[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]
DAN PETERSEN AND ORSOLA TOMMASI

Abstract. We construct a splitting of the cohomology of configuration spaces of points on a smooth proper variety with a multiplicative Chow–Künneth decomposition. Applied to hyperelliptic curves this shows that the hyperelliptic Torelli group acts trivially on the rational cohomology of ordered configuration spaces of points. Moreover, if Hg,n denotes the moduli space of n-pointed hyperelliptic curves, the Leray spectral sequence for the forgetful map Hg,n → Hg degenerates immediately, in sharp contrast to the forgetful map from Mg,n to Mg. This allows for new detailed calculations of the cohomology of M2,n for n ≤ 5, and the stable cohomology of Hg,n for n ≤ 5. We also give a detailed study of the cohomology of symplectic local systems on M2.

h(X) ≅ ⨁_{i=0}^{2d} h^{i}(X).

⨁_{i,j} h^{i}(X) ⊗ h^{j}(X) → ⨁_{k} h^{k}(X),

then $h^i(X) \otimes h^j(X) \to h^k(X)$ vanishes unless $i + j = k$. Equivalently, the class of the small diagonal $\Delta_{(3)} \subset X \times X \times X$ satisfies

\[
[\Delta_{(3)}] = \sum_{i + j + k = 2d} (\pi_i \times \pi_j \times \pi_k)_*[\Delta_{(3)}] \in CH_d(X^3).
\]

The notion of a multiplicative decomposition was isolated as an object of study in work of Voisin [Voi14] and explicitly by Shen–Vial [SV16]. Shen and Vial refer to this as a *multiplicative Chow–Künneth decomposition*, but we believe that this property is natural enough to have a more concise name.

The condition that this decomposition is multiplicative becomes that

(1) $[\Delta_{(3)}] = \Delta_{(2)} \times o_X + (\text{perm.}) - [X] \times o_X \times o_X - (\text{perm.})$

depending on the choice of a degree 1 zero-cycle $o_X \in CH_0(X)$, with corresponding projectors $\pi_0, \pi_1$ and $\pi_2$ given by

\[
h(X) \cong h^0(X) \oplus h^1(X) \oplus h^2(X)
\]

\[
\pi_0 = [X] \times o_X, \quad \pi_1 = [\Delta_{(2)}] - \pi_0 - \pi_2, \quad \pi_2 = o_X \times [X].
\]

since $\Delta_{(2)} \times o_X = \sum_{i+j=2} (\pi_i \times \pi_j \times \pi_2)_* [\Delta_{(3)}]$ and $[X] \times o_X \times o_X = (\pi_0 \times \pi_2 \times \pi_2)_*[\Delta_{(3)}]$. This condition is exactly the vanishing of the Gross–Schoen cycle [GS95] associated to $o_X$. One can show the Gross–Schoen cycle associated to $o_X$ can only vanish when $o_X = \frac{1}{2g-2} K_X$ [FLV21b, Section 3], and that in this case vanishing of the Gross–Schoen cycle is equivalent to vanishing of the Ceresa cycle [Cer83].

H⁴(F(Σ, 3), ℚ) ≅ V₂ ⊗ σ₁₁₁ ⊕ V₁ ⊗ σ₃

where V_λ denotes an irreducible representation of Sp_{2g}, and σ_μ denotes a Specht module (irreducible representation of S₃). If g ≫ 0, then H³(M_g, V₁) ≅ ℚ [Loo96]. Thus, one knows that in the Leray spectral sequence

E₂^{p,q} = H^p(M_g, R^q f_*^{(3)} ℚ) ⇒ H^{p+q}(M_{g,3}, ℚ)

one has \( E_2^{3,4} \cong \mathbb{Q} \) for sufficiently large \( g \). On the other hand the stable cohomology of \( M_{g,n} \) is concentrated in even degrees, so for \( g \) sufficiently large this class must be killed by a differential.\({ }^{2} \)

1.17. Let \( \Sigma \) be a genus \( g \) closed surface. The Totaro spectral sequence [Tot96] can be used to calculate \( H^{\bullet}(F(\Sigma, n), \mathbb{Q}) \), and the spectral sequence is \( \mathrm{Mod}(\Sigma) \)-equivariant. The action of \( \mathrm{Mod}(\Sigma) \) on the \( E_2 \)-page factors through the symplectic group \( \mathrm{Sp}_{2g}(\mathbb{Z}) \), where it is given by an algebraic representation of \( \mathrm{Sp}_{2g} \). Consequently, there is an induced filtration on the cohomology of the configuration space, such that \( \mathrm{Gr} H^{\bullet}(F(\Sigma, n), \mathbb{Q}) \) is naturally an algebraic representation of \( \mathrm{Sp}_{2g} \). According to Corollary 1.10, determining \( \mathrm{Gr} H^{\bullet}(F(\Sigma, n), \mathbb{Q}) \) as an \( \mathrm{Sp}_{2g} \)-module is equivalent to determining the action of the hyperelliptic mapping class group on \( H^{\bullet}(F(\Sigma, n), \mathbb{Q}) \). It thus follows from Corollary 1.15 that in order to calculate the cohomology groups \( H^{\bullet}(H_{g,n}, \mathbb{Q}) \), it suffices to determine \( \mathrm{Gr} H^{\bullet}(F(\Sigma, n), \mathbb{Q}) \) as an \( \mathrm{Sp}_{2g} \)-module, and to calculate \( H^{\bullet}(H_g, V_{\lambda}) \) for any irreducible representation \( V_{\lambda} \) of \( \mathrm{Sp}_{2g} \).

1.21. It is interesting to compare \( H_{\infty,n} \) and \( M_{\infty,n} \). The stable cohomology of \( M_{\infty,n} \) is algebraic, coincides with the tautological ring, and conjecturally the cycle map from Chow to cohomology is an isomorphism in a stable range (this is however widely open). By contrast, the stable cohomology of \( H_{\infty,n} \) is very complicated, not pure, and algebraic classes make up only a minuscule fragment of the cohomology. The Chow groups of \( H_{g, n} \) have been completely determined by Canning and Larson in the range \( n \leq 2g+6 \) [CL22] — in this range they coincide with the tautological rings, calculated by Tavakol [Tav18].

2. Splitting the cohomology of configuration spaces

2.1. Let us first justify the claim made in the introduction, that if a variety \( X \) admits a multiplicative decomposition then \( C^\bullet(X, \mathbb{Q}) \simeq H^\bullet(X, \mathbb{Q}) \) compatibly with mixed Hodge structures. The key point is the following commutative diagram:

Here SmComp\(_\mathbb{C}\) and ChowMotives are the 1-categories of smooth proper varieties and Chow motives over \( \mathbb{C} \), and \( D^b(\mathrm{MHS}_\mathbb{Q}) \) is the derived ∞-category of mixed Hodge structures. Importantly, all functors in the diagram are symmetric monoidal. For the two vertical arrows monoidality follows from recent work of Tubach [Tub23]. The functor \( \mathrm{SmComp}_\mathbb{C}^{\mathrm{op}} \to D^b(\mathrm{MHS}_\mathbb{Q}) \) sends \( X \) to a Hodge-theoretically enhanced version of \( C^\bullet(X, \mathbb{Q}) \); in terms of the constructions of Tubach’s paper, we are rather considering the object \( (a_X)_*(a_X)^*\mathbb{Q} \) in \( D^b(\mathrm{MHM}(\mathrm{Spec}(\mathbb{C}))) \simeq D^b(\mathrm{MHS}_\mathbb{Q}) \), where \( a_X \) is the structural morphism of \( X \). Now any \( X \) in SmComp\(_\mathbb{C}\) is a cocommutative comonoid object via the diagonal \( X \to X \times X \). This fact, and monoidality of the functors in the diagram, makes \( C^\bullet(X, \mathbb{Q}) \) into a commutative monoid object, i.e. an \( E_\infty \)-algebra, in the ∞-category \( D^b(\mathrm{MHS}_\mathbb{Q}) \). Since the category is \( \mathbb{Q} \)-linear, such an \( E_\infty \)-algebra can be rectified to an actual commutative dg algebra in mixed Hodge structures, unique up to contractible choice. But now the assumption that \( X \) admits a multiplicative decomposition implies that \( h(X) \) is isomorphic as a commutative monoid to the direct sum \( \bigoplus_i h^i(X) \), and commutativity of the diagram then implies the same for \( C^\bullet(X, \mathbb{Q}) \).

2.2. We can now prove Theorem 1.8.

Proof of Theorem 1.8. The first author has introduced an explicit functor \( A \mapsto CF(A, n) \) from non-unital commutative dg algebras to \( S_n \)-equivariant cochain complexes [Pet20]. It has the following properties:

(i) The functor \( CF(-, n) \) takes quasi-isomorphisms to quasi-isomorphisms.
(ii) If \( A \) is a cdga model for \( C^\bullet(X, \mathbb{Q}) \), where \( X \) is a locally compact Hausdorff topological space, then
\[
H^\bullet(CF(A, n)) \cong H_c^\bullet(F(X, n), \mathbb{Q}).
\]

The verification of these properties is sheaf-theoretic and uses only the six functors formalism, which implies that it makes sense purely algebraically and is compatible with all sorts of "enhanced" structure.

Now take \( X \) to be a smooth proper complex variety. We promote (a cdga model of) \( C^\bullet(X, \mathbb{Q}) \) to a cdga in mixed Hodge structures and plug it into the functor \( CF(-, n) \). Then the result

computes \( H_c^*(F(X,n), \mathbb{Q}) \) with its mixed Hodge structure. Now if \( X \) admits a multiplicative decomposition then \( C^*(X, \mathbb{Q}) \cong H^*(X, \mathbb{Q}) \), and we may instead plug \( H^*(X, \mathbb{Q}) \) into the functor, with its usual pure Hodge structure. But then the output is simply a cochain complex of pure polarizable Hodge structures, with differentials preserving the Hodge structure. Since this category is semisimple, the resulting cochain complex is quasi-isomorphic to its cohomology, and in particular \( H_c^*(F(X,n), \mathbb{Q}) \) is a direct sum of pure Hodge structures. Then the same is true for \( H^*(F(X,n), \mathbb{Q}) \) by Poincaré duality. □

2.3. A reader may be hesitant about the assertion that since the construction of \( \mathcal{CF} \) only uses the six functors, it is compatible with mixed Hodge structures. One way to think about the functor \( \mathcal{CF} \) is as follows. Fix a space \( X \) and a coefficient ring \( R \). Let \( \Pi_n \) denote the poset of partitions of the set \( \{1, \ldots, n\} \), ordered by refinement. There is a functor from \( \Pi_n \) to cochain complexes, assigning to a partition with \( k \) blocks the complex \( C_c^*(X^k, R) \). A morphism in \( \Pi_n \) corresponding to blocks being merged is mapped to a diagonal inclusion between cartesian powers of \( X \). The total homotopy fiber\(^3\) of this diagram is quasi-isomorphic to \( C_c^*(F(X,n),R) \). The point is that when \( R = \mathbb{Q} \), the whole diagram \( \Pi_n \to \text{Ch}_\mathbb{Q} \) can be reconstructed up to quasi-isomorphism from a cdga model\(^4\) for \( X \) and the Künneth formula, and the functor \( \mathcal{CF}(-) \) is a Bousfield-Kan type resolution computing this total homotopy fiber. This is really a special case of a construction introduced in [Pet17] for a general stratified space. Now the main construction of [Pet17] has been generalized to the setting of étale motivic sheaves by Dupont and Juteau [DJ20], meaning that we obtain a model of \( \mathcal{CF} \) in étale motives. With such a model in place one may then apply the realization functor to the derived category of mixed Hodge modules [Ivo16, Tub23]. This can also be carried out in families:

Proof of Theorem 1.13. We suppose as in the theorem that we are given a smooth proper family \( X \to S \) admitting a fiberwise multiplicative decomposition.

Denote the \( n \)-fold fibered power of \( X \) over \( S \) by \( X^n \). The space \( X^n \) is stratified with strata indexed by the partition lattice \( \Pi_n \). By assigning to every element of \( \Pi_n \) the closure of the corresponding stratum we get a contravariant functor from \( \Pi_n \) to smooth proper varieties over \( S \), taking a partition with \( k \) blocks to \( X^k \). We may think of this as a diagram in Chow motives over \( S \). By our assumption that there exists a fiberwise multiplicative decomposition, this diagram is isomorphic to a diagram whose values are pointwise given by direct sums of pure motives. Morphisms are morphisms of Chow motives and in particular preserving weights.

Now apply the realization to get a \( \Pi_n \)-shaped homotopy coherent diagram in the category \( \mathrm{DA}_c(S) \) of étale motives over \( S \) of geometric origin [Ayo07a, Ayo07b, Ayo14, CD16]. By [DJ20] we can form the total homotopy fiber of this diagram by an explicit procedure which outputs a Postnikov system [GM03, Chapter 4] in \( \mathrm{DA}_c(S) \), such that the output is weakly equivalent to the compactly supported motive of \( F(X/S,n) \). From the explicit construction we see that all terms in the Postnikov system are still sums of pure motives, and all morphisms preserve weights.

Now apply the realization to mixed Hodge modules [Ivo16, Tub23]. In the resulting Postnikov system in \( D^b(\mathrm{MHM}(S)) \), all terms are now sums of shifted polarizable variations of Hodge structures, with morphisms preserving the weights. By semisimplicity of the category of pure mixed Hodge modules of fixed weights, this Postnikov system must be isomorphic to one in which all objects are equal to their homology and all morphisms are zero. In particular the object of the Postnikov

---
\(^3\) Let \( D \) be a category with an initial object \( * \). The total homotopy fiber of a functor \( F : D \to M \) taking values e.g. in a model category is the homotopy fiber of the map \( F(*) \to \mathrm{holim}_{x\in D, x\not\cong *} F(x) \).

\(^4\) Over a more general ground ring one would need to know \( C_c^*(X,R) \) as an \( E_\infty \)-algebra.

system which represents the total homotopy fiber has this property, which means that the object
Rf!(n)Q is a direct sum of degree-shifted polarizable variations of Hodge structure. Then the same
is true for Rf∗(n) by duality.

2.4. Remark. In our arguments it was critical that the six operations on mixed Hodge modules
constructed by Saito [Sai90] can be lifted to the level of ∞-categories, as was recently proven
by Tubach [Tub23]. Saito only constructed these operations as functors between triangulated
categories, meaning in particular that commutative diagrams involving the six operations in Saito’s
formalism only commute up to non-specified, not necessarily coherent, homotopy. Such a diagram
does not contain enough information to form e.g. a homotopy limit. The proof of Theorem 1.13
might not at first seem to involve any higher category theory — however, the construction of [DJ20]
uses that DA(−) admits more structure than merely a system of triangulated categories; it is a
stable derivator. Moreover, we need to know that the realization functor DAc(S) → Db(MHM(S))
commutes with the six operations, which was not known until [Tub23].

3. Local systems on M2

3.1. In this section we shall explain what is known about the cohomology of the local systems Vλ
on M2, summarizing and slightly improving on results of [Pet15, Pet16, Wat18]. Our approach is to
use the Torelli map, which in genus 2 is an open embedding

M2 ↪ A2

of the moduli space of genus two curves into the moduli space of principally polarized abelian
surfaces. The complement A2 \ M2 is the locus A1,1 of products of elliptic curves, and A1,1 ≅
Sym2 M1,1. We thus get a Gysin long exact sequence:

… → Hk(A1,1, Vλ)(−1) → Hk+2(A2, Vλ) → Hk+2(M2, Vλ) → Hk+1(A1,1, Vλ)(−1) → …

As we explain shortly, all cohomology groups H•(A2, Vλ) and H•(A1,1, Vλ) are understood, and
hence if we can determine all Gysin homomorphisms

Hk(A1,1, Vλ)(−1) → Hk+2(A2, Vλ)

then we will have understood the cohomology of M2 with symplectic coefficients (at least up to
semisimplification).

3.2. A first observation is that (for λ ≠ 0), Hk(M2, Vλ) is nontrivial only for k ∈ {1, 2, 3}. This
is because the virtual cohomological dimension of the genus 2 mapping class group is 3, and H0
vanishes since the genus two mapping class group surjects onto Sp4(Z). We therefore only need
to consider two Gysin homomorphisms:

H0(A1,1, Vλ)(−1) → H2(A2, Vλ)

and

H1(A1,1, Vλ)(−1) → H3(A2, Vλ).

3.3. Let us first discuss the cohomology of A1,1 with symplectic coefficients. The computation
of H•(A1,1, Vλ) reduces to computing the cohomology of M1,1 with coefficient in an irreducible
representation of SL2, once one knows a branching formula for the restriction from Sp4 to SL2 ⋊
Σ2, which is determined in [Pet13, Proposition 3.4]. The restriction of the local systems Vλ to A1,1

decomposes as a direct sum of local systems denoted \( U_{a,b} \), \( U_a^+ \) and \( U_a^- \). The cohomologies of these local systems are given by

\[
H^{\bullet}(A_{1,1}, U_{a,b}) \cong H^{\bullet}(M_{1,1}, V_a) \otimes H^{\bullet}(M_{1,1}, V_b)
\]
\[
H^{\bullet}(A_{1,1}, U_a^+) \cong \mathrm{Sym}^2 H^{\bullet}(M_{1,1}, V_a)
\]
\[
H^{\bullet}(A_{1,1}, U_a^-) \cong \wedge^2 H^{\bullet}(M_{1,1}, V_a)
\]

where \( V_\ell \) denotes the local system on \( M_{1,1} \) associated to the \(\ell\)th symmetric power representation of \(\mathrm{SL}_2\). The cohomology \( H^\bullet(M_{1,1}, V_\ell) \) is determined by the Eichler–Shimura isomorphism:

\[
H^k(M_{1,1}, V_\ell) \cong
\begin{cases}
\mathbb{Q} & k = \ell = 0 \\
S_{\ell+2} \oplus \mathbb{Q}(-1 - \ell) & k = 1,\, \ell > 0\ \text{even} \\
0 & \text{otherwise.}
\end{cases}
\]

See [Ver61, Del69, Zuc79]. Here \( S_{\ell+2} \) denotes the "motive" (or Galois representation, or Hodge structure) attached to cusp forms for \(\mathrm{SL}(2, \mathbb{Z})\) of weight \(\ell+2\). Hence \( H^\bullet(A_{1,1}, V_\lambda) \) is completely known in all degrees for all \( \lambda \).

3.4. From the preceding paragraph we see in particular that the only local systems on \( A_{1,1} \) with nontrivial \( H^1 \) are the ones of the form \( U_{a,0} \) with \( a > 0 \) even. From the branching formula [Pet13, Proposition 3.4], each \( V_\lambda \) (where \( \lambda = (a, b) \)) contains exactly one summand of this form when restricted to \( A_{1,1} \), namely the local system \( U_{a-b,0}(-b) \), except if \( a = b \), in which case there is no such summand. Similarly \( H^0(A_{1,1}, V_\lambda) \) is nontrivial only when the restriction of \( V_\lambda \) contains a copy of the trivial representation \( U_0^+ \) as a summand, and from the branching formula this happens if and only if \( \lambda = (a, b) \) satisfies \( a = b \equiv 0 \pmod{2} \). We obtain:

3.5. Proposition. Let \( \lambda = (a, b) \) be even. The cohomology \( H^k(A_{1,1}, V_\lambda) \) in degrees \( k = 0 \) and \( k = 1 \) is given by

\[
H^0(A_{1,1}, V_\lambda) =
\begin{cases}
\mathbb{Q}(-b) & a = b \equiv 0 \pmod{2} \\
0 & \text{otherwise,}
\end{cases}
\]

and

\[
H^1(A_{1,1}, V_\lambda) =
\begin{cases}
S_{a-b+2}(-b) \oplus \mathbb{Q}(-1 - a) & a > b \\
0 & a = b,
\end{cases}
\]

respectively.

3.6. We now discuss the cohomology groups \( H^\bullet(A_2, V_\lambda) \). All these cohomology groups were completely determined in [Pet15, Theorem 2.1], building on earlier work of several people e.g. [FvdG04, Har12, Wei09, Fli05, Art04]. We do not give the explicit formula, which is rather complicated; we will be content to describe only as much of the cohomology as is needed for the computation of the relevant Gysin homomorphisms.

3.7. Starting with the Gysin map \( H^0(A_{1,1}, V_\lambda)(-1) \to H^2(A_2, V_\lambda) \), we read from [Pet15] that if \( \lambda = (a, b) \) is even then

\[
H^2(A_2, V_\lambda) =
\begin{cases}
\mathbb{Q}(-1-b)^{\oplus s_{a+b+4}} & a = b \equiv 0 \pmod{2} \\
0 & \text{otherwise.}
\end{cases}
\]

Here \( s_k \) denotes the dimension of the space of cusp forms for \(\mathrm{SL}(2, \mathbb{Z})\) of weight \( k \). Comparing with Proposition 3.5 we see that there is a possibility of a nonzero Gysin map precisely when \( a = b \equiv 0 \pmod{2} \) and \( s_{a+b+4} > 0 \). Since the domain of this Gysin map is one-dimensional we only need to determine if the homomorphism is zero or not. The Gysin map was shown to always be nonzero when \( a = b \equiv 0 \pmod{2} \) and \( s_{a+b+4} > 0 \) in [Pet16, Section 5]. Let us outline

the structure of the argument. We omit Tate twists in what follows, since they will play no role. We consider the map \( f : A_1 \times A_1 \to A_2 \), and fix an inclusion of the trivial local system \( \mathbb{Q} \) into \( f^*V_{b,b} \). It suffices to show \( H^0(A_1 \times A_1, \mathbb{Q}) \to H^2(A_2, V_{b,b}) \) is nonzero. We consider the Satake compactifications \( \overline{A}_1 \) and \( \overline{A}_2 \) and the embeddings \( j' : A_1 \times A_1 \to \overline{A}_1 \times \overline{A}_1,\, j : A_2 \to \overline{A}_2 \). There is a commutative diagram

\[
\begin{array}{ccc}
H^0(A_1 \times A_1, \mathbb{Q}) & \longrightarrow & H^2(A_2, V_{b,b}) \\
\downarrow & & \downarrow \\
(R^0 j'_*\mathbb{Q})_\xi & \longrightarrow & (R^2 j_*V_{b,b})_{f(\xi)}
\end{array}
\]

where \( \xi \) denotes the maximally degenerate point of \( \overline{A}_1 \times \overline{A}_1 \), \( f(\xi) \) is its image in \( \overline{A}_2 \) (which again is the unique maximally degenerate point), and the subscripts denote taking stalks of the respective sheaves. Since \( R^0 j'_*\mathbb{Q} \cong \mathbb{Q} \) the left vertical arrow is an isomorphism and it suffices to show that the Gysin map between stalks is nonzero. A formula due to Harder gives an expression for these stalks that can eventually be used to prove the result. The outcome is the following theorem.

\[
H^1(M_2, V_\lambda) \cong \begin{cases}
\mathbb{Q}(-3) & \text{if } \lambda = (2,2) \\
0 & \text{otherwise.}
\end{cases}
\]

**3.9.** We may now similarly consider the Gysin map \( H^1(A_{1,1}, V_\lambda)(-1) \to H^3(A_2, V_\lambda) \). We will not state the full expression for \( H^3(A_2, V_\lambda) \) from [Pet15] as it is a bit cumbersome. What will be relevant is that (up to semisimplification) \( H^3(A_2, V_{a,b}) \) contains as a direct summand the terms

\[
S_{a-b+2}(-1-b)^{\oplus s_{a+b+4}} \oplus \mathbb{Q}(-2-a)^{\oplus s_{a+b+4}};
\]

moreover, if \( b = 0 \) it contains an additional term \( \mathbb{Q}(-2-a) \). The remaining terms occurring in \( H^3(A_2, V_\lambda) \) do not occur in \( H^1(A_{1,1}, V_\lambda)(-1) \) and are therefore irrelevant for the question of determining the Gysin map. The natural expectation is that if \( a > b \) and \( s_{a+b+4} > 0 \) then \( H^1(A_{1,1}, V_{a,b})(-1) \cong S_{a-b+2}(-b) \oplus \mathbb{Q}(-1-a) \) injects into \( H^3(A_2, V_{a,b}) \), and if \( b = 0 \) then at least the summand \( \mathbb{Q}(-1-a) \) injects. Part of this was proven by Watanabe [Wat18], who showed that if \( a > b \) and \( s_{a+b+4} > 0 \) then the Gysin map is injective on the summand \( \mathbb{Q}(-2-a) \) of \( H^1(A_{1,1}, V_{a,b}) \). His argument was an adaptation of the one used in [Pet16], considering instead the commutative diagram

\[
\begin{array}{ccc}
H^1(A_1 \times A_1, V_{a-b} \boxtimes \mathbb{Q}) & \longrightarrow & H^3(A_2, V_{a,b}) \\
\downarrow & & \downarrow \\
(R^1 j'_*(V_{a-b} \boxtimes \mathbb{Q}))_\xi & \longrightarrow & (R^3 j_*V_{a,b})_{f(\xi)}
\end{array}
\]

where we use the same notation as before. For the embedding \( i : A_1 \to \overline{A}_1 \) the stalk of \( R^1 i_*V_a \) at infinity is one-dimensional for all even \( a \geq 0 \), so also \( R^1 j'_*(V_{a-b} \boxtimes \mathbb{Q})_\xi \) is one-dimensional.

**3.10.** We will now improve on this by showing moreover that if \( b = 0 \) then the summand \( \mathbb{Q}(-2-a) \) of \( H^1(A_{1,1}, V_{a,0})(-1) \) injects into \( H^3(A_2, V_{a,0}) \) under the Gysin map. (This will in fact be simpler than the arguments of [Pet16] and [Wat18].) We fix an embedding of \( V_a \boxtimes \mathbb{Q} \) into \( f^*V_{a,0} \) and consider instead the commutative diagram

\[
\begin{array}{ccc}
H^1(A_1 \times A_1, V_a \boxtimes \mathbb{Q}) & \longrightarrow & H^3(A_2, V_{a,0}) \\
\downarrow & & \downarrow \\
(R^1 j'_*(V_a \boxtimes \mathbb{Q}))_\eta & \longrightarrow & (R^3 j_*V_{a,0})_{f(\eta)}
\end{array}
\]

where we now let η ∈ 𝑨̄₁ × 𝑨̄₁ be the point at infinity of the first factor, but an arbitrary point in the interior of the second factor. Then again the stalk (R¹j'∗(Vₐ ⊠ Q))η is one-dimensional and the left vertical map is an isomorphism. The stalks occurring here are determined by Harder's formula as discussed in [Pet16, Section 5]. (From now on we freely suppose that the reader has read [Pet16, Section 5]). Let us begin by considering the stalk at f(η). Since f(η) lies on the one-dimensional boundary stratum of 𝑨̄₂ (as opposed to the zero-dimensional stratum) we need to replace the Siegel parabolic in Harder's formula with the Klingen parabolic, which we take to consist of matrices of the form

\[
\left(
\begin{array}{cccc}
* & * & * & * \\
0 & * & * & * \\
0 & 0 & * & 0 \\
0 & * & * & *
\end{array}
\right)
\]

We write MU for the Levi decomposition of the Klingen parabolic, where M ≅ SL₂ × Gₘ is given by matrices of the form

\[
\left(
\begin{array}{cccc}
* & 0 & 0 & 0 \\
0 & * & 0 & * \\
0 & 0 & * & 0 \\
0 & * & 0 & *
\end{array}
\right)
\]

and denote by u the Lie algebra of U. Harder's formula gives

\[
(R^3 j_* V_{a,0})_{f(\eta)} \cong \bigoplus_{p+q=3} H^p(M(\mathbb{Z}), H^q(u, V_{a,0})).
\]

We claim that this expression equals H³(u, Vₐ,₀). Indeed as discussed in [Pet16, Section 5] a theorem of Kostant calculates H^q(u, V_{a,b}) as an M-module, and Kostant's formula shows that as an SL₂-module it is irreducible, given by the bth symmetric power representation when q = 0, 3 and the (a + 1)st symmetric power when q = 1, 2 [Sch95, Table 2.3.4]. The result follows. Now we consider the intersection of the Klingen parabolic with the standard embedded copy of H = SL₂ × SL₂ in Sp₄, i.e.

\[
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
\times
\begin{pmatrix}
a' & b' \\ c' & d'
\end{pmatrix}
\mapsto
\begin{pmatrix}
a & 0 & b & 0 \\
0 & a' & 0 & b' \\
c & 0 & d & 0 \\
0 & c' & 0 & d'
\end{pmatrix}
\]

This intersection gives the standard Borel parabolic in the first factor and the whole group in the second factor, which means that we chose our Klingen parabolic wisely as this corresponds precisely to the stratum of the point η. We may then repeat the calculation: we get the same Levi but a smaller unipotent radical u_H which is now one-dimensional, and we find similarly (R¹j'∗(Vₐ ⊠ Q))η ≅ H¹(u_H, Vₐ ⊠ Q). Now we should determine the nonvanishing of the Gysin map H¹(u_H, Vₐ⊠Q) → H³(u, Vₐ,₀). But this map is nothing by the Poincaré dual of the restriction

\[
\mathbb{Q} \cong H^0(u, V_{a,0}) \rightarrow H^0(u_H, V_a \boxtimes \mathbb{Q}) \cong \mathbb{Q}
\]

which is evidently an isomorphism.

3.11. To recapitulate, we know that H¹(A_{1,1}, V_{a,b})(-1) ≅ S_{a-b+2}(-b) ⊕ Q(-1-a) if a > b, and the group vanishes for a = b. We know that the Gysin map is always nonzero on the second summand Q(-1-a). Our expectation is that the Gysin map restricted to the first summand S_{a-b+2}(-b) is nonzero whenever it can be, i.e. whenever s_{a-b+2} > 0 and s_{a+b+4} > 0. The smallest (a, b) satisfying both these conditions is (a, b) = (11, 1), which makes the local system V_{11,1} the unique local system of weight ≤ 12 for which there is an ambiguity.

3.12. However, this ambiguity in the cohomology of the local system \( V_{11,1} \), using recent calculations of Canning–Larson–Payne [CLP23]. Suppose by way of contradiction that the Gysin map were zero. We would obtain a nonzero summand\(^5\) \( S_{12}(-2) \) inside \( H^3(M_2, V_{11,1}) \), which under the Leray–Serre spectral sequence for the forgetful map \( M_{2,12} \to M_2 \) would give rise to a nonzero summand \( S_{12}(-2) \) inside \( H^{15}(M_{2,12}, \mathbf{Q}) \). This summand is of weight 15, i.e. of pure weight. This means [Del71, Corollaire 3.2.17] that it is in the image of the restriction map from \( H^{15}(\overline{M}_{2,12}, \mathbf{Q}) \), implying the existence of a class in \( H^{15}(M_{2,12}, \mathbf{Q}) \) not pushed forward from the Deligne–Mumford boundary. But [CLP23, Lemma 7.3] says precisely that the group \( H^{15}(M_{2,12}, \mathbf{Q}) \) is generated by cycles pushed forward from the boundary! As a consequence we obtain the following theorem:

3.13. Theorem. Let \( \lambda = (a, b) \) be even, and suppose that \( a + b \leq 12 \). Then

\[
H^2(M_2, V_\lambda) \cong
\begin{cases}
\mathbf{Q}(-a-2) & a > b > 0 \text{ and } a + b \notin \{8, 12\} \\
S_{12}(-1) & (a, b) = (10,0) \\
0 & \text{otherwise.}
\end{cases}
\]

\(^5\)Or rather, a summand up to semisimplification.

3.14. Once \( H^1(M_2, V_\lambda) \) and \( H^2(M_2, V_\lambda) \) are known one can of course leverage the knowledge of \( H^\bullet(A_{1,1}, V_\lambda) \) and \( H^\bullet(A_2, V_\lambda) \) to determine also \( H^3(M_2, V_\lambda) \). In carrying this out in practice it is easier to work with the Euler characteristics of the spaces involved (taken in a suitable Grothendieck group of Galois representations/Hodge structures) than to work with the main theorem of [Pet15], as the Euler characteristics involved admit much simpler formulae (cf. [FvdG04]). For the reader’s edification we compile in Table 1 all cohomology groups \( H^\bullet(M_2, V_\lambda) \) for \( |\lambda| \leq 12 \).

3.15. Remark. As mentioned previously, the cohomology \( H^\bullet(M_2, V_\lambda) \) was previously determined by the second named author when \( |\lambda| \leq 4 \) [Tom24], by using the Vassiliev–Gorinov method to compute the cohomology of \( M_{2,n} \) for small \( n \), and tracing the results backwards through the Leray–Serre spectral sequence.

3.16. Conjecture. Let \( \lambda = (a, b) \) be even, and suppose that \( a + b \geq 14 \). Then
\[
H^2(M_2, V_\lambda) \cong
\begin{cases}
\mathbb{Q}(-b-1)^{\oplus(s_{a+b+4}-1)} & a = b \equiv 0 \pmod{2} \\
0 & \text{otherwise}.
\end{cases}
\]

3.17. The precise status of this conjecture, after our discussion above, is the following: \( H^2(M_2, V_\lambda) \) is either given by the expression in the conjecture, or it contains an additional summand, which is itself a summand of \( S_{a-b+2}(-b-1) \).

3.18. Remark. The general strategy used in [Pet16, Wat18] and in §3.10 of reducing to a calculation on a deleted neighborhood of a boundary stratum has no chance of working when trying to prove nonvanishing of the Gysin homomorphism on the summand \( S_{a-b+2}(-1-b) \). The summand \( S_{a-b+2}(-1-b)^{\oplus s_{a+b+4}} \) of \( H^3(A_2, V_{a,b}) \) is precisely the part of cohomology coming from “endoscopy”. Unlike the other classes we have considered (which are part of the “Eisenstein cohomology”) it is genuinely supported on the interior of the Siegel variety.

3.19. Remark. We see that classes in \( H^2(M_2, V_{a,b}) \) of pure weight (i.e. weight \( a+b+2 \)) occur only for \( a = b \equiv 0 \pmod{2} \), and that the first case where such a class occurs is when \( \lambda = (10, 10) \), since 24 is the smallest weight for which the space of cusp forms of \( \mathrm{SL}(2, \mathbb{Z}) \) is of dimension \( > 1 \). The fact that \( H^2(M_2, V_{10,10}) \cong \mathbb{Q}(-11) \) is what gives rise to the non-tautological class on \( \overline{M}_{2,20} \) identified by Graber–Pandharipande [GP03] and the failure of the tautological ring of \( \overline{M}_{2,20} \) to have Poincaré duality [PT14, Pet16].

4.1. As a consequence of what we have done in this paper, we are able to compute the cohomology of \( M_{2,n} \) and \( H_{\infty,n} \) for \( n \leq 5 \). More precisely, we determine the rational cohomology groups, considered both as mixed Hodge structures and as representations of the symmetric groups \( S_n \). The cohomology of \( H_{\infty,n} \) is mixed Tate for all \( n \). It is not in general pure, but genuinely mixed; however, the mixed Hodge structure is a direct sum of pure Hodge structures (i.e. there are no nontrivial extensions). The cohomology of \( M_{2,n} \) is mixed Tate for \( n < 10 \), but not for \( n \geq 10 \), as can be seen from Table 1. When \( n < 6 \) it is even a direct sum of pure Hodge structures, as there is no room for any nontrivial extensions in the corresponding cohomology groups appearing in Table 1.

4.2. In tabulating these cohomology groups, we will find it convenient to describe them in terms of certain expressions \( f_\lambda \) and \( h_\lambda \), which we define now. Recall that if \( \lambda \) is a partition of \( n \), then \( \sigma_\lambda \) denotes the representation of \( S_n \) associated to \( \lambda \). We define \( f_\lambda = \sum_i [H^i(M_{2,n}, \mathbb{Q}) \otimes_{S_n} \sigma_\lambda] \cdot t^i \) in \( K_0(\mathrm{MHS})[t] \). We write \( L \) for the Lefschetz motive, the class of \( \mathbb{Q}(-1) \) in \( K_0(\mathrm{MHS}) \).

said in the preceding paragraph, \(f_\lambda\) is a polynomial in \(t\) and \(L\) when \(|\lambda| < 10\). In Table 2 we give the value of \(f_\lambda\) for \(|\lambda| \leq 5\).

4.3. We define similarly \(h_\lambda = \sum_i [H^i(H_{\infty, n}, \mathbb{Q}) \otimes_{S_n} \sigma_\lambda] \cdot t^i\). For a general \(\lambda\), the cohomology \(H^i(H_{\infty}, V_\lambda)\) is nonzero for infinitely many \(i\), and in particular \(h_\lambda\) is generally not a polynomial. However, \(h_\lambda\) is always a rational function. Indeed, it turns out that for any \(\lambda\), the expression

\[
\sum_i [H^i(H_\infty, V_\lambda)] \cdot t^i
\]

is a rational function of \(Lt\), whose denominator is a product of factors \((1 + L^k t^k)\) (\(k \geq 1\)) and \((1 - Lt)\), and the denominator is of degree at most \(|\lambda|\) [BDPW23, Remark 10.0.12]. When \(|\lambda| < 4\) the cohomology is in fact finite-dimensional, which means that \(h_\lambda\) is a polynomial when \(|\lambda| < 4\), tabulated in Table 3. However, \(h_\lambda\) is genuinely a rational function for \(|\lambda| \geq 4\). These rational functions for \(n = 4, 5\) are tabulated in Table 4, placed on a common denominator.

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[BDPW23] Jonas Bergström, Adrian Diaconu, Dan Petersen, and Craig Westerland. Hyperelliptic curves, the scanning map, and moments of families of quadratic L-functions. arXiv:2302.07664, 2023.

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| λ       | f                                                            |
| ------- | ------------------------------------------------------------ |
| ∅       | 1                                                            |
| (1)     | 1 + Lt2                                                      |
| (2)     | 1 + Lt2 + L5t5                                               |
| (12)    | Lt2                                                          |
| (3)     | 1 + Lt2 + L2t3 + L5t5                                        |
| (2,1)   | Lt2 + L5t5                                                   |
| (13)    | 0                                                            |
| (4)     | 1 + Lt2 + L2t3 + (L5 + L3)t5 + L6t6 + L7t8                   |
| (3,1)   | Lt2 + L2t3 + L5t5                                            |
| (22)    | L3t4 + (L5 + L3)t5                                           |
| (2,12)  | L5t6 + L7t7                                                  |
| (14)    | 0                                                            |
| (5)     | 1 + Lt2 + L2t3 + (L5 + L3)t5 + L6t6 + 2L7t8                  |
| (4,1)   | Lt2 + L2t3 + (L5 + L3)t5 + (L6 + L4)t6 + L6t7 + (L8 + 2L7)t8 |
| (3,2)   | L2t3 + L3t4 + (L5 + L3)t5 + L4t6 + L6t7 + L8t8 + L8t9        |
| (3,12)  | L4t5 + (L5 + L4)t6 + (L7 + L6)t7 + L8t8                      |
| (22,1)  | L3t4 + L3t5 + L5t6 + L7t7                                    |
| (2,13)  | L5t6 + L7t7                                                  |
| (15)    | 0                                                            |

Table 2. Cohomology of M₂,ₙ for n ≤ 5.

| λ     | h                                               |
| ----- | ----------------------------------------------- |
| ∅     | 1                                               |
| (1)   | 1 + Lt2                                         |
| (2)   | 1 − Lt + (L2 + L)t2 − (L3 + L2)t3 + L3t4 − L4t5 |
| (12)  | Lt2 − L2t3 + L3t4 − L4t5                        |
| (3)   | 1 − Lt + (L2 + L)t2 − L3t3 − L5t6               |
| (2,1) | Lt2 − L2t3 + L3t4 − L4t5                        |
| (13)  | 0                                               |

Table 3. Cohomology of H₍∞,ₙ₎ for n < 4.

| λ       | h · (1 − Lt)(1 + Lt)                                    |
| ------- | ------------------------------------------------------- |
| (4)     | 1 − Lt + (L2 + L)t2 − L3t3 + L3t5 − (L5 + L4)t6 + L5t7  |
| (3,1)   | Lt2 − (L5 − L4)t6                                       |
| (22)    | L3t4 − (L4 − L3)t5 + (L5 − L4)t6 − (L6 − L5)t7          |
| (2,12)  | 0                                                       |
| (14)    | 0                                                       |
| (5)     | 1 − Lt + (L2 + L)t2 − L3t3 + L3t5 − (L5 + L4)t6 + 2L5t7 |
| (4,1)   | Lt2 + L3t5 − (L5 − L4)t6 + L5t7 + L6t8                  |
| (3,2)   | L2t3 + L3t5 + L4t6 − (L6 − L5)t7 + L6t8                 |
| (3,12)  | L4t5 − (L5 − 2L4)t6 + (L6 − L5)t7 − (L7 − L6)t8         |
| (22,1)  | L3t4 − (L4 − L3)t5 + (L5 − L4)t6 − (L6 − 2L5)t7         |
| (2,13)  | 0                                                       |
| (15)    | 0                                                       |

Table 4. Cohomology of H₍∞,₄₎ and H₍∞,₅₎.

[BS22] Andrea Bianchi and Andreas Stavrou. Non-trivial action of the Johnson filtration on the homology of configuration spaces. Preprint available at arXiv:2208.01608, 2022.

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[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]
Let G be a planar graph and Iₛ and Iₜ be two independent sets in G, each of size k. We begin with a "token" on each vertex of Iₛ and seek to move all tokens to Iₜ, by repeated "token jumping", removing a single token from one vertex and placing it on another vertex. We require that each intermediate arrangement of tokens again specifies an independent set of size k. Given G, Iₛ, and Iₜ, we ask whether there exists a sequence of token jumps that transforms Iₛ to Iₜ. When k is part of the input, this problem is known to be PSPACE-complete. However, it was shown by Ito, Kamiński, and Ono [4] to be fixed-parameter tractable. That is, when k is fixed, the problem can be solved in time polynomial in the order of G. Here we strengthen the upper bound on the running time in terms of k by showing that the problem has a kernel of size linear in k. More precisely, we transform an arbitrary input problem on a planar graph into an equivalent problem on a (planar) graph with order O(k).

Given a graph G, a subset of V(G) is independent if it induces no edges. In this paper we study the problem of transforming one independent set into another by a sequence of small changes. We model this as follows.

to be fixed-parameter tractable. That is, when \( k \) is fixed, the problem can be solved in time polynomial in the order of \( G \).

In an excellent recent survey paper, Bousquet, Mouawad, Nishimura, and Siebertz [1] asked whether the problem admits a linear kernel. That is, given an arbitrary planar instance ISR-TJ(\( G, I_s, I_t \)), does there exist an equivalent instance on a (planar) graph \( G' \) where \( |V(G')| = O(k) \)? We answer their question affirmatively. (For a thorough history of independent set reconfiguration, and other related problems, we recommend [1].)

Theorem 1. *On the class of planar graphs, ISR-TJ parametrized by \( k \) is fixed-parameter tractable with a kernel that has size linear in \( k \).*

Proof. Fix an input graph \( G \), along with source and target independent sets, \( I_s \) and \( I_t \), each of size \( k \). We will show that either ISR-TJ(\( G, I_s, I_t \)) is trivially answered YES, or else ISR-TJ(\( G, I_s, I_t \)) is equivalent to a problem ISR-TJ(\( G', I_s, I_t \)), where \( G' \) is a subgraph of \( G \) and \( |V(G')| = O(k) \). Let \( X := I_s \cup I_t \) and note that \( |X| \leq 2k \). For each \( Y \subseteq X \), the X-projection class \( C_Y \) is defined by \( C_Y := \{ v \in V(G) \text{ s.t. } N(v) \cap X = Y \} \). Let

\[
C_1 := \bigcup_{\substack{Y \subseteq X \\ |Y| \leq 1}} C_Y \quad \text{and} \quad
C_2 := \bigcup_{\substack{Y \subseteq X \\ |Y| = 2}} C_Y \quad \text{and} \quad
C_3 := \bigcup_{\substack{Y \subseteq X \\ |Y| \geq 3}} C_Y.
\]

It is easy to show by planarity, as we do below, that \( |C_3| = O(k) \). And it is also easy to show that either \( |C_1| = O(k) \) or else the answer to ISR-TJ(\( G, I_s, I_t \)) is trivially YES. So we assume the former. Thus, in forming \( G' \) from \( G \) we will delete some (possibly empty) subset of \( C_2 \) to reach \( C'_2 \) such that \( |C'_2| = O(k) \), and also ISR-TJ(\( G', I_s, I_t \)) is equivalent to ISR-TJ(\( G, I_s, I_t \)).

Claim 1. *If \( |C_1| \geq 4k \), then the answer to ISR-TJ(\( G, I_s, I_t \)) is YES.*

Proof. Assume \( |C_1| \geq 4k \). Since \( G[C_1] \) is planar, it is 4-colorable. By Pigeonhole, \( C_1 \) contains an independent set \( I_m \) (for middle) of size \( 4k / 4 = k \). Starting with tokens on \( I_s \), for each \( v \in I_s \) with a neighbor \( w_v \in I_m \), move the token on \( v \) to some such \( w_v \). Now move all remaining tokens (in an arbitrary order) to the unoccupied vertices of \( I_m \). By symmetry, we can also move all tokens from \( I_t \) to \( I_m \). Thus, the answer to ISR-TJ(\( G, I_s, I_t \)) is YES, as claimed. ♦

Henceforth we assume \( |C_1| < 4k \).

Claim 2. \( |C_3| \leq 8k \).

Proof. First note that the number of sets \( Y \) with \( |Y| \geq 3 \) and \( C_Y \neq \emptyset \) is less than \( 2|X| \). To see this, we draw a plane graph \( G_X \) with vertex set \( X \) where each such set \( Y \) corresponds to a face of \( G_X \). (Think of restricting \( G \) to \( X \) and one vertex \( v_Y \) in \( C_Y \) for each such \( Y \). For each pair of vertices, \( u_i \) and \( u_j \), …

that appear successively around v_Y, add edge y_iy_j, if it is not already present, following the path y_iv_Yy_j. Finally, delete each v_Y; and "assign" the resulting newly created face to Y.) By Euler's Formula, the resulting plane graph has at most 3|X| - 6 edges, so at most 2|X| - 4 faces. Since G is planar, it is K_3,3-free, so |C_Y| ≤ 2 for all such Y. Thus, |C_3| ≤ 2(2|X|) ≤ 4(2k). ♦

An X-pair Y is a set {y', y''} such that y', y'' ∈ X and C_{y',y''} ≠ ∅. Let 𝒫 denote the set of all X-pairs. As in the proof of Claim 2, we can show that |𝒫| ≤ 3|X|. (Each X-pair corresponds to an edge of a plane graph with vertex set X.) We assume we are given a plane embedding¹ of G. For each X-pair Y, this embedding induces a linear order on C_Y; intuitively, this order is "left-to-right", but there are subtleties, which we highlight below in Figure 1.

Suppose Y = {y', y''}. Let H_Y := G[Y ∪ C_Y] - y'y''. Consider the outer face f_0. If f_0 contained at least three vertices of C_Y, then we could add a new vertex y''' to the interior of f_0, making it adjacent to all vertices on f_0. This would give a plane embedding of K_3,3 (with one part equal to {y', y'', y'''} and the other part contained in C_Y), a contradiction. So f_0 contains at most two vertices of C_Y. Further, since |f_0| ≥ 3 and y'y'' ∉ E(H_Y), in fact f_0 contains exactly two vertices in C_Y. Arbitrarily denote one of these z^ℓ and the other z^r (for left and right). We say z^ℓ is the leftmost vertex in C_Y. Deleting z^ℓ, we can repeat the argument on H_Y - z^ℓ. The new vertex on the outer face is now considered the "next-leftmost". Recursively, we can extend this ordering to all of C_Y. At some step in this process, we may transition from having one vertex of Y on the outer face to having two. However, at each step we will add exactly one new vertex of C_Y, and that new vertex will be the next in the linear order. Let N^{Int}(Y) := C_Y \ {z^ℓ, z^r}; this denotes the "interior" vertices in C_Y.

For distinct Y_1, Y_2 ∈ 𝒫, we would like to move tokens in C_{Y_1} independently of those in C_{Y_2}. This is made possible by restricting ourselves to only moving tokens in N^{Int}(Y_1) and N^{Int}(Y_2), as confirmed by Claim 3.

Claim 3. If Y_1, Y_2 ∈ 𝒫, with Y_1 ≠ Y_2, and z_i ∈ N^{Int}(Y_i) for each i ∈ {1,2}, then z_1z_2 ∉ E(G).

¹If not, we can find such an embedding [3, 5] in time O(|V(G)|).

Proof. Since \( N^{\mathrm{Int}}(Y_i) \neq \emptyset \), the leftmost and rightmost vertex in \( C_{Y_i} \) are distinct; call them \( z_i^\ell \) and \( z_i^r \). Denote \( Y_i \) by \(\{y_i', y_i''\}\), for each \( i \in \{1,2\}\). Let \( C_i := y_i' z_i^\ell y_i'' z_i^r \) for each \( i \in \{1,2\} \). Note that \( C_1 \) and \( C_2 \) might not be disjoint, but they intersect in at most one vertex; that is, \( |Y_1 \cap Y_2| \leq 1 \). Consider the regions with \( C_1 \) and \( C_2 \) as their boundaries; call them \( \mathcal{R}_1 \) and \( \mathcal{R}_2 \). Because \( C_1 \) and \( C_2 \) intersect at no more than one point, either \( \mathcal{R}_1 \) and \( \mathcal{R}_2 \) are disjoint or else one lies completely inside (topologically) the other; say \( \mathcal{R}_1 \) lies inside \( \mathcal{R}_2 \). In both cases, \( z_1 \) lies inside \( C_1 \), and \( z_2 \) lies outside it. Thus, \( z_1z_2 \notin E(G) \). \( \diamond \)

It is helpful to note, for each \( Y \in \mathcal{P} \), that \( G[C_Y] \) is a linear forest, since each \( z \in C_Y \) has edges only to (possibly) its predecessor and/or successor in the linear order.

Let \( \mathcal{P}_{Big} := \{\{y', y''\} \in \mathcal{P} \text{ s.t. } N^{\mathrm{Int}}(y', y'') \text{ has two non-adjacent vertices}\} \).

Claim 4. Let \( G_{\mathcal{P}_{Big}} \) be the subgraph of \( G \) induced by \( \cup_{Y \in \mathcal{P}_{Big}} N^{\mathrm{Int}}(Y) \). If \( G_{\mathcal{P}_{Big}} \) has no independent set of size \( k \), then \( |V(G)| \leq 40k \), so \( G \) is a linear kernel.

Proof. Assume \( G_{\mathcal{P}_{Big}} \) has no independent set of size \( k \). Now

\[
\begin{aligned}
|\cup_{Y \in \mathcal{P}} C_Y| &\leq |\cup_{Y \in \mathcal{P}}(2 + |N^{\mathrm{Int}}(Y)|)| \\
&= 2|\mathcal{P}| + |\cup_{Y \in \mathcal{P} \setminus \mathcal{P}_{Big}} N^{\mathrm{Int}}(Y)| + |\cup_{Y \in \mathcal{P}_{Big}} N^{\mathrm{Int}}(Y)| \\
&\leq 2|\mathcal{P}| + (\sum_{Y \in \mathcal{P} \setminus \mathcal{P}_{Big}} 2) + |\cup_{Y \in \mathcal{P}_{Big}} N^{\mathrm{Int}}(Y)| \\
&\leq 2|\mathcal{P}| + 2|\mathcal{P}| + 2(k-1) \tag{1} \\
&\leq 4|\mathcal{P}| + 2(k-1) \\
&\leq 4(3(|X|)) + 2k \leq 12(2k) + 2k = 26k.
\end{aligned}
\]

Here (1) uses that \( N^{\mathrm{Int}}(Y) \) is a linear forest (disjoint union of paths) for every \( Y \in \mathcal{P} \), that \( 2\alpha(H) \geq |V(H)| \) when \( H \) is a linear forest, and that \( \alpha(H_1 + H_2 + \cdots) = \sum_i \alpha(H_i) \), when \( H_1 + H_2 + \ldots \) is a disjoint union of linear forests. Claims 1 and 2 give \( |V(G)| = |X| + |C_1| + |C_2| + |C_3| \leq 2k + 4k + 26k + 8k = 40k \). \( \diamond \)

Claim 5. In every linear forest, every independent set of size \( k \) can be reconfigured to every other (in at most \( 2k \) steps).

Proof Sketch. We order the paths arbitrarily, and pick a “left end” for each. We now iteratively move each token on the first path as far left as possible, and then fill the remainder of the path with remaining tokens. We finish recursively on the remaining paths and tokens. This transforms an arbitrary independent set of size \( k \) to a canonical one, in at most \( k \) steps. \( \diamond \)

Henceforth, we assume that \( G_{\mathcal{P}_{Big}} \) contains an independent set of size \( k \); call it \( J_m \). We will form sets \( J_s \) and \( J_t \) (not necessarily independent, but each inducing a linear forest), each of size \( O(k) \), and form \( G' \) from \( G \) by deleting \( (\cup_{Y \in \mathcal{P}_{Big}} N^{\mathrm{Int}}(Y)) \setminus (J_m \cup J_s \cup J_t) \). Clearly \( |V(G')| = O(k) \). So it remains to specify \( J_s \) and \( J_t \), and to show that ISR-TJ\( (G', I_s, I_t) \) is equivalent to

Let \( \mathcal{P}^1 \) be the set of \( X \)-pairs with at most one vertex in \( I_s \) and let \( \mathcal{P}^2 \) be the set of \( X \)-pairs with two vertices in \( I_s \). To form \( J_s \), start by including, for all \( Y \in \mathcal{P}^1 \), each vertex in \( C_Y \) such that \( I_s \) contains at most one vertex of \( Y \), up to the point (if it occurs) where \( J_s \) contains an independent set of size \( k \) that lies entirely in \( \bigcup_{Y \in \mathcal{P}^1} N^{\mathrm{Int}}(Y) \). If this point occurs, we are done forming \( J_s \); so assume it does not. Now, we consider each \( Y \in \mathcal{P}^2 \). For each such \( Y \), add to \( J_s \) the leftmost and right vertex in \( C_Y \) as well as two vertices in \( N^{\mathrm{Int}}(Y) \), if they exist, giving preference to a non-adjacent pair. Analogously, we construct \( J_t \) with \( I_t \) in the role of \( I_s \).

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Proof. For each \( Y \in \mathcal{P}^1 \), its contribution to a maximum independent set contained in \( \bigcup_{Y \in \mathcal{P}^1} N^{\mathrm{Int}}(Y) \) is \(\alpha(G[N^{\mathrm{Int}}(Y)]) \geq |N^{\mathrm{Int}}(Y)| / 2 \), because \( N^{\mathrm{Int}}(Y) \) induces a linear forest. Since the size of such a set is at most \( k \) (by construction), we have \( |\bigcup_{Y \in \mathcal{P}^1} N^{\mathrm{Int}}(Y)| \leq 2 \sum_{Y \in \mathcal{P}^1} \alpha(G[N^{\mathrm{Int}}(Y)]) \leq 2k \). Thus,

\[
|J_s| \leq \sum_{Y \in \mathcal{P}^1} (2 + |N^{\mathrm{Int}}(Y)|) + \sum_{Y \in \mathcal{P}^2} (2 + \min(2, |N^{\mathrm{Int}}(Y)|))
\leq \sum_{Y \in \mathcal{P}^1} 2 + 2k + \sum_{Y \in \mathcal{P}^2} 4
< 4|\mathcal{P}| + 2k < 12|X| + 2k < 24k + 2k = 26k.
\]

Now \( |J_s \cup J_t \cup J_m| \leq |J_s| + |J_t| + |J_m| \leq 26k + 26k + k = 53k \). Thus, \( |V(G')| \leq |X| + |C_1| + |J_s \cup J_t \cup J_m| + |C_3| \leq 2k + 4k + 53k + 8k = 67k. \) ♦

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ISR-TJ(\( G, I_s, I_t \)). For the latter, we will show that if in \( G \) we can move a token to a vertex absent from \( G' \), then in \( G' \) we can move all tokens from \( I_s \) to \( J_s \cup J_m \), and we can also move all tokens from \( I_t \) to \( J_t \cup J_m \). Since \( G[J_s \cup J_t \cup J_m] \) is a linear forest, by Claim 4 the answer to ISR-TJ(\( G', I_s, I_t \)) is YES.

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Lastly, we show that when we restrict \( G \) to \( G' \) the problem stays equivalent.

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**Claim 7.** ISR-TJ(\( G', I_s, I_t \)) is equivalent to ISR-TJ(\( G, I_s, I_t \)).

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Proof. Since \( G' \subseteq G \), if the answer to ISR-TJ(\( G, I_s, I_t \)) is NO, then clearly the answer to ISR-TJ(\( G', I_s, I_t \)) is also NO. So it suffices to show that if the answer to ISR-TJ(\( G, I_s, I_t \)) is YES, then also the answer to ISR-TJ(\( G', I_s, I_t \)) is YES.

Suppose that the answer to ISR-TJ(\( G, I_s, I_t \)) is YES, and let \( \sigma \) be a sequence of token jumps witnessing this. If each vertex appearing in \( \sigma \) is a vertex of \( G' \) then \( \sigma \) also witnesses that the answer to ISR-TJ(\( G', I_s, I_t \)) is YES. So assume that \( \sigma \) uses some vertex that is not in \( G' \). Let \( z_s \) and \( z_t \) denote the first and last such vertex used by \( \sigma \) that are not in \( V(G') \). We will show that (a) because \( z_s \notin V(G') \), we can move all tokens from \( I_s \) to vertices of \( J_s \cup J_m \). Similarly (b) because \( z_t \notin V(G') \), we can move all tokens from \( I_t \) to vertices of \( J_t \cup J_m \). The arguments are essentially identical (we interchange the roles of \( I_s \) and \( I_t \) and run \( \sigma \) in reverse), so it suffices to prove (a).

Suppose that \( z_s \in \cup_{Y \in P^1} C_Y \). (Recall that \( P^1 \) is the set of \( X \)-pairs with at most one vertex in \( I_s \).) What caused \( z_s \) to be absent from \( J_s \)? It is because \( J_s \) contains an independent set, call it \( J_1 \) of size \( k \) that lies entirely in \( \cup_{Y \in P^1} N^{\mathrm{Int}}(Y) \). For each token on a vertex \( v \) with a neighbor \( w \) in \( J_1 \), we move the token to \( w \). This cannot create any conflicts since \( J_1 \) is an independent set and is contained in \( \cup_{Y \in P^1} N^{\mathrm{Int}}(Y) \). Now each vertex of \( J_1 \) has no neighbor with a token, so we can greedily move the remaining tokens to the unoccupied vertices of \( J_1 \). Thus, we are done if \( J_1 \) exists; so we assume it does not.

Instead assume that \( z_s \in \cup_{Y \in P^2} C_Y \). By our construction of \( G' \), since \( z_s \notin V(G') \), we know \( z_s \in \cup_{Y \in P_{\mathrm{Big}}} N^{\mathrm{Int}}(Y) \). Denote \( P_{\mathrm{Big}} \cap P^2 \) by \( \{Y_1, \ldots, Y_a\} \) in an arbitrary order (for some integer \( a \)), and denote \( Y_i \) by \( \{y_i', y_i''\} \) for each \( i \in \{1, \ldots, a\} \). Since \( \sigma \) moves a token to \( z_s \), at some (earlier) point \( \sigma \) moves a token off of some vertex in some \( Y_i \). By symmetry, assume \( \sigma \) first moves a token off of \( y_1' \). We show, by induction on \( i \), that we can move all tokens off of \( \cup_{i=1}^a Y_i \). Recall, for each \( i \), that \( N^{\mathrm{Int}}(Y_i) \) contains non-adjacent vertices that lie in \( J_m \); call such a pair \( z_i' \) and \( z_i'' \). For the base case, we move the token on \( y_1'' \) to \( z_1' \). For the induction step, we assume that no token appears on the closed neighborhood of \( z_{i-1}'' \). So we move a token from \( y_i' \) to \( z_{i-1}'' \) and then we move a token from \( y_i'' \) to \( z_i' \). This finishes the induction proof.

Once all tokens are removed from \( \cup_{i=1}^a Y_i \), by Claim 4 we move all tokens on \( \cup_{i=1}^a \{z_i', z_i''\} \) to \( J_m \). Finally, we greedily move all remaining tokens to \( J_m \). \(\lozenge\)

Claim 7 finishes the proof of Theorem 1.

In Claim 6, we showed that \( |V(G')| \leq 67k \). This upper bound can likely be strengthened, but we have not made an effort to do so. We prefer to keep the proof as simple as possible. The important point is that \( |V(G')| = O(k) \).