\begin{theorem}\label{Thm:IGP}
    For any prime $p$ for which there exists an $\ell \in \cL$ with the Kronecker symbol $\legendre{\ell}{p}$ equal to $-1$, the group $\PSL_2(\F_{p^2})$ is a \emph{regular} Galois group over $\Q$.\footnote{Concretely, this implies that $\PSL_2(\F_{p^2})$ is realisable as a regular Galois group over $\Q$ for all $p<3,267,289$ (compared to the previous implied bound $p<1,009$).}
\end{theorem}

\begin{remark}\label{Rem:shih}
Theorem~\ref{Thm:cover} should be compared to the analogous cover $X_0(p)\to X(1)$ of modular curves corresponding to the level structure $\Gamma_0(p)$.

In the one-dimensional setting, $X_0(p)$ has field of definition $\Q(\zeta_p)$ and one only obtains a $\PGL_2(\F_p)$-cover. Under certain congruence conditions, Atkin-Lehner involutions can be used to turn this into a $\PSL_2(\F_p)$-cover by what has become known as Shih's trick \cite{Shih}. In our situation such a trick is unnecessary.
\end{remark}

The following popular question, going back to at least Noether, has remained open for over a century:
\begin{question}[Inverse Galois Problem]
 Is every finite group $G$ the Galois group of a finite extension of $\Q$?
\end{question}

We also have the variant:
\begin{question}[Regular Inverse Galois Problem]
 Is every finite group $G$ the Galois group of a finite geometrically integral cover $C \to \P^1_{\Q}$?
\end{question}

By the Hilbert Irreducibility Theorem (a positive answer to) the regular Inverse Galois Problem implies (a positive answer to) the classical one. On the other hand, the regular version has various interesting implications: for instance, it gives not just one $G$-extension of $\Q$ but infinitely many linearly disjoint $G$-extensions over any number field by specialisation.

A fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\PSL_2(\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\ell\in\{2,3,5,7\}$ with $\legendre{\ell}{p}=-1$ \cite{Shih, Malle} (although non-regularly for all $p$ \cite{Zywina}) and for $k=2$ when $\legendre{5}{p}=-1$ \cite{Feit, Mestre, Mestre2}, when $\legendre{2}{p}=-1$ or $\legendre{3}{p}=-1$ \cite{Shiina}, and finally when $\legendre{-3}{p}=-1$ or $\legendre{-7}{p}=-1$ \cite{DW} (although non-regularly for more $p$ \cite{DV}).

In this paper we introduce new methods for the Inverse Galois Problem to prove the following. Let
\[
\cL:=\{\ell\ \text { prime}: \ell \leq 41, \ell \neq 31\}.
\]

A fundamental case for the questions above is when $G$ is simple. Finite simple groups have been classified in the following families: groups of Lie type, alternating groups of degree at least $5$, cyclic groups and the $27$ sporadic groups (including the Tits group). The regular Inverse Galois Problem has long been solved for alternating groups, cyclic groups, and all but one of the sporadic groups \cite[Theorem II.10.3]{IGT}. So the infinite families that remain are those of Lie type. Arguably the simplest such family, and a testing ground for any new techniques, is the series $A_1(q)=\PSL_2(\F_{q})$ for a prime power $q=p^k$. In this case, the regular Inverse Galois Problem has been solved for $k=1$ when there exists $\ell\in\{2,3,5,7\}$ with $\legendre{\ell}{p}=-1$ \cite{Shih, Malle} (although non-regularly for all $p$ \cite{Zywina}) and for $k=2$ when $\legendre{5}{p}=-1$ \cite{Feit, Mestre, Mestre2}, when $\legendre{2}{p}=-1$ or $\legendre{3}{p}=-1$ \cite{Shiina}, and finally when $\legendre{-3}{p}=-1$ or $\legendre{-7}{p}=-1$ \cite{DW} (although non-regularly for more $p$ \cite{DV}).

In particular, this realises the group $\PSL_2(\F_{p^2})$ over any Hilbertian field $k$ for the primes $p$ above. For the primes $\ell \leq 17, \ell \neq 7,11$ we apply the classical Hilbert Irreducibility Theorem to the Hilbert modular surfaces for which Elkies and Kumar \cite{EK} have proved $\Q$-rationality. For the primes $\ell=7,11,19,23,29,37$ and $41$, an additional difficulty arises: the relevant Hilbert modular surfaces are not (uni)rational. We thus prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces. More precisely, we show that the Hilbert Property holds for several low-discriminant Hilbert modular surfaces, i.e.\ their sets of rational points are not thin. These surfaces are K3 surfaces, and here our approach uses the multiple fibration method initially introduced by Corvaja-Zannier \cite{CZ}, and further studied by the first-named author \cite{Demeio} and by the second-named author in collaboration with Mezzedimi \cite{GCM}. This seems to be the first time that a non-unirational Hilbert Property is applied to the Inverse Galois Problem.

In fact, we prove an even stronger statement, namely that $X(\Q(t))$ is non-thin in $X_{\bar \Q(t)}(\bar \Q(t))$ (Theorem \ref{Thm:K3regular}), thus producing an abundance of {\em rational curves}, and not just rational points, on the Hilbert modular surfaces under investigation.
The application to the Inverse Galois Problem is found through the following
\begin{theorem}\label{Thm:cover}
 Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\mathfrak o$, and let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$.
\end{theorem}

The particular Shimura varieties under our consideration {\em already have} many (smooth) rational curves that arise as special algebraic cycles coming from the moduli structure. Namely, these come from the boundary of the compactification and from embeddings of Shimura data. However, specialising the cover $Y_p \to X$ to {\em these} rational curves does not produce surjective representations $\Gal(\overline{\Q(t)}/\Q(t)) \to \PSL_2(\F_{p^2})$ precisely because of the special origin of these curves, making this ``naive'' specialisation \emph{not} sufficient to construct Galois representations of $\PSL_2(\F_{p^2})$. Nonetheless, these smooth rational curves give rise to many genus $1$ fibrations defined over the base field $\Q$ on the surface $X$. This geometric feature of $X$, which itself does not appear to have a modular interpretation, is what ultimately allows us to propagate these rational curves and obtain new ones for which the specialised representation instead {\em does} indeed surject. More formally, this ``propagation'' is expressed by the multiple fibration method used to prove Theorem \ref{Thm:K3}.

\begin{proposition}\label{Prop331regular}
 Let $k$ be a field of characteristic $0$, and $k(V)$ be a regular finitely generated transcendental extension of $k$. Let $\phi:Y\to X$ be a Galois cover with group $G$ of geometrically integral $k(V)$-varieties. Let $S \subset X(k(V))$ be a subset whose natural image in $X(\bar k(V))=X_{\bar k(V)}(\bar k(V))$ is a non-thin set of points of the base-changed variety $X_{\bar k(V)}$. Then there are Zariski-dense $x \in S$ for which the fibre $\phi^{-1}(x)$ is irreducible with regular function field over $k$, and with $\Gal(\phi^{-1}(x)/k(V))\simeq G$.
\end{proposition}

We write $\H^+$ and $\H^-$ for the upper and lower half-plane. For each genus $g \in \Cl^+K/2$ of $\mathfrak o$ we may pick a representative fractional ideal $\mathfrak a$ (of norm $A$) and define, for each finite index subgroup $\Gamma \subseteq \PSL(\mathfrak o \oplus \mathfrak a)$, the \emph{Hilbert modular surface}
\[
X_g(\Gamma):=\Gamma \backslash (\H^+)^2,
\]
where the quotient is by the properly discontinuous action:
\[
\begin{pmatrix}
 a & b \\ c&d
\end{pmatrix}\cdot (z_1,z_2) = \left( \frac{az_1+b}{cz_1+d} , \frac{a'z_2+b'}{c'z_2+d'} \right).
\]
Let $\Lambda=\mathfrak o \oplus \mathfrak a$. We call the surface $X_g:= X_g(\PSL(\Lambda))$ of \emph{base level}.\footnote{One could take further quotients of $(\H^+)^2$, the maximum being by the symmetrised Hurwitz-Maass extension of $\PSL(\Lambda)$, but we shall not be interested in these in this paper.} The isomorphism class of $X_g$ is independent of the choice of representative $\mathfrak a$.

The classes in $\Cl(K)$ are in bijection with $\PSL(\Lambda)$-orbits of points in $\P^1(K)$ via the map sending $(x_0:x_1)\in\P^1(K)$ to the class $[x_0\cdot \mathfrak o + x_1\cdot \mathfrak a]\in\Cl(K)$. Fix a class $[\mathfrak c]\in\Cl(K)$ and write shorthand $s=s_{[\mathfrak c]}$. To $s$ we associate $M_s=\mathfrak c^{-2}\mathfrak a^{-1}$ viewed as an \emph{ordered} $\mathfrak o$-module, i.e.\ equipped with an order on $M_s\otimes_\sigma \R$ for each embedding $\sigma:K\hookrightarrow\R$. The isomorphism class of $M_s$ does not depend on the chosen representative $\mathfrak c$.
\begin{proposition}\label{Prop:CoverQ}\hfill
 \begin{enumerate}
 \item The action of $\Gamma_\Q$ on $X_g^\infty(\ov\Q)$ is trivial, in other words all cusps are defined over $\Q$.
 \item The exceptional divisor $E_s$ of the minimal resolution $\tilde X_g\to X_g$ over any cusp $s$ is a cycle of smooth rational curves intersecting transversally.
 \end{enumerate}
\end{proposition}

Combining the above, we proved the theorem for all $p$ for which there exists $D_p$ in
\[
\cD \coloneqq \{5,8,12,13,17,21,24,28,29,33,37,40,41,44,56,57,69,105\}
\]
with $\legendre{D_p}{p}=-1$. One easily checks by hand that this includes all primes $p \leq 41$. For $p > 41$, such a $D_p$ exists unless
\begin{center}
 $\legendre{D}{p}=1$ for all $D \in \cD$,
\end{center}
or, equivalently, unless $\legendre{D}{p}=1$ for all $D$ in the multiplicative subgroup $\langle \cD \rangle \subset \Q^*$ generated by $\cD$. Rewriting $\cD$ as
\[
\{5,2^3,2^2 \cdot 3,13,17,3 \cdot 7,2^3 \cdot 3,2^2 \cdot 7,29,3 \cdot 11,37,2^3 \cdot 5,41,2^2 \cdot 11,2^3 \cdot 7,3\cdot 19,3 \cdot 23,3 \cdot 5 \cdot 7\}
\]
one sees that $\langle \cD \rangle=\langle \cL \rangle$, where $\cL=\{\ell \text{ prime}: \ell \leq 41, \ell \neq 31\}$ is the list of prime factors appearing in $\cD$. So $D_p$ exists unless $\legendre{\ell}{p}=1$ for all $\ell$, concluding the proof.
\end{proof}

In particular, this realises the group $\PSL_2(\F_{p^2})$ over any Hilbertian field $k$ for the primes $p$ above. For the primes $\ell \leq 17, \ell \neq 7,11$ we apply the classical Hilbert Irreducibility Theorem to the Hilbert modular surfaces for which Elkies and Kumar \cite{EK} have proved $\Q$-rationality. For the primes $\ell=7,11,19,23,29,37$ and $41$, an additional difficulty arises: the relevant Hilbert modular surfaces are not (uni)rational. We thus prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces. More precisely, we show that the Hilbert Property holds for several low-discriminant Hilbert modular surfaces, i.e.\ their sets of rational points are not thin. These surfaces are K3 surfaces, and here our approach uses the multiple fibration method initially introduced by Corvaja-Zannier \cite{CZ}, and further studied by the first-named author \cite{Demeio} and by the second-named author in collaboration with Mezzedimi \cite{GCM}. This seems to be the first time that a non-unirational Hilbert Property is applied to the Inverse Galois Problem.

\begin{theorem}\label{Thm:K3}
 Let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. If $X$ is birational to a K3 surface, then $X(\Q)$ is not thin.
\end{theorem}

In fact, we prove an even stronger statement, namely that $X(\Q(t))$ is non-thin in $X_{\bar \Q(t)}(\bar \Q(t))$ (Theorem \ref{Thm:K3regular}), thus producing an abundance of {\em rational curves}, and not just rational points, on the Hilbert modular surfaces under investigation.
The application to the Inverse Galois Problem is found through the following
\begin{theorem}\label{Thm:cover}
    Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\mathfrak o$, and let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$.
\end{theorem}

Compared to the well-known modular curve setting, the existence of the Galois cover which is geometrically integral may be surprising at first, see Remark~\ref{Rem:shih}.

Theorem \ref{Thm:IGP} is proven by combining Theorems \ref{Thm:K3} and \ref{Thm:cover}.
The proofs of Theorems \ref{Thm:K3} and \ref{Thm:cover} rely on the theory of canonical models of compactified Shimura varieties as applied to Hilbert modular surfaces. More precisely, Theorem \ref{Thm:K3} relies on an analysis of the intersection graph of these modular surfaces, while Theorem \ref{Thm:cover} follows naturally by looking at the Shimura data involved. (Alternatively, Theorem \ref{Thm:cover} can be inferred by looking at the $p$-torsion representation of the abelian surfaces with real multiplication parametrized by Hilbert Modular Surfaces.)

The particular Shimura varieties under our consideration {\em already have} many (smooth) rational curves that arise as special algebraic cycles coming from the moduli structure. Namely, these come from the boundary of the compactification and from embeddings of Shimura data. However, specialising the cover $Y_p \to X$ to {\em these} rational curves does not produce surjective representations $\Gal(\overline{\Q(t)}/\Q(t)) \to \PSL_2(\F_{p^2})$ precisely because of the special origin of these curves, making this ``naive'' specialisation \emph{not} sufficient to construct Galois representations of $\PSL_2(\F_{p^2})$. Nonetheless, these smooth rational curves give rise to many genus $1$ fibrations defined over the base field $\Q$ on the surface $X$. This geometric feature of $X$, which itself does not appear to have a modular interpretation, is what ultimately allows us to propagate these rational curves and obtain new ones for which the specialised representation instead {\em does} indeed surject. More formally, this ``propagation'' is expressed by the multiple fibration method used to prove Theorem \ref{Thm:K3}.

The Shimura theory is particularly explicit in our case but it would be interesting to see if a similar approach could apply to other settings.

It would also be interesting to study the Hilbert Property for honestly elliptic Hilbert modular surfaces, where the multiple fibration method does not apply. If instead of proving non-thinness of rational points the goal is to only specialise a specific Galois cover like in Theorem~\ref{Thm:cover}, then one may use less than the full Hilbert Property, although knowledge of the intersection graph of the modular surface would still be necessary. We will pursue this approach in a separate forthcoming work.

The article is organised as follows: In \S\ref{Sec:hilbert}, we review and extend the multiple fibration method to nonconstant $\Q(t)$-points. We develop the necessary Shimura theory of Hilbert modular surfaces in \S\ref{Sec:shimura} and construct the cover from Theorem~\ref{Thm:cover} in \S\ref{Sec:cover}. We then study the algebraic cycles on our surfaces in \S\ref{Sec:cycles}. Finally, we prove Theorems~\ref{Thm:IGP} and \ref{Thm:K3} in \S\ref{Sec:main}.

\paragraph{Acknowledgements}
The authors thank Gerard Van der Geer and John Voight for helpful comments. They gratefully acknowledge the hospitality of ICMS Edinburgh, the Center for Mathematical Sciences at Technion, and the University of Bath, where part of this research was carried out, and thank Daniel Loughran for the invitation to the latter. The second author was supported by UKRI award UKRI094.

To prove Theorem~\ref{Thm:IGP}, the authors "combin[e] Theorems \ref{Thm:K3} and \ref{Thm:cover}." Theorem~\ref{Thm:cover} supplies, for every rational prime $p$ inert in the relevant real quadratic field $K$, a "Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$." Theorem~\ref{Thm:K3} provides the needed Hilbert irreducibility input on the Hilbert modular surface $X$: if $X$ is birational to a K3 surface then "$X(\Q)$ is not thin" (indeed they prove the stronger non-thinness statement over $\Q(t)$ in Theorem~\ref{Thm:K3regular}, yielding "an abundance of rational curves").

For small $\ell$ with $\Q$-rational Hilbert modular surfaces (Elkies--Kumar), they "apply the classical Hilbert Irreducibility Theorem." For the remaining listed primes $\ell$ where the surfaces are "not (uni)rational," they instead "prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces" by establishing that "the Hilbert Property holds" for the relevant low-discriminant surfaces via the "multiple fibration method." Concretely, although $X$ already has many rational curves coming from boundary/embedded Shimura data, "specialising the cover $Y_p \to X$ to these rational curves does not produce surjective representations" to $\PSL_2(\F_{p^2})$; instead, those curves induce many genus $1$ fibrations on $X$, and this geometric feature "allows us to propagate these rational curves and obtain new ones for which the specialised representation instead does indeed surject," with the propagation formalised by the multiple fibration method used in proving Theorem~\ref{Thm:K3}.

To prove the main theorem, the authors “combin[e] Theorems \ref{Thm:K3} and \ref{Thm:cover}.” We first record the two inputs.

\begin{theorem}\label{Thm:cover}
    Let $K$ be the real quadratic field of discriminant $D$ with ring of integers $\mathfrak o$, and let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. For every rational prime $p$ that is inert in $K$, there exists a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$.
\end{theorem}

\begin{theorem}\label{Thm:K3}
 Let $X/\Q$ be the Hilbert modular surface of discriminant $D$ and genus $g$ at base level. If $X$ is birational to a K3 surface, then $X(\Q)$ is not thin.
\end{theorem}

The first theorem supplies, for every rational prime $p$ inert in the relevant real quadratic field $K$, a Galois geometrically integral cover $Y_p \to X$ with group $\PSL_2(\mathfrak o/p)$. The second theorem provides the needed Hilbert irreducibility input on the Hilbert modular surface $X$: if $X$ is birational to a K3 surface then $X(\Q)$ is not thin (indeed they prove the stronger non-thinness statement over $\Q(t)$ in
\begin{theorem}\label{Thm:K3regular}
     Let $X/\Q$ be the Hilbert modular surface of discriminant $D\in \cS_+$ (resp.\ $\cS_-$) and principal genus (resp.\ non-principal genus) at base level. Then the natural image of $X(\Q(t))$ in $X_{\bar \Q(t)}$ is non-thin.
\end{theorem}
, yielding “an abundance of rational curves”).

For small $\ell$ with $\Q$-rational Hilbert modular surfaces (Elkies--Kumar), they “apply the classical Hilbert Irreducibility Theorem.” For the remaining listed primes $\ell$ where the surfaces are “not (uni)rational,” they instead “prove the theorem with the help of new versions of the Hilbert Irreducibility Theorem over non-unirational surfaces” by establishing that “the Hilbert Property holds” for the relevant low-discriminant surfaces via the “multiple fibration method.” Concretely, although $X$ already has many rational curves coming from boundary/embedded Shimura data, “specialising the cover $Y_p \to X$ to these rational curves does not produce surjective representations” to $\PSL_2(\F_{p^2})$; instead, those curves induce many genus $1$ fibrations on $X$, and this geometric feature “allows us to propagate these rational curves and obtain new ones for which the specialised representation instead does indeed surject,” with the propagation formalised by the multiple fibration method used in proving the theorem above.

Question: Let \(\mathcal L=\{\ell\text{ prime}: \ell\le 41,\ \ell\ne 31\}\). Suppose \(p\) is a prime such that there exists some \(\ell\in \mathcal L\) with Kronecker symbol \(\left(\frac{\ell}{p}\right)=-1\). Under this hypothesis, which statement about \(\operatorname{PSL}_2(\mathbb F_{p^2})\) holds over \(\mathbb Q\)? Here, saying that a finite group \(G\) is a regular Galois group over \(\mathbb Q\) means that there exists a finite geometrically integral cover \(C\to \mathbb P^1_{\mathbb Q}\) whose induced function-field extension is Galois with group \(G\) (equivalently, there is a Galois extension \(E/\mathbb Q(t)\) with Galois group \(G\) and \(E\cap \overline{\mathbb Q}=\mathbb Q\)).

The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a regular Galois group over \(\mathbb Q\); equivalently, there exists a Galois extension \(E/\mathbb Q(t)\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) such that \(E\cap \overline{\mathbb Q}=\mathbb Q\).

Choices:
B. The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a Galois group over \(\mathbb Q\); equivalently, there exists a finite Galois extension \(L/\mathbb Q\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\).
C. There exists a finite geometrically integral cover \(C\to \mathbb P^1_{\mathbb Q}\) whose induced function-field extension has Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\).
D. The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a regular Galois group over \(\mathbb Q\) for every prime \(p\) such that \(\left(\frac{\ell}{p}\right)=-1\) for all \(\ell\in \mathcal L\); equivalently, there exists a Galois extension \(E/\mathbb Q(t)\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) and \(E\cap \overline{\mathbb Q}=\mathbb Q\) whenever this stronger condition holds.
E. The group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) is a regular Galois group over \(\mathbb Q\); equivalently, for every real quadratic field \(K\) in which \(p\) is inert, there exists a Galois extension \(E/\mathbb Q(t)\) with Galois group \(\operatorname{PSL}_2(\mathbb F_{p^2})\) such that \(E\cap \overline{\mathbb Q}=\mathbb Q\).

The stem does not explicitly reveal the correct option. It gives hypotheses and a definition, but the answer still depends on identifying the precise existence claim among several close variants.

The correct choice is essentially a direct statement of the underlying theorem, with only equivalent phrasing added. The item mainly tests recognition of the exact theorem statement rather than selecting among genuinely competing conclusions derived from reasoning.

There is some reasoning pressure in checking quantifiers, the distinction between regular realization and mere realization over Q, and PSL2 versus PGL2. However, the task is still largely theorem recall/matching rather than substantial generative inference.

The distractors are mathematically plausible and target realistic failure modes: confusing PSL2 with PGL2, weakening regularity to mere realizability, strengthening the quantifier on the Kronecker condition, or adding an unnecessary universal quadratic-field condition.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is close to a direct restatement of the result and only moderately tests genuine generative reasoning.


[Hirose--Sato~\cite{HiroseSato2019A}]\label{thm:HiroseSato1}
For positive integers $a, b$, and $c$, we have
\begin{align*}
&\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)\\
&=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\end{align*}

\begin{theorem}[Hirose--Maesaka--Seki--Watanabe~\cite{HiroseMaesakaSekiWatanabe2025+}]
The drop 1 operator $\cD$ satisfies the following properties$:$
\label{thm:drop1}
\begin{enumerate}[label=\textnormal{(\roman*)}]
\item\label{it:HMSW1} $\{w-\cD(w) \mid w\in \cH^0\}=\Ker(Z^{\diamondsuit})\subset\Ker(Z)$,
\item\label{it:HMSW2} $\cD(w)=w$ for $w\in\cH^{\geq 2}$,
\item\label{it:HMSW3} $\cD(w)=\cD(\tau(w))$ for $w\in \cH^0$.
\end{enumerate}
\end{theorem}

\begin{definition}[Hirose--Maesaka--Seki--Watanabe~\cite{HiroseMaesakaSekiWatanabe2025+}]\label{def:drop1operator}
We define the \emph{drop 1 operator} $\cD\colon\cH^0\to\cH^{\geq 2}$ to be a $\ZZ$-linear map given as follows.
It suffices to define $\cD(w)$ for each word $w\in\cH^0$.
For $\bc=(c_1,\dots, c_{2t})$, we write $\fD(\bc)=\cD(y^{c_1}x^{c_2}\cdots y^{c_{2t-1}}x^{c_{2t}})$.
The image $\fD(\bc)$ is defined inductively as follows:
\begin{align*}
\fD(\bc)&=\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 1}}(-1)^{\#B-1}\fD(\bc_{(-A)}-\bdelta_B)x^{\#A+\#B}\\
&\quad+\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 2}}(-1)^{\#B-1}\fD(\bc_{(-A)}-\bdelta_B)z_{\#A+\#B}\\
&\quad+\sum_{\substack{A\in\cP^{\text{oe}}([2t]^1_{\bc}) \\ B\in\cP^{\smallxcancel{\text{oe}}}([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 2}}(-1)^{\#B}\fD(\bc_{(-A)}-\bdelta_B)z_{\#A+\#B}.
\end{align*}
Here the initial condition is $\fD(\varnothing)=\cD(1)=1$.
\end{definition}

\begin{corollary}\label{cor:diamond-lift}
For $w_1, w_2\in\cH^{2,3}$, we have
\[
Z^{\diamondsuit}(w_1\tau(w_2))=Z^{\diamondsuit}(w_1\star w_2).
\]
\end{corollary}

For positive integers $k_1, \dots, k_r$ with $k_r\geq 2$, the \emph{multiple zeta value} (MZV) $\zeta(k_1,\dots, k_r)$ is defined by
\[
\zeta(k_1,\dots, k_r)\coloneqq\sum_{0<n_1<\cdots <n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
Linear relations among MZVs have been studied for many years.
For each positive integer $c$, the relation
\begin{equation}\label{eq:Hoffman-conj}
\zeta(3,\{2\}^{c-1},1,2)=\zeta(\{2\}^{c+2})+2\zeta(3,3,\{2\}^{c-1})
\end{equation}
is known as \emph{Hoffman's conjectural identity} and had remained open for about seventeen years.
Here the notation $\{2\}^n$ denotes the string $2,\dots, 2$ in which the entry 2 is repeated $n$ times.
Charlton proved in \cite{Charlton2016} that there exists a rational number $q_c$ such that $\zeta(3,\{2\}^{c-1},1,2)=q_c\zeta(\{2\}^{c+2})+2\zeta(3,3,\{2\}^{c-1})$, and Hirose and Sato proved not only Hoffman's conjectural identity but also the following more general theorem.

More generally, applying the same procedure to Corollary~\ref{cor:diamond-lift} (see Section~\ref{sec:finite} for more details), we have the following.
For a word $w=z_{k_1}\cdots z_{k_r}$, we set $\overleftarrow{w}\coloneqq z_{k_r}\cdots z_{k_1}$, and $\wt(w)$ denotes the total degree of $w$, that is, $\wt(w)=k_1+\cdots + k_r$.
A $\ZZ$-linear map $Z_{\cA}\colon \ZZ+y\cH=\ZZ\langle z_1,z_2,\dots\rangle\to\cA$ is defined by $Z_{\cA}(z_{k_1}\cdots z_{k_r})=\zeta^{}_{\cA}(k_1,\dots, k_r)$ and $Z_{\cA}(1)=1$.
\begin{corollary}\label{cor:FMZV}
For words $w_1, w_2\in\cH^{2,3}$, we have
\[
(-1)^{\wt(w_2)}Z_{\cA}(w_1\overleftarrow{w_2})=Z_{\cA}(w_1\star w_2).
\]
\end{corollary}
This is a family of relations among FMZVs in which only 2 and 3 occur as entries.
Since the shuffle relation for FMZVs $(-1)^{\wt(w_2)}Z_{\cA}(w_1\overleftarrow{w_2})=Z_{\cA}(w_1\sh w_2)$ holds (see \cite{KanekoZagier2025+,Ono2017}), Corollary~\ref{cor:FMZV} can also be rewritten as
\[
Z_{\cA}(w_1\sh w_2-w_1\star w_2)=0
\]
which is of the form of a certain double shuffle relation.
\section*{Acknowledgements}
The author would like to thank Professor Minoru Hirose for kindly answering his questions about the previous works by Hirose and Sato.
This research was inspired by a question that came to mind when the author got married, and the author would like to thank his wife, Yui, for marrying him.
\section{The drop 1 operator}
We introduce the following notation to define the main operator.
Let $t$ be a positive integer, let $\bc=(c_1,\dots, c_{2t})$ be a tuple of positive integers, and let $S\subset\ZZ$.
\begin{itemize}[leftmargin=2.5em]
\item $\#A$ denotes the cardinality of a finite set $A$.
\item $[n]\coloneqq\{1,2,\dots,n\}$ for $n\in\ZZ_{\geq 0}$. $[0]=\varnothing$.
\item $[2t]_{\bc}^1\coloneqq\{i\in[2t] \mid c_i=1\}.$
\item $[2t]_{\bc}^{>1}\coloneqq\{i\in[2t] \mid c_i>1\}.$
\item $\cP^{\text{eo}}(S)\coloneqq\{A\subset S \mid i\in A\cap 2\ZZ \Longleftrightarrow i+1\in A\setminus 2\ZZ\}.$
\item $\cP^{\text{oe}}(S)\coloneqq\{A\subset S \mid i\in A\setminus 2\ZZ \Longleftrightarrow i+1\in A\cap 2\ZZ\}.$
\item $\cP^{\smallxcancel{\text{eo}}}(S)\coloneqq\{A\subset S \mid i\in A\cap 2\ZZ \Longrightarrow i+1\not\in A\setminus 2\ZZ\}.$
\item $\cP^{\smallxcancel{\text{oe}}}(S)\coloneqq\{A\subset S \mid i\in A\setminus 2\ZZ \Longrightarrow i+1\not\in A\cap 2\ZZ\}.$
\item $\bc_{(-A)}\coloneqq (c_{i_1},\dots, c_{i_r})$, where $[2t]\setminus A=\{i_1,\dots, i_r\}$ with $i_1<\cdots<i_r$, for $A\subset[2t]$.
\item $\bc_{(-\varnothing)}=\bc$, $\bc_{(-[2t])}=\varnothing$.
\item $\delta_{i\in B}\coloneqq\begin{cases}1 & \text{if } i\in B, \\ 0 & \text{otherwise}\end{cases}$ for a set $B$.
\item $\bc_{(-A)}-\bdelta_B\coloneqq(c_{i_1}-\delta_{i_1\in B},\dots, c_{i_r}-\delta_{i_r\in B})$ for the above $\bc_{(-A)}$ and $B\subset [2t]$.
\end{itemize}

We consider the case $c>1$.
Then $\cP^{\smallxcancel{\text{eo}}}([2t]^{>1}_{\bc})=\cP([2t]^{>1}_{\bc})$ holds.
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$, decompose it as $A=A_1\cup A_2 \cup A_3$, where
\[
A_1\coloneqq A \cap [2l_r-1],\quad A_2\coloneqq A\cap ([2l_r+2c-2]\setminus [2l_r]),\quad A_3\coloneqq A\setminus [2l_r+2c-1].
\]
We define a map $\iota_1\colon\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ by
\[
\iota_1(A)\coloneqq (A_1-1)\cup (A_2-1)\cup\{2l_r+2c-3,2l_r+2c-2\}\cup(A_3+1)
\]
and a map $\iota_2\colon\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ by
\[
\iota_2(A)\coloneqq (A_1-1)\cup (A_2-1)\cup(A_3+1).
\]
A bijection $\cP^{\text{eo}}([2t]^1_{\bc})\sqcup\cP^{\text{eo}}([2t]^1_{\bc})\to\cP^{\text{oe}}([2t]^1_{\bc})$ is obtained by sending an element $A$ in the first copy of $\cP^{\text{eo}}([2t]^1_{\bc})$ to $\iota_1(A)$, and an element $A$ in the second copy of $\cP^{\text{eo}}([2t]^1_{\bc})$ to $\iota_2(A)$.
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$, we have $\bc_{(-A)}=\bc_{(-\iota_2(A))}$.
Therefore, if we compute the defining recurrence for the drop 1 operator using this bijection, every term in the second sum is canceled by a term in the third sum, and we are left with
\begin{align*}
\fD(\bc)&=\sum_{\substack{A\in\cP^{\text{eo}}([2t]^1_{\bc}) \\ B\in\cP([2t]^{>1}_{\bc}) \\ \#A+\#B\geq 1}}(-1)^{\#B-1}\biggl(\fD(\bc_{(-A)}-\bdelta_B)-\fD(\bc_{(-\iota_1(A))}-\bdelta_B)z_2\biggr)x^{\#A+\#B}\\
&\quad+\fD(\bc_{(-\{2l_r+2c-3,2l_r+2c-2\})})z_2.
\end{align*}
For $A\in\cP^{\text{eo}}([2t]^1_{\bc})$ or $\cP^{\text{oe}}([2t]^1_{\bc})$, there exist $a'_1$, $\dots$, $a'_r$, $b'_1$, $\dots$, $b'_s$, and $c'$ such that
\[
\bc_{(-A)}=[a'_1,\dots, a'_r;b'_1,\dots, b'_s;c'].
\]
Here $a'_1\in[a_1]$, $\dots$, $a'_r\in[a_r]$, $b'_1\in[b_1]$, $\dots$, $b'_s\in[b_s]$, $c'\in[c]$, and $a'_1+\cdots+a'_r+b'_1+\cdots+b'_s+c'=t+1-\#A/2$.
Moreover, upon adding $-\bdelta_B$, addition occurs between the corresponding components; for example, when
\begin{align*}
\bc&=[2,1,3;4,1,1,2;3]\\
&=(\stackrel{1}{1},\stackrel{2}{1},\stackrel{3}{1},\stackrel{\color{red}{4}}{2},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{8}{1},\stackrel{9}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{14}{1},\stackrel{15}{1},\stackrel{16}{1},\stackrel{\color{red}{17}}{2},\stackrel{18}{1},\stackrel{19}{1},\stackrel{20}{1},\stackrel{21}{1},\stackrel{22}{1},\stackrel{23}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{red}{27}}{2},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{30}{1},\stackrel{31}{1},\stackrel{32}{1}),
\end{align*}
\[
A=\{2,3,8,9,14,15,18,19,22,23,30,31\}\in\cP^{\text{eo}}([32]^1_{[2,1,3;4,1,1,2;3]}),
\]
and
\[
B=\{4,17,27\}\subset[32]^{>1}_{[2,1,3;4,1,1,2;3]}=\{4,6,12,17,25,27,29\},
\]
we have
\begin{align*}
&[2,1,3;4,1,1,2;3]_{(-\{2,3,8,9,14,15,18,19,22,23,30,31\})}=[1,1,2;2,1,1,1;2]\\
&=(\stackrel{1}{1},\stackrel{\color{red}{4}}{2},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{16}{1},\stackrel{\color{red}{17}}{2},\stackrel{20}{1},\stackrel{21}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{red}{27}}{2},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{32}{1})
\end{align*}
and
\begin{align*}
&[2,1,3;4,1,1,2;3]_{(-\{2,3,8,9,14,15,18,19,22,23,30,31\})}-\bdelta_{\{4,17,27\}}\\
&=[1+1,2;1+1,1;2+2]=[2,2;2,1;4]\\
&=(\stackrel{1}{1},\stackrel{\color{blue}{4}}{1},\stackrel{5}{1},\stackrel{\color{red}{6}}{2},\stackrel{7}{1},\stackrel{10}{1},\stackrel{11}{1},\stackrel{\color{red}{12}}{2},\stackrel{13}{1},\stackrel{16}{1},\stackrel{\color{blue}{17}}{1},\stackrel{20}{1},\stackrel{21}{1},\stackrel{24}{1},\stackrel{\color{red}{25}}{2},\stackrel{26}{1},\stackrel{\color{blue}{27}}{1},\stackrel{28}{1},\stackrel{\color{red}{29}}{2},\stackrel{32}{1}).
\end{align*}
Therefore, for any $A, B$ with $\#A+\#B\geq 1$, the induction hypothesis implies that there exist words $v_1, v_2$ in $\cH^{2,3}$ such that $\bc_{(\iota_1(-A))}-\bdelta_B$ corresponds to $v_1\tau(v_2)$, and we have
\[
\fD(\bc_{(-A)}-\bdelta_B)-\fD(\bc_{(-\iota_1(A))}-\bdelta_B)z_2=\cD(v_1z_2\tau(v_2))-\cD(v_1\tau(v_2))z_2=v_1z_2\star v_2-(v_1\star v_2)z_2=0.
\]
Since we are in the case $c>1$, we can write $w_1=wz_2$ with a certain word $w$ in $\cH^{2,3}$.
Then we obtain
\begin{align*}
\fD(\bc)&=\fD(\bc_{(-\{2l_r+2c-3,2l_r+2c-2\})})z_2=\fD([a_1,\dots, a_r;b_1,\dots, b_s;c-1])z_2=\cD(w\tau(w_2))z_2\\
&=(w\star w_2)z_2=wz_2\star w_2=w_1\star w_2.
\end{align*}

Their proof can be viewed as showing that the above identity is a consequence of the \emph{confluence relation} later formulated in \cite{HiroseSato2019B}.

Hirose and Sato gave a further generalization in \cite{HiroseSato2022+}, and we set up notation in order to state their result.
Let $\mathcal{H}\coloneqq\mathbb{Z}\langle x,y\rangle$ be the non-commutative polynomial ring over $\mathbb{Z}$ in two variables $x$ and $y$.
Let $z_k\coloneqq yx^{k-1}$ for each positive integer $k$.
Subrings $\mathcal{H}^{2,3}\subset\mathcal{H}^0\subset\mathcal{H}$ are defined by $\mathcal{H}^{2,3}\coloneqq\mathbb{Z}\langle z_2,z_3\rangle$ and $\mathcal{H}^0\coloneqq\mathbb{Z}+y\mathcal{H} x$, respectively.
A $\mathbb{Z}$-linear map $Z\colon\mathcal{H}^0\to\mathbb{R}$ is defined by $Z(z_{k_1}\cdots z_{k_r})=\zeta(k_1,\dots, k_r)$ and $Z(1)=1$.
A product $\star$ on $\mathcal{H}^{2,3}$ is defined inductively by
\begin{itemize}[leftmargin=2.5em]
\item $w\star 1=1\star w=w$,
\item $wz_2\star w'=w\star w'z_2=(w\star w')z_2$,
\item $wz_3\star w'z_3=(w\star w'z_3)z_3+(wz_3\star w')z_3+(w\star w')z_2^3$
\end{itemize}
for any words $w$ and $w'$ in $\mathcal{H}^{2,3}$, and then extended by $\mathbb{Z}$-bilinearity.
Let $\tau$ be an anti-automorphism on $\mathcal{H}$ defined by $\tau(x)=y$ and $\tau(y)=x$.
Using this notation, we can now state the following theorem.
\begin{theorem}[{Hirose--Sato~\cite[Proposition~14]{HiroseSato2022+}}]\label{thm:HiroseSato2}
For $w, w'\in\mathcal{H}^{2,3}$, we have
\[
Z(w_1\tau(w_2))=Z(w_1\star w_2).
\]
\end{theorem}
By specializing Theorem~\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$, we recover Theorem~\ref{thm:HiroseSato1}.
Hirose and Sato derived Theorem~\ref{thm:HiroseSato2} from their \emph{block shuffle identity} (\cite[Theorem~10]{HiroseSato2022+}).

Hoffman conjectured in \cite{Hoffman1997} that any multiple zeta value can be expressed as a $\mathbb{Q}$-linear combination of the elements of
\[
\{\zeta(k_1,\dots, k_r) \mid r\in\ZZ_{>0}, k_1,\dots, k_r\in\{2,3\}\}.
\]
This conjecture was proved by Brown in \cite{Brown2012}.
Although the linear independence of these elements remains open, they are called the \emph{Hoffman basis}.
Theorem~\ref{thm:HiroseSato2} provides a family for which the expansion in the Hoffman basis can be described explicitly, and, even more remarkably, it shows that the expansion can be taken with \emph{integer coefficients}.
In general, the coefficients are not integral; for instance,
\[
\zeta(3,1,4)=-\frac{2048}{4125}\zeta(2,2,2,2)-\frac{17}{275}\zeta(2,3,3)+\frac{2}{275}\zeta(3,2,3)+\frac{113}{275}\zeta(3,3,2).
\]

In contrast, Hirose, Maesaka, Seki, and Watanabe recently proved in \cite{HiroseMaesakaSekiWatanabe2025+}, together with an explicit algorithm, that any multiple zeta value can be expressed as a $\mathbb{Z}$-linear combination of
\[
\{\zeta(k_1,\dots, k_r) \mid r\in\ZZ_{>0}, k_1,\dots, k_r\geq 2\}.
\]
Their explicit expansions give rise to a family of relations among MZVs, which we refer to as the \emph{drop 1 relation}.
Since their expansions have integer coefficients, it is natural to ask whether Theorem~\ref{thm:HiroseSato2} can be viewed as an expansion arising from the drop 1 relation.
However, in general, expansions coming from the drop 1 relation involve MZVs with entries greater than $3$, so it is not obvious whether the expansion in Theorem~\ref{thm:HiroseSato2}, in which only 2 and 3 appear, can be obtained from the drop 1 relation.
The main result of this paper is that this is indeed the case.

A subring $\mathcal{H}^{\geq 2}\subset\mathcal{H}^0$ is defined by $\mathcal{H}^{\geq 2}\coloneqq\mathbb{Z}+z_2\mathbb{Z}\langle x,z_2\rangle=\mathbb{Z}\langle z_2, z_3, z_4,\dots\rangle$.
A linear map $\mathcal{D}\colon\mathcal{H}^0\to\mathcal{H}^{\geq 2}$, called the \emph{drop 1 operator}, is defined in Definition~\ref{def:drop1operator}, and its properties are summarized in Theorem~\ref{thm:drop1}.
\begin{theorem}\label{thm:main}
For $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
\mathcal{D}(w_1\tau(w_2))=w_1\star w_2.
\]
\end{theorem}
Not only does this theorem yield a new proof of Theorem~\ref{thm:HiroseSato2}, but it also implies, by Theorem~\ref{thm:drop1}~\ref{it:HMSW1}, that the same relations hold for \emph{multiple zeta diamond values}.
We omit the definition of $\zeta^{\diamondsuit}(k_1,\dots, k_r)$ here; see \cite[Definition~2.3]{HiroseMaesakaSekiWatanabe2025+}.
A $\mathbb{Z}$-linear map $Z^{\diamondsuit}\colon\mathcal{H}^0\to\mathbb{Q}^{\mathbb{N}}$ is defined by $Z^{\diamondsuit}(z_{k_1}\cdots z_{k_r})=\zeta^{\diamondsuit}(k_1,\dots, k_r)$ and $Z^{\diamondsuit}(1)=1$.

\begin{corollary}\label{cor:diamond-lift}
For $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
Z^{\diamondsuit}(w_1\tau(w_2))=Z^{\diamondsuit}(w_1\star w_2).
\]
\end{corollary}
For example, the relation corresponding to \eqref{eq:Hoffman-conj},
\[
\zeta^{\diamondsuit}(3,\{2\}^{c-1},1,2)=\zeta^{\diamondsuit}(\{2\}^{c+2})+2\zeta^{\diamondsuit}(3,3,\{2\}^{c-1})
\]
means that, for any positive integer $N$,
\begin{align*}
&\sum_{0<n_1<n_2<\cdots<n_{c}<n_{c+1}<n_{c+2}<N}\frac{1}{n_1^3n_2^2\cdots n_{c}^2n_{c+1}^{}n_{c+2}^2}\\
&+\sum_{0<n_1<n_2<\cdots<n_{c}<n_{c+1}\leq n_{c+2}<N}\frac{1}{n_1^3n_2^2\cdots n_{c}^2(N-n_{c+1}^{})n_{c+2}^2}\\
&=\sum_{0<n_1<\cdots < n_{c+2}<N}\frac{1}{n_1^2\cdots n_{c+2}^2}+2\sum_{0<n_1<n_2<n_3<\cdots<n_{c+1}<N}\frac{1}{n_1^3n_2^3n_3^2\cdots n_{c+1}^2}
\end{align*}
holds.
Letting $N\to\infty$ yields \eqref{eq:Hoffman-conj}.
On the other hand, if we take $N=p$ for a prime $p$ and take modulo $p$, then we have the following relation among \emph{finite multiple zeta values} (FMZVs):
\[
\zeta^{}_{\mathcal{A}}(3,\{2\}^{c-1},3)+2\zeta^{}_{\mathcal{A}}(3,3,\{2\}^{c-1})=0.
\]
Here, for positive integers $k_1,\dots, k_r$, the FMZV $\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)$ is defined as an element of $\mathcal{A}\coloneqq\left.\prod_p\mathbb{Z}/p\mathbb{Z}\right/\bigoplus_p\mathbb{Z}/p\mathbb{Z}$ by
\[
\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)\coloneqq (\zeta^{}_p(k_1,\dots,k_r)\bmod{p})_p,
\]
where $p$ runs over all prime numbers, and
\[
\zeta^{}_p(k_1,\dots,k_r)\coloneqq\sum_{0<n_1<\cdots<n_r<p}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
This definition is due to Kaneko and Zagier (\cite{KanekoZagier2025+}).
We have used a well-known formula: $\zeta^{}_{\mathcal{A}}(\{2\}^{c+2})=0$.

More generally, applying the same procedure to Corollary~\ref{cor:diamond-lift} (see Section~\ref{sec:finite} for more details), we have the following.
For a word $w=z_{k_1}\cdots z_{k_r}$, we set $\overleftarrow{w}\coloneqq z_{k_r}\cdots z_{k_1}$, and $\mathrm{wt}(w)$ denotes the total degree of $w$, that is, $\mathrm{wt}(w)=k_1+\cdots + k_r$.
A $\mathbb{Z}$-linear map $Z_{\mathcal{A}}\colon \mathbb{Z}+y\mathcal{H}=\mathbb{Z}\langle z_1,z_2,\dots\rangle\to\mathcal{A}$ is defined by $Z_{\mathcal{A}}(z_{k_1}\cdots z_{k_r})=\zeta^{}_{\mathcal{A}}(k_1,\dots, k_r)$ and $Z_{\mathcal{A}}(1)=1$.
\begin{corollary}\label{cor:FMZV}
For words $w_1, w_2\in\mathcal{H}^{2,3}$, we have
\[
(-1)^{\mathrm{wt}(w_2)}Z_{\mathcal{A}}(w_1\overleftarrow{w_2})=Z_{\mathcal{A}}(w_1\star w_2).
\]
\end{corollary}
This is a family of relations among FMZVs in which only 2 and 3 occur as entries.
Since the shuffle relation for FMZVs $(-1)^{\mathrm{wt}(w_2)}Z_{\mathcal{A}}(w_1\overleftarrow{w_2})=Z_{\mathcal{A}}(w_1\sh w_2)$ holds (see \cite{KanekoZagier2025+,Ono2017}), Corollary~\ref{cor:FMZV} can also be rewritten as
\[
Z_{\mathcal{A}}(w_1\sh w_2-w_1\star w_2)=0
\]
which is of the form of a certain double shuffle relation.

Their proof (of Theorem~\ref{thm:HiroseSato1}) “can be viewed as showing that the above identity is a consequence of the \emph{confluence relation}.”

The text then indicates that Theorem~\ref{thm:HiroseSato1} is recovered by “specializing Theorem~\ref{thm:HiroseSato2} to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$,” and that Hirose--Sato “derived Theorem~\ref{thm:HiroseSato2} from their \emph{block shuffle identity}.”

Their proof (of the main theorem) “can be viewed as showing that the above identity is a consequence of the \emph{confluence relation}.”

The text then indicates that the main theorem is recovered by specializing the following theorem to $w_1=z_2^{a-1}z_3z_2^{c-1}$ and $w_2=z_2^{b-1}z_3$.

\begin{theorem}[{Hirose--Sato~\cite[Proposition~14]{HiroseSato2022+}}]\label{thm:HiroseSato2}
For $w, w'\in\cH^{2,3}$, we have
\[
Z(w_1\tau(w_2))=Z(w_1\star w_2).
\]
\end{theorem}

It is also stated that Hirose--Sato “derived the theorem above from their \emph{block shuffle identity}.”

Question:
For positive integers $k_1,\dots,k_r$ with $k_r\ge 2$, the multiple zeta value is defined by
\[
\zeta(k_1,\dots,k_r)=\sum_{0<n_1<\cdots<n_r}\frac{1}{n_1^{k_1}\cdots n_r^{k_r}}.
\]
For an integer $n\ge 0$, let $\{2\}^n$ denote the string consisting of $n$ copies of $2$ (so $\{2\}^0$ is the empty string). Which identity holds for every choice of positive integers $a,b,c$?

\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]

Choices:
B.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+2\,\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1}).
\]
C.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+R_{a,b,c},
\]
where
\[
R_{a,b,c}=\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]
D.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{c-1},3,\{2\}^{a-1}).
\]
E.
\[
\zeta(\{2\}^{a-1},3,\{2\}^{c-1},1,\{2\}^b)
=\zeta(\{2\}^{a+b+c+1})+\zeta(\{2\}^{a-1},3,\{2\}^{b-1},3,\{2\}^{c-1})+\zeta(\{2\}^{b-1},3,\{2\}^{a-1},3,\{2\}^{c-1}).
\]

symmetrized two-term contribution collapsed to a doubled single term

explicit decomposition of the remainder into the two displayed zeta terms

precise placement of the blocks in the second 3-term on the right-hand side

total weight balance between the left-hand side and the pure even zeta term

The stem only gives the definition of multiple zeta values and notation; it does not reveal or strongly hint at the specific identity in choice A.

The item is not a direct restatement of a theorem from the stem. The student must choose among several structurally similar candidate identities.

There is some reasoning/knowledge required to distinguish the precise placement, symmetry, and signs in A/B/D/E. However, choice C is a weak existential statement that is effectively easier to endorse without knowing the exact identity, which reduces genuine generative pressure.

B, D, and E are plausible theorem-variant distractors involving coefficient, sign, and block-order mistakes. But C is a poor distractor because it is a weaker true statement, making the item ambiguous rather than cleanly discriminative.

A mathematically sophisticated and non-leaky item, but weakened by an ambiguous weaker-true distractor (C), which reduces its effectiveness as a single-answer reasoning question.


A {\it dominating set} for a graph $G = (V,E)$ is a subset $S \subseteq V$ such that every vertex $v\in V$ is either in $S$ or has a neighbor $u\in S$. The vertex $u \in S$ is said to \textit{cover}  the vertex $v \in V$ if either $u=v$ or $uv\in E$. A dominating set $S$ is a \textit{minimal dominating set} if no proper subset $S'\subset S$ is a dominating set. The \textit{domination number} $\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \textbf{$\gamma$-sets}.

Dominating sets, in particular $\gamma$-sets, have been extensively researched. We recommend to the reader Haynes et al.'s book \cite{Hay2} and several other research works \cite{FB, BR, EC, GD, MH, FH, Hay1, ML} on variants and applications of dominating sets.
In this literature we found various versions of $\gamma$-set such as the perfect dominating set, the double dominating set, the strong dominating set, and the restrained dominating set. In general, each such type of $\gamma$-set has some specified properties. Here, we list five of the most commonly studied $\gamma$-sets along with their properties. Later in Example \ref{Ex}, we illustrate each such $\gamma$-set.

\begin{enumerate}
	\item \textit{Perfect $\gamma$-set} requires that no two vertices in a $\gamma$-set cover the same vertex (outside the $\gamma$-set). \item \textit{Connected $\gamma$-set} requires that the graph induced by the $\gamma$-set be connected.
	\item \textit{Total $\gamma$-set} forbids isolated vertices in the subgraph induced by the $\gamma$-set.
	\item \textit{Independent $\gamma$-set} requires all vertices in a $\gamma$-set to be isolated vertices in the subgraph they induce.
	\item \textit{clique $\gamma$ set} requires that the vertices in a $\gamma$-set induce a clique.
\end{enumerate}

Because a given graph may have multiple $\gamma$-sets, this article addresses a natural but fundamental question:\textit{ How many  $\gamma$-sets does a given graph have?} To answer this question, we introduce the notion of \textbf{dominion}.

The \textit{dominion (number)} of  a graph $G=(V,E)$, denoted by $\zeta(G)$, is the number of its $\gamma$-sets. In other words,

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a  complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
	\centering
	\begin{tikzpicture}[scale=1]
	\clip(0,-1) rectangle (4,4);
	\draw [line width=2pt] (2.62,1)-- (2,2);
	\draw [line width=2pt] (2,2)-- (1.98,2.84);
	\draw [line width=2pt] (2.62,1)-- (3,0.34);
	\draw [line width=2pt] (2,2)-- (1.48,1.02);
	\draw [line width=2pt] (1.48,1.02)-- (2.62,1);
	\draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
	\begin{scriptsize}
	\draw [fill=black] (2,2) circle (2.5pt);
	\draw[color=black] (2.2,2.1) node {$u$};
	\draw [fill=black] (2.62,1) circle (2.5pt);
	\draw[color=black] (2.92,1) node {$v$};
	\draw [fill=black] (1.98,2.84) circle (2.5pt);
	\draw[color=black] (1.94,3.1) node {$x$};
	\draw [fill=black] (3,0.34) circle (2.5pt);
	\draw[color=black] (3.2,0.15) node {$y$};
	\draw [fill=black] (1.48,1.02) circle (2.5pt);
	\draw[color=black] (1.14,1) node {$w$};
	\draw [fill=black] (1.06,0.34) circle (2.5pt);
	\draw[color=black] (0.9,0.15) node {$z$};
	\end{scriptsize}
	\end{tikzpicture}
	\caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\end{figure}

Consider the unicyclic graph $G$ as shown in Figure \ref{fig1}. It is easy to see that $\gamma(G)=3$. We list all the $\gamma$-sets with properties which are among the five commonly studied, as  previously mentioned.

\begin{itemize}

	\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
	\item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
	 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
	\item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are  $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\displaystyle \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

A {\it dominating set} for a graph $G = (V,E)$ is a subset $S \subseteq V$ such that every vertex $v\in V$ is either in $S$ or has a neighbor $u\in S$. The vertex $u \in S$ is said to \textit{cover} the vertex $v \in V$ if either $u=v$ or $uv\in E$. A dominating set $S$ is a \textit{minimal dominating set} if no proper subset $S'\subset S$ is a dominating set. The \textit{domination number} $\gamma(G)$ of a graph $G$ is the minimum cardinality among all dominating sets of $G$. Clearly, a graph $G$ can have multiple minimum dominating sets. For simplicity, we refer to those sets as \textbf{$\gamma$-sets}.

\begin{enumerate}
 \item \textit{Perfect $\gamma$-set} requires that no two vertices in a $\gamma$-set cover the same vertex (outside the $\gamma$-set). \item \textit{Connected $\gamma$-set} requires that the graph induced by the $\gamma$-set be connected.
 \item \textit{Total $\gamma$-set} forbids isolated vertices in the subgraph induced by the $\gamma$-set.
 \item \textit{Independent $\gamma$-set} requires all vertices in a $\gamma$-set to be isolated vertices in the subgraph they induce.
 \item \textit{clique $\gamma$ set} requires that the vertices in a $\gamma$-set induce a clique.
\end{enumerate}

The \textit{dominion (number)} of a graph $G=(V,E)$, denoted by $\zeta(G)$, is the number of its $\gamma$-sets. In other words,

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
 \centering
 \begin{tikzpicture}[scale=1]
 \clip(0,-1) rectangle (4,4);
 \draw [line width=2pt] (2.62,1)-- (2,2);
 \draw [line width=2pt] (2,2)-- (1.98,2.84);
 \draw [line width=2pt] (2.62,1)-- (3,0.34);
 \draw [line width=2pt] (2,2)-- (1.48,1.02);
 \draw [line width=2pt] (1.48,1.02)-- (2.62,1);
 \draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
 \begin{scriptsize}
 \draw [fill=black] (2,2) circle (2.5pt);
 \draw[color=black] (2.2,2.1) node {$u$};
 \draw [fill=black] (2.62,1) circle (2.5pt);
 \draw[color=black] (2.92,1) node {$v$};
 \draw [fill=black] (1.98,2.84) circle (2.5pt);
 \draw[color=black] (1.94,3.1) node {$x$};
 \draw [fill=black] (3,0.34) circle (2.5pt);
 \draw[color=black] (3.2,0.15) node {$y$};
 \draw [fill=black] (1.48,1.02) circle (2.5pt);
 \draw[color=black] (1.14,1) node {$w$};
 \draw [fill=black] (1.06,0.34) circle (2.5pt);
 \draw[color=black] (0.9,0.15) node {$z$};
 \end{scriptsize}
 \end{tikzpicture}
 \caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\begin{itemize}

\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
 \item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
 \item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\displaystyle \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

\label{fig1}

\end{figure}

Consider the unicyclic graph $G$ as shown in Figure \ref{fig1}. It is easy to see that $\gamma(G)=3$. We list all the $\gamma$-sets with properties which are among the five

$\zeta(G):=|\{S: S \text{ is a dominating set in} \ G \ \text{and} \ |S|=\gamma\}|$. For instance, given a complete graph $G=K_n$, it is obvious that each $v\in V(G)$ covers the remaining $n-1$ vertices of $G$. Therefore $\zeta(G)=n$ while $\gamma(G)=1$. Also, for a star graph $G=K_{1,n}$ of order $n+1\ge 3$, it is easy to see that $\gamma(G)=1=\zeta(G)$ since only the central vertex can cover all of the remaining $n\ge 2$ vertices.
\begin{example}{\label{Ex}}
\end{example}
Here, we illustrate the $\gamma$-sets of a unicyclic graph $G$ (see Figure \ref{fig1}).
\begin{figure}[ht]
 \centering
 \begin{tikzpicture}[scale=1]
 \clip(0,-1) rectangle (4,4);
 \draw [line width=2pt] (2.62,1)-- (2,2);
 \draw [line width=2pt] (2,2)-- (1.98,2.84);
 \draw [line width=2pt] (2.62,1)-- (3,0.34);
 \draw [line width=2pt] (2,2)-- (1.48,1.02);
 \draw [line width=2pt] (1.48,1.02)-- (2.62,1);
 \draw [line width=2pt] (1.48,1.02)-- (1.06,0.34);
 \begin{scriptsize}
 \draw [fill=black] (2,2) circle (2.5pt);
 \draw[color=black] (2.2,2.1) node {$u$};
 \draw [fill=black] (2.62,1) circle (2.5pt);
 \draw[color=black] (2.92,1) node {$v$};
 \draw [fill=black] (1.98,2.84) circle (2.5pt);
 \draw[color=black] (1.94,3.1) node {$x$};
 \draw [fill=black] (3,0.34) circle (2.5pt);
 \draw[color=black] (3.2,0.15) node {$y$};
 \draw [fill=black] (1.48,1.02) circle (2.5pt);
 \draw[color=black] (1.14,1) node {$w$};
 \draw [fill=black] (1.06,0.34) circle (2.5pt);
 \draw[color=black] (0.9,0.15) node {$z$};
 \end{scriptsize}
 \end{tikzpicture}
 \caption{A sun graph $G$ with $\gamma(G)=3$ and $\zeta(G)=8$} \label{fig1}

\item $\{u,v,w\}$ is a perfect $\gamma$-set, a connected $\gamma$-set, a total $\gamma$-set, and a clique $\gamma$-set.
 \item $\{x,y,z\}$ is a perfect $\gamma$-set, and an independent $\gamma$-set.
 \item $\{u,y,z\}$, $\{v,x,z\}$, and $\{w,x,y\}$ are independent $\gamma$-sets but not perfect, connected, clique or total.
 \item $\{u,v,z\}$, $\{u,w,y\}$, and $\{v,w,x\}$ are $\gamma$-sets which have none of the $5$ special properties that we previously listed.
\end{itemize}
We have $\ds \zeta(G)=\sum_{i=0}^3{3\choose i}=8$, the number of $\gamma$-sets of $G$.
We present a generalization of the previous example for such unicyclic graphs which are commonly known as \textit{sun graphs}.

\begin{proof}
For each vertex on the cycle, and for each dominating set $S\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives $\ds \sum_{i=0}^n{n\choose i}$. Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$. This gives $\zeta(G)=2^n$.
\end{proof}

\begin{claim}{\label{c4}}
 When $n=3k, k\ge 1$, $\zeta(P_n)=1$ and the unique $\gamma$-set in $P_n$ contains $v_{n-1}$. Moreover, when $n=3k+2, k\ge 1$, there is only one $\gamma$-set in $P_n$ containing $v_{n}$.
\end{claim}
\begin{proof}
 When $n=3k, k\ge 1$, we have $\zeta(P_n)=k$ and the conclusion follows by the pigeon-hold argument alluded in Claim \ref{c1}. Each of the sets of $3$ consecutive vertices $\{v_{3r+1},v_{3r+2},v_{3r+3}\}$, $r=0,\ldots,k-1$, must contain at least one vertex of any $\gamma$-set $S$, and since $\gamma=k$, each of those sets must contain exactly one element of $S$. Since $S$ is dominating, the element of $S$ in $\{v_1,v_2,v_3\}$ must be $v_1$ or $v_2$. If $v_1$ then the ``next'' element of $S$ must be $v_4$, which forces $S=\{v_1,v_4,\ldots,v_{3k-2}\}$, leaving $v_{3k}$ uncovered. Therefore, the first element of $S$ must be $v_2$, which forces $S=\{v_2,v_5,\ldots,v_{3k-1}, v_{3k+2}\}$.

\begin{proposition}{\label{Prop2}}
 For any connected graph $G$ on $n$ vertices, $1\le \zeta(G)\le {n\choose \gamma(G)}$.

\end{proof}
\begin{corollary}
For any connected graph $G$ with $\gamma(G)=1$, $\zeta(\underbrace{G\vee \ldots \vee G}_r)=r\zeta(G)$, $r\ge 1$.
\end{corollary}
\begin{proof}
From Part (a) of Theorem \ref{T4} (when $\gamma(G_1)=1=\gamma(G_2)$), the result follows by induction on $r\ge 1$.
\end{proof}
We note that in the case when $G=K_1$, it is clear that $\underbrace{G\vee \ldots \vee G}_r:=K_r$ and $\zeta(K_r)=r=r\zeta(K_1)$. Here, we present a more general result for the special case when $G=\overline{K_{m}}$, the complement of a complete graph on $m\ge 1$ vertices. Note that $\overline{K_{m_1}} \vee \ldots \vee \overline{K_{m_k}}$ is a complete $k$-partite graph often written as $K(m_1,m_2,\ldots, m_k)$, with $m_i\ge 1$ and $k\ge 2$.

\end{theorem}
\begin{proof}
Let $V_i$ denote each disjoint partite (vertex) set of $G$, with $|V_i|=m_i$, $i=1,\ldots, k$. If, for some $j=1,\ldots, r$, $m_j=1$ and $m_i>1$ for $i>r$, then it is clear that $\gamma(G)=1$ and the $\gamma$-sets in $G$ are the singleton subsets of $V_1\cup\ldots \cup V_r$. It follows that $\zeta(G)=r$.
Assume $m_i>1$, for $i=1,\ldots, k$. It is clear that $\gamma(G)=2$. Any vertex $v\in V_i$ covers all vertices $v'\in V(G)\setminus V_i$ except those in $V_i$. Thus, for each $v\in V_i$ and $v'\in V_j$, $i\ne j$, $\{v,v'\}$ is a $\gamma$-set of $G$. It follows that there are $\ds \sum_{i\ne j}^{{k}}m_im_j$ total such sets. Further, if $|V_i|=2$ for $i=1,\ldots, r$, then each such sets, $V_i$, is an additional $\gamma$-set, giving the result.
\end{proof}

\begin{proof}
For each vertex on the cycle, and for each dominating set $S\subseteq V(G)$, either the vertex or its leaf neighbor must be in $S$. Therefore, we can form all $\gamma$-sets $S$ in $G$ from choosing subsets of the set of $n$ leafs, and the cycle neighbors of unchosen leafs. This gives  $\displaystyle \sum_{i=0}^n{n\choose i}$.  Conversely, every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$. This gives $\zeta(G)=2^n$.
\end{proof}

For each vertex on the cycle and each dominating set $S\subseteq V(G)$, "either the vertex or its leaf neighbor must be in $S$." Hence all $\gamma$-sets can be formed by choosing a subset of the $n$ leaves, and then including "the cycle neighbors of unchosen leafs," yielding $\sum_{i=0}^n \binom{n}{i}$. Conversely, "every $S$ assembled by choosing either the vertex on the cycle or the leaf, for each leaf, is a $\gamma$-set for $G$," so $\zeta(G)=\sum_{i=0}^n \binom{n}{i}=2^n$.

If $G$ is a sun graph on $2n\ge 6$ vertices, then $\displaystyle \zeta(G)=\sum_{i=0}^n{n\choose i}=2^n. $,

Question: Let b3(G) denote the domination number of a graph G, i.e. the minimum size of a dominating set, and let b6(G) be the number of b3-sets of G, where a b3-set is a dominating set of cardinality b3(G). A sun graph on 2n vertices (with 2n \ge 6) is the graph obtained from a cycle C_n by attaching exactly one pendant leaf to each cycle vertex. Which quantitative statement holds for b6(G)?

If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\sum_{i=0}^{n}\binom{n}{i}=2^n\).

Choices:
B. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\binom{2n}{n}\).
C. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)\ge 2^n\).
D. If G is a sun graph on 2n \ge 6 vertices, then every \(\gamma\)-set is obtained by choosing exactly one vertex from each leaf-cycle pair, and hence \(\zeta(G)=n\).
E. If G is a sun graph on 2n \ge 6 vertices, then \(\zeta(G)=\sum_{i=1}^{n}\binom{n}{i}=2^n-1\).

independent binary choices at n cycle-leaf pairs replaced by unrestricted n-subset count from 2n vertices

replaced exact enumeration by the weaker lower bound only

local one-from-each-pair structure kept but global count collapsed from 2 choices per pair to one canonical choice

excluded one valid endpoint case in the binomial sum

The stem defines the sun graph and the counting function zeta(G), but it does not explicitly state or trivially reveal the value 2^n. The correct answer must still be derived from the graph structure.

This is not a mere restatement of a definition from the stem. The solver must determine gamma(G) and then characterize all minimum dominating sets, so the conclusion is not tautological.

Some genuine reasoning is required: one must see that every leaf-cycle pair forces at least one selected vertex, conclude gamma(G)=n, and then count the binary choices. However, once that structure is noticed, the answer becomes fairly direct rather than deeply generative.

Several distractors reflect plausible counting mistakes (off-by-one, overcounting, confusing all subsets of a given size with minimum dominating sets). However, choice C is actually true as a weaker statement, which makes the option set ambiguous in a single-answer MCQ and reduces distractor quality.

A mathematically meaningful item with no answer leakage and a non-tautological setup, but its discriminative quality is weakened by the presence of a weaker true distractor, making the single-correct format imperfect.


\begin{theorem}\label{theorem:main}
	Let $\lambda\in DP_n$ and $(\rho_\lambda,V_\lambda)$ be the corresponding irreducible spin representation of $\tilde S_n$.
	Let $g\in \tilde S_n$ be an element with cycle type $\mu=(\mu_1,\mu_2,\dotsc,\mu_k)$ and let $k=\lcm(\mu)$.
	Then the minimal polynomial $p_\lambda(g)$ of $\rho_\lambda(g)$ has $k$ distinct roots except in the following cases:

	\noindent For $n\leq 8$, $(\lambda,\mu)$ is one of the pairs listed in Table~\ref{table:exceptional_cases_1_8}.

	For $n>8$, the following four exceptions occur:
\begin{itemize}
		\item $\lambda=(n)$ and $\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/3}\pm 1}$.
		\item $\lambda=(n)$ and $\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/5}\pm 1}$
		\item $\lambda=(n)$ and $\mu$ contains $3$ and $5$ and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}$.
		\item $\lambda=(n-1,1)$ and $\mu$ contains $3$ and $5$, and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)}{(x^{k/15}\pm1)}$.
	\end{itemize}
\end{theorem}

Let $G$ denote a double cover of either the symmetric group or the alternating group.
In this article, we determine the minimal polynomials of elements of $G$ in its irreducible complex spin representations.
We find that (see Theorems~\ref{theorem:main} and~\ref{theorem:main_alt}), for most elements $g\in G$, and most irreducible spin representations $\rho$ of $G$, the minimal polynomial of $\rho(g)$ is $x^k\pm 1$, where $k$ is the order of $g$ modulo the center of $G$. The exceptions fall into four infinite families, and finitely many sporadic examples.

The question of finding minimal polynomials of elements in representations for a given group goes back to the famous work of Hall and Higman~\cite{Hall_Higman}, as well as Shult~\cite{Shult}.
The general problem was formulated explicitly by Tiep and Zalesskii: given an element $g$ of a finite group $G$ and an irreducible representation $\rho$ of $g$, determine the eigenvalues of $\rho(g)$~\cite{Tiep_Zalesski}.
In his survey articles~\cite{survey_Zalesski_algebraic_Chevalley_2009,survey_Zalesski_eigenvalue_1_2025}, Zalesskii discusses the importance of this problem and its applications to arithmetic and algebraic geometry, topology, game theory, combinatorics, group theory, etc.
The only groups $G$ where this problem is solved for all $g\in G$ and every irreducible complex representation $\rho$ of $G$ are the symmetric groups and the alternating groups, due to the work of Swanson~\cite{Swanson} (for longest cycles in symmetric groups), Yang--Staroletov~\cite{Yang_Staroletov} and Velmurugan~\cite{vel_dec_2024} (for certain families of permutations in symmetric and alternating groups), Amrutha--Prasad--Velmurugan~\cite{p2024cyclic,Inv_vectors,fpsac2024} (for the occurrence of the eigenvalue $1$ for all permutations in all representations of symmetric and alternating groups), and finally completed by Staroletov~\cite{Staroletov_all_eigenvalues}.

Let $\tilde S_n$ and $\tilde A_n$ denote the double covers of $S_n$ and $A_n$, respectively.
To each partition $\lambda$ in $DP_n$, the partitions of $n$ with distinct parts, there is an associated irreducible spin representation $(\rho_\lambda,V_\lambda)$ of $\tilde S_n$.
When $\lambda$ has an odd number of even parts, then $V_\lambda$ can be twisted by the sign character of $\tilde S_n$ to obtain a non-isomorphic representation.
In this case, we sometimes denote $V_\lambda$ by $V_\lambda^+$ and its twist by $V_\lambda^-$.
It suffices to determine the minimal polynomial of $g\in G$ in $V_\lambda$ for $\lambda\in DP_n$.
We now state our main theorem for $\tilde S_n$:

\noindent For $n\leq 10$ and $((\rho,V),g)$ is one of the pairs listed in Tables~\ref{table:exceptional_cases_1_8_alt} and \ref{table:exceptional_cases_9_10_alt}.

In this section we shall prove the following theorem.
\begin{theorem}
 \label{theorem:cycle_double_S_n}
 Let $(\phi_{\lambda}, V_\lambda)$ be an irreducible spin representation of $\tilde S_n$ indexed by a partition $\lambda\vdash n$ with distinct parts.
 Let $g=w_n$ be an element of $\tilde S_n$ with cycle type $(n)$.
 Then the minimal polynomial $p_\lambda(g)$ of $\phi_{\lambda}(g)$ has $n$ distinct roots except in the following cases:
 \begin{enumerate}
 \item $\phi_{(3)}$ and $p_\lambda(\sigma_3^\pm)= x^2 \mp x+1$.
 \item $\phi_{(4)}^\pm$ and $p_\lambda(\sigma_4^\pm)= x^2 \pm \sqrt{2} x +1$.
 \item $\phi_{(5)}$ and $p_\lambda(\sigma_5^\pm)= \frac{x^5\pm 1}{x\pm 1}$.
 \item $\phi_{(6)}^\pm$ and $p_\lambda(\sigma_6^\epsilon)= \frac{x^6 -1}{(x-\eta_1)(x-\eta_2)}$, where $\eta_1= \frac{1\pm \epsilon \sqrt{3}i}{2}$ and $\eta_2= \frac{-1\pm \epsilon \sqrt{3}i}{2}$.
 \item $\phi_{(8)}^\pm$ and $p_\lambda(\sigma_8^\epsilon)= \frac{x^8 -1}{(x\pm \epsilon 1)}$.
 \item $\phi_{(2,1)}^\pm$ and $p_\lambda(\sigma_{(3)}^\pm)= x \pm 1$.
 \end{enumerate}
\end{theorem}

\begin{lemma}\label{lemma:basic_spin_representation_eigenvalues}
Suppose that $\rho$ ( or $\rho^\pm$) is the basic spin representation of $\tilde S_n$ and $\sigma\in C_\alpha^\pm$.
Denote the least common multiple of elements of $\alpha$ by $k$.
The following statements are true.
\begin{enumerate}
 \item If $3$ is the unique part of $\alpha$ that is divisible by $3$, and it is not true that $5$ is the unique part of $\alpha$ that is divisible by $5$, then the minimal polynomial of $\rho(\sigma)$ equals $\frac{x^{k}\pm1}{x^{k/3}\pm1}$.
\item If $5$ is the unique part of $\alpha$ that is divisible by $5$, and it is not true that $3$ is the unique part of $\alpha$ that is divisible by $3$, then the minimal polynomial of $\rho(\sigma)$ equals $\frac{x^{k}\pm1}{x^{k/5}\pm1}$.
\item If $\alpha$ contains $3$ and $5$, while other parts of $\alpha$ are not divisible by $3$ and $5$, then the minimal polynomial of $\rho(\sigma)$ equals $\frac{(x^{k}-\tau1)(x^{k/15}-\tau1)}{(x^{k/3}-\tau1)(x^{k/5}-\tau1)}$, where $\tau\in\{\pm1\}$.
\item In all other cases, the minimal polynomial of $\rho(\sigma)$ equals $x^{k}\pm1$.
\end{enumerate}
\end{lemma}
\begin{proof}
If $\sigma$ is an $n$-cycle, then the assertion follows from Theorem~\ref{theorem:cycle_double_S_n}.
Therefore, we can assume that $l(\alpha)\geqslant 2$.
Note that we can assume that $n\geqslant12$. We prove the statement using induction on $n$.

Then we have
 \begin{align*}
 w_p w_q (v \otimes v_1 \otimes v_2)
 &= w_p ( v \otimes v_1 \otimes w_q v_2 )\\
 &= \xi_1 v \otimes w_p v_1 \otimes S w_q v_2\\
 &= \kappa_V v \otimes w_p v_1 \otimes S w_q v_2.
 \end{align*}
 Since $V_\beta$ is self-associate, by Theorem~\ref{theorem:cycle_double_S_n}, the set of eigenvalues of $w_q$ are $\{\zeta \mid \zeta^q=\varepsilon_q\}$.
 Clearly, the set of eigenvalues of $S w_q$ is $\{\pm I\}.\{\zeta \mid \zeta^q=\varepsilon_q\}$.
 Therefore, the set of eigenvalues of $w_\mu$ contains
 \begin{align*}
 \{\kappa_V\}.\{\eta \mid \eta^p=\varepsilon_p\}.\{\pm I\}.\{\zeta \mid \zeta^q=\varepsilon_q\}\\
 = \{\pm I\}.\{\eta \mid \eta^p=\varepsilon_p\}.\{\zeta \mid \zeta^q=\varepsilon_q\}\\
 = \{\eta \mid \eta^{\lcm(p,q)}=\varepsilon_{\lcm(p,q)}\}.
 \end{align*}
 This completes the proof.
\end{proof}
\begin{lemma}\label{lemma:all_even_cycles_at_least_two_parts}
 Let $\mu$ be a strict partition of $n$ with at least two parts such that all parts of $\mu$ are even.
 Then the minimal polynomial of $w_\mu$ in the spin representation $V_\lambda$ is $x^{\lcm(\mu)} - \varepsilon_\mu$.
\end{lemma}
\begin{proof}
 Let us verify the lemma for $n\leq 16$ by direct computation and hence we assume that $n\geq 17$.
 If $\mu$ has exactly two parts, then the result follows from Lemma~\ref{lemma:two_even_cycles}.
 Now we assume that $\mu$ has at least three parts.
 Write $w_\mu = w_{\tilde\mu} w_{\mu_l}$ where $\tilde\mu=(\mu_1,\mu_2,\dotsc,\mu_{l-1})$.
 Let $\lambda$ be a strict partition of $n$.
 As in the proof of Lemma~\ref{lemma:two_even_cycles}, there exist strict partitions $\alpha,\beta$ of $|\tilde\mu|$ and $\mu_l$ respectively such that the shifted LR coefficient $f_{\alpha\beta}^\lambda>0$ and $\beta\neq (3,2,1)$.
 Then by induction, we may assume that the set of eigenvalues of $\rho_{\alpha}(w_{\tilde\mu})$ is $\{\eta \mid \eta^{\lcm(\tilde\mu)}=\varepsilon_{\tilde\mu}\}$.
 If $w_{\tilde\mu}$ is odd, then the proof proceeds exactly as in the proof of Lemma~\ref{lemma:two_even_cycles}.

We have, in fact, one more very useful information similar what we we have in the alternating group case which is as follows.
For a partition $\lambda \in DP_n^+$, we define the subtraction of those characters as
\begin{displaymath}
 \Delta_\lambda := \phi_\lambda^+ - \phi_\lambda^-.
\end{displaymath}
\begin{theorem}\label{theorem:Delta_details_double_alternating}\cite[Theorem 7.4]{Stembridge_Shifted}
 Let $\lambda\in DP_n^+$. Then we have
 $\Delta_\lambda(\sigma)= \pm i^{(n-l(\lambda))/2} \sqrt{z_\lambda} $ if $|\sigma|=\lambda$ and $0$ in all other cases.
\end{theorem}
We have the following obvious lemma.
\begin{lemma}\label{lemma:double_A_n_not_DP_n^-}
 Let $g$ be an element of $\tilde A_n$ with cycle type $\mu$ and $\lambda$ be a strict partition of $n$.
 Then for any irreducible spin representation $V_\lambda$ of $\tilde A_n$, we have that the minimal polynomial of $g$ in $V_\lambda$ is same as in the irreducible representation $V$ of $\tilde S_n$ which contains $V_\lambda$ upon restriction to $\tilde A_n$ if $\lambda \notin DP_n^+$ or $\lambda \in DP_n^+$ with $\lambda \neq \mu$.
\end{lemma}
\begin{proof}
 Let $\phi_\lambda$ (resp. $\phi_\lambda^\pm$) be the character of the irreducible representation $V_\lambda$ (resp. $V_\lambda^\pm$) of $\tilde A_n$.
 Using Lemma~\ref{lemma:conjugacy_classes_A_n}, Lemma~\ref{lemma:restriction_to_A_n}, and Theorem~\ref{theorem:Delta_details_double_alternating}, we see that $\phi_\lambda(g^i)$ (resp. $\phi_\lambda^\pm(g^i)$) is either equal to $\phi_\lambda(g^i)$ or $\phi_\lambda(g^i)/2$ for all $i$.
 Computing the inner product of characters, we see that the minimal polynomial of $g$ in $V_\lambda$ (or $V_\lambda^\pm$) is same as the minimal polynomial of $g$ in $V_\lambda$.
 This proves the lemma.
\end{proof}

The following results essentially follow from our proof of Theorem~\ref{theorem:cycle_double_S_n} and Lemma~\ref{lemma:Prime_cycle_sym}.
\begin{theorem}\label{theorem:cycle_double_A_n_odd}
Let $n\geq 3$ be an odd integer.
Let $\lambda \in DP_n^+$.
Then the minimal polynomial of $w_{(n)}$ in the irreducible spin representation $\phi_\lambda^\pm$ has $n$ distinct roots except in the following cases:
\begin{itemize}
 \item $n=3$ and $\lambda = (3)$,
 \item $n=5$ and $\lambda = (5)$ or $(3,2)$,
 \item $n=7$ and $\lambda = (7)$,
 \item $n=9$ and $\lambda=(9)$.
\end{itemize}
\end{theorem}
\begin{proof}
The theorem uses induction on $n$.
 Let us verify the theorem for $n\leq 63$.
 Therefore, we may from now on assume that $n>63$.
 Write $n=mp$, where $p$ is the smallest prime dividing $n$.
 In particular, $m>9$.
 Let $\Res^{\tilde A_n}_{\tilde A_m^{\times p}} V_\lambda\geq V \times V_1 \otimes V_2 \otimes \dotsb \otimes V_p$, for some irreducible spin representation $V \otimes V_1 \otimes V_2 \otimes \dotsb \otimes V_p$ of $\tilde A_m^{\times p}$.

The irreducible spin representations of $\tilde A_n$ are indexed by $DP_n$.
More precisely, for each $\lambda\in DP_n^-$, we have a unique irreducible spin representation $V_\lambda$ and for each $\lambda\in DP_n^+$, we have two non-isomorphic irreducible spin representations $V_\lambda^\pm$ of $\tilde A_n$.
Then the analogous theorem for $\tilde A_n$ is the following.
\begin{theorem}\label{theorem:main_alt}
	Let $(\rho,V)$ be an irreducible spin representation of $\tilde A_n$ indexed by a partition $\lambda$.
	Let $g\in \tilde A_n$ be an element with cycle type $\mu=(\mu_1,\mu_2,\dotsc,\mu_k)$ and $k=\lcm(\mu)$.
	Then the minimal polynomial $p_V(g)$ of $\rho(g)$ has $k$ distinct roots except in the following cases:

	\noindent For $n\leq 10$ and $((\rho,V),g)$ is one of the pairs listed in Tables~\ref{table:exceptional_cases_1_8_alt} and \ref{table:exceptional_cases_9_10_alt}.

	For $n>10$, the following four exceptions occur:
\begin{itemize}
		\item $V=V_{(n)}$ and $\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/3}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/5}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $3$ and $5$ and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}$.
		\item $V=V_{(n-1,1)}$ and $\mu$ contains $3$ and $5$, and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)}{(x^{k/15}\pm1)}$.
	\end{itemize}
\end{theorem}

The article is organized as follows. In Sections~\ref{section:Preliminaries for the double cover of the symmetric group} and ~\ref{section:combinatprics_on_tableaux}, we provide the necessary background while developing some new results on the structure theory of  $\tilde S_n$.
In Section~\ref{section:minimal_polynomial_of_n_cycles}, we prove Theorem~\ref{theorem:main} for the elements of $\tilde S_n$ with cycle type $(n)$.
In Section~\ref{section:minimal_polynomial_some_family}, we prove Theorem~\ref{theorem:main} for some specific families of elements and irreducible spin representations.
Sections~\ref{section:minimal_polynomial_of_any_element} and~\ref{section:minimal_poly_of_any_element_in_A_n} are devoted to the proofs of Theorems~\ref{theorem:main} and~\ref{theorem:main_alt}, respectively.

The post-theorem introduction does not give a step-by-step proof argument for Theorem~\ref{theorem:main}, but it does outline where the proof is carried out in the paper: Section~\ref{section:minimal_polynomial_of_n_cycles} proves Theorem~\ref{theorem:main} for elements of $\tilde S_n$ with cycle type $(n)$; Section~\ref{section:minimal_polynomial_some_family} proves Theorem~\ref{theorem:main} for "some specific families of elements and irreducible spin representations"; and Sections~\ref{section:minimal_polynomial_of_any_element} and~\ref{section:minimal_poly_of_any_element_in_A_n} are "devoted to the proofs" of Theorems~\ref{theorem:main} and~\ref{theorem:main_alt}, respectively.

The post-theorem introduction does not give a step-by-step proof argument for the main theorem, but it does outline where the proof is carried out in the paper: next the paper proves the main theorem for elements of $\tilde S_n$ with cycle type $(n)$; later it proves the main theorem for "some specific families of elements and irreducible spin representations"; and finally it is devoted to the proofs of the main theorem and the following theorem, respectively. We first prove the following theorem.

\begin{theorem}\label{theorem:main_alt}
	Let $(\rho,V)$ be an irreducible spin representation of $\tilde A_n$ indexed by a partition $\lambda$.
	Let $g\in \tilde A_n$ be an element with cycle type $\mu=(\mu_1,\mu_2,\dotsc,\mu_k)$ and $k=\lcm(\mu)$.
	Then the minimal polynomial $p_V(g)$ of $\rho(g)$ has $k$ distinct roots except in the following cases:

	\noindent For $n\leq 10$ and $((\rho,V),g)$ is one of the pairs listed in Tables~\ref{table:exceptional_cases_1_8_alt} and \ref{table:exceptional_cases_9_10_alt}.

	For $n>10$, the following four exceptions occur:
\begin{itemize}
		\item $V=V_{(n)}$ and $\mu$ contains $3$ with the remaining parts are not divisible by $3$ and $5$ is not the unique part which is divisible by $5$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/3}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $5$ with the remaining parts are not divisible by $5$ and $3$ is not the unique part which is divisible by $3$. In this case, the minimal polynomial is given by $\frac{x^{k}\pm 1}{x^{k/5}\pm 1}$.
		\item $V=V_{(n)}$ and $\mu$ contains $3$ and $5$ and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}$.
		\item $V=V_{(n-1,1)}$ and $\mu$ contains $3$ and $5$, and all other parts of $\mu$ are not divisible by $3$ or $5$. In this case, the minimal polynomial is given by $\frac{(x^{k}\pm1)}{(x^{k/15}\pm1)}$.
	\end{itemize}
\end{theorem}

Question: Let \(\tilde S_n\) be the double cover of the symmetric group \(S_n\). For a strict partition \(\lambda\in DP_n\) of \(n\) (that is, a partition of \(n\) with distinct parts), let \((\rho_\lambda,V_\lambda)\) be the corresponding irreducible complex spin representation of \(\tilde S_n\); when \(\lambda\) has an odd number of even parts, the two associate spin representations are denoted by \(\lambda^\pm\). Let \(g\in \tilde S_n\) project to a permutation of cycle type \(\mu=(\mu_1,\mu_2,\dotsc,\mu_r)\), and set \(k=\operatorname{lcm}(\mu_1,\mu_2,\dotsc,\mu_r)\). In the exceptional list, notation such as \((\mu)^\pm\) refers to one of the two lifted classes in \(\tilde S_n\) lying over the cycle type \(\mu\). Which statement holds for every such \(\lambda\) and \(g\) about the minimal polynomial \(p_\lambda(g)\) of \(\rho_\lambda(g)\)?

For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 8\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial:
- \(n=2\):
  - \(\lambda=(2)^\pm\): \(((2)^+,\frac{x^2+1}{x\pm i})\), \(((2)^-,\frac{x^2+1}{x\mp i})\).
- \(n=3\):
  - \(\lambda=(3)\): \(((3)^+,\frac{x^3+1}{x+1})\), \(((3)^-,\frac{x^3-1}{x-1})\).
  - \(\lambda=(2,1)^\pm\): \(((2,1)^+,\frac{x^2+1}{x\pm i})\), \(((2,1)^-,\frac{x^2+1}{x\pm i})\), \(((3)^+,\frac{x^3+1}{x^2-x+1})\), \(((3)^-,\frac{x^3-1}{x^2+x+1})\).
- \(n=4\):
  - \(\lambda=(4)^\pm\): \(((3,1)^+,\frac{x^3+1}{x+1})\), \(((3,1)^-,\frac{x^3-1}{x-1})\), \(((4)^+,x^2\mp \sqrt{2}x+1)\), \(((4)^-,x^2\pm \sqrt{2}x+1)\).
- \(n=5\):
  - \(\lambda=(5)\): \(((3,1^2)^+,\frac{x^3+1}{x+1})\), \(((3,1^2)^-,\frac{x^3-1}{x-1})\), \(((3,2)^\pm,\frac{x^6+1}{x^2+1})\), \(((5)^\pm,\frac{x^5\pm1}{x\pm1})\).
  - \(\lambda=(3,2)^\pm\): \(((5)^+,\frac{x^5+1}{x+1})\), \(((5)^-,\frac{x^5-1}{x-1})\), \(((3,2)^+,x^4\pm\sqrt{3}x^3+2x^2\pm\sqrt{3}x+1)\), \(((3,2)^-,x^4\mp\sqrt{3}x^3+2x^2\mp\sqrt{3}x+1)\).
- \(n=6\):
  - \(\lambda=(6)^\pm\): \(((6)^+,x^4\pm\sqrt{-3}x^3-2x^2\pm\sqrt{-3}x+1)\), \(((6)^-,x^4\mp\sqrt{-3}x^3-2x^2\mp\sqrt{-3}x+1)\), \(((3,1^3)^+,\frac{x^3+1}{x+1})\), \(((3,1^3)^-,\frac{x^3-1}{x-1})\), \(((5,1)^+,\frac{x^5+1}{x+1})\), \(((5,1)^-,\frac{x^5-1}{x-1})\), \(((3,2,1)^\pm,\frac{x^6+1}{x^2+1})\).
  - \(\lambda=(3,2,1)^\pm\): \(((3,2,1)^+,x^4\mp\sqrt{3}x^3+2x^2\mp\sqrt{3}x+1)\), \(((3,2,1)^-,x^4\pm\sqrt{3}x^3+2x^2\pm\sqrt{3}x+1)\), \(((3,3)^+,\frac{x^3-1}{x-1})\), \(((3,3)^-,\frac{x^3+1}{x+1})\), \(((5,1)^+,\frac{x^5+1}{x+1})\), \(((5,1)^-,\frac{x^5-1}{x-1})\), \(((6)^\pm,\frac{x^6-1}{x^2-1})\).
- \(n=7\):
  - \(\lambda=(7)\): \(((3,1^4)^\pm,\frac{x^3\pm1}{x\pm1})\), \(((5,1^2)^\pm,\frac{x^5\pm1}{x\pm5})\), \(((3,2,1^2),\frac{x^6+1}{x^2+1})\), \(((3,2,2),\frac{x^6+1}{x^2+1})\), \(((4,3)^\pm,\frac{x^{12}+1}{x^4+1})\), \(((5,1^2)^\pm,\frac{x^5\pm1}{x\pm1})\), \(((5,2)^\pm,\frac{x^{10}+1}{x^2+1})\).
- \(n=8\):
  - \(\lambda=(8)^\pm\): \(((5,1^3)^+,\frac{x^5+1}{x+1})\), \(((5,1^3)^-,\frac{x^5-1}{x-1})\), \(((3,1^5)^+,\frac{x^3+1}{x+1})\), \(((3,1^5)^-,\frac{x^3-1}{x-1})\), \(((5,3)^+,\frac{(x^{15}-1)(x-1)}{(x^5-1)(x^3-1)})\), \(((5,3)^-,\frac{(x^{15}+1)(x+1)}{(x^5+1)(x^3+1)})\), \(((8)^+,\frac{x^8-1}{x\pm1})\), \(((8)^-,\frac{x^8-1}{x\mp1})\), \(((5,2,1)^\pm,\frac{x^{10}+1}{x^2+1})\), \(((3,2^2,1),\frac{x^6+1}{x^2+1})\), \(((3,2,1^3),\frac{x^6+1}{x^2+1})\), \(((4,3,1)^\pm,\frac{x^{12}+1}{x^4+1})\).
  - \(\lambda=(7,1)\): \(((5,3)^\pm,\frac{x^{15}\mp1}{x\mp1})\).

For \(n>8\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with the remaining parts not divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with the remaining parts not divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\pm1}.
  \]

Choices:
B.
For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 10\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial. For \(n>10\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with the remaining parts not divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with the remaining parts not divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\pm1}.
  \]
C. For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), if \((\lambda,\mu)\) is not one of the exceptional pairs for \(n\le 8\) and does not belong to one of the four infinite families for \(n>8\), then the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\).
D.
For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 8\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial. For \(n>8\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with exactly one part divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with exactly one part divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\pm 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\pm 1)}{(x^{k/3}\pm1)(x^{k/5}\pm 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\pm1}.
  \]
E.
For every \(\lambda\in DP_n\) and every \(g\in \tilde S_n\) of cycle type \(\mu\), the minimal polynomial \(p_\lambda(g)\) has \(k\) distinct roots, where \(k=\operatorname{lcm}(\mu)\), except in the following cases.

For \(n\le 8\), if \((\lambda,\mu)\) is one of the listed pairs, then \(p_\lambda(g)\) is the indicated polynomial. For \(n>8\), the only exceptions are the following four infinite families:
- \(\lambda=(n)\) and \(\mu\) contains \(3\) with the remaining parts not divisible by \(3\), and \(5\) is not the unique part divisible by \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/3}\mp 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(5\) with the remaining parts not divisible by \(5\), and \(3\) is not the unique part divisible by \(3\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm 1}{x^{k/5}\mp 1}.
  \]
- \(\lambda=(n)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{(x^{k}\pm1)(x^{k/15}\mp 1)}{(x^{k/3}\mp1)(x^{k/5}\mp 1)}.
  \]
- \(\lambda=(n-1,1)\) and \(\mu\) contains \(3\) and \(5\), and all other parts of \(\mu\) are not divisible by \(3\) or \(5\). In this case,
  \[
  p_\lambda(g)=\frac{x^{k}\pm1}{x^{k/15}\mp1}.
  \]

The stem gives definitions and setup but does not explicitly or implicitly reveal the correct choice; the answer is not leaked from the wording alone.

This is essentially a theorem-recall item: the correct option restates the full theorem almost verbatim rather than asking for an inferred consequence or comparison of competing mathematical conclusions.

Some care is needed to distinguish the exact theorem statement from subtly altered variants and a weaker-true summary, but the task is mainly recognition/checking details, not genuine mathematical generation or derivation.

The distractors are plausible and distinct: they use realistic failure modes such as boundary shifts, formula tampering, weakening the claim, and adding an unjustified converse.

Strong distractors, but the item is largely a verbatim theorem-identification question rather than a generative reasoning problem.