivation coming from the {\sf NP}-completeness of $3$--colorability of planar graphs.
Our main theorem confirms this conjecture and, more generally, shows that the evaluation spectrum is exponential in $n$ for every fixed
$q\notin\{0,1,2\}$.

In this paper we study the asymptotics of the \emph{evaluation spectra} of the chromatic polynomial. First, let $\mG_n$ be the set of all simple graphs on $n$ vertices. For each fixed real number $q$ and integer $n\ge 1$, we can now define
\[
\mathsf{S}_n(q)\;\coloneqq \;\{\,P_G(q)\,:\,G \in \mG_n\,\}.
\]
In words, $\mathsf{S}_n(q)$ represents the set of distinct values attained by $P_G(q)$ as $G$ ranges over all simple graphs on $n$ vertices. Furthermore, let
\begin{equation*}
 \mathsf{S}_n^{\pl}(q) \; \coloneqq \; \{\, P_G(q) \, : \, G \in \mG_n^{\pl} \, \}
\end{equation*}
where $\mG_n^{\pl}$ is the set of all $n$--vertex, simple, planar graphs. Trivially, $\abs{\mathsf{S}_n(q)} \geq \abs{\smash{\mathsf{S}_n^{\pl}}(q)}$.

\medskip
\noindent\textbf{Our results.}
We establish lower bounds for $\mathsf{S}_n(q)$ and $\mathsf{S}_n^{\pl}(q)$ for arbitrary fixed $q$. For integer $q\ge 3$ the value $P_G(q)$ counts proper $q$--colorings, and Agol conjectured that the spectrum $\mathsf{S}_n^{\pl}(3)$ should also grows exponentially over $\mG_n^{\pl}$; see~\cite{TreumannMO} for more context, in particular a motivation coming from the {\sf NP}-completeness of $3$--colorability of planar graphs.
Our main theorem confirms this conjecture and, more generally, shows that the evaluation spectrum is exponential in $n$ for every fixed
$q\notin\{0,1,2\}$.

In particular, Theorem \ref{thm:planar} and the trivial inequality $|\mathcal{P}_n| \geq \abs{\mathsf{S}_n(-1)} \geq \abs{\smash{\mathsf{S}_n^{\pl}}(-1)}$ together imply that $|\mathcal{P}_n| \geq 2^{(n-2)/2}$. This fact also improves quantitatively upon the lower bound from \cite{AKFlow}. Furthermore, the lower bounds from Theorem \ref{thm:planar} hold true even if we additionally require the family of (planar) graphs in $\mG_n^{\pl}$ to be connected. See Remark \ref{rmk:2-connec} for more details.

\subsection{Exponential growth for every non-integer \texorpdfstring{$q>0$}{q > 0}}
We can then prove the following lower bound for the chromatic polynomial evaluation spectra at non-integer positive numbers.
\begin{theorem}\label{thm:noninteger-positive}
Let $q>0$ be a non-integer real and set $m\coloneqq \lceil q\rceil$.
Then $q-m\in(-1,0)$ and $(q)_m\neq 0$, and for all $n\ge m+2$,
\[
|\mathsf{S}_n (q)| \ \ge\ 2^{(n-m-2)/2}.
\]

In this section, we prove the last bound of Theorem \ref{thm:main} for $\mathsf{S}_n(q)$ by showing the following
\begin{theorem}\label{thm:q_greater_than_2}
 Let $q > 2$ be a real number and $n \geq 3$, then
 \[\abs{\mathsf{S}^{\pl}_n(q)} \geq \sqrt{F_{n-2}} =\Theta(\varphi^{n/2}),\]
 where $(F_n)_n$ is the Fibonacci sequence and $\varphi = (1+\sqrt{5})/2$ is the golden ratio. In particular, we get the bound for integers $q \geq 3$.
\end{theorem}

\begin{lemma}[Square-root lemma]\label{lem:sqrt-pos}
If there are at least $N$ distinct $n$--attainable vectors, then
\[
|\mathsf{S}^{\pl}_n(q)| \;\ge\; \sqrt{N}.
\]
\end{lemma}
\begin{proof}
 Since $A_n$ is invertible, there are at least $N$ distinct $n$-feasible vectors. By Lemma~\ref{lem:sqrt-neg}, $|\mathsf{S}^{\pl}_n(q)| \ge \sqrt{N}.$
\end{proof}
The same basic operations $B$ and $S$ defined in Section~\ref{sec:negative} will be the building blocks of our construction of exponentially many $n$--attainable vectors.
\begin{lemma}[Matrix action]\label{lem:matrices-pos}
Let $e$ be an edge of a planar graph $G$, and write $w(G,e)=\binom{t}{u}$.
Then
\[
(w\circ S)(G,e)
=
\begin{pmatrix}
-1 & 1\\
0 & q-1
\end{pmatrix}
\binom{t}{u},
\qquad
(w\circ B)(G,e)
=
\begin{pmatrix}
q-1 & 0\\
1 & q-2
\end{pmatrix}
\binom{t}{u}.
\]
\end{lemma}
\begin{proof} Let $n = |V(G)|$ and observe that Lemma~\ref{lem:matrices-neg} and \eqref{eq:feasible-to-attainable} imply
 \[(w\circ S)(G,e)
=A_{n+1}\begin{pmatrix}
1 & 1\\
0 & -q+1
\end{pmatrix}A_n^{-1}
\binom{t}{u} = \begin{pmatrix}
-1 & 1\\
0 & q-1
\end{pmatrix}
\binom{t}{u}\]
and
\[(w\circ B)(G,e)
=A_{n+1}\begin{pmatrix}
-q+1 & 0\\
1 & -q+2
\end{pmatrix}A_n^{-1}
\binom{t}{u} = \begin{pmatrix}
q-1 & 0\\
1 & q-2
\end{pmatrix}
\binom{t}{u}. \]

\begin{proof}[Proof of Theorem~\ref{thm:planar_0_2}]
Let $(G_0,e_0,K,L)$ be given by Lemma~\ref{lem:ping_pong_02} and let $k=k(q)$ and $\ell=\ell(q)$ be the number of basic operations in blocks $K$ and $L$, respectively. Let $t = \smash{ \lfloor \frac{n-|V(G_0|)}{(k+\ell)}\rfloor}$ and notice that $n_0 = |V(G_0)|+t(k+\ell) \le n$. By Lemma~\ref{lem:ping_pong_02}, there are at least
\begin{equation}\label{eq:attainable_02}
\binom{2t}{t} \ge \frac{2^{2t}}{2t+1} = 2^{\Omega_{q}(n)}
\end{equation}
distinct $n_0$--attainable vectors, since words in $\{K,L\}^\ast$ consisting of $t$ $K$'s and $t$ $L$'s produce distinct $n_0$--attainable vectors. Thus,
\[
|\mathsf{S}^{\mathrm{pl}}_n(q)| \;\ge\; |\mathsf{S}^{\mathrm{pl}}_{n_0}(q)| \;=\;2^{\Omega_q(n)},
\]
where the left inequality comes from the fact that adding a leaf to a connected planar graph preserves planarity and alters the chromatic polynomial by a factor of $x-1$, and the right identity is given by~\eqref{eq:attainable_02} and Lemma~\ref{lem:sqrt-pos}.
\end{proof}
\begin{remark}[The dependence on $q$ or lack thereof]
 \label{rem:q_no_dependence}
 Theorem~\ref{thm:planar_0_2} as proven only gives that for every $q \in (0, 2) \setminus \set{1}$ the existence of $c(q) > 0$ and a positive integer $n_0(q)$ such that $\abs{\mathsf{S}_n^{\pl} (q)} \geq \exp\of{(c(q) +o(1)) n}$. However, a more detailed analysis easily can show that we may choose $c(q)$ to be an absolute constant, namely $1/8$.
 \begin{itemize}
 \item For $q \in (0,1.458)$, the construction in Case~1 of Lemma~\ref{lem:ping_pong_02} works with $m = 1$. This construction provides the bound $\abs{\mathsf{S}_n^{\pl} (q)} \geq 2^{n/4-o(n)}$;
 \item For $q \in (1,3/2]$, one can use the construction in Case~3 of Lemma~\ref{lem:ping_pong_02}. This construction provides the bound $\abs{\mathsf{S}_n^{\pl} (q) }\geq 2^{n / 6 - o(n)}$;

\noindent {\bf{Superexponential spectra.}} For every fixed real $q\notin\{0,1,2\}$ we proved that the evaluation spectrum
\[
\mathsf{S}_n(q)\;=\;\{P_G(q):|V(G)|=n\}
\]
grows exponentially in $n$, with explicit lower bounds of different qualities in various regimes. For positive integers $q \not\in \left\{0,1,2\right\}$, we also noted that exponential growth for the evaluation spectrum of $P_{G}(q)$ is best possible (up to the correct base of the exponent), due to the fact that $|\mathsf{S}_n(q)| \leq q^{n}+1$. Nevertheless, at other values of $q$, the size of $\mathsf{S}_n(q)$ could be significantly larger, in principle. For example, when $q=-1$, $|P_{G}(-1)|$ counts the number of acyclic orientations of $G$, so the natural upper bound for $|\mathsf{S}_n(-1)|$ is $|P_{K_n}(-1)|=n!$, rather than $q^n+1$. In this direction, we would like to make the appealing following conjecture for the spectrum of acyclic orientations.

In particular, Theorem \ref{thm:planar} and the trivial inequality $|\mathcal{P}_n| \geq \abs{\mathsf{S}_n(-1)} \geq \abs{\smash{\mathsf{S}_n^{\pl}}(-1)}$ together imply that $|\mathcal{P}_n| \geq 2^{(n-2)/2}$. This fact also improves quantitatively upon the lower bound from \cite{AKFlow}. Furthermore, the lower bounds from Theorem \ref{thm:planar} hold true even if we additionally require the family of (planar) graphs in $\mG_n^{\pl}$ to be connected. See Remark \ref{rmk:2-connec} for more details.

Before we say more about the proof of Theorem \ref{thm:planar}, let us first remark that for $q\in\{0,1,2\}$ the spectrum is not exponential even if we allow $G$ to vary over {\it{all}} graphs on $n$ vertices.
Indeed,
\begin{itemize}
\item $P_G(0)=0$ for every graph with at least one vertex, so $|\mathsf{S}^{\pl}_n(0)|=|\mathsf{S}_n(0)|=1$;
\item $P_G(1)=1$ if $G$ is edgeless and $P_G(1)=0$ otherwise, so $|\mathsf{S}^{\pl}_n(1)|=1$ and $|\mathsf{S}_n(1)|= 2$;
\item $P_G(2)$ counts proper $2$--colorings, so it is $0$ for non-bipartite $G$, and equals $2^{c(G)}$ for bipartite $G$ with $c(G)$ connected components.
Hence $|\mathsf{S}^{\pl}_n(2)|\leq |\mathsf{S}_n(2)|= n+1$.
\end{itemize}

The proof of Theorem \ref{thm:planar} is more delicate when $q \in (0,2)$ (and precise quantitative bounds are also more difficult to extract), so, as a prequel to that side of the story, we will first use an elementary argument to show that $|\mathsf{S}_n(q)|$ is exponential in that range, by reducing the problem to the case when $q < 0$. We record this as a second theorem, in greater generality.

\begin{theorem}\label{thm:main}
For every non-integer real number $q > 0$, and $n\ge \lceil q\rceil+2$,
$$|\mathsf{S}_n(q)| \;\ge\; 2^{(n-\lceil q\rceil-2)/2}.$$
\end{theorem}

In particular, if $q \in (0,2)$, then Theorem \ref{thm:main} immediately implies that $|\mathsf{S}_n(q)| \ge 2^{(n-4)/2}$ for all $n\ge 4$.

Last but not least, let us also emphasize that for positive integers $q \not\in \left\{0,1,2\right\}$, exponential growth for $\abs{\mathsf{S}_n(q)}$ is best possible, up to the correct base of the exponent. Indeed, if $q\in\mathbb{Z}_{\ge 1}$ then every $n$--vertex graph has at most $q^n$ proper $q$--colorings, so $|\mathsf{S}_n(q)|\le q^n+1$.

When $q$ is \emph{not} a nonnegative integer, there is however no comparable a priori bound of the form $|P_G(q)|\le C^n$.
When $q=-1$, Stanley's theorem~\cite{Stanley1973} gives $|P_{K_n}(-1)|=n!$, and, more generally, it is not difficult to show that for any fixed $q\in\mathbb{R}\setminus\mathbb{Z}_{\ge 0}$ the complete graph satisfies
$P_{K_n}(q)=(q)_n=q(q-1)\cdots(q-n+1)$, whose magnitude grows on the order of $n!$.
These factorial-scale values suggest that, at least for some non-integer or negative points, the spectrum itself should be \emph{superexponential}. We will discuss this some more in Section \ref{sec:conclusion}.

\vspace{+3mm}
\noindent\textbf{Related work and proof ideas.}
A closely related spectrum problem was recently resolved for the spanning-tree enumerator.
Let $\tau(G)$ be the number of spanning trees of $G$, and set
\[
\mathcal{T}(n)\coloneqq \{\tau(G): G \in \mG_n \},
\]
as well as
\begin{equation*}
    \mathcal{T}^{\pl}(n) \coloneqq \{\, P_G(q) \, : \, G \in \mG_n^{\pl} \}.
\end{equation*}

In \cite{ChanKontorovichPak24trees}, Chan, Kontorovich and Pak proved that $|\mathcal{T}^{\pl}(n)|$ grows exponentially in $n$, via a connection to continued fractions, thin orbits, and a deep result of Bourgain-Kontorovich \cite{BourgainKontorovich14} on the so-called Zaremba conjecture \cite{Zaremba72}. See \cite{ChanKontorovichPak24trees} and the references therein for more information. Soon after, Alon--Buci\'c--Gishboliner~\cite{ABG} gave a much simpler alternative proof, which also came with improved quantitative bounds and results for other classes of graphs. Both arguments made crucial use of the deletion--contraction formula for the spanning-tree enumerator $\tau$ and a handful of basic operations that increase the number of vertices of the graph, and which allow one to track the evolution of the parameters $\tau(G-e)$ and $\tau(G/e)$ (for various edges $e$ of $G$), as the graph $G$ undergoes these changes. In \cite{ABG}, however, Alon, Buci\'c and Gishboliner noted that the parameters $\tau(G-e)$ and $\tau(G/e)$ of $G$ can sometimes instantly uniquely decode the entire process, an observation which drastically simplified the analysis.

For the chromatic polynomial spectrum, Agol and Krushkal \cite{AKTutte} also previously employed a similar strategy to show that $|\mathcal{P}_n|$ grows exponentially with $n$. An important difference in their argument is that the basic operations employed are less simple than the ones in ~\cite{ABG} or \cite{ChanKontorovichPak24trees}, and so one cannot readily decode the process if their operations are used freely to construct their $n$--vertex graphs. In order to circumvent this barrier, they consider only graphs obtained through a sequence of block operations, each of which combines some of their set of basic operations. To decode the sequence of block operations from the final parameters obtained, Agol and Krushkal then appeal to a simple application of the ping--pong lemma, a basic tool from geometric group theory. See for example \cite{Loh} and the references therein or \cite{BrennerCharnow78}.

Our argument, in some sense, unifies the two strategies from \cite{AKTutte} and \cite{ABG}. To prove Theorem \ref{thm:planar}, we make use of block operations that consist of sequences of two of the basic operations used in~\cite{ABG}, and then establish various ping--pong lemma type criteria, in order to decode.

\vspace{+2mm}

\noindent\textbf{Organization of the paper.} The proof of Theorem \ref{thm:planar} is most pleasant when $q<0$, so we set the stage by discussing this case first in Section \ref{sec:negative}.

After that, in Section \ref{sec:noninteger}, we take a first dip into the more problematic case when $q \in (0,2)$, which we will first address for the $|\mathsf{S}_n(q)|$. Using a different kind of combinatorial argument, we will derive a lower bound by reducing the problem to the case when $q \in (-1,0)$, for which the result from Section \ref{thm:planar} applies. This argument is the content of Theorem \ref{thm:main} and works in greater generality for every non-integer real number $q > 0$, and $n\ge \lceil q\rceil+2$. It does not apply directly to lower bound $|\mathsf{S}^{\pl}_n(q)|$ but it provides a better quantitative bound for $|\mathsf{S}_n(q)|$ than Theorem \ref{thm:planar} does in this range.

In Section \ref{sec:integers}, we continue the proof of Theorem \ref{thm:planar} with a more complicated ping--pong argument showing that $|\mathsf{S}^{\pl}_n(q)|$ is exponential in $n$ for each $q >2$.

In Section \ref{sec:planar_0_2}, we then complete the proof of Theorem \ref{thm:planar} by showing that $|\mathsf{S}^{\mathrm{pl}}_n(q)|$ is exponential in $n$ for every $q\in(0,2)\setminus\{1\}$ as well, by using a more intricate combination of block constructions.

In Section \ref{sec:conclusion}, we end with some concluding remarks and a couple of natural open problems.

The introduction indicates that the proof of Theorem~\ref{thm:planar} uses a “block operations + decoding” strategy: “To prove Theorem~\ref{thm:planar}, we make use of block operations that consist of sequences of two of the basic operations used in~\cite{ABG}, and then establish various ping--pong lemma type criteria, in order to decode.”

It also sketches how the argument is organized by ranges of $q$:
- “The proof of Theorem~\ref{thm:planar} is most pleasant when $q<0$,” treated first (Section~\ref{sec:negative}).
- For $q\in(0,2)$ the proof is “more delicate”; as a “prequel” they “first use an elementary argument to show that $|\mathsf{S}_n(q)|$ is exponential in that range, by reducing the problem to the case when $q<0$,” recorded as Theorem~\ref{thm:main}. Later they “complete the proof of Theorem~\ref{thm:planar}” for planar graphs in this range “by using a more intricate combination of block constructions” (Section~\ref{sec:planar_0_2}).
- For $q>2$, they “continue the proof of Theorem~\ref{thm:planar} with a more complicated ping--pong argument” to show exponential growth (Section~\ref{sec:integers}).

The introduction indicates that the proof of the main theorem uses a “block operations + decoding” strategy: “To prove the main theorem, we make use of block operations that consist of sequences of two of the basic operations used in~\cite{ABG}, and then establish various ping--pong lemma type criteria, in order to decode.”

It also sketches how the argument is organized by ranges of $q$:
- “The proof of the main theorem is most pleasant when $q<0$,” treated first (Section~\ref{sec:negative}).
- For $q\in(0,2)$ the proof is “more delicate”; as a “prequel” they “first use an elementary argument to show that $|\mathsf{S}_n(q)|$ is exponential in that range, by reducing the problem to the case when $q<0$,” recorded as the following theorem.

\begin{theorem}\label{thm:main}
For every non-integer real number $q > 0$, and $n\ge \lceil q\rceil+2$,
$$|\mathsf{S}_n(q)| \;\ge\; 2^{(n-\lceil q\rceil-2)/2}.$$
\end{theorem}

Later they “complete the proof of the main theorem” for planar graphs in this range “by using a more intricate combination of block constructions” (Section~\ref{sec:planar_0_2}).
- For $q>2$, they “continue the proof of the main theorem with a more complicated ping--pong argument” to show exponential growth (Section~\ref{sec:integers}).

\label{thm:planar}
For every real number $q\notin\{0,1,2\}$,
the evaluation spectrum $|\mathsf{S}_n^{\pl}(q)|$ grows exponentially in $n$.

Moreover, we can show the following uniform lower bounds:
\[
|\mathsf{S}^{\pl}_n(q)| \;\ge\;
\begin{cases}
2^{(n-2)/2}, & q<0,\\[4pt]
2^{n/8 - o(n)}, & \hspace{-4mm} 0 < q < 2,\\[4pt]
\sqrt{F_{n-2}}= \Theta(\varphi^{(n-2)/2}), & q>2,\
\end{cases}
\]
where $(F_n)_n$ is the Fibonacci sequence and $\varphi = (1+\sqrt{5})/2$ is the golden ratio.,

Question:
For a simple graph $G$, let $P_G(x)$ denote its chromatic polynomial. For each integer $n\ge 1$, let $\mathcal G_n^{\mathrm{pl}}$ be the set of all simple planar graphs on $n$ vertices, and for a real number $q$ define the planar evaluation spectrum
\[
\mathsf S_n^{\mathrm{pl}}(q)=\{P_G(q):G\in \mathcal G_n^{\mathrm{pl}}\}.
\]
Let $(F_n)_n$ be the Fibonacci sequence and let $\varphi=(1+\sqrt5)/2$. Which statement holds for every real number $q\notin\{0,1,2\}$?

For every fixed real $q\notin\{0,1,2\}$, the cardinality $|\mathsf S_n^{\mathrm{pl}}(q)|$ grows exponentially with $n$. More precisely,
\[
|\mathsf S_n^{\mathrm{pl}}(q)|\ge
\begin{cases}
2^{(n-2)/2}, & q<0,\\[4pt]
2^{n/8-o(n)}, & 0<q<2\text{ and }q\ne 1,\\[4pt]
\sqrt{F_{n-2}}=\Theta\!\big(\varphi^{(n-2)/2}\big), & q>2.
\end{cases}
\]

Choices:
B.
For every fixed real $q\notin\{0,1,2\}$, the cardinality $|\mathsf S_n^{\mathrm{pl}}(q)|$ grows exponentially with $n$. More precisely,
\[
|\mathsf S_n^{\mathrm{pl}}(q)|\ge
\begin{cases}
2^{(n-2)/2}, & q<0,\\[4pt]
2^{(n-2)/2}, & 0<q<2\text{ and }q\ne 1,\\[4pt]
\sqrt{F_{n-2}}=\Theta\!\big(\varphi^{(n-2)/2}\big), & q>2.
\end{cases}
\]
C. For every fixed real $q\notin\{0,1,2\}$, the cardinality $|\mathsf S_n^{\mathrm{pl}}(q)|$ grows exponentially with $n$.
D.
For every real $q\notin\{0,1,2\}$, there exists an absolute constant $c>0$ such that
\[
|\mathsf S_n^{\mathrm{pl}}(q)|\ge 2^{cn}
\]
for all sufficiently large $n$.
E.
For every fixed real $q\notin\{0,1,2\}$, the cardinality $|\mathsf S_n^{\mathrm{pl}}(q)|$ grows exponentially with $n$. More precisely,
\[
|\mathsf S_n^{\mathrm{pl}}(q)|\ge
\begin{cases}
2^{(n-2)/2}, & q\le 0,\\[4pt]
2^{n/8-o(n)}, & 0<q<2,\\[4pt]
\sqrt{F_{n-2}}=\Theta\!\big(\varphi^{(n-2)/2}\big), & q\ge 2.
\end{cases}
\]

The stem does not explicitly reveal the correct choice. Mentioning the Fibonacci sequence and  does hint that the q>2 regime involves a Fibonacci-type bound, but that feature appears in multiple options, so it does not uniquely leak the answer.

The item is largely asking the test-taker to recognize the exact theorem statement about the planar evaluation spectrum. Although the options introduce subtle variants, the question is still close to a direct restatement rather than an application or consequence-based problem.

Some reasoning is needed to compare the piecewise bounds and detect endpoint issues or overstrong claims, but the task is mainly theorem recognition. It does not strongly force generative mathematical reasoning beyond matching the correct quantitative statement.

Several distractors are plausible and mathematically aligned with common errors: including q=2, overstating the Fibonacci growth, or transplanting the negative-q bound into 0<q<2. However, choice C is a weaker statement that is also true, so the distractor set does not support a clean single-correct MCQ.

A mathematically sophisticated but only moderately well-designed MCQ: it avoids direct answer leakage and has plausible false variants, but it is too close to theorem recall and is fundamentally weakened by the presence of a weaker true alternative, making the single-answer format problematic.


\label{conj:PTB}
Let $\pi$ be a generic representation of $G$, and $\phi=\phi_\pi$.
\begin{itemize}
\item[(1)] If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\Sp_n(\mathbb{C})$.
\item[(2)] Suppose $\phi$ takes values in $\Sp_n(\mathbb{C})$ and we write it in the following form:
    \[
    \phi = \bigoplus_{i\in I_\phi} \phi_i \oplus \bigoplus_{j\in J_\phi}(\phi_j\oplus\phi_j^\vee),
    \]
where $I_\phi,  J_\phi$ are finite sets and $\phi_k\,\colon WD_F\to\GL_{n_k}(\mathbb{C})$ is a discrete $L$-parameter such that
\begin{itemize}
\item for each $i\in I_\phi$,  $n_i$ is even and $\phi_i$ takes values in $\Sp_{n_i}(\mathbb{C})$,
\item for any $i,  i'\in I_\phi$,  we have $\phi_i\not\simeq\phi_{i'}$ if $i\neq i'$.
\end{itemize}
Then $\pi$ is $E$-distinguished if and only if we have
    \[
    \e_E(\phi_i) = (-1)^{n_i/d},   \qquad  i\in I_\phi.
    \]
\end{itemize}

Let $F$ be a a finite extension of $\Q_p$, and $D$ be a central division $F$-algebra of dimension $d^2$. Let $m$ be a positive integer and set $G=G_m=\GL_m(D)$ with a positive integer $m$. Let $E$ be a quadratic extension of $F$, and let $\eta_{E/F}$ be the quadratic character of $F^\times$ with kernel $N_{E/F}(E^\times)$.
Assume that $n:=md$ is even so that $E$ embeds in $\CM_m(D)$, uniquely up to conjugation according to Skolem-Noether theorem.
Let $H$ denote the centralizer of $E^\times$ in $G$.
We say that a smooth representation $\pi$ of $G$ is \emph{$E$-distinguished} if $\Hom_H(\pi, \mathbb{C})\neq 0$. We also say that $\pi$ is \emph{$F\times F$-distinguished} if $m$ is even and $\Hom_{\GL_{m/2}(D)\times \GL_{m/2}(D)}(\pi, \mathbb{C})\neq 0$.

When $p>2$, Theorem \ref{conj:PTB} below, generalizing the Saito--Tunnell criterion of \cite{Sai} and \cite{Tu} for local toric periods on $\GL_2(F)$, follows from the works of Suzuki \cite{Sjnt}, Xue \cite{Xue21}, S\'echerre \cite{Sec24}, Suzuki and Xue \cite{SX24}, with an input from \cite{Ch}. In this paper we give a new proof of its cuspidal case, independent of the previous proofs, and which removes the restriction $p>2$ coming from type theory. The general version below then follows from the cuspidal case thanks to \cite{Sjnt}, \cite{SX24}, and \cite{Ch}: we observe that these papers only rely on the Mackey theory, intertwining periods, and globalization arguments, hence so does the proof presented in this paper.

We refer to \cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\mathrm{JL}(\pi)$, which is a generic representation of $\GL_n(F)$, and we define the L-parameter $\phi_\pi$ of $\pi$ by setting \[\phi_\pi:=\phi_{\mathrm{JL}(\pi)}:\WD_F\to \GL_n(\mathbb{C}).\] Let $\psi:F\to \mathbb{C}^\times$ be a non trivial character, and $\psi_E:=\psi\circ \tr_{E/F}$. If $\phi:\WD_F\to \Sp_n(\mathbb{C})$ is a symplectic L-parameter, we set \[\e_E(\phi):=\eta_{E/F}(-1)^{n/2}\epsilon(1/2,\phi_{|\WD_E},\psi_E)=
\eta_{E/F}(-1)^{n/2}\epsilon(1/2,\phi,\psi)\epsilon(1/2,\eta_{E/F}\otimes\phi,\psi),\] where the $\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \cite{Tate}. This quantity does not depend on the choice of the character $\psi$ since $\phi$ is symplectic.

\begin{theorem}\label{conj:PTB}
Let $\pi$ be a generic representation of $G$, and $\phi=\phi_\pi$.
\begin{itemize}
\item[(1)] If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\Sp_n(\C)$.
\item[(2)] Suppose $\phi$ takes values in $\Sp_n(\C)$ and we write it in the following form:
    \[
    \phi = \bigoplus_{i\in I_\phi} \phi_i \oplus \bigoplus_{j\in J_\phi}(\phi_j\oplus\phi_j^\vee),
    \]
where $I_\phi,  J_\phi$ are finite sets and $\phi_k\,\colon WD_F\to\GL_{n_k}(\C)$ is a discrete $L$-parameter such that
\begin{itemize}
\item for each $i\in I_\phi$,  $n_i$ is even and $\phi_i$ takes values in $\Sp_{n_i}(\C)$,
\item for any $i,  i'\in I_\phi$,  we have $\phi_i\not\simeq\phi_{i'}$ if $i\neq i'$.
\end{itemize}
Then $\pi$ is $E$-distinguished if and only if we have
    \[
    \e_E(\phi_i) = (-1)^{n_i/d},   \qquad  i\in I_\phi.
    \]
\end{itemize}
\end{theorem}

We refer to \cite[Definition 2.3]{Sign} for the definition of a generic representation of $G$. If $\pi$ is generic, then it has a well-defined Jacquet-Langlands transfer $\JL(\pi)$, which is a generic representation of $\GL_n(F)$, and we define the L-parameter $\phi_\pi$ of $\pi$ by setting \[\phi_\pi:=\phi_{\JL(\pi)}:\WD_F\to \GL_n(\C).\] Let $\psi:F\to \C^\times$ be a non trivial character, and $\psi_E:=\psi\circ \tr_{E/F}$. If $\phi:\WD_F\to \Sp_n(\C)$ is a symplectic L-parameter, we set \[\e_E(\phi):=\eta_{E/F}(-1)^{n/2}\epsilon(1/2,\phi_{|\WD_E},\psi_E)=
\eta_{E/F}(-1)^{n/2}\epsilon(1/2,\phi,\psi)\epsilon(1/2,\eta_{E/F}\otimes\phi,\psi),\] where the $\epsilon$ factor above is the usual Deligne-Langlands epsilon factor as defined in \cite{Tate}. This quantity does not depend on the choice of the character $\psi$ since $\phi$ is symplectic.

Let $F$ be a local field of characteristic zero, possibly Archimedean. We fix a nontrivial character $\psi:F\to \C^\times$. Let $A$ be an $F$-separable algebra of dimension two, i.e. either $A/F$ is a quadratic extension, of $A\simeq F\times F$. We set $\psi_A:=\psi\circ \tr_{A/F}$, and denote by $\eta_{A/F}$ the character of $F^\times$ with kernel the norms of $A^\times$, in particular $\eta_{A/F}$ has order two when $A/F$ is quadratic, whereas it is trivial when $A\simeq F\times F$. We note that $A$ can be written $F[u]$, with $\tr_{A/F}(u)=0$, i.e. $u^2\in F$. Such an element $u$ is unique up to scaling by an element in $F^\times$. Let $D$ be a central division algebra over $F$, such that $[D:F]=d^2$ with $d\geq 1$. We set $\CM:=\CM_m(D)$ for $m\geq 1$, and $n:=md$. We put $\psi_{\CM}:=\psi\circ \tr_{\CM/F}$. We assume that $A$ embeds in $\CM$ as an $F$-subalgebra, in such a way that $\tr_{\CM/F}(u)=0$, and we realize the embedding explicitly, as explained in \cite[Section 2.5]{MOY}. In particular $n$ is even, and if $A/F$ is quadratic, this parity condition is also sufficient for the assumption $\tr_{\CM/F}(u)=0$ to hold. If $A\simeq F\times F$, the assumption holds if and only if $m$ is even and $u$ is conjugate to $\lambda\diag(I_{m/2},-I_{m/2})$ with $\lambda\in F^\times$. We set $G_m=G=\CM^\times$, and $H_{A,m}=H_A=Z_{\CM}(A)^\times$. The group $H_A$ is the subgroup of $G$ fixed by the inner involution $\theta:=\ad(u)$ of $G$. We denote by $\nu:G\to \C^\times$ the restriction of the composition of the normalized absolute value $|\ |$ on $F$ with the reduced norm on $\CM$. We denote by $P_{m,m}$ the block upper triangular parabolic subgroup of $G_{2m}$ corresponding to the partitition $(m,m)$, and by $M_{m,m}$ its block diagonal Levi subgroup. We denote by $x_0$ the representative of the unique open double coset $Px_0H$ fixed in \cite[Section 4.1]{MOY}. In particular the involution $\theta_o(g):=x_o\theta(x_o^{-1}gx_o)x_o^{-1}$ of $G_{2m}$ is such that $\theta_o(P_{m,m})$ is opposite to $P_{m,m}$ with respect to $M_{m,m}$. We denote by $\FL(G)$ the objects of the category of smooth admissible representations of $G$ of finite length, where admissible means admissible (yes) but also Fr\'echet and of moderate growth, when $F$ is Archimedean (see \cite[Chapter 11]{RRG2}). For $\pi\in \FL(G)$, we write $\pi^\vee$ for its smooth contragredient, and we define $\ell\in \Hom(\pi\otimes \pi^\vee,\C)$ by
\[\ell(v\otimes v^\vee)=\langle v, v^\vee \rangle\] for $v \in \pi,\ v^\vee \in \pi^\vee$. We use the Bernstein-Zelevinsky product notation $\times$ for normalized parabolic induction.

An element $\pi\in \Irr(G)$ is called generic if it satisfies the conditions of \cite[Definition 2.3]{Sign}. We denote by $\gen(G)$ the set of isomorphism classes of generic representations of $G$. In particular, for $\pi\in \gen(G)$, its Jacquet-Langlands transfer $\JL(\pi)\in \gen(\GL_n(F))$ is well-defined. Let $\WD_F$ be the Weil-Deligne group of $F$. By definition, the L-parameter
\[\phi_\pi:\WD_F\to \GL_n(\C)\] of $\pi$ is that of its Jacquet-Langlands transfer $\JL(\pi)$ (\cite{HT},\cite{Hen},\cite{DKV}), and we say that $\pi$ is symplectic if $\phi_\pi(\WD_F)\subseteq \Sp_n(\C)$. We denote by $\symp(G)$, resp. $\sdg(G)$, the subset of $\gen(G)$, the elements of which are symplectic, resp. self-dual; $\pi\in \sdg(G)$ if $\pi^\vee= \pi$. In particular $\symp(G)\subseteq \sdg(G)$. When $F$ is $p$-adic and only in this case, we set $\sympc(G)$ to be the cuspidal part of $\symp(G)$. In particular in many statements below, if we mention cuspidal representations, it will be implicit that for this part of the statement, the field $F$ has to be $p$-adic.

\begin{lemma}\label{lm:globlz}
Let $\pi\in \sympc(G)$. There exist:
\begin{enumerate}
\item a number fields $\mathfrak{F}$ and a place $v_0$ such that $\mathfrak{F}_{v_0}\simeq F$,
\item an $\mathfrak{F}$-central division algebra $\mathfrak{D}$ such that:
\begin{enumerate}
\item $\mathfrak{D}_{v_0}/\mathfrak{F}_{v_0}\simeq D/F$,
\item $\mathfrak{D}_{w_0}$ is a division algebra of Hasse invariant opposite to that of $D$, for some finite place $w_0\neq v_0$ of $\mathfrak{F}$ lying over an odd prime number,
\item $\mathfrak{D}_v$ is a split matrix algebra for any place $v$ of $\mathfrak{F}$ different from $v_0$ and $w_0$,
\end{enumerate}
\item a cuspidal automorphic representation $\Pi$ of $\GL_m(\mathfrak{D}_{\A_\mathfrak{F}})$ such that:
\begin{enumerate}
\item $\JL(\Pi)$ is cuspidal and symplectic,
\item $\Pi_{v_0}\simeq \pi$,
\item $\Pi_{w_0}$ is cuspidal and $E'$-distinguished for some quadratic extension $E'/\mathfrak{F}_{w_0}$,
\end{enumerate}
\item a quadratic extension $\mathfrak{E}/\mathfrak{F}$ such that:
\begin{enumerate}
\item $\mathfrak{E}_{v_0}/\mathfrak{F}_{v_0}\simeq E/F$ and $\mathfrak{E}_{w_0}/\mathfrak{F}_{w_0} \simeq E'/\mathfrak{F}_{w_0}$,
\item $\mathfrak{E}_v/\mathfrak{F}_v\simeq \mathfrak{F}_v\times \mathfrak{F}_v$ splits at the finite number of places $v\notin \{v_0,w_0\}$ such that $\pi_v$ is not isomorphic to a principal series induced from a Borel subgroup of $\GL_m(\mathfrak{D}_v)\simeq \GL_n(\mathfrak{F}_v)$,
\item $\mathfrak{E}/\mathfrak{F}$ embeds into $\CM_m(\mathfrak{D})/\mathfrak{F}$.
\end{enumerate}
\end{enumerate}
\end{lemma}
\begin{proof}
(1) The existence of $\mathfrak{F}$ is well-known.

\begin{fact}\label{fct}
By construction, the cuspidal automorphic representation $\Pi$ in Lemma \ref{lm:globlz} is such that $\Pi_v$ is $\mathfrak{E}_v$-distinguished
for all $v\neq v_0$.
\end{fact}
\begin{proof}
It suffices to consider a place $v\neq v_0,w_0$. Note that $\mathfrak{D}_v$ is split by our choice of $\mathfrak{D}$, and that $\Pi_v=\JL(\Pi_v)=\JL(\Pi)_v$ is symplectic by the discussion at the beginning of Section \ref{sec:conv}. Hence if $\mathfrak{E}_v\simeq \mathfrak{F}_v\times \mathfrak{F}_v$, the result follows from the equality $\gadist(\GL_n(\mathfrak{F}_v))=\symp(\GL_n(\mathfrak{F}_v))$, with $A=\mathfrak{F}_v\times \mathfrak{F}_v$, recalled at the beginning of Section \ref{sec:not}. If not, i.e. if $\mathfrak{E}_v/\mathfrak{F}_v$ is quadratic, then by assumption $\Pi_v$ is a principal series induced from a Borel subgroup. The result then follows from \cite[Theorem 10.4]{MOY}.
\end{proof}

\begin{corollary}\label{cor:main}
Let $\pi\in\cusp(G)$ and let $\phi=\phi_\pi$ the corresponding $L$-parameter.
The following two conditions are equivalent.
\begin{itemize}
\item[(i)] $\pi$ is $H$-distinguished.
\item[(ii)] $\phi$ takes values in $\Sp_n(\C)$ and satisfies $\e_E(\phi)=(-1)^m$.
\end{itemize}
\end{corollary}
\begin{proof}
We only need to prove that (ii) implies (i). We assume that $\phi$ takes values in $\Sp_n(\C)$ and satisfies $\e_E(\phi)=(-1)^m$. Then Theorem \ref{thm:key} asserts that $\beta_{\pi,E,\psi}(0)=1$, further Theorem \ref{thm:key1} implies that $J_{\pi,E}(s)$ has a pole at $s=0$, and we conclude thanks to Proposition \ref{prop:key} that $\pi$ is $E$-distinguished.
\end{proof}

The above statement was predicted by Prasad and Takloo-Bighash in \cite[Conjecture 1]{PTB}, in a slightly more general form involving a character twist. The heart of the conjecture is its cuspidal case although the reduction from the discrete to the cuspidal case, done in \cite{SX24}, uses local intertwining periods in a non trivial way. As written above, the general case follows from the cuspidal case by \cite{Sjnt}, \cite{SX24}, and \cite{Ch}.

We now review the techniques involved in the previous proof of the conjecture in the cuspidal case. The proof of the direct implication is based on establishing special cases of the direct implication of the Guo--Jacquet conjecture, via the global trace formula approach proposed by Guo in \cite{Guo96}. Once this global conjecture established, the direct implication of the Prasad and Takloo-Bighash conjecture follows from an argument of Prasad in \cite{Pr07}, as we shall recall in Section \ref{sec:direct}. We observe in passing that the direct implication of the Guo-Jacquet conjecture is now a consequence of the recent \cite[Theorem 1.4]{MOY}, established by a totally new method, which can be thought of as a global analogue of the approach developed in this paper. In \cite{FMW}, Feigon, Martin and Whitehouse prove the direct implication of the split cuspidal case of Theorem \ref{conj:PTB}, by establishing a special case of the Guo-Jacquet conjecture. More on the Guo-Jacquet conjecture is further established by Xue in \cite{Xue21}, which provides two proofs of the direct implication in the cuspidal case. Xue also proves the converse implication in the cuspidal case, under the assumption $d\leq 2$, by the global trace formula approach as well. Then, by type theoretic methods, S\'echerre \cite{Sec24} provides a full proof of the converse implication, under the assumption that $p\neq 2$, which we remove here with our new proof. One common point of our proof of the converse implication and that of S\'echerre, probably the only one, is that they rely on the direct implication.

Theorem \ref{conj:PTB} should also follow from a local trace formula approach proving a special case of the conjecture in \cite{Wan}, as explained and started by Suzuki in \cite{Strace}. The advantage of this approach is that the dichotomy conjecture is put into a very general framework. On the other hand it seems to be expected that the remaining open problems there are technically highly challenging.

Our method here looks much simpler, and also very natural. However, it uses two local statements from \cite{MOY}, the proof of which is quite technical.

The statement of Theorem \ref{conj:PTB} also makes sense over Archimedean local fields, and its more general version involving the character twist was recently settled by Tamori and Suzuki \cite{ST23}.

As a next step, following a strategy explained to us by Miyu Suzuki, we intend to deduce the conjecture for non Archimedean local fields of odd positive characteristic, via the theory of close local fields.

Let's discuss and summarize the proof of the converse implication of Theorem \ref{conj:PTB} in the cuspidal case. It was observed in \cite{MatJFA} that the functional equation of local intertwining periods contains much information about distinction of quotients of induced representations, and this was used in \cite{SX24} to reduce the discrete case of Theorem \ref{conj:PTB} to its cuspidal case. Our main and new discovery, is that this functional equation actually also encodes most of the information about the distinction of cuspidal representations. Namely, let $\pi$ be a cuspidal representation of $G$. It is well-known to experts that $\pi$ is $E$-distinguished if and only if the open intertwining period \[J_{\pi,E}(s):\pi[s/2]\times \pi^\vee[-s/2]\to \mathbb{C}\]  has a pole at $ s=0$  (see Section \ref{sec:signFE} for this fact, and Section \ref{sec:OIP} for unexplained notations).
Now assume that $\pi$ has a symplectic Langlands parameter, so that in particular $\pi=\pi^\vee$. Then by multiplicity at most one, we have a functional equation
\[J_{\pi,E}(-s)\circ M_{\pi}(s)=\alpha_{\pi,E}(s) J_{\pi,E}(s),\] where
\[M_{\pi}(s):\pi[s/2]\times \pi[-s/2]\to \pi[-s/2]\times \pi[s/2]\] is the standard intertwining operator. One first main observation, proved in Theorem \ref{thm:key1}, is that if we normalize $\alpha_{\pi,E}(s)$ by the Shahidi exterior and symmetric square gamma factors as \[\beta_{\pi,E,\psi}(s)=\gamma(-s,\mathrm{JL}(\pi),\wedge^2,\psi)\gamma(s,\mathrm{JL}(\pi),\mathrm{sym}^2,\psi)\alpha_{\pi,E}(s),\] then $J_{\pi,E}(s)$ has a pole at $s=0$ if and only if $\beta_{\pi,E,\psi}(0)=1$, in other words:
\begin{equation}\label{eq:iff} \pi \mbox{ is distinguished} \iff \beta_{\pi,E,\psi}(0)=1 .\end{equation} It remains to compute $\beta_{\pi,E,\psi}(0)$, which we do by a local to global argument in Section \ref{sec:conv}. In order to achieve this, in Lemma \ref{lm:globlz}, we globalize $E/F$ as the completion at an inert place $v_0$ of a quadratic extension $\mathfrak{E}/\mathfrak{F}$ of number fields, and with the help of new results of Takanashi and Wakatsuki \cite{TW}, we globalize $\pi$ as the local component $\pi=\Pi_{v_0}$ of a cuspidal automorphic representation $\Pi$ with symplectic Jacquet-Langlands transfer, in such a way that $\Pi_v$ is moreover $\mathfrak{E}_v$-distinguished for all places $v\neq v_0$ of $\mathfrak{F}$. Then, in order to put the globalization argument to good use, we prove in Proposition \ref{prop:sign dist} that the direct implication in \eqref{eq:iff} holds as soon as $\pi$ is generic and $G$ is $F$-split, even if we  allow $E=F\times F$ (see Sections \ref{sec:OIP} and \ref{sec:signFE} for the precise meaning of this latter assertion). From this we deduce a formula for $\beta_{\Pi_v,\mathfrak{E}_v,\Psi_v}(0)$ for all places $v\neq v_0$ of the number field globalizing $F$ in Corollary \ref{cor:key}. The combination of Proposition \ref{prop:sign dist} and Corollary \ref{cor:key}, i.e. the formula for $\beta_{\Pi_v,E_v,\Psi_v}(0)$, is our second main observation, and we appeal to the main local result of \cite{MOY} in its proof, as well as to the direct implication of Theorem \ref{conj:PTB}. Finally, the product of all $\beta_{\Pi_v,E_v,\Psi_v}(0)$ including $\beta_{\pi,E,\psi}(0)$ being equal to one, as we prove in Lemma \ref{lm:product of c is one} thanks to another result of \cite{MOY}, we deduce in Theorem \ref{thm:key} the expected formula for $\beta_{\pi,E,\psi}(0)$. The final formula that we obtain is that if $\pi$ is cuspidal with symplectic L-parameter:
\[\beta_{\pi,E,\psi}(0)=(-1)^m\e_E(\phi_\pi).\] The converse implication of the cuspidal case of Theorem \ref{conj:PTB} follows; it is Corollary \ref{cor:main}.

\begin{Ac}
We thank Miyu Suzuki for sharing her insights and notes on the positive characteristic case, which served as an impetus for completing the proof in characteristic zero. The idea of the proof occurred to the author when he was visiting Huajie Li at the YMSC of Tsinghua university. We thank him and the YMSC for the hospitality.
\end{Ac}

For Theorem~\ref{conj:PTB}, the text reviews prior approaches and then sketches a new proof strategy for the *converse implication in the cuspidal case*.

- **Prior reductions / direct implication context.** The “heart of the conjecture is its cuspidal case”; the “reduction from the discrete to the cuspidal case” uses “local intertwining periods”, and “the general case follows from the cuspidal case” by \cite{Sjnt}, \cite{SX24}, \cite{Ch}. The *direct implication* was previously obtained by proving special cases of the Guo–Jacquet conjecture via “the global trace formula approach”.

- **Key characterization via poles of open intertwining periods.** For cuspidal $\pi$, it is “well-known” that $\pi$ is $E$-distinguished **iff** the open intertwining period
  \[J_{\pi,E}(s):\pi[s/2]\times \pi^\vee[-s/2]\to \mathbb{C}\]
  “has a pole at $s=0$.”

- **Functional equation and normalization.** Assuming $\pi$ has symplectic parameter (so “in particular $\pi=\pi^\vee$”), multiplicity-one gives a functional equation
  \[J_{\pi,E}(-s)\circ M_{\pi}(s)=\alpha_{\pi,E}(s)\,J_{\pi,E}(s),\]
  with $M_{\pi}(s)$ the standard intertwining operator. The first main observation (Theorem~\ref{thm:key1}) is that after normalizing
  \[\beta_{\pi,E,\psi}(s)=\gamma(-s,\mathrm{JL}(\pi),\wedge^2,\psi)\,\gamma(s,\mathrm{JL}(\pi),\mathrm{sym}^2,\psi)\,\alpha_{\pi,E}(s),\]
  one has
  \[\pi\ \text{distinguished}\iff \beta_{\pi,E,\psi}(0)=1\tag{\ref{eq:iff}}\]
  i.e. “$J_{\pi,E}(s)$ has a pole at $s=0$ if and only if $\beta_{\pi,E,\psi}(0)=1$.”

- **Compute $\beta_{\pi,E,\psi}(0)$ by local-to-global.** “It remains to compute $\beta_{\pi,E,\psi}(0)$,” done by a “local to global argument”: globalize $E/F$ to $\mathfrak{E}/\mathfrak{F}$ with $E/F$ appearing at an inert place $v_0$, and (using \cite{TW}) globalize $\pi$ as $\pi=\Pi_{v_0}$ for a cuspidal automorphic $\Pi$ with symplectic Jacquet–Langlands transfer, arranged so that “$\Pi_v$ is moreover $\mathfrak{E}_v$-distinguished for all places $v\neq v_0$.”

- **Input from a split generic direct implication and MOY.** To make the globalization effective, they prove (Proposition~\ref{prop:sign dist}) that the direct implication in \eqref{eq:iff} holds when “$\pi$ is generic and $G$ is $F$-split,” even allowing $E=F\times F$. From this they “deduce a formula for $\beta_{\Pi_v,\mathfrak{E}_v,\Psi_v}(0)$ for all places $v\neq v_0$” (Corollary~\ref{cor:key}). This combination is called the “second main observation” and it uses “the main local result of \cite{MOY} … as well as … the direct implication of Theorem~\ref{conj:PTB}.”

- **Global product formula forces the local value at $v_0$.** They show “the product of all $\beta_{\Pi_v,E_v,\Psi_v}(0)$ including $\beta_{\pi,E,\psi}(0)$” equals $1$ (Lemma~\ref{lm:product of c is one}, using another result of \cite{MOY}); hence one solves for $\beta_{\pi,E,\psi}(0)$ and obtains in Theorem~\ref{thm:key} the formula (for cuspidal $\pi$ with symplectic parameter)
  \[\beta_{\pi,E,\psi}(0)=(-1)^m\,\e_E(\phi_\pi).\]
  Combined with \eqref{eq:iff}, this yields the “converse implication of the cuspidal case of Theorem~\ref{conj:PTB}” (Corollary~\ref{cor:main}).

For Theorem~\ref{conj:PTB}, the text reviews prior approaches and then sketches a new proof strategy for the *converse implication in the cuspidal case*.

- **Prior reductions / direct implication context.** The “heart of the conjecture is its cuspidal case”; the “reduction from the discrete to the cuspidal case” uses “local intertwining periods”, and “the general case follows from the cuspidal case” by \cite{Sjnt}, \cite{SX24}, \cite{Ch}. The *direct implication* was previously obtained by proving special cases of the Guo–Jacquet conjecture via “the global trace formula approach”.

- **Key characterization via poles of open intertwining periods.** For cuspidal $\pi$, it is “well-known” that $\pi$ is $E$-distinguished **iff** the open intertwining period
  \[J_{\pi,E}(s):\pi[s/2]\times \pi^\vee[-s/2]\to \mathbb{C}\]
  “has a pole at $s=0$.”

- **Functional equation and normalization.** Assuming $\pi$ has symplectic parameter (so “in particular $\pi=\pi^\vee$”), multiplicity-one gives a functional equation
  \[J_{\pi,E}(-s)\circ M_{\pi}(s)=\alpha_{\pi,E}(s)\,J_{\pi,E}(s),\]
  with $M_{\pi}(s)$ the standard intertwining operator. We first use the following theorem.

  \begin{theorem}\label{thm:key1}
Let $\pi\in\cusp(G)$ be symplectic. Assume that $A/F=E/F$ is quadratic. Then:
\begin{enumerate}
    \item if $J_{\pi,E}(s)$ has a pole at $s=0$, then $\beta_{\pi,E,\psi}(0)=1$,
    \item if $J_{\pi,E}(s)$ is holomorphic at $s=0$, then $\beta_{\pi,E,\psi}(0)=-1$.
    \end{enumerate}
\end{theorem}

  More precisely, after normalizing
  \[\beta_{\pi,E,\psi}(s)=\gamma(-s,\mathrm{JL}(\pi),\wedge^2,\psi)\,\gamma(s,\mathrm{JL}(\pi),\mathrm{sym}^2,\psi)\,\alpha_{\pi,E}(s),\]
  one has the key equivalence
  \begin{equation}\label{eq:iff} \pi \mbox{ is distinguished} \iff \beta_{\pi,E,\psi}(0)=1 .\end{equation}
  i.e. “$J_{\pi,E}(s)$ has a pole at $s=0$ if and only if $\beta_{\pi,E,\psi}(0)=1$.”

- **Compute $\beta_{\pi,E,\psi}(0)$ by local-to-global.** “It remains to compute $\beta_{\pi,E,\psi}(0)$,” done by a “local to global argument”: globalize $E/F$ to $\mathfrak{E}/\mathfrak{F}$ with $E/F$ appearing at an inert place $v_0$, and (using \cite{TW}) globalize $\pi$ as $\pi=\Pi_{v_0}$ for a cuspidal automorphic $\Pi$ with symplectic Jacquet–Langlands transfer, arranged so that “$\Pi_v$ is moreover $\mathfrak{E}_v$-distinguished for all places $v\neq v_0$.”

- **Input from a split generic direct implication and MOY.** To make the globalization effective, they prove the following proposition.

  \begin{proposition}\label{prop:sign dist}
Assume that either $G=\GL_n(F)$ is split and $\pi\in \gadist(G)$, or that $\pi\in \cadist(G)$, then $\beta_{\pi,A,\psi}(0)=1$.
\end{proposition}

  From this they “deduce a formula for $\beta_{\Pi_v,\mathfrak{E}_v,\Psi_v}(0)$ for all places $v\neq v_0$” via the following corollary.

  \begin{corollary}\label{cor:key}
Let $\pi$ be as in Proposition \ref{prop:sign dist}, but assume moreover that it is unitary, and let $c(\pi,\psi)$ be as in Theorem \ref{thm:almost explicit constant}. Then $c(\pi,\psi)=(-1)^m\eta_{A/F}(-1)^{n/2}$.
\end{corollary}

  This combination is called the “second main observation” and it uses “the main local result of \cite{MOY} … as well as … the direct implication in establishing the main theorem.”

- **Global product formula forces the local value at $v_0$.** They show “the product of all $\beta_{\Pi_v,E_v,\Psi_v}(0)$ including $\beta_{\pi,E,\psi}(0)$” equals $1$ by the following lemma.

  \begin{lemma}\label{lm:product of c is one}
Let $\mathfrak{E}/\mathfrak{F}$ be a quadratic extension of number fields, and $\Psi:\mathfrak{F}\backslash \A_\mathfrak{F}\to \C^\times$ be a non trivial character. Let $\mathfrak{D}$ be an $\mathfrak{F}$-central division algebra. Assume moreover that
$\mathfrak{E}$ is contained inside $\CM_m(\mathfrak{D})$.
 If $\Pi$ is a self-dual cuspidal automorphic representation of $\GL_m(\mathfrak{D}_{\A_\mathfrak{F}})$ such that $\JL(\Pi)$ is cuspidal, then \[\prod_v c(\Pi_v,\Psi_v)=1,\] where the product is over all places of $F$. More specifically \[\prod_{v\in S} c(\Pi_v,\Psi_v)=1,\] where $S$ is the set of places such that of $\mathfrak{F}$ outside of which $\mathfrak{D}_v$ is split, and $\mathfrak{E}_v/\mathfrak{F}_v$, $\pi_v$ and $\Psi_v$ are all unramified.
\end{lemma}

  Hence one solves for $\beta_{\pi,E,\psi}(0)$ and obtains the value asserted in the following theorem:
  \begin{theorem}\label{thm:key}
Let $\pi\in \sympc(G)$. Then
\[c(\pi,\psi)=(-1)^{m}\eta_{E/F}(-1)^{n/2}.\]
\end{theorem}
  Together with the equation above \eqref{eq:iff}, this yields the “converse implication of the cuspidal case in establishing the main theorem,” formulated as:

  \begin{corollary}\label{cor:main}
Let $\pi\in\cusp(G)$ and let $\phi=\phi_\pi$ the corresponding $L$-parameter.
The following two conditions are equivalent.
\begin{itemize}
\item[(i)] $\pi$ is $H$-distinguished.
\item[(ii)] $\phi$ takes values in $\Sp_n(\C)$ and satisfies $\e_E(\phi)=(-1)^m$.
\end{itemize}
\end{corollary}

Biconditional or Equivalence
Implication

Question:
Let $F$ be a finite extension of $\mathbb{Q}_p$, let $D$ be a central division $F$-algebra of dimension $d^2$, let $m\ge 1$, set $G=\mathrm{GL}_m(D)$, and write $n=md$. Assume $n$ is even, so that a quadratic extension $E/F$ embeds in $\mathrm{M}_m(D)$. Let $H$ be the centralizer of $E^\times$ in $G$, and say that a smooth representation $\pi$ of $G$ is $E$-distinguished if $\operatorname{Hom}_H(\pi,\mathbb{C})\ne 0$. Let $\pi$ be a generic representation of $G$, and let $\phi=\phi_\pi:\mathrm{WD}_F\to \mathrm{GL}_n(\mathbb{C})$ be its $L$-parameter, defined via the Jacquet--Langlands transfer. For any symplectic $L$-parameter $\theta:\mathrm{WD}_F\to \mathrm{Sp}_r(\mathbb{C})$, define
\[
\varepsilon_E(\theta):=\eta_{E/F}(-1)^{r/2}\,\epsilon\!\left(\tfrac12,\theta_{|\mathrm{WD}_E},\psi_E\right)
=\eta_{E/F}(-1)^{r/2}\,\epsilon\!\left(\tfrac12,\theta,\psi\right)\epsilon\!\left(\tfrac12,\eta_{E/F}\otimes\theta,\psi\right),
\]
where $\eta_{E/F}$ is the quadratic character of $F^\times$ with kernel $N_{E/F}(E^\times)$, $\psi:F\to\mathbb{C}^\times$ is nontrivial, and $\psi_E=\psi\circ\operatorname{tr}_{E/F}$. Suppose further, when applicable, that $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and decomposes as
\[
\phi = \bigoplus_{i\in I_\phi} \phi_i \oplus \bigoplus_{j\in J_\phi}(\phi_j\oplus\phi_j^\vee),
\]
with $I_\phi,J_\phi$ finite, each $\phi_k:\mathrm{WD}_F\to\mathrm{GL}_{n_k}(\mathbb{C})$ a discrete $L$-parameter, each $n_i$ even and each $\phi_i$ taking values in $\mathrm{Sp}_{n_i}(\mathbb{C})$ for $i\in I_\phi$, and $\phi_i\not\simeq \phi_{i'}$ whenever $i\ne i'$. Which statement about $E$-distinction holds?

If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished if and only if
\[
\varepsilon_E(\phi_i)=(-1)^{n_i/d}\qquad\text{for every }i\in I_\phi.
\]

Choices:
B.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished if and only if
\[
\varepsilon_E(\phi_i)=(-1)^{n_i}\qquad\text{for every }i\in I_\phi.
\]
C.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished whenever
\[
\varepsilon_E(\phi_i)=(-1)^{n_i/d}\qquad\text{for every }i\in I_\phi.
\]
D.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished if and only if there exists a choice of nontrivial character $\psi:F\to\mathbb{C}^\times$ such that
\[
\varepsilon_E(\phi_i)=(-1)^{n_i/d}\qquad\text{for every }i\in I_\phi.
\]
E.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished if and only if
\[
\varepsilon_E(\phi_k)=(-1)^{n_k/d}\qquad\text{for every }k\in I_\phi\sqcup J_\phi.
\]

sign exponent depends on $n_i/d$, not $n_i$

dropped the converse direction of the criterion

$\varepsilon_E(\theta)$ is independent of $\psi$ for symplectic parameters; no existential choice of $\psi$ is needed

criterion only constrains the symplectic self-dual summands indexed by $I_\phi$, not the paired summands in $J_\phi$

The stem gives the setup and relevant notation, but it does not state or strongly hint at the specific criterion \(\varepsilon_E(\phi_i)=(-1)^{n_i/d}\). The correct answer is not leaked by the wording alone.

This is essentially a direct theorem-recall item: it asks for the statement equivalent to \(E\)-distinction and the correct option is the theorem's exact criterion. It does not substantially reframe the result into a new problem.

There is some pressure to distinguish between subtle variants involving exponents, quantifiers, and uniform sign twists, so careful reading is needed. However, the item primarily rewards memory of the exact theorem rather than independent mathematical generation or derivation.

The distractors are mathematically plausible and target realistic errors: confusing \(n_i\) with \(n_i/d\), weakening a universal condition to existence, inserting an unjustified global sign, or replacing the blockwise criterion with an ambient parity condition.

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


Question:
Let $F$ be a finite extension of $\mathbb{Q}_p$, let $D$ be a central division $F$-algebra of dimension $d^2$, let $m\ge 1$, set $G=\mathrm{GL}_m(D)$, and write $n=md$. Assume $n$ is even, so that a quadratic extension $E/F$ embeds in $\mathrm{M}_m(D)$. Let $H$ be the centralizer of $E^\times$ in $G$, and say that a smooth representation $\pi$ of $G$ is $E$-distinguished if $\operatorname{Hom}_H(\pi,\mathbb{C})\ne 0$. Let $\pi$ be a generic representation of $G$, and let $\phi=\phi_\pi:\mathrm{WD}_F\to \mathrm{GL}_n(\mathbb{C})$ be its $L$-parameter, defined via the Jacquet--Langlands transfer. For any symplectic $L$-parameter $\theta:\mathrm{WD}_F\to \mathrm{Sp}_r(\mathbb{C})$, define
\[
\varepsilon_E(\theta):=\eta_{E/F}(-1)^{r/2}\,\epsilon\!\left(\tfrac12,\theta_{|\mathrm{WD}_E},\psi_E\right)
=\eta_{E/F}(-1)^{r/2}\,\epsilon\!\left(\tfrac12,\theta,\psi\right)\epsilon\!\left(\tfrac12,\eta_{E/F}\otimes\theta,\psi\right),
\]
where $\eta_{E/F}$ is the quadratic character of $F^\times$ with kernel $N_{E/F}(E^\times)$, $\psi:F\to\mathbb{C}^\times$ is nontrivial, and $\psi_E=\psi\circ\operatorname{tr}_{E/F}$. Suppose further, when applicable, that $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and decomposes as
\[
\phi = \bigoplus_{i\in I_\phi} \phi_i \oplus \bigoplus_{j\in J_\phi}(\phi_j\oplus\phi_j^\vee),
\]
with $I_\phi,J_\phi$ finite, each $\phi_k:\mathrm{WD}_F\to\mathrm{GL}_{n_k}(\mathbb{C})$ a discrete $L$-parameter, each $n_i$ even and each $\phi_i$ taking values in $\mathrm{Sp}_{n_i}(\mathbb{C})$ for $i\in I_\phi$, and $\phi_i\not\simeq \phi_{i'}$ whenever $i\ne i'$. Which statement about $E$-distinction holds?

Choices:
B.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished if and only if
\[
\varepsilon_E(\phi_i)=(-1)^{n_i}\qquad\text{for every }i\in I_\phi.
\]
C.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished whenever
\[
\varepsilon_E(\phi_i)=(-1)^{n_i/d}\qquad\text{for every }i\in I_\phi.
\]
D.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished if and only if there exists a choice of nontrivial character $\psi:F\to\mathbb{C}^\times$ such that
\[
\varepsilon_E(\phi_i)=(-1)^{n_i/d}\qquad\text{for every }i\in I_\phi.
\]
E.
If $\pi$ is $E$-distinguished, then $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$. Moreover, if $\phi$ takes values in $\mathrm{Sp}_n(\mathbb{C})$ and has the displayed decomposition, then $\pi$ is $E$-distinguished if and only if
\[
\varepsilon_E(\phi_k)=(-1)^{n_k/d}\qquad\text{for every }k\in I_\phi\sqcup J_\phi.
\]

The stem does not explicitly reveal the correct option. It sets up notation and hypotheses, but the decisive criterion distinguishing the correct answer from the near-miss alternatives is not stated in the prompt itself.

This is essentially a theorem-recall item: the correct choice is the full statement of a specific distinction criterion, while the other options are slight perturbations of that statement. It functions mainly as recognition of the theorem rather than application to a new situation.

There is some reasoning pressure because the distractors differ in subtle ways: the exponent n_i/d versus n_i, 'if and only if' versus a one-way implication, unnecessary dependence on ψ, and whether the condition applies to I_φ or all summands. However, the item still leans much more toward precise recall than genuine generative mathematical reasoning.

The distractors are strong: each alters a mathematically meaningful component of the criterion in a plausible way, reflecting realistic failure modes such as sign mistakes, quantifier weakening, spurious auxiliary-data dependence, and overextending the condition to the wrong summands.

A technically well-crafted theorem-recognition MCQ with high-quality distractors, but it is largely a restatement/recall task rather than a genuinely generative reasoning question.