[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
Abstract. In this paper, we study multiplication formula of \( F \)-polynomial of representations of Hernandez and Leclerc’s quivers with potentials. Since the truncated \( q \)-characters of some real simple modules over a quantum affine group \( U_q(\widehat{\mathfrak{g}}) \) can be expressed in terms of such \( F \)-polynomials, one can describe the product of two simple modules over \( U_q(\widehat{\mathfrak{g}}) \) using this multiplication formula.

Consider a simple Lie algebra \( \mathfrak{g} \). The quantum affine algebra for \( \mathfrak{g} \), denoted as \( U_q(\widehat{\mathfrak{g}}) \), holds significant importance across various disciplines. Central to understanding representations of \( U_q(\widehat{\mathfrak{g}}) \) is the concept of \( q \)-characters, introduced by Frenkel and Reshetikhin [FR98]. However, detailing the \( q \)-characters of all simple modules over \( U_q(\widehat{\mathfrak{g}}) \) proves intricate. To address this complexity, Hernandez and Leclerc introduced the truncated \( q \)-character for simple modules over \( U_q(\widehat{\mathfrak{g}}) \) [HL10]. Subsequently, they expressed these truncated \( q \)-characters of Kirillov–Reshetikhin modules in terms of \( F \)-polynomials of modules over their quivers \( Q_{HL} \) with potentials [HL16]. Extending this to a broader class of simple modules over \( U_q(\widehat{\mathfrak{g}}) \), Kashiwara, Kim, Oh, and Park demonstrated that the truncated \( q \)-characters of real simple modules corresponding to cluster monomials can be described using \( F \)-polynomials of representations of \( Q_{HL} \) with potentials [KKOP24].

Consequently, understanding the product of real simple modules over \( U_q(\widehat{\mathfrak{g}}) \) necessitates examining the product of \( F \)-polynomials of two corresponding representations of \( Q_{HL} \) with potentials.

On a separate note, the multiplication formula has garnered attention from numerous experts [CK08, GLS07, Pal12, XX09, KPQ23]. However, much of the focus lies on triangular categories or 2-Calabi–Yau categories. In this study, we delve into the category of representations of \( Q_{HL} \) with potentials. It’s worth mentioning that \( Q_{HL} \) can be identified with the Gabriel quiver of an endomorphism ring of a tilting object \( T \) of the Frobenius category of modules over a preprojective algebra. However, whether this endomorphism ring is the Jacobian algebra for \( Q_{HL} \) remains unknown, rendering our findings novel.

As a result, we establish a connection between the head of the product of two real simple modules and the generic extension of modules over the Jacobian algebra for \( Q_{HL} \), akin to [LM21, Conjecture 5.1].

Acknowledge. The author expresses gratitude to Professor Fan Qin for engaging discussions and valuable insights.

In this section, we delve into the representations of quivers with potentials as introduced in [HL16]. Let \( C = (c_{ij})_{i,j \in I} \) represent an indecomposable \( n \times n \) Cartan matrix of finite type. Accompanying \( C \), there exists a diagonal matrix \( D = (d_i)_{i \in I} \) such that their product, denoted as \( B = DC = (b_{ij})_{i,j \in I} \), yields a symmetric matrix. For the purposes of this paper, we assume \( C \) to be simply-laced, implying its symmetry, where \( C = B \). Here, \( I \) signifies the vertex set of the associated quiver linked to the Cartan matrix \( C \), and \( r \) denotes the lacing number of \(\mathfrak{g}\).

2.1. Semi-infinite quiver. Before introducing Hernandez and Leclerc's quiver, let's revisit a semi-infinite quiver. For further elaboration, we refer readers to [HL16].

Define \( \widetilde{V} = I \times \mathbb{Z} \), introducing a quiver \( \widetilde{\Gamma} \) with the vertex set \( \widetilde{V} \). The arrows of \( \widetilde{\Gamma} \) are determined by
\[
((i, r) \to (j, s)) \iff (b_{ij} \neq 0 \text{ and } s = r + b_{ij})
\]

Following [HL16], \( \widetilde{\Gamma} \) encompasses two isomorphic connected components, of which we select one and term it \( \Gamma \). The vertex set of \( \Gamma \) is denoted by \( V \).

For further analysis, we adopt a secondary labeling of the vertices of \( \Gamma \). This labeling is derived from the initial one through the function \( \psi \) defined as
\[
\psi(i, r) = (i, r + d_i), \quad ((i, r) \in V). \tag{2.1}
\]

Let \( W \subset I \times \mathbb{Z} \) denote the image of \( V \) under \( \psi \). \( \Gamma \) can thus be denoted by \( G \) with vertices labeled by \( W \). Define \( W^- := W \cap (I \times \mathbb{Z}_{\leq 0}) \), and let \( G^- \) be the full subquiver of \( G \) with the vertex set \( W^- \).

Consider an infinite set of indeterminates \( z^- = z_{i,r} \mid (i, r) \in W^- \) over \( \mathbb{Q} \). Let \( \mathcal{A} \) denote the cluster algebras defined by the initial seed \( (z^-, G^-) \).

Define \( V^- := \psi^{-1}(W^-) \), and denote by \( \Gamma^- \) the full subquiver of \( \Gamma \) with the vertex set \( V^- \). For every \( i \neq j \) in \( I \) with \( c_{ij} \neq 0 \), and every \( (i, m) \) in \( V^- \), we observe in \( \Gamma^- \) an oriented

MULTIPLICATION FORMULA FOR HERNANDEZ AND LECLERC'S QUIVERS WITH POTENTIALS3

cycle as depicted in

We establish a potential \(S\) as the formal sum of all these oriented cycles up to cyclic permutations, referencing [DWZ08]. Let \(R\) represent the list of all cyclic derivatives of \(S\). Define \(J\) as the two-sided ideal of the path algebras \(\mathbb{C}\Gamma^{-}\) generated by \(R\). Let \(A\) denote the infinite-dimensional \(\mathbb{C}\)-algebra \(\mathbb{C}\Gamma^{-}/J\).

## 2.2. Generalized preprojective algebras

Consider a finite quiver \(\widetilde{Q} = (\widetilde{Q}_0, \widetilde{Q}_1, s, t)\) associated with any \(\mathfrak{g}\) as follows:

\(\widetilde{Q}_0 = I\), \(\widetilde{Q}_1 = \{\alpha_{ij} \mid (i, j) \in I, i \sim j\} \cup \{\epsilon_i \mid i \in I\}\)
\(s(\alpha_{ij}) = i, t(\alpha_{ij}) = j, s(\epsilon_i) = t(\epsilon_i) = i\)

We equip the quiver \(\widetilde{Q}\) with a bigrading by

\[
\deg(\alpha_{ij}) := (b_{ij}, 1), \quad \deg(\epsilon_{i}) = (b_{ii}, 0) \tag{2.3}
\]

Next, choose a sign \(\omega_{ij} \in \{-1, 1\}\) for \(i \sim j\) such that \(\omega_{ij} = -\omega_{ji}\). Define the \(k\)-algebra \(\Pi\) as the quotient of \(k\widetilde{Q}\) by the following two types of relations:

(1) \(\epsilon_i^{-c_{ij}} \alpha_{ij} = \alpha_{ij} \epsilon_j^{-c_{ji}}\) for any \(i, j \in I\) with \(i \sim j\).

(2) \[
\sum_{j \in I; j \sim i} \sum_{k=0}^{-c_{ij}-1} \omega_{ij} \epsilon_i^k \alpha_{ij} \alpha_{ji} \epsilon_i^{-c_{ij}-1-k} = 0
\]
for each \(i \in I\).

We naturally identify a graded module \(M\) over the path algebra \(k\widetilde{Q}\) with a graded representation of \(\widetilde{Q}\), comprising an \(I\)-tuple of graded \(k\)-vector spaces \((e_i M)_{i \in I}\) alongside linear maps

\[
M(\epsilon_i) \in \operatorname{Hom}_k(q^{2d_i} e_i M, e_i M), \quad M(\alpha_{ij}) \in \operatorname{Hom}_k(q^{b_{ij}} e_j M, e_i M)
\]

for each \(i \sim j \in I\) with \(i \sim j\). Here \((q^k e_i M)_l = e_i M_{l-k}\) for any \(k, l \in \mathbb{Z}\).

cycle as depicted in

\[
\begin{tikzcd}
(i, m) & (j, m+b_{ij}) \\
(i, m-b_{ii}) \ar[u] \ar[ru] & \\
\vdots \\
(i, m+2b_{ij}+b_{ii}) \\
(i, m+2b_{ij}) \ar[u]
\end{tikzcd}
\]
(2.2)

For a graded \( k\widetilde{Q} \)-module \( M \), we associate the representation \( \Phi(M) \) of \( A \) over \( k \) as \( \Phi(M)_{(i,p)} = e_i M_p \) and

\[
\Phi(M)(\epsilon_i(p)) := M(\epsilon_i)|_{e_i M_p}, \quad \Phi(M)(\alpha_{ij}(p)) = M(\alpha_{ij})|_{e_j M_p}.
\]

This functor induces an isomorphism of categories, as given by [FM23]:

\[
\Phi : \widetilde{\Pi} \text{-gmod} \cong A \text{-mod} \tag{2.4}
\]

We present some fundamental properties of \( \widetilde{\Pi} \)-gmod. Define

\[
\epsilon = \sum_{i \in I} \epsilon_i^{r/d} e_i
\]

and the algebra \( \Pi(l) \) by

\[
\Pi(l) = \widetilde{\Pi}/\epsilon^l \widetilde{\Pi}.
\]

Proposition 2.1. Let \( M \) and \( N \) be two modules over \( \widetilde{\Pi} \), locally free over \( \Pi(l) \) for some \( l \), then we have:

\[
\operatorname{Ext}_A^1(\Phi(M), \Phi(N)) = D \operatorname{Ext}_A^1(\Phi(N), \Phi(M)) \tag{2.5}
\]

Proof. This follows from the isomorphism \( \Phi \) and [GLS17, Theorem 1.8]:

\[
\operatorname{Ext}^1_{\widetilde{\Pi}}(M, N) \cong D \operatorname{Ext}^1_{\widetilde{\Pi}}(N, M).
\]
\(\quad\) □

2.3. GLS quivers. In this section, we aim to realize the algebra \( A \) as the endomorphism ring of a tilting module over the preprojective algebra of \( Q \). For detailed discussions, we refer readers to [KKOP24].

Let \( w_0 \) represent the longest element of the Weyl group of \( \mathfrak{g} \), and let \( w_0 = s_{i_1} \cdots s_{i_l} \) be a reduced expression of \( w_0 \). We extend the definition of \( (i_k)_{k \in [1,l]} \) to \( (i_k)_{k \in \mathbb{Z}} \) by

\[
i_{k+l} = (i_k)^* \text{ for any } k \in \mathbb{Z}.
\]

where \( i^* \) satisfies \( \alpha_{i^*} = -w_0(\alpha_i) \). Given a sequence \( (i_k)_{k \in \mathbb{Z}} \), we frequently employ the notations:

\[
s^+ = \min \{ t \mid s < t, i_t = i_s \}, \quad s^- = \max \{ t \mid t < s, i_t = i_s \}
\]
\[
s(j)^+ := \min \{ t \mid s \leq t, i_t = j \}, \quad s(j)^- := \max \{ t \mid s \geq t, i_t = j \} \tag{2.6}
\]

for \( s \in \mathbb{Z} \) and \( j \in I \).

A function \( \xi : I \rightarrow \mathbb{Z} \) is termed a height function on the Dynkin diagram \( \Delta \) if it satisfies the following condition:
(1) For any \( i, j \in I \) such that \( i \sim j \), we have \( |\xi_i - \xi_j| = 1 \).

MULTIPLICATION FORMULA FOR HERNANDEZ AND LECLERC’S QUIVERS WITH POTENTIALS

From a height function on Δ, we can construct a Dynkin quiver Q as follows: there exists an arrow i → j ∈ Q₁ if i ∼ j and ξᵢ > ξⱼ.

For a height function ξ, we define
$$
\widehat{I} := \{(i, p) \in I \times \mathbb{Z} \mid p - \xi_i \in 2\mathbb{Z}\}
$$

We can select Γ as the component of $\widetilde{\Gamma}$ such that $\widehat{I} \subset \Gamma$.

Let $\widehat{\Phi} := \Phi^+ \times \mathbb{Z}$, where $\Phi^+$ is the set of positive roots. According to [HL15], there exists a bijection $\phi : \widehat{I} \to \widehat{\Phi}$. Let $I_\xi$ denote the inverse image $\phi^{-1}(\Phi \times 0)$. If a sequence $(i_k, p_k)_{k \in [1, l]}$ of elements of $I_\xi$ satisfies

(1) $I_\xi = \{(i_k, p_k) \mid k \in [1, l]\};$
(2) $(i_k, p_k) \rightarrow (i_j, p_j)$ in Γ implies $k < j$,

then $w_0 = s_{i_1}\cdots s_{i_l}$ is an adapted reduced expression of $w_0$.

We say that an infinite sequence $c := (i_k, p_k)_{k \in \mathbb{Z}}$ in $I \times \mathbb{Z}$ is admissible if the following conditions are satisfied:

(1) $p_{s^+} = p_s + 2$,
(2) $p_t = p_s + 1$ if $i_t \sim i_s$ and $t^- < s < t < s^+$,
(3) $s_{i_k}\cdots s_{i_{k+l-1}} = w_0$ for every $k \in \mathbb{Z}$.

For an admissible sequence $c$, the map $s \mapsto (i_s, p_s)$ gives a bijection $\mathbb{Z} \to \widehat{I}$.

For a sequence $c$ of $w_0$, let us define $Q_{GLS}(w_0)$ as the quiver structure on $\mathbb{Z}$ with two types of arrows:
vertical arrows: $s \to t$ if $s^- < t^- < s < t$ and $i_s \sim i_t$,
horizontal arrows: $s \to s^-$                                                        (2.7)

According to [KKOP24, Proposition 7.27], the bijection $\mathbb{Z} \to \widehat{I}$ yields an isomorphism:
$$
Q_{GLS}(c) \simeq \Gamma. \hspace{2cm} (2.8)
$$

We define the subquiver $Q^-_{GLS}(c)$ as the inverse image of $\Gamma^-$. The cluster algebra $\mathcal{A}$ is identified with the cluster algebra with initial seed $\left((z_k)_{k \in \mathbb{Z}_{\leq 0}}, Q_{GLS}(c), \mathbb{Z}_{\leq 0}\right)$.

Following [KKOP24, Definition 4.14], we say that $c = [a, b]$ is $i$-box if $-\infty < a \leq b < +\infty$ and $i_a = i_b$. We denote by
$$
[a, b\} := [a, b(i_a)^-]
$$

Definition 2.2. For an $i$-box $c = [a, b]$, one can define the module $M[a, b]$ as the generic kernel of maps $\operatorname{Hom}_A(I_{(i_a, p_a)}, I_{(i_b, p_b)})$, namely,
$$
0 \rightarrow M[a, b] \rightarrow I_{(i_a, p_a)} \rightarrow I_{(i_b, p_b)} \rightarrow 0
$$

By the isomorphism (2.4) and [FM23, Proposition 5.1], M[a, b] is equal to the image of the \( q^{p_a} K^{(i_a)}_{\frac{p_a-p_b}{2}} \) under the map \(\Phi\), where \( K^{(i)}_k \) is the kernel of the following map:

\( I_i(\infty) \xrightarrow{\epsilon^k_i} q^{-2k} I_i(\infty) \to 0 \)

Thus, we have the following short exact sequence:

\( 0 \to \Phi^{-1}(M[a, b]) \to q^{p_a} I_{i_a} \xrightarrow{\epsilon^{p_a-p_b/2}_i} q^{p_b} I_{i_a} \to 0. \)

Theorem 2.3. Consider a \(\xi\)-adapted admissible sequence \(c\) in \(\tilde{I}_g\) and \(a \in \mathbb{Z}\), we set the module
\( M[a,0] \).
The module \( M^{-\infty} := \bigoplus_{a \in \mathbb{Z}_{<0}} M[a,0] \) is a rigid module, and the Gabriel quiver \( Q^{-\infty} \) of the endomorphism ring \( \text{End}_A(M^{-\infty}) \) equals \( Q_{GLS} \).

Here we conjecture the following statement.

Conjecture 2.4. We conjecture that \( \text{End}_A(M^{-\infty}) \) is the Jacobian algebra for \( Q_{GLS} \) with the potential (2.2).

Remark 2.5. If the above conjecture holds, then our multiplication formula 3.2 is a consequence of the results of [KPQ23].

## 3. MULTIPLICATION FORMULA FOR A

### 3.1. Cluster algebras A.
Let \( M \) denote a finite-dimensional module over the algebra \( A \), and let \( e \) be a dimension vector in \( \mathbb{N}[V^-] \). The variety \( \mathrm{Gr}_e(M) \) represents submodules of \( M \) with dimension vector \( e \), and \( \chi(\mathrm{Gr}_e(M)) \) denotes its Euler characteristic. Inspired by [DWZ10], we introduce the polynomial

\[
F_M = \sum_{e \in \mathbb{N}[V^-]} \chi(\mathrm{Gr}_e(M)) \prod_{(i, r) \in V^-} v_{i, r}^{e_{i, r}}
\]
(3.1)

Consider two modules \( M_1 \) and \( M_2 \) of \( A \). Building on [DWZ10], we find that

\[
F_{M_1} F_{M_2} = F_{M_1 \oplus M_2}
\]

### 3.2. Multiplication formula.
This section focuses on proving our multiplication formula. First, let’s review some facts about extension groups of modules over \(A\).

Let \( D(M_1, M_2) \) denote \( \widetilde{\mathrm{Ker}} \, f_{M_1, M_2} \) as in [GLS17, formula (12.5)]. We decompose \( D(M_1, M_2) \) into

\[
D(M_1, M_2) = \mathrm{Im} \, \tilde{g}_{M_1, M_2} \oplus E(M_1, M_2)
\]

Notably, \( \mathrm{Ext}^1_\Pi(M_1, M_2) \cong E(M_1, M_2) \).

MULTIPLICATION FORMULA FOR HERNANDEZ AND LECLERC'S QUIVERS WITH POTENTIALS

Using isomorphism (2.4), we derive
\[
\operatorname{Ext}^1_A(M_1, M_2) \cong D\operatorname{Ext}^1_A(M_2, M_1) \tag{3.2}
\]

From here on, let \( E(M_1, M_2) \cong \operatorname{Ext}^1_A(M_1, M_2) \).

Consider an extension \( \eta \) of \( M_1 \) and \( M_2 \), and let \( m(\eta) \) be the corresponding module over \( A \), forming a short exact sequence:
\[
\eta : 0 \to M_2 \to m(\eta) \to M_1 \to 0
\]

For a dimension vector \( \beta \) such that \( \beta \leq \dim m(\eta) = \gamma \), define
\[
\operatorname{Gr}_\beta(m(\eta)) := \{ W' \in \operatorname{Gr}(\beta, \gamma) \mid m(\eta)(W') \subset W' \} \tag{3.3}
\]
where \( \operatorname{Gr}(\beta, \gamma) = \prod_i \operatorname{Gr}(\beta_i, \gamma_i) \).

Choosing a decomposition \( W \oplus T = V \) with \( \dim V = \gamma \) and \( \dim W = \dim M_2 \), we express \( m(\eta) \) as a tuple of matrices
\[
x_h = \begin{pmatrix} x_{2,h} & u_h \\ 0 & x_{1,h} \end{pmatrix} \tag{3.4}
\]
for each arrow \( h \in Q \). For any \( W' \in \operatorname{Gr}_\beta(m(\eta)) \), define subspaces
\[
W'_2 = W' \cap W \qquad W'_1 = W'/W' \cap W
\]

This induces a map:
\[
q_\eta : \operatorname{Gr}_\beta(m(\eta)) \to \bigsqcup_{\alpha_1 + \alpha_2 = \beta} \operatorname{Gr}_{\alpha_1}(M_1) \times \operatorname{Gr}_{\alpha_2}(M_2) \tag{3.5}
\]

To understand the map \( q_\eta \), recall the following map:
\[
p : \operatorname{Gr}(\beta, \gamma) \to \bigsqcup_{\beta_1 + \beta_2 = \beta} \operatorname{Gr}(\beta_1, \gamma_1) \times \operatorname{Gr}(\beta_2, \gamma_2)
\]
sending \( W' \) to \( (V'/W \cap V', \; W \cap V') \). For a pair vector space \( (T', W') \), it's shown that:
\[
p^{-1}(T', W') = \operatorname{Hom}_I(T', W/W')
\]
For an element \( z \) in \( \operatorname{Hom}_I(T', W/W') \), define \( V_z \) by
\[
V_z = (W', 0) + \{ (z(t), t) \mid t \in T' \}
\]
If there exists a short exact sequence:
\[
0 \to W' \to V' \to T' \to 0,
\]
it yields a map \( z \in \operatorname{Hom}_I(T', V'/\ominus W') \) such that
\[
V' = (W', 0) + \{ (z(t), t) \mid t \in T' \}
\]

Hence, we obtain formula (3.6). Consider \( q_\eta^{-1}(T', W') \) as in (3.5).

Lemma 3.1. Given a pair of vector spaces \((T', W')\), for an element \(z \in \mathrm{Hom}_I(T', W/W')\) we have \(V_z \in \mathrm{Gr}_\beta(m(\eta))\) if and only if

\[
x_{2,h} z(t) - z x_{1,h}(t) + u_h(t) \in W' \quad \text{for all } t \in T' \tag{3.7}
\]

Moreover, \(q_\eta^{-1}(T', W') = \varnothing\) or \(q_\eta^{-1}(T', W')\) is isomorphic to an affine space.

Proof. By equation (3.4), the condition \(x(V_z) \subset V_z\) is equivalent to

\[
(x_{2,h} z - z x_{1,h} + u_h)|_{T'} \text{ is equal to 0 in } W/W'.
\]

This proves the claim.
If \(q_\eta^{-1}(T', W') \neq \varnothing\), any two \(z_1, z_2 \in q_\eta^{-1}(T', W')\) satisfy

\[
(x_{2,h}(z_1 - z_2) - (z_1 - z_2)x_{1,h})(T') \subset W'.
\]

It implies

\[
z_1 - z_2 \in \mathrm{Hom}_A(M_2, M_{1| W/W'})
\]

Hence, \(q_\eta^{-1}(T', W') \cong \mathrm{Hom}_A(M_2, M_{1| W/W'})\).

Theorem 3.2. Let \(M_1, M_2\) be two modules over \(A\), then

\[
\chi(\mathbb{P} \mathrm{Ext}_A^1(M_1, M_2)) F_{M_1} \cdot F_{M_2} = \sum_Y \left( \chi(\mathbb{P} \mathrm{Ext}_A^1(M_1, M_2)_Y) + \chi(\mathbb{P} \mathrm{Ext}_A^1(M_2, M_1)_Y) \right) F_Y
\]

Proof. Let \(L_1\) be the image of \(q\), then by Lemma 3.1 we see that

\[
\chi(\mathrm{Gr}_\beta(m(\eta))) = \chi(L_1)
\]

Define

\[
EF(M_1, M_2) := \{(V', \eta) \in \mathrm{Gr}(\beta, \gamma) \times_{\mathbb{C}^*} E(M_1, M_2) \mid m(\eta)(V') \subset V'\} \tag{3.8}
\]

and a map

\[
\psi : EF(M_1, M_2) \to \mathbb{P}E(M_1, M_2)
\]

sending \((V', \eta)\) to \(\eta\).
Following [Dim04, Proposition 4.1.31], \(\mathbb{P}E(M_1, M_2)\) can be decomposed into

\[
\mathbb{P}E(M_1, M_2) = \bigsqcup_Y \mathbb{P}E(M_1, M_2)_{\langle Y \rangle}
\]

such that \(\chi(\mathrm{Gr}_\beta(m(\eta_1))) = \chi(\mathrm{Gr}_\beta(m(\eta_2)))\) for any two elements \(\eta_1, \eta_2\) in \(\mathbb{P}E(M_1, M_2)_{\langle Y \rangle}\).
Here \(\mathbb{P}E(M_1, M_2)_{\langle Y \rangle}\) is a constructible subset for each \(Y\). By [GLS07, Proposition 7.4.1],
we see

\[
\chi(EF(M_1, M_2)) = \sum_Y \chi(\mathrm{Gr}_\beta(m(Y))) \chi(\mathbb{P}E(M_1, M_2)_{\langle Y \rangle}) \tag{3.9}
\]

We consider the following map

\[
\Phi : EF(M_1, M_2) \to \bigsqcup_{\beta_1 + \beta_2 = \beta} \mathbb{P}E(M_1, M_2) \times \mathrm{Gr}_{\beta_1}(M_1) \times \mathrm{Gr}_{\beta_2}(M_2) \tag{3.10}
\]

MULTIPLICATION FORMULA FOR HERNANDEZ AND LECLERC'S QUIVERS WITH POTENTIALS

sending $(V', \eta)$ to $(\eta, V'/V'\cap W, V'\cap W)$. Let $L_1$ be its image. For any point $(\eta, T', W') \in L_1$, the fiber $\Phi^{-1}(\eta, T', W')$ is equal to $q_\eta^{-1}(T', W')$, an affine space by Lemma 3.1. Hence we obtain
$$\chi(EF(M_1, M_2)) = \chi(L_1) \tag{3.11}$$

Let $L_2$ be the complement of $L_1$. Define
$$C := \{(\eta, T', W', \eta', V') \in L_2 \times EF(M_2, M_1) \mid V' \cap T = T', V'/V'\cap T = W', (\eta,\eta') \neq 0\} \tag{3.12}$$

Fix a pair of subvector spaces $(T', W') \in \operatorname{Im} q_\eta$. This means there exists a vector subspace $V'$ such that the following diagram is exact

\[
\begin{array}{ccccccccc}
\eta\colon & 0 & \to & M_2 & \to & m(\eta) & \to & M_1 & \to 0 \\
\eta \circ \iota_W\colon & 0 & \to & M_2|_{W'} & \to & m(\eta)|_{V'} & \to & M_1 & \to 0 \\
\iota_T \circ \eta \circ \iota_W: & 0 & \to & M_2|_{W'} & \to & m(\eta)|_{V'} & \to & M_1|_{T'} & \to 0
\end{array}
\]

Comparing this diagram and the one in [GLS07, lemma 2.4.1], we see that ours is a special case. Denote $E(M_1, M_2)_{T', W'}$ as the subspace of $\operatorname{Ext}^1_A(M_1, M_2)$ consisting of elements $\eta$ such that there exists a subspace $V'$ with $m(\eta)(V') \subseteq V'$ and $V'\cap W = W'$, $V'/V'\cap W = T'$. By [GLS07, Proposition 2.5.2], we have
$$E(M_1, M_2)^{\perp}_{T', W'} = E(M_2, M_1)_{W', T'} \tag{3.13}$$

This implies that the map $C \to L_2$, defined by mapping $(\eta, T', W', \eta', V')$ to $(\eta, T', W')$, is surjective. Using the proof of [GLS07, Proposition 5.6.1], we see that the fibers of this map are extensions of affine spaces. This implies
$$\chi(L_2) = \chi(C) \tag{3.14}$$

On the other hand, there’s another map
$$pr: C \to EF(M_2, M_1)$$

sending $(\eta, T', W', \eta', V')$ to $(\eta', V')$. Equation (3.13) implies
$$pr^{-1}(\eta', V') = \mathbb{P}(E(M_1, M_2) \setminus {}^\perp d'),$$
which is an affine space. Hence, we obtain
$$\chi(C) = \chi(EF(M_2, M_1)) \tag{3.15}$$

To sum up, we get

\[
\chi(\mathbb{P}\mathrm{Ext}^1_A(M_1, M_2)) \sum_{\beta_1 + \beta_2 = \beta} \chi(\mathrm{Gr}_{\beta_1}(M_1)) \chi(\mathrm{Gr}_{\beta_2}(M_2))
= \sum_Y \big( \chi(\mathbb{P}E(M_1, M_2)_Y) + \chi(\mathbb{P}E(M_2, M_1)_Y ) \big) \chi(\mathrm{Gr}_\beta(Y))
\tag{3.16}
\]

In particular, when \(\mathrm{Ext}^1_A(M_1, M_2) = 1\), we obtain

\[
F_{M_1} \cdot F_{M_2} = F_{Y_1} + F_{Y_2}
\tag{3.17}
\]

where \(Y_1\) subjects to \(0 \to M_1 \to Y_1 \to M_2 \to 0\) and \(Y_2\) subjects to \(0 \to M_2 \to Y_2 \to M_1 \to 0\).

Let \(\mathfrak{g}\) be the simple Lie algebra over \(\mathbb{C}\). We denote by \(\alpha_i\) (\(i \in I\)) the simple roots of \(\mathfrak{g}\), and by \(\omega_i\) the fundamental weights of \(\mathfrak{g}\). Let \(\hat{\mathfrak{g}}\) be the corresponding untwisted affine Lie algebra and \(U_q(\hat{\mathfrak{g}})\) be the Drinfeld-Jimbo quantum enveloping algebra of \(\hat{\mathfrak{g}}\), see e.g. [CP95].

Frenkel and Reshetikhin [FR98] have attached to every complex finite-dimensional representation \(M\) of \(U_q(\hat{\mathfrak{g}})\) a q-character \(\chi_q(M)\). By definition, the q-character \(\chi_q(M)\) is a Laurent polynomial with positive integer coefficients in the infinite set of variables

\[
Y = \{Y_{i,a} \mid i \in I, a \in \mathbb{C}^* \},
\]

which should be seen as a quantum affine analogue of \(\{ e^{\omega_i} \mid i \in I \}\). In this paper, we will be concerned only with polynomials involving the subset of variables

\[
Y_{i, q^r}, \; ((i, r) \in W).
\]

For simplicity of notation, we shall therefore write \(Y_{i,r}\) instead of \(Y_{i,q^r}\).

Let \(m\) be a dominant monomial in the variables \(Y_{i,r} \in Y\), that is, a monomial with nonnegative exponents. We denote by \(L(m)\) the corresponding irreducible representation of \(U_q(\hat{\mathfrak{g}})\), and by \(\chi_q(m) = \chi_q(L(m))\) its q-character. For example, if \(m\) is of the form

\[
m = \prod_{j=0}^{k-1} Y_{i, r+j b_i}, \qquad (i \in I, r \in \mathbb{Z}, k \geq 1)
\tag{4.1}
\]

\(L(m)\) is called Kirillov-Reshetikhin module, and usually denoted by \(W^{(i)}_{k,r}\). \(\mathcal{C}^-\) denotes the monomial subcategory of modules generated by fundamental module \(V_{i,a}\) for all \((i, a) \in \Gamma^-\).

MULTIPLICATION FORMULA FOR HERNANDEZ AND LECLERC'S QUIVERS WITH POTENTIALS

Finally, following [FR98], for $(i, r) \in V$ we introduce the following quantum affine analogue of $e^{\alpha_i}$:

\[
A_{i,r} := Y_{i, r-d_i} Y_{i, r+d_i}
\left( \prod_{j : c_{ji} = -1} Y_{j,r} \right)^{-1}
\left( \prod_{j : c_{ji} = -2} Y_{j, r-1} Y_{j, r+1} \right)^{-1}
\left( \prod_{j : c_{ji} = -3} Y_{j, r-2} Y_{j, r} Y_{j, r+2} \right)^{-1}
\qquad (4.2)
\]

To relate the $q$-character of simple modules with the $F$-polynomials for the algebra $A$, we need more notations.

Let
\[
Y^- = \{ Y_{i,r} \in Y \mid (i, r) \in W^- \}.
\]

For $(i, r) \in W^-$, we perform the substitution
\[
z_{i,r} = \prod_{k \geq 0, \, r + k b_{ii} \leq 0} Y_{i, r + k b_{ii}}
\]

In the sequel, we shall evaluate the variables of the $F$-polynomials as the inverses of the variables $A_{i,r}$ introduced in (4.2), namely:
\[
v_{i, r} = A_{i, r}^{-1} \qquad (4.3)
\]

Our $F$-polynomials are given by the equation (3.1).

Next, we will introduce a class of modules over $A$: Let $I_{i, m}$ be the (infinite-dimensional) injective $A$-module with socle isomorphic to $S_{i,m}$. The $\mathbb{C}$-vector space $I_{i, m}$ has a basis indexed by classes modulo $J$ of paths in $\Gamma^-$ with final vertex $(i, m)$. Let $K_{k, m}^{(i)}$ be the kernel of a generic $A$-module homomorphism from $I_{i, m}$ to $I_{i, m - k b_{ii}}$.

One also defines such modules on generalized preprojective algebra $\tilde{\Pi}$. For each $i \in I$ and $k \in \mathbb{Z}_{>0}$, we define the graded $\tilde{\Pi}$-module $K_k^{(i)}$ by
\[
K_k^{(i)} := q^{k d_i} \mathbb{D} \left( (\tilde{\Pi} / \tilde{\Pi} \epsilon_i^k) e_i \right) \qquad (4.4)
\]

From the definition, the module $K_k^{(i)}$ fits into the following short exact sequence:
\[
0 \to K_k^{(i)} \to q^{k d_i} I_i(\infty) \xrightarrow{\epsilon_i^k} q^{-k d_i} I_i(\infty) \to 0. \qquad (4.5)
\]

Following [FM23, Remark 5.2], we obtain
\[
\Phi \left( q^{r - k d_i} K_k^{(i)} \right) = K_{k, r}^{(i)}
\]

In [HL16], we have the following theorem.

Theorem 4.1. [HL16, Theorem 4.8] Let $(i, r) \in V^-$ and $k \in \mathbb{N}$. We have
\[
\chi_q^-\left( W_{k, r - (2k-1)d_i}^{(i)} \right) = \left( \prod_{s=1}^k Y_{i, r - (2s-1) d_i} \right) F_{K_{i, r}^{(i)}}
\]

where the variable \( v_{i,r} \) of the F-polynomial are evaluated as in (4.3).

In general, Hernandez and Leclerc [HL16] showed that

**Theorem 4.2.** [HL16, Theorem 5.1; Conjecture 5.2] The cluster algebra \( \mathcal{A} \) is identified with the image of the injective ring homomorphism from \( K_0(C^-) \) to \( \mathbb{Z}[Y_{i,r} \mid (i,r) \in Y^-] \) given by \( L(m) \mapsto \chi_q^-(m) \). Moreover, the cluster monomials get identified with the truncated \( q \)-characters of real simple modules.

Note that the last statement was proved by Kashiwara, Kim, Oh and Park in [KKOP24].

**Proposition 4.3.** Let \( m_1, m_2 \) be two cluster monomials in \( \mathcal{A} \), \( L_1, L_2 \) be corresponding simple modules over \( U_q(\widehat{\mathfrak{g}}) \), and \( M_1, M_2 \) be the corresponding rigid modules over \( \mathcal{A} \). The tensor product \( L_1 \otimes L_2 \) is simple if and only if

\[
\mathrm{Ext}_{\mathcal{A}}^1(M_1, M_2) = 0
\]

*Proof.* If \( L_1 \otimes L_2 \) is a simple module, then there exists a rigid module \( T \) over \( \mathcal{A} \) such that

\[
F_{M_1} \cdot F_{M_2} = F_T \text{ and } g_T = g_{M_1} + g_{M_2}
\]

where \( g_M \) is the g-vector for module \( M \). It implies that there exists a short exact sequence

\[
0 \to M_1 \to T \to M_2 \to 0 \text{ or } 0 \to M_2 \to T \to M_1 \to 0
\]

If \( \mathrm{Ext}_{\mathcal{A}}^1(M_1, M_2) \neq 0 \), then by theorem 3.2, we get

\[
\chi(\mathbb{P}\mathrm{Ext}_{\mathcal{A}}^1(M_1, M_2)) F_T = \sum_Y ( \chi(\mathbb{P}\mathrm{Ext}_{\mathcal{A}}^1(M_1, M_2)_Y ) + \chi(\mathbb{P}\mathrm{Ext}_{\mathcal{A}}^1(M_2, M_1)_Y ) ) F_Y
\]

since \( T \) appears in the right hand of the above equation, we have \( Y = T \) for all \( Y \) in the right hand of the above equation, which implies that

\[
F_T = 2F_T
\]

It is a contradiction. Hence, we prove our claim.

If \( \mathrm{Ext}_{\mathcal{A}}^1(M_1, M_2) = 0 \), then \( M_1 \oplus M_2 \) is a rigid module. The truncated \( q \)-character of \( L_1 \otimes L_2 \) is equal to

\[
z^{g_1+g_2} F_{M_1 \oplus M_2}
\]

which is a cluster monomial. It means that \( L_1 \otimes L_2 \) is a simple module. □

### 4.1. Proof of Theorem 2.3.

In order to prove theorem 2.3, we need a lemma. Following [KKOP24], we say that i-boxes \([a_1, b_1]\) and \([a_2, b_2]\) commute if we have either

\[
a_1^- < a_2 \leq b_2 < b_1^+ \text{ or } a_2^- < a_1 \leq b_1 < b_2^+
\]

**Lemma 4.4.** If two i-boxes \([a_1,b_1]\) and \([a_2,b_2]\) commute, then
\[
\mathrm{Ext}_{\mathcal{A}}^1(M[a_1,b_1], M[a_2,b_2]) = 0
\]

MULTIPLICATION FORMULA FOR HERNANDEZ AND LECLERC'S QUIVERS WITH POTENTIALS

Theorem 4.5. For any two real simple modules \( L_1, L_2 \) over \( U_q(\widehat{\mathfrak{g}}) \) for two cluster monomials \( m_1, m_2 \), let \( M_1 \) and \( M_2 \) be the rigid module over \( A \) for \( m_1, m_2 \). If the head of \( L_1 \otimes L_2 \) is a simple module corresponding to some cluster monomial \( m \), then its corresponding module \( M \) over \( A \) is the generic extension \( M_1 \) by \( M_2 \) or \( M_2 \) by \( M_1 \), namely,
\[
M = M_1 * M_2 \text{ or } M = M_2 * M_1.
\]

Proof. Since \( \chi_q^-(L_i) = z^{g_i} F_{M_i} \) for any \( i = 1, 2 \), we get
\[
\chi_q^-(L_1) \cdot \chi_q^-(L_2) = z^{g_1 + g_2} F_{M_1 \oplus M_2}
\]

If the head of \( L_1 \otimes L_2 \) is a monomial simple module, then there exists a rigid module \( T \) such that
\[
\chi_q^-(L_1 \otimes L_2) = \chi_q^-(L_1) \cdot \chi_q^-(L_2) = z^{g'} F_T + P
\]
for some \( P \). The dominant monomial of \( L_1 \otimes L_2 \) is unique, it implies that
\( g' = g_1 + g_2. \)

Proof. Following [KKOP24, Definition 4.14], for any \( i \)-box \([a, b]\), they define the determinantal module \( \mathcal{M}[a, b] \) over \( U_q(\widehat{\mathfrak{g}}) \). Using [HL16, Theorem 4.18] and [KKOP24, Theorem 8.2], we see that the truncated \( q \)-character \( \chi_q^-(\mathcal{M}[a, b]) = z^g F_M \) for some vector \( g \) and a rigid module \( M \) over \( A \). Since the dominant monomial for \( \mathcal{M}[a, b] \) is equal to
\[
\prod_{k, p_a < p_a + k b_i \leq p_b} Y_{i, p_a + k b_i} = z_{i_a, p_a} / z_{i_b, p_b}
\]
According to the proof of [HL16, Proposition 4.16], the vector \( g \) is identified with
\[
g_{i_k, p_k} =
\begin{cases}
1 & \text{if } k = a \\
-1 & \text{if } k = b \\
0 & \text{otherwise}
\end{cases}
\]
Following [DWZ10], \( M \) is the generic kernel of \( I_{i_a, p_a} \to I_{i_b, p_b} \to 0 \), which is equal to \( M[a, b] \) by our definition.

By [KKOP24, Theorem 4.21], the tensor product of \( \mathcal{M}[a_1, b_1] \otimes \mathcal{M}[a_2, b_2] \) is a real simple module. Using Proposition 4.3, we see that
\[
\operatorname{Ext}_A^1(M[a_1, b_1], M[a_2, b_2]) = 0
\]
\(\Box\)

Proof of Theorem 2.3. Following the above Lemma and [KKOP24, Theorem 7.20], \( M^{-\infty} \) is a tilting module over \( A \). We will calculate the Gabriel quiver of \( \operatorname{End}_A(M^{-\infty}) \). It is directly deduced from [BIRS09, Theorem II.4.1] \(\Box\)

It follows that

\[
0 \rightarrow M_i \rightarrow I_1^{g_i^-} \rightarrow I_2^{g_i^+} \rightarrow 0
\]

for each \( i = 1, 2 \) and

\[
0 \rightarrow T \rightarrow I_1^{g'_-} \rightarrow I_2^{g'_+} \rightarrow 0
\]

It is easy to see that there exists an exact sequence

\[
0 \rightarrow M_{L_1} \rightarrow T \rightarrow M_{L_2} \rightarrow 0 \text{ or } 0 \rightarrow M_{L_2} \rightarrow T \rightarrow M_{L_1} \rightarrow 0.
\]

Since \( T \) is a rigid module, it follows that \( T = M_1 * M_2 \) or \( T = M_2 * M_1 \). \(\Box\)

**Remark 4.6.** This theorem gives an answer to the [LM21, Conjecture 5.1] in the sense of generalized preprojective algebras.

## References

[BIRS09] Aslak Bakke Buan, Osamu Iyama, Idun Reiten, and Jeanne Scott. *Cluster structures for 2-Calabi–Yau categories and unipotent groups.* Compositio Mathematica 145.4 (2009): 1035–1079.

[CK08] Philippe Caldero and Bernhard Keller. *From triangulated categories to cluster algebras.* Inventiones mathematicae, 172(1):169–211, 2008.

[CP95] Vyjayanthi Chari and Andrew N Pressley. *A guide to quantum groups.* Cambridge university press, 1995.

[Dim04] Alexandru Dimca. *Sheaves in topology.* Springer Science Business Media, 2004.

[DWZ08] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky. *Quivers with potentials and their representations I: Mutations.* Selecta Mathematica, 14(3):59–119, 2008.

[DWZ10] Harm Derksen, Jerzy Weyman, and Andrei Zelevinsky. *Quivers with potentials and their representations II: applications to cluster algebras.* Journal of the American Mathematical Society, 23(3):749–790, 2010.

[FM23] Ryo Fujita and Kota Murakami. *Deformed cartan matrices and generalized preprojective algebras I: Finite type.* International Mathematics Research Notices, 2023(8):6924–6975, 2023.

[FR98] E Frenkel and N Reshetikhin. *The q-characters of representations of quantum affine algebras and deformations of w-algebras.* Recent developments in quantum affine algebras and related topics, Contemp. Math, (248):163–205, 1998.

[GLS07] Christof Geiss, Bernard Leclerc, and Jan Schröer. *Semicanonical bases and preprojective algebras II: A multiplication formula.* Compositio Mathematica, 143(5):1313–1334, 2007.

[GLS17] Christof Geiss, Bernard Leclerc, and Jan Schröer. *Quivers with relations for symmetrizable cartan matrices I: Foundations.* Inventiones mathematicae, 209:61–158, 2017.

[HL10] David Hernandez and Bernard Leclerc. *Cluster algebras and quantum affine algebras.* Duke Mathematical Journal, 154(2):265–341, 2010.

[HL15] David Hernandez and Bernard Leclerc. *Quantum grothendieck rings and derived hall algebras.* Journal für die reine und angewandte Mathematik (Crelle's Journal), 2015(701):77–126, 2015.

[HL16] David Hernandez and Bernard Leclerc. *A cluster algebra approach to q-characters of Kirillov–Reshetikhin modules.* Journal of the European Mathematical Society, 18(5):1113–1159, 2016.

[KKOP24] Masaki Kashiwara, Myungho Kim, Se-jin Oh, and Euiyong Park. *Monoidal categorification and quantum affine algebras II.* Inventiones mathematicae, pages 1–88, 2024.

[KPQ23] Bernhard Keller, Pierre-Guy Plamondon, and Fan Qin. *A refined multiplication formula for cluster characters.* arXiv preprint arXiv:2301.01059, 2023.

DEPARTMENT OF MATHEMATICS, HARBIN ENGINEERING UNIVERSITY
Email address: yingjinbi@mail.bnu.edu.cn


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

- **Do not** include any other keys or prose outside these two tagged blocks.
- **Do not** report stylistic or citation issues.
- Be as precise as possible about **where** (section ID or figure/table) and **what** the scientific flaw is.
- Each description for the errors must be rooted in a scientific rationale explaining why it is 'wrong' (not how it could be improved).

Begin your analysis now.

[User]
A single aperture Canted-Cosine-Theta (CCT) quadrupole magnet, made of NbTi superconductors, has been developed for the final focus region of the FCC-ee study. The conductor layout is optimised to mitigate edge effects on one of the two sides of the magnet that typically lead to undesired higher-order multipoles. Experimental results of a prototype, including paraffin wax impregnation and cryogenic temperature measurements, are presented. The magnet exhibits no training behaviour, surpassing the nominal current during the initial ramp. Field quality is excellent, with higher-order multipoles below 1 × 10⁻⁴ units, consistent with simulations and room temperature tests. These findings confirm the potential of superconducting CCT magnets to offer compact solutions for applications demanding stringent field quality.

The Future Circular Collider (FCC) study plans on building a Higgs and electroweak factory through collisions of positrons and electrons (FCC-ee) [1]. Once these particles have been accelerated to their desired energy, strong so-called final focus quadrupole magnets, focus the two individual beams around the so-called interaction region where collisions occur.

A Canted-Cosine-Theta (CCT) magnet topology has been proposed for the realization of the final-focus quadrupole, following a similar work done at CERN [2]. The conductor layout has been designed such that the edge effects are reduced on one side and kept unchanged on the other side for comparison. A prototype of shorter axial length was designed and built in 2019. Its specifications are summarized in Table 1. The integrated quadrupole component B₂ is 2.2 times less than the full-size final magnet of length 0.7 m (initially 1.2 m-long but recently adjusted in the project baseline). The magnet is wound with 8 individually insulated superconducting wires. The wires are then connected in series in a joint box positioned at one end of the magnet.

Superconducting magnets need impregnation for better strand-to-strand thermal management and better mechanical integrity. This consists of filling the gaps between windings with an electrically insulating material, improving the overall thermal conductivity of the pack, and better distributing the stress across it. Traditional epoxy-based impregnation often exhibits a training phenomenon, requiring multiple cycles before optimal performance. Recent studies demonstrate that paraffin wax impregnation eliminates this issue [3–5], solving a bothersome problem of superconducting magnets. Here, we describe a dedicated impregnation setup developed for this purpose. Magnetic measurements of the demonstrator were initially conducted at room temperature (i.e. warm) [6] and are now presented at cryogenic temperatures (i.e. cold) to assess superconducting magnet performance accurately. The cold tests were conducted in Nov. 2023.

The magnet impregnation process involves injecting hot melted wax into the empty gaps between windings, which then solidifies to ensure mechanical integrity. Paraffin wax undergoes a liquid-solid phase transition at around 55 °C, contracting by approximately 15% upon solidification.

A. Thabuis, M. Koratzinos, G. Kirby, M. Liebsch, C. Petrone
European Organization for Nuclear Research (CERN), Geneva, Switzerland

Figure 1 shows the impregnation station and a cross-section of the top part of the magnet. Internal wax reservoirs at the top of the magnet provide additional impregnation volume to compensate for crystallization shrinkage. The system is placed within an oven to achieve the temperature necessary for the wax to be melted. The magnet is air-tight and a weak vacuum is created within the magnet to avoid trapping air bubbles which would degrade the impregnation quality. Wax flow is facilitated by a peristaltic pump, filling the magnet from the bottom. A copper shaft is inserted inside the magnet and connected to a copper plate lying at the bottom of it. This induces two temperature gradients across the magnet: one from the inner layer to the outer layer, and another from the top to the bottom part of the magnet. A water cooling circuit is added below the magnet enhancing temperature control of the heat exchanger and accelerating wax cooling.

Figure 2 shows the temperature profiles measured at various positions across the magnet and downsampled by averaging over ten points. An average 3 °C and 5 °C temperature difference was observed between the inner side (in contact with the heat exchanger shaft) and outer side (in contact with the magnet housing); and between the top and bottom part of the magnet respectively.

prevent the formation of trapped empty spaces within the winding pack, temperature gradients are applied across the magnet during progressive cooling, following an approach similar to Ref. [3].

Multiple taps within the joint box monitor voltage between wires to detect quenches and protect the magnet by discharging energy into a dump resistor. A 50 Ω crowbar connected to a 75 Ω dump resistor is employed for energy extraction. A 100 mV threshold with 10 ms validation time is used for quench detection. Inductive voltages are canceled by comparing the first half of the magnet with the second half, and quarters of the magnets are also compared for redundancy and in the case of symmetric quenches.

Once impregnated, the magnet was tested at cryogenic temperatures for characterization. The prototype was placed inside a cryostat with rotating coils, as depicted on the right side of Fig. 3. The cryostat is filled with helium and cooled to 1.9 K and later to 4.5 K. The test took place in November 2023 at the Siegtag cryostat of the SM18 cryogenic facility of CERN.

Various ramping tests determine the critical current of the magnet, as illustrated in Fig. 4. Notably, no training is observed. The short-sample critical current already measured on previous occasions is reached on the first attempt. A margin exceeding 20% at 1.9 K is validated for the nominal operating current. Additional tests have been performed at 4.5 K for demonstration, also reaching the short sample limit associated with this operating temperature.

The field quality is determined by the normalized coefficients resulting from the Fourier analysis of the flux density distribution across the magnet cross-section [8, Chapter 6]. These coefficients are determined by inductive passive PCB coils rotating around the axis inside the magnet. Three coils cover the entire axial length: one at the center for nominal magnetic strength characterization, and two others to investigate edge effects. A distinctive conductor layout on one side corrects undesired higher-order multipole content. This is illustrated in Fig. 3 with the grooves hosting the conductor diverging from the usual sinusoidal profile that leads to undesired edge effects. The correction is confirmed by comparing coefficients from the coils positioned on the corrected and non-corrected sides as shown in Table 2. The field quality analysis results, although worse than the warm tests, are still below 1 unit in the central region. We have not repeated the extra step done at warm which reduced multipole errors and consisted of rotating the magnet by about 45° with respect to the measuring system (before repeating the measurements) [6].

The magnitude of undesired higher-order harmonics is significantly reduced on the corrected side compared to the non-corrected one, reaching values below 5 units (the normalization uses the quadrupole field of that side and not of the whole magnet). One should note that when evaluating the overall performance of the magnet, the total length would be considered, resulting in a larger normalizing quadrupole component which mainly comes from the central part. This dilutes the contribution of the edges to the field quality of the prototype, and even more for the full-scale magnet.

This validates the possibility of tuning the conductor layout of CCT magnets in the edge to remove undesired higher-order multipole content. This approach can also get rid of crosstalk from the adjacent magnet in a twin aperture arrangement as envisaged in FCC-ee.

This work presents the first prototype built for the final focus quadrupole of the FCC-ee study. A CCT superconducting magnet impregnated with paraffin wax is built and tested. The magnet exhibits no training behavior and reaches the short-sample critical current (1000 A) for various current rates up to 100 A/s. The field quality is excellent. Additionally, this prototype validates the possibility of tuning the conductor layout of a CCT magnet to greatly reduce edge effects for improved field quality. Future work will present a similar approach for crosstalk compensation of a twin-aperture arrangement. This will be demonstrated with a CCT sextupole wound with High-Temperature-Superconducting (HTS) tapes.

We would like to acknowledge the help of Michael Daly, Colin Müller, and Jaap Kosse from the Paul Scherrer Institute (PSI), as well as Daniel Barna from the Wigner Research Centre for Physics for their support with the impregnation station. Additionally, many thanks to the team at SM18 Arnaud Devred, Gerard Willering, Jerome Feuvrier, and Franco Julio Mangiarotti for their help with the cold tests. Finally, we would like to thank Austin Ball, Günther Dissertori, Markus Klute, and Maf Alidra for their invaluable help during various stages of the project.

Table 2: Comparison of the field quality between the warm and cold (1.9 K) measurements of the CCT quadrupole powered with a current of 5 A and 765 A respectively. The normal \(b_n\) and skew \(a_n\) components of different higher order multipoles \(n\) are evaluated at a \(R_{ref} = 10\) mm. They are expressed in \(10^{-4}\) units.


[System]
You are a **scientific‐rigor auditor**. You will receive the parsed contents of a research paper. Your job is to identify **only** those errors or flaws that directly undermine the **scientific validity** of the paper’s methods, analyses, or conclusions.

Your sole focus is to identify flaws—such as errors in experimental design, data integrity, calculations, statistical inference, or reproducibility—that directly call into question the validity of a specific claim, paragraph, or the paper as a whole.

**Do not** report issues that are purely presentational, rhetorical, stylistic, or related to citation practices.

---

After you’ve done a **detailed walkthrough** of the paper, output exactly in this format—no extra keys or commentary:

```
<analysis>
{detailed walk‑through of how you checked each section/figure and why you flagged (or did not flag) any flaw}
</analysis>

<response>
{
  "has_error": <true|false>,
  "errors": [
    {
      "location": "Section 2.1",
      "description": "Claim that ‘all X are Y’ is contradicted by citation [5], which shows examples of non‑Y X."
    },
    {
      "location": "Figure 3",
      "description": "XAxis labeled ‘Time (s)’ but units appear to be milliseconds; scale bar mismatches caption."
    }
    // …more entries…
  ]
}
</response>
```

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We investigate the explicit expression for the principal eigenvalue $\lambda_1^X(D)$ for a large class of compound Poisson processes $X$ on a bounded open set $D$ by examining its spectral heat content. When the jump density of the compound Poisson process is radially symmetric and strictly decreasing, we demonstrate that balls are the unique minimizers for $\lambda_1^X(D)$ among all sets with equal Lebesgue measure. Furthermore, we show that this uniqueness fails if the jump density is not strictly decreasing.

The principal eigenvalue of stochastic processes on domains is of fundamental importance. One of the many practical implications of the principal eigenvalue is its role as a large deviation parameter, determining the decay rate of survival probability ([6, Equation 2.4]). An intriguing question in this context is what shape of domains minimizes the principal eigenvalue, ensuring the highest survival probability among all sets with equal Lebesgue measure. The answer lies in the celebrated Faber-Krahn inequality, which asserts that balls are the unique minimizers for the principal eigenvalue. However, except in very few cases, the explicit expression for the principal eigenvalue is not available, and one must rely on other objects such as torsion functions (expected lifetime) or the inradius of the underlying domains to estimate the principal eigenvalue (see [6]). Hence, calculating the precise value or finding the concrete expression for the principal eigenvalue becomes an important inquiry.

The primary objective of this paper is to study the principal eigenvalue of a broad class of compound Poisson processes and derive its explicit expression. Furthermore, when the jump density of the compound Poisson process is radially symmetric and strictly decreasing, we demonstrate that the principal eigenvalue is minimized on balls among all sets with equal Lebesgue measure (see Corollary 2.2). On the other hand, we provide a counterexample that this uniqueness of the minimal eigenvalue for balls breaks down in the absence of a strictly decreasing jump density.

The main device to prove the main results, Theorems 2.1 and 2.3, is the spectral heat content (SHC). For a Lévy process $Y = \{Y_t\}_{t \ge 0}$ and a bounded open set $D$ in $\mathbb{R}^d$, we define the spectral

heat content on \(D\) with respect to \(Y\) by

\[
Q_D^Y(t) = \int_D \mathbb{P}_x(\tau_D^Y > t) \, dx,
\]

where \(\tau_D^Y = \inf\{ t > 0 : Y_t \notin D \}\) is the first exit time of \(Y\) from \(D\) and \(\mathbb{P}_x\) is the law of \(Y\) started at \(x\). The spectral heat content for jump processes has been studied intensively in the last decade (see [7, 9, 10, 11] and references therein). When there exists a discrete spectrum for the infinitesimal generator of the process, the spectral heat content has the spectral representation (3.1), and one can obtain the principal eigenvalue from this representation. For compound Poisson processes, the spectral heat content also admits the jump decomposition (3.3), since there are only a countable number of jumps of which the duration of the next jump is strictly positive. This is a crucial property that holds only for compound Poisson processes. For general Lévy processes whose Lévy measures are infinite, the jump times are dense on \([0, \infty)\) and the jump decomposition (3.3) of the spectral heat content does not hold. Extension to more general Lévy processes with infinite Lévy measure is certainly an interesting and natural question and this will be investigated in a future project.

The strategy to establish the main result is as follows. We first establish two technical lemmas, Lemmas 4.4 and 4.5. In Lemma 4.4, we prove that the jump chain, \(S_n = J_1 + \cdots + J_n\), conditioned on \(\{ S_n \in D \}\) converges to a uniform distribution on \(D\) as \(n \to \infty\). Intuitively speaking, if \(D\) is bounded, then it is difficult for the jump chain to stay inside a bounded open set \(D\) for large \(n\), meaning that there should be no preference for \(S_n\) to be in a certain region in \(D\) than another. Hence, the distribution of \(S_n\) conditioned on \(\{ S_n \in D \}\) should converge to a uniform distribution on \(D\). Then, in Lemma 4.5, we compute the probability that the jump chain \(S_{n+1}\) started at \(x \in D\) remains inside \(D\) conditioned on it has stayed inside \(D\) up to \(n\) and show that the conditional probability converges to the probability that a single jump \(J_1\) stays inside \(D\) averaged on all starting points \(x \in D\). From these two facts, together with the comparison between the spectral decomposition (3.1) and the jump decomposition (3.3), we derive the precise expression for the principal eigenvalue in Theorem 2.1.

As an application of Theorem 2.1, we show that the principal eigenvalue for symmetric compound Poisson processes becomes minimized on balls among all sets with equal Lebesgue measure when the probability density function \(j(x)\) of an individual jump \(J_1\) of the underlying compound Poisson process is radially symmetric and strictly decreasing. The fact that balls are minimizers is proved in [1, Corollary 5.5] for a more general class of Lévy processes, but the ball being the unique minimizer for symmetric compound Poisson processes has not been studied in the past to the best of the authors’ knowledge. A natural question arises for the necessity of the condition that \(j(x)\) being radially symmetric or strictly decreasing. In Theorem 2.3, we show that the uniqueness fails by considering a compound Poisson process whose jump distribution is uniformly distributed over some set \(A \subset \mathbb{R}^d\), whose Lévy density is obviously not strictly decreasing. In fact, in Theorem 2.3, we characterize the maximal spectral heat content under volume constraint and observe that this

characterization has two scenarios depending on the relative size of the support of the jump and the underlying set D. When the Lebesgue measure |A| of the support of the jump is relatively small compared to the Lebesgue measure |D|, the spectral heat content becomes maximized when the domain D and the support of jump A are congruent ellipsoids. On the other hand, when |A| is relatively large compared to |D|, the spectral heat content becomes maximized as long as D − D ⊂ A. The intuition behind this fact is that when the support of jump is much larger than the size of D, the probability that the next jump stays inside D does not depend on the location from which it jumps, and each jump lands on D with the same probability |D|/|A|. The rigorous proof of the two cases depends on the characterization of the equality cases of Riesz’s inequality in [3]. When there exist discrete and countable eigenvalues, the non-uniqueness of minimal eigenvalues among all sets with equal Lebesgue measure follows from the non-uniqueness of maximal spectral heat contents and (3.2).

The paper is organized as follows. In Section 2, we state the main results of this paper. In Section 3, we recall necessary preliminaries such as Lévy processes, their semigroups, and rearrangement inequalities. We provide the proofs of the main results in Section 4.

The indicator function of a set A is denoted by 1_A. For a bounded set A, its Lebesgue measure is denoted by |A|. Let A, B ⊂ R^d be subsets of R^d, a ∈ R and z ∈ R^d. We define aA := {ax : x ∈ A}, z + A := {z + x : x ∈ A}, and A + B := {x + y : x ∈ A, y ∈ B}. We use P_x and E_x to denote the probability measure and expectation for processes starting at x ∈ R^d. When x = 0, we use P and E instead of P_0 and E_0. We reserve notation X = {X_t}_{t≥0} for compound Poisson processes and Y = {Y_t}_{t≥0} for generic Lévy processes.

Let D be a bounded open set in R^d and λ_1^X(D) be the principal eigenvalue for the semigroup for a compound Poisson process X on D and j(x) be the probability density function for J_1 (see Section 3 for definitions).

**Theorem 2.1**  Let X = {X_t}_{t≥0} be a compound Poisson process with jump rate r > 0. Assume that X satisfies Condition 4.1. For a bounded open set D, its principal eigenvalue λ_1^X(D) is given by

λ_1^X(D) = (r / |D|) ∫_D ∫_{D^c} j(y − x) dy dx.

It follows from the Riesz's rearrangement inequality (see [8, Theorem 3.9])

∫_D ∫_D j(y − x) dy dx ≤ ∫_{D^*} ∫_{D^*} j^*(y − x) dy dx,

where j^* is the symmetric decreasing rearrangement of j (see Section 3 for the definition). Then, Theorem 2.1 implies

λ_1^X(D) ≥ (r / |D|) ∫_{D^*} ∫_{(D^*)^c} j^*(y − x) dy dx = λ_1^{X^*}(D^*),  (2.1)

where \( D^* \) is a centered ball with \( |D^*| = |D| \) and \( \lambda_1^X(D) \) and \( \lambda_1^{X^*}(D^*) \) are the principal eigenvalues for the compound Poisson process \( X \) in \( D \) and the rotationally symmetrized process \( X^* \) in \( D^* \) (see Subsection 4.2 for the definition), respectively. Indeed, this was shown in [1] for more general Lévy processes. The following result characterizes equality cases when the probability density function \( j(x) \) is radially symmetric and strictly decreasing.

Corollary 2.2 Let \( D \) be a bounded open set in \( \mathbb{R}^d \) and \( X^* \) be the symmetrization of \( X \). If the probability density function \( j(x) \) of \( J_1 \) is radially symmetric and strictly decreasing, then the following Faber-Krahn inequality holds;
\[
\lambda_1^X(D) \geq \lambda_1^{X^*}(D^*).
\]

Furthermore, the equality above holds if and only if \( D \) is a ball.

Proof. The inequality follows from (2.1). The proof for the equality cases follows from the equality case of strict rearrangement inequality [8, Theorem 3.9].

Corollary 2.2 asserts that the ball is the unique minimizer for the principal eigenvalue for the compound Poisson process when \( j(x) \) is radially symmetric and strictly decreasing. In Theorem 2.3, we show that if the jump density is not strictly decreasing, then the uniqueness of the minimizers for the principal eigenvalue fails. In fact, we investigate cases that guarantee the maximal spectral heat content for the compound Poisson process whose jump distribution is uniformly distributed on some subset \( A \subset \mathbb{R}^d \) without assuming Condition 4.1. When there exist discrete eigenvalues, the non-uniqueness of minimal principal eigenvalue follows from the non-uniqueness of maximal spectral heat content and (3.2).

Theorem 2.3 Let \( X = \{X_t\}_{t \geq 0} \) be a compound Poisson process, where the sequence of individual jumps \( \{J_i\}_{i=1}^{\infty} \) is uniformly distributed on a bounded set \( A \subset \mathbb{R}^d \) and \( X^* \) be the symmetrization of \( X \). If \( |A| < 2^d|D| \), then the followings are equivalent:

On the other hand, if \( |A| \geq 2^d|D| \), then the followings are equivalent:

Furthermore, assume that the spectral heat content for \( X \) has the spectral representation (3.1). If \( |\mathcal{A}| < 2^d|D| \), then \( \lambda_1^X(D) \) is minimized if \( D = a + \alpha\mathcal{E} \) and \( A = \gamma\mathcal{E} \) for some \( a \in \mathbb{R}^d \) and some ellipsoid \( \mathcal{E} \subset \mathbb{R}^d \) up to measure zero. On the other hand, if \( |\mathcal{A}| \geq 2^d|D| \), then \( \lambda_1^X(D) \) is minimized if \( D - D \subset A \). Hence, the uniqueness of the minimal principal eigenvalue for balls does not hold.

We close this section by introducing further questions related to our main results.

Let \( (\Omega, \mathcal{F}, \mathbb{P}) \) be a probability space. A \( d \)-dimensional Lévy process \( Y = \{Y_t\}_{t \geq 0}\) on \( (\Omega, \mathcal{F}, \mathbb{P}) \) is a stochastic process with càdlàg paths and independent and stationary increments. The characteristic exponent \( \psi(\xi) \) of a Lévy process \( Y = \{Y_t\}_{t \geq 0} \) is defined by \( \mathbb{E}[e^{i \xi \cdot Y_t}] = e^{-t \psi(\xi)} \) for \( \xi \in \mathbb{R}^d \). It follows from the Lévy–Khintchine theorem that \( Y = \{Y_t\}_{t \geq 0} \) is a Lévy process with characteristic exponent \( \psi(\xi) \) if and only if there exists a triplet \( (b, A, \nu) \) such that

\[
\psi(\xi) = ib \cdot \xi + \frac{1}{2} \xi \cdot A \xi + \int_{\mathbb{R}^d} (1 - e^{i \xi \cdot y} + (i \xi \cdot y) 1_{\{x:|x|<1\}}(y)) \nu(dy)
\]

where \( b \in \mathbb{R}^d \), \( A \) is a positive semi-definite \( d \times d \) matrix, and \( \nu \) is a \( \sigma \)-finite Borel measure on \( \mathbb{R}^d \setminus \{0\} \) satisfying \( \int_{\mathbb{R}^d \setminus \{0\}} (1 \wedge |y|^2) \nu(dy) < \infty \).

Let \( N = \{N_t\}_{t \geq 0} \) be a Poisson process with rate \( r \), that is \( \mathbb{P}(N_t = k) = e^{-rt} \frac{(rt)^k}{k!},\ k \geq 0 \).
Let \( S_n = J_1 + \dots + J_n \) be the jump chain, where \( \{J_i\}_{i=1}^{\infty} \) is a sequence of i.i.d. random variables independent of \( N \) whose characteristic exponent is given by \( \phi(\xi) := \mathbb{E}[e^{i\xi J_1}] = \int_{\mathbb{R}^d} e^{i \xi \cdot y} \mu(dy) \), where \( \mu \) is a probability measure on \( \mathbb{R}^d \) with \( \mu(\{0\}) = 0 \). We define a compound Poisson process \( X = \{X_t\}_{t \geq 0} \) with rate \( r > 0 \) and a jump measure \( \mu \) by \( X_t := S_{N_t} \). When each \( J_i \) is symmetric, that is \( \mu(A) = \mu(-A) \) for all Borel sets \( A \), then \( X \) is also symmetric and we will call \( X \) a symmetric compound Poisson process. Note that

\[
\mathbb{E}[e^{i \xi X_t}] = \mathbb{E}[e^{i\xi S_{N_t}}] = \sum_{n=0}^{\infty} \mathbb{E}[e^{i\xi S_n}, N_t = n] = \sum_{n=0}^{\infty} e^{-rt} \frac{(rt)^n}{n!} \phi(\xi)^n = \exp \left(-rt \int_{\mathbb{R}^d} (1 - e^{i\xi \cdot y}) \mu(dy)\right).
\]

By the Lévy-Khintchine formula, this shows that the Lévy measure of \( X \) is \( r\mu \).

Let \( D \) be a bounded open set in \( \mathbb{R}^d \) and \( Y = \{ Y_t \}_{t \geq 0} \) a symmetric Lévy process. Let \( P_t^D f(x) = \mathbb{E}_x [ f(Y_t); t < \tau_D ] \) be a semigroup associated with the killed process \( Y^D \), where \( Y_t^D = Y_t \) if \( t < \tau_D^Y := \inf \{ t > 0 : Y_t \notin D \} \) and \( Y_t^D = \partial \) for \( t \geq \tau_D^Y \), where \( \partial \) is a cemetery state. The infinitesimal generator \( \mathcal{L} \) for the semigroup \( (P_t^D)_{t \geq 0} \) is given by

\[
\mathcal{L} f(x) = \lim_{t \downarrow 0} \frac{P_t^D f(x) - f(x)}{t},
\]

whenever the limit exists, where the limit is taken in the supremum norm. The generator of the strongly continuous semigroup is non-positive definite and self-adjoint. We assume that the generator \( \mathcal{L} \) and the semigroup \( P_t^D \) admit discrete spectra \( \{ \lambda_n^Y(D) \}_{n=1}^\infty \) and \( \{ e^{- t \lambda_n^Y(D)} \}_{n=1}^\infty \) such that \( P_t^D \varphi_n^Y = e^{-t \lambda_n^Y(D)} \varphi_n^Y \) and \( \mathcal{L} \varphi_n^Y = \lambda_n^Y(D) \varphi_n^Y \), where \( \{ \varphi_n^Y \} \) is an orthonormal basis of eigenfunctions in \( L^2(D) \).

Assume that \( Y \) admits a transition density \( p_D^Y(t, x, y) \) on \( D \). It is well-known that \( p_D^Y(t, x, y) \) has the following representation

\[
p_D^Y(t, x, y) = \sum_{n=1}^\infty e^{-t \lambda_n^Y} \varphi_n^Y(x) \varphi_n^Y(y).
\]

From this representation, the spectral heat content can be expressed as

\[
Q_D^Y(t) = \int_D \mathbb{P}_x(\tau_D^Y > t) \, dx = \int_D \int_D p_D^Y(t, x, y) \, dy \, dx = \sum_{n=1}^\infty e^{-t \lambda_n^Y} \left( \int_D \varphi_n^Y(y) dy \right)^2. \tag{3.1}
\]

We call the above expression the spectral representation of the spectral heat content. This spectral representation will play an important role in proving Theorem 2.1. In particular, from (3.1) one can recover the principal eigenvalue \( \lambda_1^Y(D) \) by the following expression

\[
\lambda_1^Y(D) = - \lim_{t \to \infty} \frac{\ln Q_D^Y(t)}{t}. \tag{3.2}
\]

Besides the spectral decomposition, which holds for a large class of Lévy processes, the spectral heat content for a compound Poisson process has the following jump decomposition. Note that \( X_t = S_{N_t} \), where \( S_n = J_1 + \cdots + J_n \), \( N_t \) is an independent Poisson process with rate \( r \) and \( \{ J_i \}_{i=1}^\infty \) is a sequence of i.i.d. random variables. Since \( \{J_i\}_{i=1}^\infty \) and \( \{N_t\}_{t > 0} \) are independent,

\[
\begin{align*}
Q_D^X(t) &= \int_D \mathbb{P}_x(\tau_D^X > t) dx \\
&= \sum_{n=0}^\infty \int_D \mathbb{P}_x(\tau_D^X > t,\; N_t = n) dx \\
&= \sum_{n=0}^\infty \int_D \mathbb{P}_x \left( \bigcap_{k=1}^n \{ S_k \in D \},\; N_t = n \right) dx \\
&= \sum_{n=0}^\infty e^{-rt} \frac{(r t)^n}{n!} \int_D A(x, n, D) dx,
\end{align*} \tag{3.3}
\]

where \( A(x, n, D) = \mathbb{P}_x\left( \bigcap_{k=1}^n \{ S_k \in D \} \right) \).

For a subset \( D \subset \mathbb{R}^d \) with \( |D| < \infty \), its symmetric rearrangement \( D^* \) is a centered ball with the same volume as \( D \). That is, \( D^* = B(0, r) \), where \( \omega_d r^d = |D| \) and \( \omega_d = \frac{\pi^{d/2}}{\Gamma(1+d/2)} \) is the volume of a unit ball in \( \mathbb{R}^d \). For a nonnegative function \( f \) with \( |\{x \in \mathbb{R}^d : f(x) > t\}| < \infty \) for all \( t > 0 \), the symmetric decreasing rearrangement \( f^* \) is defined by

\[
f^*(x) = \int_0^\infty 1_{\{f(x)>t\}}^* dt.
\]

Riesz's rearrangement inequality [8, Theorems 3.7 and 3.9] states that

\[
\int_{\mathbb{R}^d} \int_{\mathbb{R}^d} f(x)g(x-y)h(y) \, dx \, dy \leq \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} f^*(x)g^*(x-y)h^*(y) \, dx \, dy.
\]

More generally, if \( f_1, f_2, \ldots, f_m \) are nonnegative functions with \( |\{x \in \mathbb{R}^d : f_i(x) > t\}| < \infty \) for all \( t > 0 \) and for \( i=1,2,\ldots,m \), and \( B = (b_{ij}) \) is a \( k \times m \) matrix for \( k \leq m \), then

\[
\int \cdots \int \prod_{i=1}^m f_i \left( \sum_{j=1}^k b_{ij}x_j \right) dx_1 \cdots dx_k \leq \int \cdots \int \prod_{i=1}^m f_i^* \left( \sum_{j=1}^k b_{ij}x_j \right) dx_1 \cdots dx_k. \tag{3.4}
\]

This inequality is called the Brascamp–Lieb–Luttinger inequality [2, Theorem 1.2]. We use the following notations

\[
\mathcal{J}(f,g,h) := \int (f * g)(x)h(x)dx, \quad \mathcal{J}(A,B,C) := \mathcal{J}(1_A, 1_B, 1_C).
\]

A natural question arises for the characterization of equality cases in Riesz’s rearrangement inequality. It follows from [8, Theorem 3.9] that if \( g = g^* \) and strictly decreasing, then \( \mathcal{J}(f,g,h) = \mathcal{J}(f^*,g^*,h^*) \) implies that \( f \) and \( h \) are radially symmetric up to the same translation, that is, there exists \( y \in \mathbb{R}^d \) such that \( f(x) = f^*(x-y) \) and \( h(x) = h^*(x-y) \). In [3], the author characterized the equality cases for Riesz’s rearrangement inequality when \( f, g, h \) are indicator functions.

**Theorem 3.1 [3, Theorem 1]**
Let \( A, B \) and \( C \) be measurable sets of finite positive measure in \( \mathbb{R}^d \). Denote by \( A^*, B^*, C^* \) the centered balls of the same measure as \( A, B, C \), respectively, and let \( \alpha, \beta, \gamma \) be the radii of these balls. If \( |\alpha - \beta| < \gamma < \alpha + \beta \), then there is equality in

\[
\mathcal{J}(A,B,C) \leq \mathcal{J}(A^*, B^*, C^*) \tag{3.5}
\]

if and only if, up to sets of measure zero,

\[
A = a + \alpha \mathcal{E}, \quad B = b + \beta \mathcal{E}, \quad C = c + \gamma \mathcal{E},
\]

where \( \mathcal{E} \) is a centered ellipsoid in \( \mathbb{R}^d \), and \( a, b \) and \( c = a+b \) are vectors in \( \mathbb{R}^d \). Otherwise, permute the three sets so that \( \gamma \geq \alpha + \beta \). Then, the equality in (3.5) holds if and only if \( A + B \subset C \) up to a set of measure zero.

In this section, we prove Theorem 2.1. Let \( X = \{X_t\}_{t \geq 0} \) be a compound Poisson process with rate \( r > 0 \) and a jump measure \( \mu \). Recall that \( \varphi(\xi) = \mathbb{E}[e^{i\xi J_1}] = \int_{\mathbb{R}^d} e^{i\xi y} \mu(dy) \), where \( J_i \) are jumps of the compound Poisson process. In this section, we assume the following conditions.

Condition 4.1  1. \( J_1 \) has a symmetric probability density function \( j(x) \).

Hence, \( p(t, x) := \sum_{n=0}^\infty e^{-rt} \frac{(rt)^n}{n!} j^{\otimes n}(x) \) is the probability density function for \( X_t \).

\[
\mathbb{P}(X_t \in A) = \mathbb{P} \bigg( \sum_{i=1}^{N_t} J_i \in A \bigg) = \sum_{n=0}^\infty \mathbb{P} \bigg( \sum_{i=1}^n J_i \in A, N_t = n \bigg)
= \sum_{n=0}^\infty e^{-rt} \frac{(rt)^n}{n!} \int_A j^{\otimes n}(x) dx.
\]

Remark 4.2  1. Under the condition (2) in (4.1), \( X_t \) has a symmetric probability density function. Indeed, for any Borel set \( A \)

We recall the following version of the central limit theorem. The proof is given in [5, XV.5 Theorem 2] and the comment below it.

Lemma 4.3 ([5, XV.5 Theorem 2]) Let \( \{J_i\}_{i=1}^\infty \) be a sequence of i.i.d. random variables with probability density function \( j(x) \). Suppose that \( \operatorname{Var}(J_1) < \infty \) and there exists a positive integer \( k \)

such that \(|\phi(\xi)|^k\) is integrable on \(\mathbb{R}^d\), where \(\phi(\xi) = \mathbb{E}[e^{i\xi J_1}]\) is the characteristic function of \(J_1\). Let \(j_n(x)\) be the density of \(\frac{\sum_{i=1}^n J_i}{\sqrt{n \mathrm{Var}(J_1)}}\). Then, uniformly on \(x \in \mathbb{R}^d\),

\[
\lim_{n \to \infty} j_n(x) = (2\pi)^{-d/2} e^{-\frac{|x|^2}{2}}.
\]

Recall that \(S_n = \sum_{i=1}^n J_i\) is the jump chain of the underlying compound Poisson process. We show that conditional on \(S_n \in D\), the distribution of \(S_n\) converges weakly to the uniform distribution on \(D\).

Furthermore, the convergence is uniform for all \(x \in E\).

\[
\lim_{n \to \infty} \mathbb{E}_x \left[ e^{i\xi \cdot S_n} \mid S_n \in D \right] = \frac{1}{|D|} \int_{\mathbb{R}^d} e^{i\xi \cdot w} 1_D(w) dw, \quad \xi \in \mathbb{R}^d
\]

**Lemma 4.4** Let \(D\) be a bounded open set. Suppose the Condition 4.1 holds true. For any \(x \in \mathbb{R}^d\)

**Proof.** Let \(j_n(u)\) be the transition density of \(\frac{S_n}{\sigma \sqrt{n}}\) where \(\mathrm{Var}(J_1) = \sigma^2\). By the change of variable \(\sigma \sqrt{n} u = v\):

\[
\begin{aligned}
\mathbb{E}_x \left[ e^{i\xi \cdot S_n} \mid S_n \in D \right]
&= \frac{\mathbb{E}_x \left[ e^{i\xi \cdot S_n}; S_n \in D \right]}{\mathbb{P}_x (S_n \in D)}
= \frac{\mathbb{E} \left[ e^{i\xi \cdot (S_n + x)}; S_n \in D \right]}{\mathbb{P}(S_n + x \in D)} \\[2ex]
&= \frac{\mathbb{E} \left[ e^{i\xi \cdot x} e^{i\xi \cdot \sigma \sqrt{n} \frac{S_n}{\sigma\sqrt{n}}}; \frac{S_n}{\sigma\sqrt{n}} \in \frac{D-x}{\sigma\sqrt{n}} \right]}{\mathbb{P} \left( \frac{S_n}{\sigma\sqrt{n}} \in \frac{D-x}{\sigma\sqrt{n}} \right) } \\[2ex]
&= \frac{\int_{\mathbb{R}^d} e^{i\xi \cdot x} e^{i \sigma \sqrt{n} \xi \cdot u} 1_{ \frac{D-x}{\sigma\sqrt{n}}}(u) j_n(u) du}{\int_{\mathbb{R}^d} 1_{ \frac{D-x}{\sigma\sqrt{n}}}(u) j_n(u) du} \\[2ex]
&= \frac{\int_{\mathbb{R}^d} e^{i\xi \cdot x} e^{i\xi \cdot v} 1_{D-x}(v) j_n\left(\frac{v}{\sigma\sqrt{n}}\right) (\sigma\sqrt{n})^{-d} dv}{\int_{\mathbb{R}^d} 1_{D-x}(v) j_n\left(\frac{v}{\sigma\sqrt{n}}\right) (\sigma\sqrt{n})^{-d} dv} \\[2ex]
&= \frac{ \int_{\mathbb{R}^d} e^{i\xi \cdot (x+v)} 1_D(x+v) j_n\left(\frac{v}{\sigma\sqrt{n}}\right) dv }{ \int_{\mathbb{R}^d} 1_D(x+v) j_n\left(\frac{v}{\sigma\sqrt{n}}\right) dv }.
\end{aligned}
\]

Let \(\gamma(u) = (2\pi)^{-d/2}e^{-\frac{|u|^2}{2}}\). By the change of variable \(w = v + x\), we obtain

\[
\begin{aligned}
&\left| \int_{\mathbb{R}^d} e^{i\xi \cdot (x+v)} 1_D(x + v) j_n\left(\frac{v}{\sigma\sqrt{n}}\right) dv
- \int_{\mathbb{R}^d} e^{i\xi \cdot w} 1_D(w) \gamma(0) dw \right| \\
&\leq \int_{\mathbb{R}^d} e^{i\xi \cdot (x+v)} 1_D(x+v) \left| j_n\left(\frac{v}{\sigma\sqrt{n}}\right) - \gamma(0) \right| dv \\
&\leq |D| \sup_{v \in D-x} \left| j_n\left(\frac{v}{\sigma\sqrt{n}}\right) - \gamma(0) \right| \\
&\leq |D| \sup_{v \in D-D} \left| j_n\left(\frac{v}{\sigma\sqrt{n}}\right) - \gamma(0) \right| ,
\end{aligned}
\]

where \( D - D = \{x - y : x, y \in D\} \). It follows from Lemma 4.3 that the last expression converges to 0 as \( n \to \infty \). Using the same argument, uniformly for all \( x \in E \), we have

\[
\lim_{n \to \infty} \int_{\mathbb{R}^d} 1_D(x + v) j_n \left( \frac{v}{\sigma \sqrt{n}} \right) dv = |D| \gamma(0).
\]

Hence, uniformly for all \( x \in E \),

\[
\lim_{n \to \infty} \frac{ \int_{\mathbb{R}^d} e^{i\xi(x+v)} 1_D(x+v) j_n \left( \frac{v}{\sigma \sqrt{n}} \right) dv } { \int_{\mathbb{R}^d} 1_D(x+v) j_n \left( \frac{v}{\sigma \sqrt{n}} \right) dv }
= \frac{1}{|D|} \int_{\mathbb{R}^d} e^{i\xi \cdot w} 1_D(w) dv.
\]
\(\Box\)

**Lemma 4.5** Let \( D \) be a bounded open set. Suppose that the Condition 4.1 holds true. For any \( x \in \mathbb{R}^d \) we have
\[
\lim_{n \to \infty} \mathbb{P}_x \left( S_{n+1} \in D \Bigg| \bigcap_{k=1}^n \{ S_k \in D \} \right)
= \frac{1}{|D|} \int_D \mathbb{P}_x(J_1 \in D) dx = \frac{1}{|D|} \int_D \int_D j(x+u) du\, dx.
\]
Furthermore, the convergence is uniform for all \( x \in E \).

**Proof.**
By the strong Markov property, we have
\[
\mathbb{P}_x \left( S_{n+1} \in D \Bigg| \bigcap_{k=1}^n \{ S_k \in D \} \right)
= \frac{ \mathbb{P}_x (S_{n+1} \in D, \bigcap_{k=1}^n \{ S_k \in D \}) }{ \mathbb{P}_x (\bigcap_{k=1}^n \{ S_k \in D \}) }
\]
\[
= \frac{ \mathbb{E}_x [ \mathbb{P}_{S_n}(J_{n+1} \in D);\, \bigcap_{k=1}^n \{ S_k \in D\} ] }{ \mathbb{P}_x (\bigcap_{k=1}^n \{ S_k \in D \}) }.
\]

From Lemma 4.4 the distribution of \( S_n \), conditioned on \( S_n \in D \), converges weakly to the uniform distribution on \( E \). Hence, the independence and the weak convergence yield that
\[
\lim_{n \to \infty} \mathbb{P}_{S_n}(J_{n+1} \in D) = \frac{1}{|D|} \int_D \mathbb{P}_z(J_1 \in D) dz,
\]
as it is conditioned on the event \( S_n \in D \) in the numerator. Now, the conclusion follows from the bounded convergence theorem and cancellation between the same expressions \( \mathbb{P}_x (\bigcap_{k=1}^n \{ S_k \in D \}) \) in the numerator and denominator.
\(\Box\)

We are now ready to prove Theorem 2.1.

**Proof of Theorem 2.1.**
Recall from (3.3) that the spectral heat content \( Q_D^X(t) \) has the following jump decomposition

\[
Q_D^X(t) = \int_D \mathbb{P}_x(\tau_D^X > t) dx = \sum_{n=0}^{\infty} e^{-rt} \frac{(rt)^n}{n!} \int_D A(x, n, D) dx,
\]

10

where \( A(x, n, D) = \mathbb{P}_x\left( \bigcap_{k=1}^n \{ S_k \in D \} \right) \). Let \( \alpha = \frac{1}{|D|} \int_D \mathbb{P}_z(J_1 \in D) dz \) for simplicity. From (3.2), it suffices to show that

\[
\lim_{t \to \infty} \frac{\log Q_D^X(t)}{t} = -r(1-\alpha).
\]

Given \( \varepsilon > 0 \) with \( \varepsilon < \alpha \), it follows from Lemma 4.5 that there exists \( N = N(\varepsilon) \) such that

\[
\alpha - \varepsilon < \mathbb{P}_x \left( S_n \in D \mid \bigcap_{k=1}^{n-1} \{ S_k \in D \} \right) < \alpha + \varepsilon \tag{4.1}
\]

for all \( n \geq N \) and \( x \in D \). Hence, for all \( n \geq N \) and \( x \in D \)

\[
\begin{aligned}
A(x, n, D) &= \mathbb{P}_x \left( \bigcap_{k=1}^{N-1} \{ S_k \in D \} \right) \prod_{k=N}^n \mathbb{P}_x \left( S_k \in D \mid \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right) \\
&\leq \prod_{k=N}^n \mathbb{P}_x \left( S_k \in D \mid \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right) \\
&\leq (\alpha + \varepsilon)^{n-N+1}.
\end{aligned}
\]

Using a trivial upper bound \( A(x, n, D) \leq 1 \),

\[
\begin{aligned}
Q_D^X(t) &= \sum_{n=0}^\infty e^{-rt} \frac{(rt)^n}{n!} \int_D A(x, n, D) dx \\
&\leq |D| \left( \sum_{n=0}^{N-1} e^{-rt} \frac{(rt)^n}{n!} + \sum_{n=N}^\infty e^{-rt} \frac{(rt)^n}{n!} (\alpha + \varepsilon)^{n-N+1} \right) \\
&\leq |D| \left( \sum_{n=0}^{N-1} e^{-rt} \frac{(rt)^n}{n!} + (\alpha + \varepsilon)^{-N+1} e^{rt(\alpha + \varepsilon - 1)} \right).
\end{aligned}
\]

Taking logarithm and letting \( t \to \infty \)

\[
\limsup_{t \to \infty} \frac{\log Q_D^X(t)}{t} \leq -r(1 - \alpha) + r \varepsilon.
\]

Since \( \varepsilon > 0 \) is arbitrary we conclude that

\[
\limsup_{t \to \infty} \frac{\log Q_D^X(t)}{t} \leq -r(1 - \alpha).
\]

For the lower bound, it follows from (4.1) that for all \( n \geq N \) and \( x \in D \),

\[
A(x, n, D) = \mathbb{P}_x \left( \bigcap_{k=1}^{N-1} \{ S_k \in D \} \right) \prod_{k=N}^n \mathbb{P}_x \left( S_k \in D \mid \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right)
\]

1.
In this section, we prove Theorem 2.3 when the distribution of \( J_1 \) of the underlying compound Poisson process is uniformly distributed on a set \( A \subset \mathbb{R}^d \). Note that we do not impose the Condition 4.1 for the current section, which were used to derive Lemmas 4.4 and 4.5. For a compound Poisson process \( X_t = S_{N_t} \), we define the symmetrized compound Poisson process \( X_t^* = S_{N_t}^* \) where \( \{J_i^*\}_{i=1}^\infty \) is a sequence of i.i.d. random variables with probability density function \( j^* \) and \( S_n^* = \sum_{i=1}^n J_i^* \). We call \( X^* \) the symmetrization of \( X \). Then, we define

2.
\[
A(x, n, D) = \mathbb{P}_x \left(\bigcap_{k=1}^n \{S_k \in D\}\right), \qquad
A^*(x, n, D) = \mathbb{P}_x \left(\bigcap_{k=1}^n \{S_k^* \in D\}\right).
\]

3.
\[
\liminf_{t \to \infty} \frac{\log Q_D^X(t)}{t} \geq -r(1-\alpha) - r\varepsilon.
\]

4.
Furthermore, since \( \alpha > \varepsilon \) and \( N \) is fixed, we may decrease the value of \( c \), if necessary, so that

5.
\[
Q_D^X(t) \geq ce^{-rt} \left( e^{rt(\alpha-\varepsilon)} - \sum_{n=0}^{N-1} \frac{(rt)^n}{n!}(\alpha-\varepsilon)^n \right).
\]

6.
\[
\liminf_{t \to \infty} \frac{\log Q_D^X(t)}{t} \geq -r(1 - \alpha)
\]

7.
\[
\geq \mathbb{P}_x\left(\bigcap_{k=1}^{N-1} \{S_k \in D\}\right) (\alpha - \varepsilon)^{n-N+1}
\]

8.
for all sufficiently large \( t > 0 \). Taking logarithm and letting \( t \to \infty \) yield

9.
Since \( \varepsilon > 0 \) is arbitrary we conclude that

10.
Hence,

11.
\[
Q_D^X(t) \geq c e^{-rt(1-\alpha+\varepsilon)}.
\]

12.
Hence, for any constant \( c < (\alpha - \varepsilon)^{-N+1} \int_D \mathbb{P}_x \left(\bigcap_{k=1}^{N-1} \{S_k \in D\}\right) dx \), we have

13.
\[
Q_D^X(t) \geq \sum_{n=N}^{\infty} e^{-rt} \frac{(rt)^n}{n!} \int_D \mathbb{P}_x \left(\bigcap_{k=1}^{N-1} \{S_k \in D\}\right) (\alpha - \varepsilon)^{n-N+1} dx
\]
\[
\geq e^{-rt} (\alpha-\varepsilon)^{-N+1} \left( e^{rt(\alpha-\varepsilon)} - \sum_{n=0}^{N-1} \frac{(rt)^n}{n!} (\alpha-\varepsilon)^n \right) \int_D \mathbb{P}_x \left(\bigcap_{k=1}^{N-1} \{S_k \in D\}\right) dx.
\]

Note that

\[
\int_D A(x,n,D)\,dx = \int_{\mathbb{R}} 1_D(x) \int_{(\mathbb{R})^n} 1_D(x+x_1)\cdots 1_D(x+x_1+\cdots+x_n) \prod_{i=1}^n j(x_i) \,dx_1\cdots dx_n dx
\]

\[
= \int_{\mathbb{R}^{n+1}} \prod_{l=0}^{n} 1_D\left(x + \sum_{k=1}^l x_k\right) \prod_{i=1}^n j(x_i)\, dx_1\cdots dx_n dx.
\]

It follows from Brascamp–Lieb–Luttinger inequality (3.4) that

\[
\int_D A(x,n,D)\,dx \leq \int_{\mathbb{R}^{n+1}} \prod_{l=0}^{n} 1_{D^*}\left(x + \sum_{k=1}^l x_k\right) \prod_{i=1}^n j^*(x_i)\,dx_1\cdots dx_n dx
\]

\[
= \int_{D^*} A^*(x,n,D^*)\,dx.
\]

We observe that the spectral heat content is invariant under linear isomorphisms. Let \(T : \mathbb{R}^d \to \mathbb{R}^d\) be a linear isomorphism with \(|\det(T)| = 1\) and \(D,A\subset\mathbb{R}^d\) be open sets. Let \(\{J_i\}_{i=1}^\infty, \{\tilde{J}_i\}_{i=1}^\infty\) be sequences of random variables with densities \(j\) and \(\tilde{j} = T \circ j\) respectively. Let \(N_t\) be a Poisson process with rate \(r>0\), \(S_n = \sum_{i=1}^n J_i\), and \(\tilde{S}_n = \sum_{i=1}^n \tilde{J}_i\). Define \(X_t = S_{N_t}\) and \(\tilde{X}_t = \tilde{S}_{N_t}\). Then, we have \(Q_D^X(t) = Q_{T(D)}^{\tilde{X}}(t)\) for all \(t>0\). Indeed, the result follows from

\[
Q_D^X(t) = \sum_{n=0}^\infty e^{-rt} \frac{(rt)^n}{n!} \int_D A(x,n,D)\,dx
\]

where \(A(x,n,D) = \mathbb{P}_x\left(\bigcap_{k=1}^n\{S_k \in D\}\right)\) and

\[
\int_D A(x,n,D)\,dx = \int_{T(D)} \tilde{A}(x,n,T(D))\,dx,
\]
by the change of variables.

## Proposition 4.6
Let \(X_t = \sum_{i=1}^{N_t} J_i\) be a compound Poisson process, where \(J_i\) is uniformly distributed on a set \(A\subset \mathbb{R}^d\) and \(X^*\) be the symmetrization of \(X\). Assume that \(|A| < 2^d |E|\). Then, the followings are equivalent:

1. \(Q_D^X(t) = Q_{D^*}^{X^*}(t)\) for all \(t>0\).

2. \(Q_D^X(t) = Q_{D^*}^{X^*}(t)\) for some \(t>0\).

3. \(D = a + bE\) and \(A = cE\), where \(a \in \mathbb{R}^d\), \(b,c\in \mathbb{R}\), and \(E\) is a centered ellipsoid in \(\mathbb{R}^d\).

### Proof.
Clearly, the first condition implies the second.

Secondly, assume that \(Q_E^X(t) = Q_{E^*}^{X^*}(t)\) for some \(t > 0\). From [1, Corollary 5.5] \(Q_E^X(t) \leq Q_{E^*}^{X^*}(t)\) and from the jump decomposition (3.3) of the spectral heat content, we must have

\[
\int_D A(x,n,D)\,dx = \int_{D^*} A^*(x,n,D^*)\,dx
\]

for all $n \in \mathbb{N}$. By the change of variable $y = -x$

$$
\int_D A(x, 1, D) dx = \int_D \mathbb{P}_x(J_1 \in D) dx = \int_{-D} \mathbb{P}_{-y}(J_1 \in D) dy
$$

$$
= \int_{\mathbb{R}^d} 1_{-D}(y) \int_{\mathbb{R}^d} 1_D(-y + z) \frac{1_A(z)}{|A|} dz dy
$$

$$
= \int_{\mathbb{R}^d} \frac{1_A(z)}{|A|} \int_{\mathbb{R}^d} 1_{-D}(y) 1_D(-y + z) dy dz
$$

$$
= \int_{\mathbb{R}^d} \frac{1_A(z)}{|A|} (1_{-D} * 1_D)(z) dz
$$

$$
= \mathcal{J}(1_{-D}, 1_D, 1_A).
$$

Similarly, $\int_{D^*} A^*(x, 1, D^*) dx = \mathcal{J}(1_{(-D)^*}, 1_{D^*}, 1_{A^*})$, which in turn implies $\mathcal{J}(1_{-D}, 1_D, 1_A) = \mathcal{J}(1_{(-D)^*}, 1_{D^*}, 1_{A^*})$. Recall that in Theorem 3.1 $\alpha, \beta,$ and $\gamma$ represent the radii of underlying three sets arranged in an increasing order. In our setting, $\alpha$ and $\beta$ are the radius of $D^*$ and $\gamma$ is the radius of $A^*$. Hence, the condition $|\alpha - \beta| < \gamma < \alpha + \beta$ in Theorem 3.1 is equivalent to $|A| < |2D| = 2^d |D|$. It follows from Theorem 3.1 $D = a + b\mathcal{E}$ and $A = c\mathcal{E}$ for some centered ellipsoid $\mathcal{E}$.

Finally, suppose that $D = a + b\mathcal{E}$ and $A = c\mathcal{E}$, where $a \in \mathbb{R}^d$ and $\mathcal{E}$ is a centered ellipsoid in $\mathbb{R}^d$. Since the spectral heat content is translation invariant, we may assume that $a = 0$. Then, there exists a linear isomorphism $T$ such that $T(D) = D^*$ and $T(A) = A^*$. By the change of variables, for any $t > 0$ we have

$$
Q_D^X(t) = \sum_{n=0}^{\infty} e^{-rt} \frac{(rt)^n}{n!} \int_D A(x, n, D) dx
$$

$$
= \sum_{n=0}^{\infty} e^{-rt} \frac{(rt)^n}{n!} \int_{T(D)} \tilde{A}(x, n, T(D)) dx = Q_{D^*}^{X^*}(t).
$$

$\Box$

Now we study the case $|A| \geq 2^d|D|$.

## Proposition 4.7
Let $X_t = S_{N_t}$ be a compound Poisson process whose jump distribution is uniform on a set $A \subset \mathbb{R}^d$ and $X^*$ be the symmetrization of $X$. Assume that $|A| \geq 2^d |D|$. Then, the followings are equivalent:

1. $Q_D^X(t) = Q_{D^*}^{X^*}(t)$ for all $t > 0$.

2. $Q_D^X(t) = Q_{D^*}^{X^*}(t)$ for some $t > 0$.

3. $D - D \subset A$ up to measure zero, where $D - D := \{x - y : x, y \in D\}$.

4. $Q_D^X(t) = |D| \exp\left(-rt\left(1 - \frac{|D|}{|A|}\right)\right)$ for all $t > 0$.

Proof. Clearly, the first condition implies the second.

Secondly, suppose that \( Q_D^X(t) = Q_{D^*}^{X^*}(t) \) for some \( t > 0 \). It follows from the proof of Proposition 4.6 we have \( \mathcal{J}(1_{-D}, 1_D, 1_A) = \mathcal{J}(1_{D^*}, 1_{D^*}, 1_{A^*}) \). The condition \( |A| \geq 2^d |D| \) is equivalent to \( \gamma \geq \alpha + \beta \) in Theorem 3.1 and from Theorem 3.1, we conclude \( D - D \subset A \).

Thirdly, assume that \( D - D \subset A \). Without loss of generality, we may assume \( x = 0 \) and \( 0 \in D \). Since \( D = D - 0 \subset D + (-D) \subset A \)

\[
A(0,1,D) = \int_{\mathbb{R}^d} 1_D(y) \frac{1}{|A|} 1_A(y) dy = \frac{1}{|A|} \int_{\mathbb{R}^d} 1_{D \cap A}(y) dy = \frac{1}{|A|} \int_{\mathbb{R}^d} 1_D(y) dy = \frac{|D|}{|A|}.
\]

Similarly, by the chain rule for conditional probability, we have

\[
A(0,n,D) = \mathbb{P}\left( \bigcap_{k=1}^n \{ S_k \in D \} \right) = \prod_{k=1}^n \mathbb{P} \left( S_k \in D \bigg| \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right).
\]

We will show that \( \mathbb{P} \left( S_k \in D \bigg| \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right) = \frac{|D|}{|A|} \) for each \( 1 \leq k \leq n \). Let \( j_{k-1}(y_1, \cdots, y_{k-1}) \) be the joint density for \( (S_1, \cdots, S_{k-1}) \). Since \( \{S_j\}_{j=1}^{k-1} \) and \( J_k \) are independent, their joint density is \( j_{k-1}(y_1, \cdots, y_{k-1}) \cdot \frac{1}{|A|} 1_A(z) \). Hence,

\[
\begin{aligned}
\mathbb{P}\left( S_k \in D \bigg| \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right)
&= \frac{\mathbb{P}\left( \bigcap_{j=1}^{k-1} \{ S_j \in D \}, S_{k-1} + J_k \in D \right)}{\mathbb{P} \left( \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right)} \\
&= \frac{1}{\mathbb{P} \left( \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right)} \int_{(\mathbb{R}^d)^{k-1}} \int_{\mathbb{R}^d} 1_{D^{k-1}}(y_1,\cdots,y_{k-1}) 1_{D-y_{k-1}}(z) j_{k-1}(y_1,\cdots,y_{k-1}) \frac{1_A(z)}{|A|} dz dy_1 \cdots dy_{k-1} \\
&= \frac{1}{\mathbb{P} \left( \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right)} \int_{(\mathbb{R}^d)^{k-1}} 1_{D^{k-1}}(y_1,\cdots,y_{k-1}) j_{k-1}(y_1,\cdots,y_{k-1}) \int_{\mathbb{R}^d} 1_{D-y_{k-1}}(z) \frac{1_A(z)}{|A|} dz dy_1\cdots dy_{k-1} \\
&= \frac{1}{\mathbb{P} \left( \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right)} \int_{(\mathbb{R}^d)^{k-1}} 1_{D^{k-1}}(y_1,\cdots,y_{k-1}) j_{k-1}(y_1,\cdots,y_{k-1}) \frac{|(D-y_{k-1})\cap A|}{|A|} dy_1\cdots dy_{k-1} \\
&= \frac{|D-y_{k-1}|}{|A|} = \frac{|D|}{|A|},
\end{aligned}
\]

where we used the fact \( D-y_{k-1} \subset D-D \subset A \). This establishes \( \mathbb{P} \left( S_k \in D \bigg| \bigcap_{j=1}^{k-1} \{ S_j \in D \} \right) = \frac{|D|}{|A|} \) for each \( 1 \leq k \leq n \) and \( A(0,n,D) = \left( \frac{|D|}{|A|} \right)^n \) for all \( n \geq 1 \). Then,

\[
Q_D^X(t) = \sum_{n=0}^{\infty} e^{-rt} \frac{(rt)^n}{n!} \int_D A(x,n,E) dx = \sum_{n=0}^\infty |D|\left( \frac{|D|}{|A|} \right)^n \cdot e^{-rt} \frac{(rt)^n}{n!} = |D| \exp\left( -rt \left( 1 - \frac{|D|}{|A|} \right) \right).
\]

Finally, assume the spectral heat content is given by \( Q_D^X(t) = |D| \exp\left( -rt \left( 1 - \frac{|D|}{|A|} \right) \right) \). By repeating the same argument for \( X^* \) and \( D^* \), we know that \( Q_{D^*}^{X^*}(t) = |D| \exp\left( -rt \left( 1 - \frac{|D|}{|A|} \right) \right) \) and this implies \( Q_D^X(t) = Q_{D^*}^{X^*}(t) \) for all \( t > 0 \).
\(\Box\)

We are now ready to prove Theorem 2.3.

Proof of Theorem 2.3. The first part on the equivalent conditions follows from Propositions 4.6 and 4.7.

Assume that the spectral heat content for the process \( X \) has the spectral representation (3.1). From [1, Corollary 5.5], \( Q_D^X(t) \leq Q_{D^*}^{X^*}(t) \) for all \( t > 0 \), and the principal eigenvalue \( \lambda_1^X \) can be obtained using (3.2). Hence, any equivalent condition on the maximality of the spectral heat content guarantees the minimality of the principal eigenvalue. \(\Box\)

Acknowledgment: The second-named author is grateful to Professor Younghwan Son (POSTECH, South Korea) for the discussion and helpful comments on Lemmas 4.4 and 4.5.

Daesung Kim
School of Mathematics, Georgia Institute of Technology, GA 30332, USA
E-mail: dkim3009@gatech.edu

Hyunchul Park
Department of Mathematics, State University of New York at New Paltz, NY 12561, USA
School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea
E-mail: parkh@newpaltz.edu