[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. We answer in the affirmative the surprisingly difficult questions: If a complex Banach space possesses a real predual \(X\), then is \(X\) a complex Banach space? If a complex Banach space possesses a real predual, does it have a complex predual? We also answer the analogous questions for operator spaces, that is, spaces of operators on a Hilbert space, up to complete isometry. Indeed, we use operator space methods to solve these Banach space questions.

In 1932 Banach's famous monograph [2] appeared, in which Banach spaces and their dual (or 'conjugate' or 'adjoint') space play a central role. Shortly thereafter F. J. Murray of 'Murray and von Neumann' fame provided the well known \(\varphi(x) - i\varphi(ix)\) trick (see the second page of [25]) to deal with complex dual spaces \(X^*\). It then becomes clear that the real Banach space dual \(X^\dagger\) of a complex Banach space \(X\) is itself a complex Banach space. That is, this real Banach space has isometric complex structure. Indeed it is easy to see that  \(X^\dagger \cong X^*\) real linearly isometrically and weak* homeomorphically (see e.g. [23, Proposition 1.1.6]). However there remain a couple of obvious questions:

**Problem 1.** If \(Y\) is a complex Banach space possessing a real predual \(X\), then is \(X\) (isometrically) a complex Banach space?

That is, does \(X\) have (isometric) complex structure? Rephrasing, Problem 1 asks if \(X\) is a real Banach space such that its real Banach space dual \(X^\dagger\) has (isometric) complex structure, then does \(X\) have (isometric) complex structure?

In Problem 1, one would also desire that the sought-for complex structure on \(X\) is a complex predual of \(Y\), if mercy prevails. Indeed we have:

**Problem 2.** If a complex Banach space possesses a real predual \(X\), then does it have a complex predual? Is \(X\) a complex predual?

We note that by 'predual' in Problems 1 and 2 we mean an isometric predual. We will discuss the isomorphic question later.

Problems 1 and 2 do not appear explicitly in any textbook. However related questions have been discussed by Dixmier, Dieudonné, Bourgain, Szarek, Kalton, Godefroy, Ferenczi, and many others (see e.g. [15, 12, 36, 17, 18] and references therein). Some of these papers are connected to the question above of whether mercy and desire prevail. Indeed if a complex Banach space \(X\) is a real predual of complex Banach space \(Y\) then the complex dual \(X^*\) is real isometric to \(Y\), as in the first paragraph above. We claim that this need not imply in general that \(X\) is a complex predual of \(Y\). So mercy does not prevail in this generality, although it

does for us, as we shall see. Note that if the claim were false then Problem 2 would follow immediately from an affirmative solution to Problem 1. In fact the claim (and its isomorphic variant) is related to nasty examples in some of the works just cited. For example, let \( Y \) be a complex dual Banach space which is not complex isometric to its complex conjugate \( \bar{Y} \) (e.g. Bourgain's reflexive example [12]). If \( E \) is a complex predual of \( \bar{Y} \) then \( E \) is a real predual of \( Y \) and \( \bar{E} \) is a complex predual of \( Y \). These complex preduals of \( Y \) and \( \bar{Y} \) are real isometrically isomorphic to each other, but not complex isometrically so, and \( E \) is not a complex predual of \( Y \). This establishes the claim.

Problems 1 and 2 turn out to be quite difficult and subtle, as any direct attack eventually reveals. Duality is often a slippery business, particularly when higher duals are involved, as they turn out to be. We are able to answer Problems 1 and 2 in the affirmative. Indeed the complex dual of this complex structure on \( X \) agrees with \( Y \) complex isometrically and weak* homeomorphically, as one would wish. This seems a useful contribution to Banach space theory, clarifying an aspect of the relationship between real and complex Banach spaces. Thus it should be helpful in certain problems in the theory of both of these classes of Banach spaces, in the sense that it should sometimes allow each of them to better access the other.

As a corollary we have the equivalent reformulation of Problem 1:

**Theorem 1.1.** *If \( X \) is a real Banach space such that its real Banach space bidual \( X^{\dagger\dagger} \) has complex structure (isometrically), then \( X \) has complex structure (isometrically).*

To see that this is equivalent to an affirmative solution to Problem 1: Indeed if \( X^{\dagger} \) has (isometric) complex structure then its real dual is isometrically a complex Banach space by the facts in the first paragraph of our paper (but with \( X \) replaced by \( X^{\dagger} \)). Thus \( X \) is isometrically a complex Banach space by the theorem. The converse is trivial.

Our techniques use the theory of operator spaces and \( C^* \)-algebras, although we would guess that at some later point this will be removed, i.e. there will be a later solution only involving Banach spaces. Indeed at the end of our paper we make some remarks on how our methods suggest the direction in which this might be done. The reader unfamiliar with operator spaces should browse early chapters of e.g. [6, 16, 27] or [29]. While our proofs may often appear petite, this is usually because they include only the modifications to be made in the real case of selected points in existing rather technical and lengthy arguments in the complex case. One would have to master the latter to completely understand some of our ‘short proofs’.

Operator spaces may be viewed as Banach spaces \( X \) with a specified norm on the \( n \times n \) matrices over \( X \) satisfying (Ruan’s) conditions. The category of operator spaces is very similar to the category of Banach spaces, indeed there are canonical functors between these categories, e.g. Min, which turns every Banach space into an operator space in a ‘minimal way’ (see e.g. 1.2.21 and 1.4.12 in [6], and Section 2 below). Real operator spaces have been studied in e.g. [33, 34, 35, 9, 4], and recent progress here led to the present paper.

**Theorem 1.2.** *Let \( X \) be a real operator space such that the real operator space bidual \( X^{\dagger\dagger} \) has complex operator space structure. Then \( X \) has complex operator space structure, such that if \( G \) is \( X \) with this complex operator space structure then \( G^{**} \cong X^{\dagger\dagger} \) real completely isometrically and weak* homeomorphically.*

(This result was advertised in [4, Section 3] without proof, as a future application of Corollary 3.2 from that paper.) Due to the nature of operator space duality, one cannot expect that if the real operator space dual of X has complex operator space structure (completely isometrically) then so does X (see Example 2 below). However the correct result matching our solution to Problem 1 is a small modification of the last statement (namely, Theorem 3.3 (3) below). We also note that a real operator space can have complex Banach space structure isometrically but not complex operator space structure completely isometrically (e.g. the quaternions [4, Remark 1 after Theorem 3.1]).

Valentin Ferenczi has shown us that, interestingly, the isomorphic version of Problem 1 has a quick negative answer. It is well known that a bounded map J : X → X on a real Banach space corresponds isomorphically to a complex Banach space structure on X (with ix = Jx) if and only if J² = −I and J is an isomorphism. Thus for example, by this criterion (real) ℓ₁ has isomorphic complex structure using J(α) = (−α₂, α₁, −α₄, α₃, ···). From this it is easy to see for example that a real Banach space which is real isomorphic to a complex Banach space, need not have isometric complex structure. There are various formulations of the isomorphic version of Problem 1, e.g.:

Problem 3. If Y is a real Banach space with real predual X, and Y is isomorphic to a complex Banach space, then is X isomorphic to a complex Banach space?

Example 1. (Ferenczi) In the last fifteen years many authors have studied real Banach spaces X on which every operator is a scalar multiple of the identity plus a compact. Considering the operator J above Problem 3, one sees immediately that such X cannot be isomorphic to a complex space. (For J² = (aI + k)² = −I, for k compact and a real, gives the contradiction a² = −1.)

One such space X is the Argyros-Haydon space [1], whose dual X* is isomorphic to ℓ₁, so is isomorphic to a complex Banach space as we stated above. This answers Problem 3 in the negative, and it also shows that the issue (and the various reformulations of Problem 3) cannot be fixed by any kind of renorming trick, etc. If Problem 1 has an affirmative solution then it follows that no renorming of the real dual of X to have complex structure can be the real dual space of a space isomorphic to X.

It is easy to see using real Min-Max duality (see [35, Proposition 2.6]), that Max(X) is a counterexample to the operator space (completely isomorphic) version of Problem 3.

Ferenczi and his coauthors have many very striking results about complexifications, e.g. an example of complex Banach spaces that are isometric as real spaces but ‘totally incomparable’ as complex spaces (so no infinite-dimensional subspace of the one is complex isomorphic to a subspace of the other) [17].

We will give our solution to Problems 1 and 2 in Section 4, and give some immediate corollaries or sample applications there. Since the solution to Problems 1 and 2 is complicated we suggest that the reader might warm up to it, and start to get a feeling for the technique, by first mastering the proof of the relatively easy Theorem 3.3 (1). Indeed Section 3 is mostly devoted to the operator space version of Problems 1 and 2, but also contains some applications and some preparatory material for Section 4. In the rest of Section 1 and in Section 2 we provide necessary background and lemmata.

Turning to definitions, we write \( i_X \) for the canonical map \( X \) into its bidual. Because there are many ‘i’s around we sometimes write \( \iota \) for the complex number \( i \). It is well known and obvious that a map \( J : X \to X \) on a real Banach space corresponds isometrically to a complex Banach space structure on \( X \) (with \( ix = Jx \)) if and only if \( J^2 = -I \) and \( x \mapsto sx + tJ(x) \) is an isometry whenever \( s = \cos \theta \), \( t = \sin \theta \), that is whenever \( s^2 + t^2 = 1 \).

We say that an algebra is unital if it has an identity 1 of norm 1, and a map \( T \) is unital if \( T(1) = 1 \). We say that a map \( T \) is selfadjoint if \( T(x^*) = T(x)^* \). For us, a contraction is a linear map with norm \( \leq 1 \). For some theory of real or complex \( C^* \)-algebras we refer the reader to e.g. [29, 23]. We reserve the letters \( H \) and \( K \) for Hilbert spaces. A TRO (or ternary system) is a closed linear subspace \( Z \subset B(K, H) \), for Hilbert spaces \( K \) and \( H \), satisfying \( ZZ^*Z \subset Z \).

We refer the reader to [24] for a survey of relationships between real and complex Banach spaces. If \( X \) is a complex Banach space we will sometimes write \( X_r \) for \( X \) viewed as a real Banach space. We usually will write \( X^\dagger \) for the real dual and \( X^* \) for the complex dual. The canonical map \( X^\dagger \to X^* \) described at the start of our paper is given by the canonical duality pairing

\[
\langle \varphi, x \rangle = \varphi(x) - i \varphi(ix), \quad \varphi \in X^\dagger, \, x \in X.
\]

**Lemma 1.3.** If \( Y \) is a complex Banach space then \( Y^{**} \cong Y^{\dagger\dagger} \). Writing \( Y_r \) as \( E \), and \( E^\dagger \) for the real dual and \( Y^* \) for the complex dual, we have

\[
Y^{**} \cong E^{\dagger\dagger}
\]

real linear isometrically via an isomorphism which is a homeomorphism for the two weak* topologies, and takes \( i_Y(y) \) to \( i_E(y) \) for \( y \in Y \).

**Proof.** By [23, Proposition 1.1.6] we have \( Y^* \cong E^\dagger \) real linear isometrically via the map \( j_Y : \varphi \mapsto \mathrm{Re}\, \varphi \). The inverse map is \( \varphi \mapsto \tilde{\varphi} \) where \( \tilde{\varphi}(x) = \varphi(x) - i \varphi(ix) \). Moreover this map is clearly a homeomorphism for the two weak* topologies. Similarly

\[
Y^{**} \cong ((Y^*)_r)^\dagger \cong (E^\dagger)^\dagger = E^{\dagger\dagger}
\]

isometrically, via a homeomorphism \( \mu \) for the two weak* topologies. Indeed \( \mu = ((j_Y)^\dagger)^{-1} \circ j_{Y^*} \), where \( (j_Y)^\dagger : E^{\dagger\dagger} \to (Y^*)^\dagger \), and \( j_{Y^*} : Y^{**} \to (Y^*)^\dagger \). Note that \( i_Y(y) \) maps to \( i_E(y) \) here (exercise). □

**Remarks.** 1) Since \( Y^{**} \cong ((Y^*)_r)^\dagger \), the canonical real linear isometry \( E \to Y \to Y^{**} \) taking \( x \mapsto i_Y(x) \) extends uniquely to a weak* continuous real linear contraction \( \rho : E^{\dagger\dagger} \to Y^{**} \) with \( i_E(x) \) mapping to \( i_Y(x) \). By uniqueness this is the isometry \( \mu \) in the proof, and we will need this fact later.

2) Any complex Banach space \( Y \) with an (isometric) complex predual \( X \) clearly has an (isometric) real predual. Indeed \( (X_r)^\dagger \cong X^* \cong Y \).

## 2. Some results on real operator spaces

If \( T : X \to Y \) is a linear map between operator spaces we write \( T_n \) for the canonical ‘entrywise’ amplification taking \( M_n(X) \) to \( M_n(Y) \). We say that \( T \) is completely bounded if \( \|T\|_{cb} = \sup_n \|T_n\| < \infty \), and write \( CB(X, Y) \) for the space of such completely bounded maps. Then \( T \) is completely contractive (resp. completely isometric, completely positive) if \( T_n \) is a contraction (resp. is an isometry, is positive) for all \( n \). We write UCP for ‘unital completely positive’. A real operator system is a unital selfadjoint subspace of a unital real \( C^* \)-algebra (or of \( B(H) \)).

The following result improves on Proposition 4.1 in [33], and is in [9] (but not stated in this form).

## Proposition 2.1.
Let X be a real operator system and T : X → B(H) a real linear unital map. Then T is completely positive if and only if T is completely contractive, and in this case T is selfadjoint.

**Proof.** If T is completely positive then it is selfadjoint and completely contractive by [9, Lemma 2.3]. The other direction formally follows from [9, Corollary 2.5 and Proposition 2.4], but in fact as indicated there is immediate by going to the complexifications (i.e. by the complex result applied to the complexified map Tc).

Many well known results about completely positive maps generalize to the real case. In [9, 4] we verified the real case of the results 1.3.4–1.3.8 and 1.3.11–1.3.13 in [6] (that Ran (u) ⊂ Δ(Y) in 1.3.8 follows since u is selfadjoint by [9, Lemma 5.16]). Indeed items 1.3.4, 1.3.5 and 1.3.9 there had previously been verified by Ruan in [33] for selfadjoint unital completely positive maps, and he also checked 1.3.10. However [9, Lemma 2.3] shows that any completely positive map is selfadjoint, so that in fact 1.3.9, 1.3.11, 1.3.12 and 1.3.13 in [6] hold verbatim in the real case for UCP maps or equivalently for unital completely contractive maps. In particular we will need the powerful ‘multiplicative domain’ trick:

## Theorem 2.2.
Suppose that u : A → B is a real linear UCP map (or equivalently is a unital completely contractive map) between unital real C*-algebras, and that there is a C*-subalgebra C of A with 1_A ∈ C, such that π = u|_C is a *-homomorphism. Then u is a ‘bimodule map’:

u(ac) = u(a)π(c) and u(ca) = π(c)u(a), a ∈ A, c ∈ C.

We remind the reader of the minimal operator space structure Min (see e.g. 1.2.21 and 1.4.12 in [6]). Here (in both the real and complex case) the matrix norm ‖[y_ij]‖ on M_n(Y) is the supremum of ‖[ψ(y_ij)]‖ over all contractive functionals ψ on Y. Min has the universal property that any contraction T : X → Y on an operator space T is a complete contraction into Min(Y).

## Proposition 2.3.
Let T : X → Y be a continuous real linear map between complex Banach spaces, and let T_m be T regarded as a real linear map from Min_C(X) to Min_C(Y). Then ‖T_m‖_cb = ‖T‖.

**Proof.** Certainly ‖T‖ ≤ ‖T_m‖_cb. Conversely, assume that ‖T‖ ≤ 1. Since φ ∘ T is a contractive real functional for all φ ∈ Ball(Y†), it is clear that T : Min_R(X) → Min_C(Y) is a complete contraction. Call it R. On the other hand the identity contraction Min_C(X) → X induces a complete contraction Min_C(X) → Min_R(X). Composing this with R we obtain T_m. So T_m is a complete contraction.

We believe that the correct tool for Problems 1 and 2 (and their operator space version) is the largely forgotten tool of ‘multipliers of linear spaces’. Indeed the main technical part of our proof uses the operator space multipliers which we introduced in 2001. An operator T : X → X on an operator space is a left operator space multiplier, or is in the left multiplier operator algebra M_ℓ(X), if there is a complete isometry σ : X → B for a C*-algebra B, and an operator a ∈ B, with σ(Tx) = aσ(x) for all x ∈ X. Of course there are two definitions hidden here, corresponding

to the real and complex case. We will not review the theory of multipliers in detail here: the reader is referred to [10] for a very short introduction that does a far better job than we could do here. In fact we will only need to know that one particular map, multiplication by i, is an operator space multiplier. However we will then need to feed this one map into the rather complicated operator space multiplier machinery. See the second half of [6, Chapter 4] for a more detailed account of this theory, and see [35] and [4, Section 4] for the real case. These multipliers may be defined in terms of the injective envelope, which we shall mention in more detail above Lemma 3.5.

The space CB(X, Y) defined at the start of the last section has canonical matrix norms with respect to which it is an operator space. In particular for 𝔽 = ℝ or ℂ the dual Banach space B𝔽(Y, 𝔽) = CB𝔽(Y, 𝔽) of an operator space X has a canonical operator space structure. This is called the operator space dual. It turns out that the canonical map i_X of X into its operator space bidual is completely isometric. This and other very basics of the duality theory of operator spaces are explained in e.g. [6, Section 1.4]. The real case of these results were checked in [33, 34, 35, 9, 4]. Thus a dual operator space is an operator space completely isometric to the operator space dual of another operator space.

A key fact for us will be the following result from [7] concerning the multipliers mentioned at the end of the last section:

Theorem 3.1. Every operator space left multiplier on a dual operator space is weak* continuous.

Lemma 3.2. If E is a complex operator space and F = E_r then E** ≅ F†† real linear completely isometrically, where E** here is the complex operator space bidual, and F†† is the real operator space bidual. This isomorphism is a homeomorphism for the two weak* topologies, and takes i_{F}(x) to i_{F††}(x) for x ∈ X.

Proof. By Lemma 1.3 we have E** ≅ F†† isometrically, via a homeomorphism μ for the two weak* topologies, with i_{F}(x) mapping to i_{F††}(x).

By the Remark 1 after Lemma 1.3 we have that μ⁻¹ is the unique weak* continuous real linear extension ρ of the canonical complete isometry F → E → E** taking i_{F}(x) to i_{E}(x). Now Mₙ(E)_r = Mₙ(F) isometrically, so taking the complex bidual we obtain a unique real linear isometric weak* homeomorphism (Mₙ(E))** ≅ Mₙ(F)†† taking i_{Mₙ(E)}([x_{ij}]) to i_{Mₙ(F)}([x_{ij}]). So by [6, Theorem 1.4.11] and its real version there is a unique real linear isometry Mₙ(E**) ≅ Mₙ(F††) taking [i_{E}(x_{ij})] to [i_{F}(x_{ij})]. By uniqueness this must be the inverse of ρₙ : Mₙ(F††) → Mₙ(E**). Thus ρ is a real linear complete isometry. □

As mentioned in the introduction, due to the nature of operator space duality one cannot expect that: if the real operator space dual of X has complex operator space structure (completely isometrically) then so does X. See the simple Example 2 below. To correct this we need to change the matrix norms to match those in the bidual. The correct statement is the following modification of the expectation mentioned above:

Theorem 3.3. Let X be a real operator space such that the real operator space dual X† has complex operator space structure (completely isometrically). Let Y = X† with this complex operator space structure.

Proof. (1) Let u = L_i, multiplication by i on Y, which is almost trivially a (complex) operator space multiplier of Y. (We do not need this here but since it will be important in future work we mention that clearly u is a (complex) operator space centralizer of Y in the sense of e.g. [11, 4].) Since ℳ^ℝ(Y) = ℳ^ℂ(Y) (this is Theorem 4.5 in [4]), the operator L_i is a real operator space multiplier on (X*)_r. So it is σ(Y, X)-continuous by Theorem 3.1. Let J : X → X be the predual map of L_i. It is easy to see by duality theory that J² = I (since (J²)* = -I). So X has (isometric) complex structure by the criterion mentioned after Example 1. Indeed if s, t are real with s² + t² = 1 then (sI + tJ)* = sI + tL_i is an isometric isomorphism, so sI + tJ is isometric. We remark that sI + tJ is clearly seen to have inverse sI - tJ. Write W for X viewed as a complex Banach space. Then by [23, Proposition 1.1.6] the complex dual W* ≅ Y, weak* homeomorphically and real isometrically via the duality ⟨φ, x⟩ = φ(x) - iφ(ix) for φ ∈ Y, x ∈ W = X. However because J* = L_i it is easy to check that this isomorphism is a complex linear isometry.

To do this we view W ⊂ Y* as usual via i_W : W → Y*, and W is given the operator space structure making i_W a complete isometry. We use the second criterion in 1.6.4 in [6] to check that W has complex operator space dual Y. Suppose that [y_ij^λ] is a net in Ball(M_n(Y)) with y_ij^λ → y_ij in σ(Y, W). Thus

y_{ij}^λ(x) - ι y_{ij}^λ(ιx) → y_{ij}(x) - ι y_{ij}(ιx),   x ∈ X.

Taking real parts, we see that y_{ij}^λ → y_{ij} in σ(X*, X). So since Y is the real operator space dual of X, by the second criterion in 1.6.4 in [6] in the real case, [y_{ij}] is in Ball(M_n(Y)). Thus by the same criterion in the complex case, W* ≅ Y completely

isometrically as complex operator spaces. We checked in (1) that this is a weak* homeomorphism.

We claim that the map Id : W → X is a real linear isometry and complete contraction with completely bounded inverse. For n, m ∈ N and [ϕ_{ij}] ∈ Ball(M_m(Y)), [x_{kl}] ∈ M_n(X) we have

$$\|[\varphi_{ij}(x_{kl})]\| \leq \|[\varphi_{ij}(x_{kl}) - \iota \varphi_{ij}(\iota x_{kl})]\| \leq 2 \|[\varphi_{ij}]\| \|[x_{kl}]\|_{M_n(X)},$$

and $\|[\varphi_{ij}]\|_{M_m(Y)} = \|[\varphi_{ij}]\|_{M_m(X^{\dagger})} \leq 1$. Here we used the fact that $Jx = \iota x$ is completely isometric on $X$ (see e.g. the proof of (2) above). We also used the fact that for real matrices $\alpha, \beta$ we have $\|\alpha\| \leq \|\alpha + i\beta\|$. Taking the supremum over such $[\varphi_{ij}]$, and using that $X \subset X^{\dagger\dagger}$ completely isometrically we obtain

$$\|[x_{kl}]\|_{M_n(X)} \leq \|[x_{kl}]\|_{M_n(W)} \leq 2\|[x_{kl}]\|_{M_n(X)}$$

for all $n$ and $x_{kl} \in X$. Thus $W$ is a real isometric copy of $X$ but with an equivalent operator space renorming.

The following complement matches the Remark at the end of Section 2:

## Example 2.
Let $X$ be the real operator space dual of the complex numbers $\mathbb{C}$. Then the real operator space dual $X^{\dagger}$, which is $\mathbb{C}$, certainly has complex operator space structure. However $X$ does not have complex operator space structure completely isometrically. Indeed if it did it would be a one dimensional complex operator space, hence is complex linearly completely isometric to $\mathbb{C}$. However the real operator space dual of the complex numbers $\mathbb{C}$ is known (see [35, Proposition 2.8]) to be not completely isometric to $\mathbb{C}$. Thus the real operator space dual (or predual) of a real operator space $E$ having complex operator space structure (completely isometrically) does not imply that $E$ does.

### Corollary 3.4.
A complex operator space with a (completely isometric) complex operator space predual $X$ has a (completely isometric) real operator space predual. Indeed we may find a real operator space predual that is real isometric and completely boundedly isomorphic to $X$.

**Proof.** This proof is similar to that of Theorem 3.3 (3). Suppose that $X^* = Y$ as complex operator spaces. We view $X \subset Y^\dagger$ as usual via the duality pairing

$$\langle \varphi, x \rangle = \operatorname{Re} \varphi(x), \quad \varphi \in Y, x \in W = X.$$

and let $W$ be $X$ but with the operator space structure making $i_X : W \to Y^\dagger$ a complete isometry. We use the second criterion in 1.6.4 in [6] to check that $W$ has real operator space dual $Y$. Suppose that $[y^\lambda_{ij}]$ is a net in $Ball(M_n(Y))$ with $y^\lambda_{ij} \to y_{ij}$ in $\sigma(Y, W)$. Thus $\operatorname{Re} y^\lambda_{ij}(x) \to \operatorname{Re} y_{ij}(x)$ for all $x \in X$. Replacing $x$ by $ix$ we see that in fact $y^\lambda_{ij}(x) \to y_{ij}(x)$. That is $y^\lambda_{ij} \to y_{ij}$ in $\sigma(Y, X)$. So since $Y$ is the operator space dual of $X$, by the second criterion in 1.6.4 in [6], $[y_{ij}]$ is in $Ball(M_n(Y))$. Thus by the same criterion in the real case, $W^\dagger \cong Y$ completely isometrically as real operator spaces. It is easy to see that this is a weak* homeomorphism.

We claim that the map $Id : X \to W$ is a real linear isometry and complete contraction with completely bounded inverse. For $n, m \in \mathbb{N}$ and $[\varphi_{ij}] \in Ball(M_m(Y)), [x_{kl}] \in M_n(X)$ we have

$$\|[\operatorname{Re} \varphi_{ij}(x_{kl})]\| \leq \|[\varphi_{ij}(x_{kl})]\| \leq \|[\varphi_{ij}]\| \|[x_{kl}]\|_{M_n(X)}.$$

So \( Id : X \to W \) is completely contractive. That its inverse is completely bounded follows from

\[
\|[\varphi_{ij}(x_{kl})]\| \leq \|[\mathrm{Re}\, \varphi_{ij}(x_{kl})]\| + \|[\mathrm{Re}\, \varphi_{ij}(i x_{kl})]\| \leq 2\|[x_{kl}]\|_{M_n(W)},
\]

and taking the supremum over such \([\varphi_{ij}] \in \mathrm{Ball}(M_m(Y))\). \(\Box\)

We end this section with some results which are needed in Section 4.
Let \(B\) be a real unital \(C^*\)-algebra, such as \(B(H)\). The real Paulsen system of a real operator space \(X \subset B\), is the real operator system

\[
S_{\mathbb{R}}(X) = \begin{bmatrix} \mathbb{R}I & X \\ X^* & \mathbb{R}I \end{bmatrix} = \left\{ \begin{bmatrix} \lambda I & x \\ y^* & \mu I \end{bmatrix} : x,y \in X, \lambda, \mu \in \mathbb{R} \right\}
\]

in \(M_2(B)\), where \(I = I_B\). The well known ‘Paulsen trick’ (see e.g. [6, Lemma 1.3.15] or [27, Chapter 8]) holds in the real case (see e.g. the proof of [33, Theorem 4.3]).

We will also need the theory of injective envelopes \(I(\cdot)\) of operator spaces and operator systems. For the uninitiated, probably an efficient way to get hold of the main details and facts here is to read e.g. Section 4.2 up to and including 4.2.7, 4.4.2, and 4.4.7, in [6].

If \(B\) is a \(C^*\)-algebra then the multiplier algebra \(M(B)\) contains a copy of every \(C^*\)-algebra containing \(B\) as an essential ideal (2.6.10 in [6]). We checked this in the real case in [4]. This is one ingredient for the real case of [8, Corollary 1.8], a result for nonzero operator spaces \(X\). Indeed the latter has the same proof, using the real versions from e.g. [4] of the complex facts and results used in [8, Corollary 1.8]. We shall need the first assertion of [8, Corollary 1.8 (iii)] in the real case in the next proof. In both the real and complex case we remark that 1) this first assertion also easily follows from the second, and 2) the interested reader can find a proof of the second in the real case in our Appendix to a recently published paper of Cecco (again \(X \neq (0)\)). We sketch a proof of 1) in the complex case (the real case is the same): Let \(W\) be the complex TRO \(I(X)\), and \(\mathcal{L}(W)\) its usual ‘TRO linking \(C^*\)-algebra’, with \(WW^*, W, W^*,\) and \(W^*W\) as its four corners. Let \(\mathcal{L} = \mathcal{L}(W)\). Since \(\mathcal{L}\) is an ideal in \(I(S(X))\), by that second assertion there is a canonical completely contractive unital homomorphism

\[
I(\mathcal{L}) \cong I(S(X)) \to LM(\mathcal{L}) \subset I(\mathcal{L}),
\]

that compose to the identity, and factors through a *-homomorphism \(\pi : I(S(X)) \to M(\mathcal{L})\). Alternatively, \(\pi\) is faithful by the ‘essential property’ of the injective envelope. So \(I(\mathcal{L}) \cong M(\mathcal{L})\), the desired first assertion of [8, Corollary 1.8 (iii)].

**Lemma 3.5.** Let \(E\) be a nonzero complex operator space, and \(X = E_r\), the underlying real operator space. The complex ‘injective linking algebra’ \(I(S(E))\) (see [6, 4.4.2 ff.]) is a real injective envelope of the real Paulsen system of \(X\). That is (loosely speaking), \(I(S(E)) = I_{\mathbb{R}}(S_{\mathbb{R}}(X))\).

Proof. We will use notation from [6, 4.4.2 ff.] and [8]. Let \(B = I(S(E))\). Let \(W\) be the complex TRO \(I_{\mathbb{C}}(E)\), and \(\mathcal{L}(W)\) its usual ‘TRO linking \(C^*\)-algebra’ as discussed above the lemma. By [5, Corollary 4.14] we have that \(W\) is a real injective envelope of \(X\): that is \(W = I_{\mathbb{C}}(E) = I_{\mathbb{R}}(X)\). By [8, Corollary 1.8 (iii)] and its ‘real case’ discussed above we have

\[
I_{\mathbb{R}}(S_{\mathbb{R}}(X)) = M_{\mathbb{R}}(\mathcal{L}(W)) = M_{\mathbb{C}}(\mathcal{L}(W)) = B.
\]

We are also using the fact that the complex multiplier algebra $M_C(\cdot)$ of a nonzero complex $C^*$-algebra is its real multiplier algebra $M_R(\cdot)$. The latter is easily seen since every complex Hilbert space $H$ is a real Hilbert space, and a faithful nondegenerate complex $*$-representation on $H$ may be viewed as a real $*$-representation. □

**Theorem 4.1.** *If the real dual $Y$ of a real Banach space $X$ has complex Banach space structure (isometrically) then $X$ has complex Banach space structure (isometrically). Moreover, there exists a canonical complex structure on $X$ whose complex dual is $Y$ (with its given complex structure) complex isometrically and weak$^*$ homeomorphically.*

**Proof.** We adapt the difficult proof of Theorem 3.1, as it is presented in [6, Theorem 4.7.1]. Let $Y$ be a complex Banach space having an isometric real predual $X$ via an isometry $\nu : Y \to X^\dagger$. There is a canonical weak$^*$ continuous real linear contractive projection $Y^{\dagger\dagger} \to Y$, the dual of the inclusion $i_X : X \to Y^\dagger$. Via the isomorphism $Y^{**} \cong Y^{\dagger\dagger}$ in Lemma 1.3 we obtain a canonical weak$^*$ continuous real linear contraction $q : Y^{**} \to Y$, which is a projection (taking $i_Y(y)$ to $y$ for $y \in Y$).

We endow $Y$ with its (complex) minimal operator space structure (see e.g. 1.2.21 and 1.4.12 in [6]). Then $Y$ is a complex operator space, and $q$ is a weak$^*$ continuous completely contractive real linear projection by Proposition 2.3.

Let $u = L_i$, multiplication by $i$ on $Y$. As in the first line of the proof of Theorem 3.3 (1), this is a (complex) multiplier (indeed centralizer, but we do not need this) of $Y$. To prove the first assertion of the theorem it suffices to prove that $u$ is weak$^*$ continuous, by the ideas in the proof of Theorem 3.3 (1). We will not repeat that argument here. Of course $u^{**}$ is in $B(Y^{**})$ and is weak$^*$ continuous. By the argument in the four lines below equation (4.14) in [6, Theorem 4.7.1] it suffices to prove that equation, namely that

$$q(u^{**}(\eta)) = u q(\eta), \quad \eta \in Y^{**}.$$

Let $A = I(S(Y))$ (see 4.4.2 ff. in [6]), the complex 'injective linking algebra' of $Y$. We now follow the proof in [6, Theorem 4.7.1] verbatim (but with $X$ there being our $Y$), up to the point where we view $S(Y^{**}) \subset A^{**}$. We then adapt the rest of the proof of [6, Theorem 4.7.1] as follows:

By the 'Paulsen trick' in the remark below Corollary 4.5, $q$ induces a canonical real linear unital complete contraction $Q : S_R(Y^{**}) \to S_R(Y) \subset A$. It is evident that $Q|_{S(Y)} = I|_{S(Y)}$. Since $A$ is real injective, we can extend $Q$ to a real linear unital complete contraction $R : A^{**} \to A$. Since $A$ is a real injective envelope $I_R(S_R(Y))$ by Lemma 3.5, and $R|_{S_R(Y)} = I|_{S_R(Y)}$, we see that $R|_A = I_A$ by the rigidity property of the real injective envelope. So $R$ is a 'conditional expectation' from $A^{**}$ onto its $C^*$-subalgebra $A$ (it is UCP by Proposition 2.1). By Theorem 2.2 $R$ is an $A$-bimodule map (and in particular $R$ is complex linear). The rest of the proof of the displayed equation above, and hence of the weak$^*$ continuity of $u$ and the first assertion of our theorem, is as in [6, Theorem 4.7.1].

The last assertion follows as in the proof of the last assertion of Theorem 3.3 (1). □

**Remarks.** 1) The complex subspace of $Y^*$ corresponding to the $\sigma(Y, X)$-continuous complex valued functionals is real linearly isometric to $X$ of course,

and is complex linearly isometric to X with the complex structure that we found. This uses the last assertion of the theorem. Thus all is as one would desire.

2) Special cases of Theorem 4.1 are sometimes valid with a simpler proof. Gilles Godefroy reminded us that if Y has (strongly) unique real predual X then every invertible isometry of Y is weak*-continuous. In particular, multiplication by i is weak* continuous, so Theorem 4.1 holds. As he kindly communicated, this applies to a score of spaces, e.g. when X has RNP, or Y does not contain l₁, or when X is isomorphic to a subspace of a wsc Banach lattice, spaces which are isomorphic to subspaces of L-embedded spaces in the sense of [22, Chapter 4], etc. See [19] and e.g. [30, Corollary 4]. He also mentioned that Sections 2, 6, and 9 of [20, Chapter VII] have particular relevance to our paper. This book should be forthcoming in English in 2025. We thank him warmly for these and other interesting historical comments.

3) Continuing this theme, we give an interesting reduction in the case that the bidual of a real commutative C*-algebra A has complex Banach space structure. Indeed the bidual is a real commutative W*-algebra so by [5, Theorem 3.1] it is a C*-algebra real *-isomorphic to M = L∞(Ω, ℂ) ⊕∞ L∞(X, ℝ), where we allow the two summands to possibly be (0). If M has complex Banach space structure then Lᵢ is a complex Banach space centralizer, and the latter commute with every M-projection (see e.g. [22, Section I.3]). In particular if the second summand is not (0) then Lᵢ commutes with (0, 1) ∈ M, which forces L∞(X, ℝ) to have complex Banach space structure. However a C(K, ℝ) cannot have complex structure isometrically. Indeed if J : C(K, ℝ) → C(K, ℝ) is a surjective isometric antisymmetry then by the Banach-Stone theorem it is of form u·(f∘τ) for a unimodular u. We also need |1/√2 + 1/√2 u(·)| ≤ 1, which forces u = −1. Since J²(1) = −1 we have u·(u∘τ) = −1, a contradiction. Thus A^{††} is real *-isomorphic (and real completely isometric by e.g. [9, Theorem 2.6]) to L∞(Ω, ℂ).

We end with some quick 'typical applications'. By the well known Dixmier-Ng characterization of dual spaces [26, 15] we deduce:

Corollary 4.2. Let X be a complex normed space. Suppose that there exists a real Hausdorff locally convex topology on X with respect to which Ball(X) is compact. Then X is (isometrically) a complex dual Banach space.

Corollary 4.3. Let X be a complex normed space. Suppose that there exists a total subspace V of the real dual X† of X such that Ball(X) is σ(X,V)-compact. Then X is (isometrically) a complex dual Banach space.

It is tempting to think that the Dixmier-Ng conditions could be used to prove Theorem 4.1. However there appears to be a subtle but serious problem with this.

Corollary 4.4. Let A be a complex C*-algebra. Then A is a complex von Neumann algebra if and only if it has a real Banach space predual.

Proof. By Remark 2 after Lemma 1.3 any complex von Neumann algebra has a real predual. Conversely, if A has a real predual then it has a complex predual by Theorem 4.1 (or by Remark 2 above), and hence it is a von Neumann algebra by Sakai's theorem (characterizing von Neumann algebras). □

Remark. A similar result holds for TRO's, or indeed in any category that has a 'Sakai theorem' of the type used above. There is a similar result for unital operator

algebras in the operator space category using our characterization of dual operator algebras in [6, Theorem 2.7.9].
We now give a quick proof of Theorem 1.2:

Proof. By Theorem 4.1 applied twice, X is a complex Banach space, and (L_i)^†† = L_i. It is easy to see that ι_{i_X}(X) ⊂ i_X(X) in X^††. Indeed this is a Banach space statement which we leave as a pleasant exercise for the reader. So X is a complex operator subspace of X^{**}. The last assertion is clear by Lemma 3.2. □

Remark. It is shown in [4, Section 3] that a real operator space has at most one complex operator space structure up to complete isometry.

In the spirit of deducing complex structure from complex structure in the bidual, we offer the following result:

Corollary 4.5. Let A be a real C*-algebra. The following are equivalent:
(i) A is a complex C*-algebra.
(ii) The operator space bidual A^†† has (completely isometrically) complex operator space structure.
(iii) A^†† is (real *-isomorphic to) a complex von Neumann algebra.

Proof. (iii) ⇒ (ii) Use e.g. [9, Theorem 2.6].
(ii) ⇒ (i) If A^†† has complex operator space structure then A has complex operator space structure by Theorem 1.2. So A is a complex C*-algebra by [5, Theorem 4.14].
(i) ⇒ (iii) This can be seen in several ways. Of course A^{**} is a complex von Neumann algebra. There is a weak* continuous complete isometry A^{**} ≅ A^†† from Lemma 3.2, and it is easy to see that it is unital. Hence by an operator space version of the Kadison-Banach-Stone theorem (see e.g. [34, Theorem 4.4]) the map is also a *-isomorphism. □

Remark. A similar result holds for TRO's, or for approximately unital operator algebras (using again [5, Theorem 4.14]). We leave these to the interested reader to state. We also remark that one may not drop the words ‘completely’ and ‘operator space’ in (ii). A counterexample is the quaternions (see [4, Remark 1 after Theorem 3.1]).

Closing Remark. It is natural to ask for a proof of Theorem 4.1 only involving Banach spaces. We believe that ‘multipliers’ of linear spaces, and of their bidual, are essentially the correct tool for any proof. Indeed since the key to our proof involves centralizers, and Banach space centralizers correspond to certain maps on the extreme points of the ball of the dual space (see [22, Section I.3] or [3]), we expect that there should be an ingenious extreme point/multiplier argument. Unfortunately multiplication L_i by i is not a real Banach space centralizer or multiplier of a complex space X in the sense of the just cited literature. That is, L_i ∉ Cent_ℝ(X), unlike in the operator space case where L_i ∈ Z_ℝ(X) = Z_ℂ(X) for operator space centralizers [4, Theorem 4.5]. Thus it appears that one cannot access the classical Banach space centralizer/multiplier theory. We would guess that the main idea should be a clever and technical manipulation of φ and restrictions of its real part,

for extreme points of the ball of the dual space, and the third dual space. However if such a proof is found it would only effect a small part of our paper, namely providing a simpler (but not typographically shorter) proof of Theorem 4.1.

Acknowledgements. We have spoken about the operator space results (such as Theorems 3.3 and 1.2), and have advertised Problem 1, at several forums, e.g. SUMIRFAS 2023. We thank Bill Johnson and Florent Baudier for encouraging conversations on the topic. We are deeply indebted to Valentin Ferenczi for Example 1 mentioned after Theorem 1.2, for allowing us to include this, and for spending some time in 2023 thinking about adapting this example to obtain a counterexample to Problem 1. We thank Caleb McClure for sitting through a presentation of most of the proofs, and for comments, questions, catching typos and inconsistencies, etc. We also thank Gilles Godefroy, Mehrdad Kalantar, and Arianna Cecco, for several comments (see Remark 2 after Theorem 4.1).

DEPARTMENT OF MATHEMATICS, UNIVERSITY OF HOUSTON, HOUSTON, TX 77204-3008, USA
Email address, David P. Blecher: dpblecher@central.uh.edu


[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. J. Silverman proved that a dynamical system on 𝑃¹ descends to the field of moduli if it is polynomial or it has even degree, but for non-polynomial ones of odd degree the picture is less clear. We give a complete characterization of which dynamical systems over 𝑃¹ descend to the field of moduli.

We work over a perfect field 𝑘 with algebraic closure 𝐾.

Consider 𝜙 : 𝑃¹_𝐾 → 𝑃¹_𝐾 a finite map of degree 𝑑 ≥ 2. The interplay between the number-theoretic and dynamical properties of 𝜙 has been the subject of a lot of research in the last decades, see e.g. [Sil07].

The dynamics of 𝜙 do not change if we conjugate it by a projective linear transformation 𝑓 ∈ PGL₂(𝐾). Define 𝜙^𝑓 ≝ 𝑓⁻¹ ∘ 𝜙 ∘ 𝑓; this gives an action of PGL₂(𝐾) on the set of rational functions. A dynamical system ξ on 𝑃¹_𝐾 is an equivalence class for this action.

The Galois group Gal(𝐾/𝑘) acts on the set of dynamical systems by acting on the coefficients of the corresponding rational functions in 𝐾(𝑡). The field of moduli 𝑘_ξ is the subfield of 𝐾 fixed by the elements of Gal(𝐾/𝑘) mapping ξ to itself. If ξ descends to a dynamical system on a subextension 𝑘' ⊂ 𝐾, then clearly 𝑘_ξ ⊂ 𝑘'. It is then natural to ask whether ξ descends to a dynamical system on 𝑘_ξ.

Recall that the stabilizer 𝔄_𝜙 ⊂ PGL₂(𝐾) of 𝜙 is the subgroup {𝑓 ∈ PGL₂(𝐾) | 𝜙^𝑓 = 𝜙}; its isomorphism class only depends on ξ, so that 𝔄_ξ is well defined as an abstract group, and it is finite [Sil95, Proposition 4.1]. J. Silverman proved the following.

Theorem 1 ([Sil95, Theorem 5.1]). Let ξ be a dynamical system on 𝑃¹_𝐾, assume that char 𝑘 does not divide |𝔄_ξ|. If deg ξ is even or if ξ contains a polynomial, then ξ descends to a dynamical system on 𝑃¹_{𝑘_ξ}.

Furthermore, Silverman proved that ξ descends to a Brauer–Severi curve if the stabilizer 𝔄_ξ is trivial [Sil95, Theorem 2.1]. We prove that every dynamical system with non-trivial stabilizer descends to 𝑃¹_{𝑘_ξ}. This is somewhat counter-intuitive, since the case of non-trivial stabilizer is more complex.

We thus get a complete characterization of descent properties for dynamical systems on 𝑃¹_𝐾.

Theorem 2. Every dynamical system ξ = [𝜙] on 𝑃¹_𝐾 descends to a dynamical system on some Brauer–Severi curve defined over the field of moduli 𝑘_ξ. Furthermore, the following are equivalent.

(1) Diagram image:
\[
\begin{array}{ccc}
\mathbb{P}^1_K & \xrightarrow{\varphi} & \mathbb{P}^1_K \\
\downarrow{j} & & \downarrow{j} \\
B & \longrightarrow & B
\end{array}
\]

(2) Text crops (in reading order):

where \( B \) is a Brauer–Severi curve over a subextension \( K/k'/k \) with \( B(k') = \emptyset \), and \( j_K : \mathbb{P}^1_K \to B_K \) is an isomorphism.

R. Hidalgo proved that, in characteristic 0, every dynamical system \( \xi \) descends to a dynamical system on \( \mathbb{P}^1_{k'} \), where \( k'/k_\xi \) is an extension of degree at most 2 [Hid14]. As a consequence of Theorem 2, Hidalgo's theorem holds in any characteristic, since every dynamical system descends to a Brauer–Severi curve over \( k_\xi \) and every Brauer–Severi curve is split by an extension of degree at most 2.

Theorem 2 implies the following generalization of Silverman's result.

**Theorem 3.** *A dynamical system* \( \xi = [\varphi] \) *on* \( \mathbb{P}^1_K \) *descends to a dynamical system on* \( \mathbb{P}^1_{k_\xi} \) *if any of the following conditions holds.*

- The stabilizer \( A_\xi \) is non-trivial.
- The degree \( d \) is even.
- There exists an integer \( e \geq 2 \) such that the number of points with ramification index \( e \) is odd.

If \( \xi \) contains a polynomial, either \( \xi \) contains \( x^d \) (which is clearly defined over the field of moduli) or there is a unique \( K \)-point with ramification index \( d \), so that \( \xi \) satisfies the third condition of Theorem 3.

We mention that analogous problems for dynamical systems on \( \mathbb{P}^n \) are studied in [HM14, DS19, Breb]. Furthermore, recently F. Veneziano and S. Vishkautsan have studied similar problems for iterates of a dynamical system on \( \mathbb{P}^1 \) [VV24].

**A remark about Silverman's results.** Silverman does not assume the base field to be perfect, so our results seem to be less general in this regard. However, if we drop this hypothesis there is a technical flaw in Silverman's paper. Let \( k \) be an arbitrary field, \( \overline{K} \) an algebraic closure, \( K \subset \overline{K} \) the separable closure, \( k^p \subset \overline{K} \) the perfect closure. There is a natural identification of Galois groups \( \mathrm{Gal}(K/k) = \mathrm{Gal}(\overline{K}/k^p) \), call them \( G_k \).

In [Sil95, Proposition 1.2], it is stated that \( H^1(G_k, \mathrm{PGL}_2(\overline{K})) \) classifies Brauer–Severi curves over \( k \). However, the proof uses a reference [Sil09, §X, Theorem 2.2] which assumes the base field to be perfect. In general, Brauer–Severi curves over \( k \) are classified by \( H^1(G_k, \mathrm{PGL}_2(K)) \) [GS17, Theorem 5.2.1], while \( H^1(G_k, \mathrm{PGL}_2(\overline{K})) \) classifies Brauer–Severi curves over \( k^p \).

If \( \mathrm{char}\, k \neq 2 \), the natural map

\[
H^1(G_k, \mathrm{PGL}_2(K)) = \operatorname{Br}(k)[2] \to H^1(G_k, \mathrm{PGL}_2(\overline{K})) = \operatorname{Br}(k^p)[2]
\]

is an isomorphism, since \( \operatorname{Br}(k) \to \operatorname{Br}(k^p) \) is surjective [Jac96, Theorem 4.1.5] with char \( k \)-primary kernel. Still, Silverman's article remains flawed even if we assume \( \mathrm{char}\, k \neq 2 \). In fact, in the proof of [Sil95, Theorem 2.1(b)], the morphism \( \Phi = j g \varphi g^{-1} j^{-1} \) is defined over \( \overline{K} \) (because \( g \) is defined over \( \overline{k} \)) as opposed to \( K \): in order to verify that \( \Phi \) is defined over \( k \), Silverman checks that \( \Phi \) is Galois-invariant, but since \( \Phi \) is a priori defined over \( \overline{K} \) this only proves that \( \Phi \) is defined over \( k^p \).

Finally, let us mention that there is a way to study the problem even when the base field is not perfect. This is done by replacing Galois cohomology either with fppf cohomology or stack theory, such as in [BV24]. We don't know whether all the various results remain completely true in this generality. Still, most of their content remains true, see §4.

Acknowledgements. I would like to thank J. Silverman, A. Vistoli and F. Veneziano for very fruitful discussions.

A model of φ over a subfield k′ is a dynamical system b : B → B on a Brauer–Severi curve B/k′ with an isomorphism ℙ¹_K ≅ B_K such that the diagram

      ℙ¹_K  --φ-->   ℙ¹_K
        |                |
        v                v
         B  --b-->      B

commutes.

Let 𝒩_φ be the group of k-linear isomorphisms f : ℙ¹_K → ℙ¹_K such that f∘φ = φ∘f.
An element of 𝒩_φ induces an homomorphism Γ(𝒪_{ℙ¹_K}) = K → Γ(𝒪_{ℙ¹_K}) = K, hence we get a short exact sequence

1 → 𝒜_φ → 𝒩_φ → Gal(K/k)

Lemma 4. The image of 𝒩_φ → Gal(K/k) is Gal(K/k_ξ), hence we have a short exact sequence

1 → 𝒜_φ → 𝒩_φ → Gal(K/k_ξ) → 1.

Proof. Consider φ ∈ K(t) as a rational function. An element σ ∈ Gal(K/k) acts on the coefficients of φ, write φ^σ ∈ K(t) for the resulting rational function. If τ_σ : ℙ¹_K → ℙ¹_K is the standard Galois action of σ on ℙ¹_K, then we have φ^σ = τ_σ ∘ φ ∘ τ_σ⁻¹.

If σ is the image of f ∈ 𝒩_φ, then
φ^σ = τ_σ ∘ φ ∘ τ_σ⁻¹ = (τ_σ ∘ f⁻¹) ∘ φ ∘ (τ_σ ∘ f⁻¹)⁻¹.

Since τ_σ ∘ f⁻¹ ∈ PGL₂(K) is K-linear, this implies that σ ∈ Gal(K/k_ξ).

On the other hand, if σ ∈ Gal(K/k_ξ), then there exists f ∈ PGL₂(K) such that φ^σ = f ∘ φ ∘ f⁻¹. Since φ^σ = τ_σ ∘ φ ∘ τ_σ⁻¹, this implies that τ_σ ∘ f⁻¹ is an element of 𝒩_φ mapping to σ. □

Recall that, since d ≥ 2, the stabilizer 𝒜_φ is finite [Sil95, Proposition 4.1], hence 𝒩_φ is pro-finite. There are natural actions of 𝒩_φ on ℙ¹_K and Spec K; denote by 𝒫_ξ^D, 𝒢_ξ^D the quotient stacks [ℙ¹_K/𝒩_φ], [Spec K/𝒩_φ].

If the reader finds quotients by pro-finite groups uncomfortable, notice that we can find a large finite Galois extension k′/k where everything is defined: we can then proceed to repeat the above with k′ instead of K, so that all the groups involved are finite. In particular, this shows that 𝒫_ξ^D and 𝒢_ξ^D are Deligne–Mumford stacks over k_ξ. Furthermore, 𝒢_ξ^D is gerbe over k_ξ, since k_ξ is perfect and 𝒩_φ maps surjectively on Gal(K/k_ξ) by Lemma 4.

We use the superscript •^D to distinguish the Deligne–Mumford stacks 𝒫_ξ^D and 𝒢_ξ^D from their non-Deligne–Mumford siblings 𝒫_ξ and 𝒢_ξ that will appear in §4, and which will be used to study the problem over non-perfect fields.

The morphisms \(\mathbb{P}^1_K \to \text{Spec } K\), \(\phi : \mathbb{P}^1_K \to \mathbb{P}^1_K\) induce morphisms \(\pi : \mathcal{P}^D_\xi \to \mathcal{G}^D_\xi\), \(\varphi : \mathcal{P}^D_\xi \to \mathcal{P}^D_\xi\) with a commutative diagram

The restriction of \(\varphi : \mathcal{P}^D_\xi \to \mathcal{P}^D_\xi\) to the tautological section \(\text{Spec } K \to \mathcal{G}^D_\xi = [\text{Spec } K / \mathcal{N}_\phi]\) identifies canonically with \(\phi : \mathbb{P}^1_K \to \mathbb{P}^1_K\). Since \(\mathcal{G}^D_\xi\) is a gerbe over \(k_\xi\), given any section \(s : \text{Spec } k_\xi \to \mathcal{G}^D_\xi\) the base change \(s_K\) is isomorphic to the tautological section and hence the restriction \(\varphi|_s : \mathcal{P}^D_{\xi, s} \to \mathcal{P}^D_{\xi, s}\) is a model of \(\phi\) over \(k_\xi\).

On the other hand, if \(b : B \to B\) is a model of \(\phi\) over \(k_\xi\), the natural Galois action on \(B_K \cong \mathbb{P}^1_K\) defines a section \(\text{Gal}(K/k_\xi) \to \mathcal{N}_\xi\), which in turn induces a section \(s : \text{Spec } k_\xi \to \mathcal{G}^D_\xi\) by [Brea, Lemma 5]. It is straightforward to check that these constructions are inverses to each other; we have thus proved the following.

**Lemma 5.** The dynamical system \(\xi\) descends to a Brauer–Severi curve over \(k_\xi\) if and only if \(\mathcal{G}^D_\xi(k_\xi) \neq \emptyset\). Furthermore, it descends to a dynamical system on \(\mathbb{P}^1_{k_\xi}\) if and only if \(\mathcal{P}^D_\xi(k_\xi) \neq \emptyset\). \(\Box\)

If \(k_\xi\) is finite, then \(\mathcal{G}^D_\xi(k_\xi) \neq \emptyset\) by [Bre24, Lemma 13]. Since Brauer–Severi curves over finite fields always have rational points [GS17, Remark 5.1.6], then \(\xi\) is defined over \(\mathbb{P}^1_{k_\xi}\) by Lemma 5. From now on, we assume that \(k_\xi\) is infinite.

Let \(\mathcal{P}^D_\xi \to \mathbb{P}^D_\xi\) be the coarse moduli space of \(\mathcal{P}^D_\xi\), it is a Brauer–Severi curve over \(k_\xi\) with a natural identification \(\mathbb{P}^D_{\xi, K} \cong \mathbb{P}^1_K / \mathcal{A}_\phi \cong \mathbb{P}^1_K\). Since the action of \(\mathcal{A}_\phi\) on \(\mathbb{P}^1_K\) is faithful, the stack \(\mathcal{P}^D_\xi\) is generically a scheme, thus we have a birational inverse \(\mathbb{P}^D_\xi \dashrightarrow \mathcal{P}^D_\xi\). Since we are assuming that \(k_\xi\) is infinite, we get the following corollary of Lemma 5.

**Corollary 6.** The dynamical system \(\xi\) descends to a dynamical system on \(\mathbb{P}^1_{k_\xi}\) if and only if \(\mathbb{P}^D_\xi(k_\xi) \neq \emptyset\).

By [Sil95, Theorem 2.1], it is enough to prove that \(\xi\) descends to \(\mathbb{P}^1_{k_\xi}\) if the stabilizer \(\mathcal{A}_\xi\) is non-trivial.

So let us assume that \(\mathcal{A}_\xi\) is non-trivial; equivalently, \(\mathcal{G}^D_\xi \neq \text{Spec } k_\xi\). Assume by contradiction that \(\xi\) does not descend to \(\mathbb{P}^1_{k_\xi}\); by Corollary 6, this is equivalent to assuming that \(\mathbb{P}^D_\xi(k_\xi) = \emptyset\).

The composition \(\mathbb{P}^D_\xi \dashrightarrow \mathcal{P}^D_\xi \to \mathcal{G}^D_\xi\) defines a morphism whose geometric fibers are irreducible curves of genus 0. In fact, if \(\text{Spec } K \to \mathcal{G}^D_\xi = [\text{Spec } K / \mathcal{N}_\xi]\) is the preferred section and we base change to \(K\), we have a commutative diagram

which is cartesian on the locus of points where the action is free. Since \( \mathbf{P}^D_\xi(k_\xi) = \emptyset \), by [Bre24, Proposition 28] we get that \( \mathcal{A}_\phi \) is cyclic of order prime with char \( k \).

Choose coordinates on \( \mathbf{P}^1_K \) so that \( 0, \infty \) are the two fixed points for the action of \( \mathcal{A}_\xi \). Let \( n \) be the order of \( \mathcal{A}_\phi \), then \( \mathcal{A}_\xi \) acts on \( \mathbf{P}^1_K \) by \( x \mapsto \zeta_n x \), where \( \zeta_n \) is a primitive \( n \)-th root of 1. The fact that \( \mathcal{A}_\phi \) stabilizes \( \phi \) implies that \( \phi \) has the form

\[
\phi(x) = \lambda x^{a} \prod_{i=0}^{n-1} \frac{q_1(\zeta_n^i x)}{q_2(\zeta_n^i x)}
\]

for some \( \lambda \in K \), \( a \in \mathbb{Z} \) and \( q_1, q_2 \in K[x] \).

Notice that, since \( x \mapsto \zeta_n x \) stabilizes \( \phi \), we get the equation \( \phi(\zeta_n x) = \zeta_n \phi(x) \). Applying this to the formula above, we get

\[
\zeta_n = \zeta_n^a,
\]

i.e. \( a \equiv 1 \pmod{n} \).

Let us now prove that

\[
2a + n \deg q_1 - n \deg q_2 = 0.
\]

Equivalently, we want to prove that the orders of vanishing of \( \phi \) on \( 0 \) and \( \infty \) are equal.

Notice that the subset \( \{0, \infty\} \subset \mathbf{P}^1_K \) is distinguished in the sense of [Bre23, §7]: if \( h \in \mathcal{N}_\phi \) and \( g \in \mathcal{A}_\phi \), then \( g(h(0)) = h(h^{-1} g h(0)) = h(0) \) and hence \( h(0) \) is a fixed point for \( \mathcal{A}_\phi \). If the orders of vanishing of \( \phi \) on \( 0 \) and \( \infty \) are different, then \( \{0\} \) is a distinguished subset as well, hence by [Bre23, Lemma 17] the image of \( 0 \) in \( \mathbf{P}^1_K/\mathcal{A}_\xi \) descends to a \( k_\xi \)-rational point of \( \mathbf{P}^D_\xi \), which is absurd.

Recall that we are assuming that \( \mathcal{A}_\phi \) is non-trivial, or equivalently \( n \neq 1 \). Since \( a \equiv 1 \pmod{n} \) and \( 2a + n\deg q_1 - n\deg q_2 = 0 \), the only possibility is \( n = 2 \). Hence, we get the equality \( a = \deg q_2 - \deg q_1 \) and the formula

\[
\phi(x) = \lambda x^{\deg q_2 - \deg q_1} \frac{q_1(x)q_1(-x)}{q_2(x)q_2(-x)}.
\]

This implies that \( \phi \) has degree equal to \( 2 \max(\deg q_1, \deg q_2) \). By Silverman's theorem, this implies that \( \phi \) descends to \( \mathbf{P}^1_{k_\xi} \), giving the desired contradiction.

The fact that \( \xi \) descends to \( \mathbf{P}^1_K \) under the first condition is a particular case of Theorem 2. If the second condition holds, then this follows by Theorem 2 plus the well-known fact that a finite morphism \( B \to B \) has odd degree if \( B \) is a non-trivial Brauer–Severi curve; Silverman gives a proof of this fact in [Sil95, Corollary 2.2].

Assume that there exists an integer \( e > 1 \) such that the number of geometric points with ramification index \( e \) is odd. By Theorem 2, we may assume that \( \mathcal{A}_\xi \) is trivial, hence \( \mathscr{P}^D_\xi = \mathbf{P}^D_\xi \) is a Brauer–Severi curve over \( k_\xi \) and we get a commutative diagram

\[
\begin{CD}
\mathbf{P}^1_K @>\phi>> \mathbf{P}^1_K \\
@VVV  @VVV \\
\mathbf{P}^D_\xi @>>> \mathbf{P}^D_\xi
\end{CD}
\]

with \( \mathbf{P}^1_K \to \mathbf{P}^D_{\xi,K} \) an isomorphism.

Let $S \subset \mathbb{P}^D_\xi$ be the image of the points $p \in \mathbb{P}^1_K(K)$ with ramification index $e$, considered as a closed scheme with the reduced structure. We have that $k_\xi$ is perfect since the same holds for $k$; this implies that the degree of $S$ coincides with the number of geometric points with ramification index $e$, and hence it is odd. We conclude that $\mathbb{P}^D_\xi(k_\xi) \neq \emptyset$ by [Bre24, Lemma 27].

Let $k$ be any field with algebraic closure $\bar{K}$, separable closure $K \subset \bar{K}$ and perfect closure $k^p \subset \bar{K}$. A dynamical system is an equivalence class for the action of $\text{PGL}_2(\bar{K})$ on the set of finite morphisms $\mathbb{P}^1_{\bar{K}} \to \mathbb{P}^1_{\bar{K}}$. Fix $\xi$ a dynamical system of degree $d \geq 2$. We have already seen in §1 that if the base field is finite then dynamical systems always descend to the field of moduli. We may then assume that $k$ is infinite.

**Definition 7.** Define categories fibered in groupoids $\mathcal{G}_\xi$, $\mathcal{P}_\xi$ as follows.
If $S$ is a scheme over $k$, then $\mathcal{G}_\xi(S)$ is the groupoid of commutative diagrams
\[
\begin{array}{ccc}
P & \xrightarrow{\psi} & P \\
\downarrow{\pi} & & \downarrow{\pi} \\
S & = & S
\end{array}
\]
where $\pi : P \to S$ is a projective bundle of relative dimension 1 such that, for every geometric point $s \in S$, the restriction $\psi|_s : P_s \to P_s$ is an element of $\xi$.
We define $\mathcal{P}_\xi(S)$ to be the groupoid of commutative diagrams as above plus a section $S \to P$. There is an obvious induced commutative diagram
\[
\begin{array}{ccc}
P_\xi & \longrightarrow & P_\xi \\
\downarrow & & \downarrow \\
G_\xi & = & G_\xi
\end{array}
\]
Arrows in $\mathcal{G}_\xi(S)$ are isomorphisms $P' \simeq P$ over $S$ commuting with the diagrams. Arrows in $\mathcal{P}_\xi(S)$ are isomorphisms $P' \simeq P$ over $S$ commuting with the diagrams and mapping the section $S \to P'$ to the section $S \to P$.

**Lemma 8.** The fibered categories $\mathcal{G}_\xi$, $\mathcal{P}_\xi$ are fppf stacks locally of finite presentation.

**Proof.** This follows from [Vis05, Theorem 4.38, Example 4.39]. □

Notice that the Galois groups $\text{Gal}(\bar{K}/k^p)$ and $\text{Gal}(K/k)$ naturally coincide, and that they act on the set of dynamical systems by acting on the coefficients of the corresponding functions.

**Definition 9.** Denote by $H \subset \text{Gal}(\bar{K}/k^p) = \text{Gal}(K/k)$ the subgroup of elements $\sigma$ such that $\xi^\sigma = \xi$; it is an open subgroup since $\xi$ contains a function defined over a finite extension of $k$.
Assume that $\xi$ contains a morphism defined over $K$. The field of moduli of $\xi$ is the fixed field $k_\xi = K^H \subset K$.

**Proposition 10.** Assume that $\xi$ contains a morphism defined over $K$. The stacks $\mathcal{G}_\xi$, $\mathcal{P}_\xi$ are algebraic with finite inertia, and $\mathcal{G}_\xi$ is a finite gerbe over the field of moduli $k_\xi$.

Denote by \(\mathcal{A}_\xi\) the automorphism group scheme of any representative of \(\xi\), its isomorphism class is well defined as an abstract finite group scheme over \(\overline{K}\) (as opposed to the automorphism group of a fixed representative, which has a natural embedding in \(\mathrm{PGL}_2\)).

Notice that \(\mathcal{P}_\xi\) is generically a scheme, since its base change to \(\overline{K}\) is the quotient stack \([\mathbb{P}^1_{\overline{K}} / \mathrm{Aut}_{\overline{K}}(\phi)]\). Furthermore, \(\mathcal{P}_\xi\) is a Brauer–Severi curve over \(k_\xi\), since its base change to \(\overline{K}\) is \(\mathbb{P}^1_{\overline{K}} / \mathrm{Aut}_{\overline{K}}(\phi)\). Since we are assuming that \(k\) is infinite, then \(\mathcal{P}_\xi(k_\xi)\) is non-empty if and only if \(P_\xi(k_\xi)\) is non-empty. The first statement then follows from the definition of \(\mathcal{P}_\xi\).

Furthermore, \(\xi\) descends to a unique Brauer–Severi curve over \(k_\xi\) if the group scheme \(\mathcal{A}_\xi\) is trivial. Finally, \(\xi\) always descends to \(\mathbb{P}^1_{k'}\), where \(k'\) is a finite extension of \(k_\xi\) of degree at most 2.

Proof. Let \(\phi \in \xi\) be a morphism defined over \(K\). Notice that \(\mathrm{Aut}_K(\phi) \subset \mathrm{PGL}_2\) is a finite group scheme over \(K\): in fact, it is clearly of finite type, and it has a finite number of geometric points by [Sil95, Proposition 4.1]. The statement then follows from [BV24, Proposition 3.9, Proposition 3.13]. \(\Box\)

Assume condition (2). The morphism \(\mathcal{P}_\xi \to P_\xi\) has degree \(d\) as well, since its base change to \(\overline{K}\) is the morphism \(\mathbb{P}^1_{\overline{K}} / \mathrm{Aut}_{\overline{K}}(\phi) \to \mathbb{P}^1_{\overline{K}} / \mathrm{Aut}_{\overline{K}}(\phi)\) induced by \(\phi\). If \(d\) is even, then \(\mathcal{P}_\xi(k_\xi) \neq \emptyset\) (this is shown in the proof of [Sil95, Corollary 2.2]), hence we conclude by Lemma 11.

The gerbe \(\mathcal{G}_\xi\) is called residual gerbe of \(\xi\), and \(\mathcal{P}_\xi \to \mathcal{G}_\xi\) is the universal family. Denote by \(P_\xi\) the coarse moduli space of \(\mathcal{P}_\xi\), it is called the compression of \(\xi\).

Under condition (1), the proof of Theorem 2 still works by replacing \(\mathcal{G}_\xi\) with the maximal étale quotient of \(\mathcal{G}_\xi\), i.e. the rigidification of \(\mathcal{G}_\xi\) [AGV08, Appendix C] by the connected component of the inertia stack; condition (1) ensures that this maximal quotient is non-trivial.

Assume condition (4), i.e. there exists an integer \(e \geq 2\) such that the number of points with ramification index \(e\) is odd, and let \(S \subset \mathbb{P}\) be the image of said points

Under condition (3), i.e. \(\xi\) contains a polynomial, either \(\xi\) contains \(x^d\) (which is clearly defined over \(k_\xi\)) or \(\xi\) satisfies condition (4).

Lemma 11. The dynamical system \(\xi\) descends to \(\mathbb{P}^1_{k_\xi}\) if and only if \(P_\xi(k_\xi) \neq \emptyset\). It descends to a Brauer–Severi curve over \(k_\xi\) if and only if \(\mathcal{G}_\xi(k_\xi) \neq \emptyset\).

If \(\mathcal{A}_\xi\) is trivial, then \(\mathcal{G}_\xi = \mathrm{Spec}\, k_\xi\) and \(\mathcal{P}_\xi = P_\xi\) is a Brauer–Severi curve. By construction, \(\mathcal{P}_\xi\) is equipped with a model of \(\xi\) over \(k_\xi\), since \(\mathcal{G}_\xi = \mathrm{Spec}\, k_\xi\) this is the only model over \(k_\xi\).

Proof. In general, there exists an extension \(k'/k_\xi\) of degree at most 2 with \(P_\xi(k') \neq \emptyset\), hence \(\xi\) descends to \(\mathbb{P}^1_{k'}\) by Lemma 11.

Theorem 12. Assume that \(\xi\) contains a morphism defined over \(K\). Then \(\xi\) descends to \(\mathbb{P}^1_{k_\xi}\) if any of the following conditions holds.

Proof. The second statement follows from the definition of \(\mathcal{G}_\xi\).

with the reduced structure. The assumption is equivalent to saying that |S(𝐾)| is odd; if char k ≠ 2, this is equivalent to saying that S has odd degree. We conclude that P(kξ) ≠ ∅ by [Bre24, Lemma 27]. □

Remark 13. The gerbe 𝔾^𝔻_ξ constructed in §1 is the rigidification of 𝔾_ξ [AGV08, Appendix C] by the connected component of the inertia stack. It follows that the induced morphism 𝔾_ξ → 𝔾^𝔻_ξ is a relative gerbe with connected relative inertia. If k is perfect, it can be proved that such a morphism induces a bijection on isomorphism classes of rational points: this is the reason why the simpler formalism used in the previous sections works when k is perfect.

Remark 14. It is natural to ask if we can drop the assumption that ξ contains a morphism defined over K. We only use this assumption to define the field of moduli and to prove that 𝔾_ξ is an algebraic stack. Even if ξ does not contain a morphism defined over K, if we know that 𝔾_ξ is an algebraic stack then the field of moduli can be defined as the coarse moduli space of 𝔾_ξ [BV24, Proposition 3.10] and Theorem 12 holds in this generality. See [BV24, Proposition 3.8] for a criterion to check whether 𝔾_ξ is an algebraic stack.

[AGV08] Dan Abramovich, Tom Graber, and Angelo Vistoli, Gromov-Witten theory of Deligne-Mumford stacks, Amer. J. Math. 130 (2008), no. 5, 1337–1398.

[Brea] Giulio Bresciani, Real versus complex plane curves, arxiv:2309.12192.

[Breb] ——, Uniform bounds for fields of definition in projective spaces, arxiv.

[Bre23] ——, The field of moduli of a variety with a structure, Boll. Unione Mat. Ital. (2023), https://doi.org/10.1007/s40574--023--00399--z.

[Bre24] ——, The field of moduli of a divisor on a rational curve, Journal of Algebra 647 (2024), 72–98.

[BV24] Giulio Bresciani and Angelo Vistoli, Fields of moduli and the arithmetic of tame quotient singularities, Compositio Mathematica 160 (2024), no. 5, 982–1003.

[DS19] John R. Doyle and Joseph H. Silverman, A uniform field-of-definition/field-of-moduli bound for dynamical systems on ℙ^N, J. Number Theory 195 (2019), 1–22.

[GS17] Philippe Gille and Tamás Szamuely, Central simple algebras and Galois cohomology, second ed., Cambridge Studies in Advanced Mathematics, vol. 165, Cambridge University Press, Cambridge, 2017.

[Hid14] Rubén A. Hidalgo, A simple remark on the field of moduli of rational maps, Q. J. Math. 65 (2014), no. 2, 627–635.

[HM14] Benjamin Hutz and Michelle Manes, The field of definition for dynamical systems on ℙ^N, Bull. Inst. Math. Acad. Sin. (N.S.) 9 (2014), no. 4, 585–601.

[Jac96] Nathan Jacobson, Finite-dimensional division algebras over fields, Springer-Verlag, Berlin, 1996.

[Sil95] Joseph H. Silverman, The field of definition for dynamical systems on ℙ^1, Compositio Math. 98 (1995), no. 3, 269–304.

[Sil07] ——, The arithmetic of dynamical systems, Graduate Texts in Mathematics, vol. 241, Springer, New York, 2007.

[Sil09] ——, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Dordrecht, 2009.

[Vis05] Angelo Vistoli, Grothendieck topologies, fibered categories and descent theory, Fundamental algebraic geometry, Math. Surveys Monogr., vol. 123, Amer. Math. Soc., Providence, RI, 2005, pp. 1–104.

[VV24] Francesco Veneziano and Solomon Vishkautsan, The field of iterates of a rational function, 2024.

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy
Email address: giulio.bresciani@gmail.com


[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]
This is a partial extension of the characterisation of completely join-irreducible torsionfree classes given by Barnard, Carroll and Zhu.

ABSTRACT. We show that every join-irreducible torsionfree class in the category of finitely generated modules over an artinian ring is cogenerated by a single (not necessarily finitely generated) brick.

Torsion pairs have been a central topic in the representation theory of artin algebras in the last few years.

In particular, one of the main aspects which was thoroughly investigated is the complete lattice structure of the set of torsion (or equivalently torsionfree) classes in the category of finitely generated modules over an artin algebra, see [6].

Recall that an object in some lattice is said to be completely join-irreducible if it cannot be written as an arbitrary join of strictly smaller elements.

The following result, holding in general in every abelian length category, gives a complete classification of completely join-irreducible torsion(free) classes in terms of bricks: i.e. objects whose endomorphism ring is a skew-field:

Theorem 1.1 ([3, Theorem 1.0.5]). There is a bijection between the set of completely join-irreducible torsionfree classes in an abelian length category A and the set of bricks of A.

However, in general we might have torsionfree classes which are join-irreducible, but not completely join-irreducible. Clearly, such classes can only occur when the lattice of torsionfree classes is infinite.

In fact, as a consequence of [11], we always have at least one such torsionfree class in the lattice of torsionfree classes of a τ-tilting infinite algebra.

The aim of this short note is to give a partial extension of Theorem 1.1, showing that every join-irreducible torsionfree class in the category of finitely generated modules over an artinian ring is determined by some brick.

Notation Throughout the paper, Λ will be a finite-dimensional k-algebra over some field k, Λ-Mod (resp. Λ-mod) the category of (finite dimensional) left Λ-modules. Uniqueness of some object is always to be intended up to isomorphism. All the subcategories appearing in the work are assumed to be full and closed under isomorphism.

For a subcategory E of some abelian category we denote by filt(E) the closure of E under extensions and by cogen(E) the subcategory of objects which can be embedded in some finite coproduct of objects in E.

Monobricks were introduced by Enomoto in [7] to study left Schur subcategories of some abelian length category. This class of subcategories contains in particular the sets of torsionfree classes and wide subcategories.

They play a prominent role in the proof of our main result, thus we give a quick overview of the subject.

**Definition 2.1.** A collection of (isomorphism classes of) Λ-bricks {Bi}i∈I is a monobrick if for every f ∈ HomΛ(Bi, Bj) we have f = 0 or f is a monomorphism.

**Remark 2.2.** We can see every monobrick as a poset with the following order relation Bi ≺ Bj if there exists a monomorphism f: Bi → Bj. This relation is clearly reflexive and transitive. Antisymmetry is a consequence of the definition of brick.

**Definition 2.3.** Let (t, f) be a torsion pair in some abelian category A. An object 0 ≠ B in f is f-simple if all its proper quotients are in t.

**Examples 2.4.**
- Every semibrick is a monobrick, the corresponding poset is just a discrete set.
- The collection of f-simple modules for a torsionfree class f in Λ-mod is a monobrick and the maximal elements of the corresponding poset are the minimal coextending modules for f [7, Remark 4.6].

Notice that the poset corresponding to any monobrick in Λ-mod has at most countable height (only countably many dimensions of finite-dimensional modules) but it might have uncountable width (consider the case of some torsionfree class with uncountably many minimal coextending modules).

**Theorem 2.5** ([7, Theorem 2.11]). *Let f be a torsionfree class in Λ-mod and simp(f) be the collection of f-simple modules. Then f = filt(simp(f)).*

The result above tells us that every torsionfree object admits a filtration in terms of f-simples.

**Corollary 2.6.** *Let f be a torsionfree class. Assume there exist torsionfree classes v, w with f = v ∨ w. Then simp(f) ⊆ simp(v) ∪ simp(w).*

Proof. Any f-simple object B is contained in the join, which can be expressed as filtcogen(v ∪ w). However, f-simples are not filtered by other torsionfree modules, thus it must be that B ∈ cogen(v ∪ w) that is B ∈ v or B ∈ w. This shows the inclusion, as an f-simple contained in v is automatically v-simple (the torsion class corresponding to v contains the torsion class corresponding to f). The same for w. □

This result is enough to give a characterisation of join-irreducibility of a torsionfree class in terms of properties of the corresponding monobrick of simples.

**Definition 2.7.** Let L be a lattice, l ∈ L is join-irreducible if l is not the 0 of L and whenever l = x ∨ y we have l = x or l = y.

**Definition 2.8.** A nonempty poset P is directed if every finite subset of P has an upper bound.

Proposition 2.9. Let f be a torsionfree class. Then f is join irreducible if and only if the monobrick of f-simple objects is directed.

Proof. “⇒” We prove the contrapositive: assume that the monobrick is not directed. If the monobrick is empty, then f = 0 thus it is not join irreducible. So assume it is non-empty.

Then we can find f-simples B₁, B₂ such that B₁↑ := {B ∈ simp(f) | B₁ ≺ B} and B₂↑ are disjoint.

Consider the two torsionfree classes v := filtcogen(B₁↑) and w := filtcogen(C) where C := simp(f) \ B₁↑. Clearly, we have that f = v ∨ w as every f-simple is contained in v or in w.

Moreover v ≠ f. In fact f contains B₂. If B₂ ∈ v i.e. if B₂ is in filtcogen(B₁↑) using f-simplicity we would have B₂ is in cogen(B₁↑) but this would contradict our hypothesis on the disjointness of B₁↑ and B₂↑.

In a similar way, w ≠ f as it does not contain B₁: C does not contain, by definition, any module B such that B₁ embeds in B, so using f-simplicity we can conclude that B₁ ∉ filtcogen(C). This shows that f is not join-irreducible.

“⇐” Assume that the monobrick of simples is directed. Let v, w be torsionfree classes such that f = v ∨ w.

By the corollary above, every f-simple is either a v-simple or a w-simple. So assuming that v ⊊ f and w ⊊ f we can find an f-simple module B_v which is in v \ w and an f-simple B_w which is in w \ v. However, using the directness of the poset, we can find an f-simple B_vw such that B_w ≺ B_vw and B_v ≺ B_vw. Now it must be the case that this new simple is an element of v or of w and in both cases this yields a contradiction, as these classes are closed under submodules.

It follows that either f = v or f = w, proving join-irreducibility. □

We can also restate the description of completely join irreducible elements in terms of monobricks:

Corollary 2.10 ([3], [6]). Let f be a torsionfree class in Λ-mod. Then f is completely join irreducible if and only if the monobrick of f-simple modules is directed and has finite height (i.e. it has a maximum).

Proof. Being completely join irreducible is equivalent to the existence of a (finitely-generated) brick B such that f = filtcogen(B), moreover a standard computation shows that the brick B is f-simple. Using this characterisation, it is immediate to see that the equivalence holds as the monobrick being directed and of finite height means that there exists a top element. □

So by exclusion we have:

Corollary 2.11. Let f be a torsionfree class. Then f is join irreducible, but not completely join irreducible if and only if the monobrick of f-simple modules is directed and has infinite height.

To obtain the connection with large bricks we need to use some tools from the theory of cosilting modules. We recall a couple of fundamental results and refer the reader to the survey [1] for the basic notions of silting/cosilting theory.

The first result associates with every torsion pair in the small module category a nice torsion pair in the large one. We use Cosilt(R) for the set of torsion pairs cogenerated by a cosilting module.

Theorem 3.1. When R is a left noetherian ring, there is a bijection

tors(R) ↔ Cosilt(R).

It associates to a torsion pair (t, f) in R - mod the limit closure (T, F) := (lim→ t, lim→ f), which coincides with the torsion pair (Gen t, t⊥0) generated by t. The inverse of this map sends a cosilting torsion pair (T, F) to its restriction (T ∩ R - mod, F ∩ R - mod).

The second result relates cosilting torsion pairs with t-structures in the derived category:

Theorem 3.2 ([8], [9, Theorem 1.2], [10, Theorem 7.1]). Let R be a left noetherian ring, (t, f) a torsion pair in R - mod and (T, F) = (lim→ t, lim→ f) the corresponding cosilting torsion pair in R - Mod.

Then (T, F) determines, via the HRS-construction, a t-structure (U, V) in the unbounded derived category.

The heart H of this t-structure is a locally coherent Grothendieck category and (F, T[−1]) is a torsion pair in it.

Moreover, the category of finitely presented objects of H is precisely the intersection of H with the bounded derived category Db(R - mod).

We will make use of the following computational lemma:

Lemma 3.3 ([2, Lemma 2.3]). Let F, V be two modules in a (cosilting) torsionfree class F. Then a morphism f : F → V is an epimorphism in the corresponding heart H if and only if the cokernel of f is a torsion module.

Finally, we recall the definition of a torsionfree, almost torsion module and a characterisation of those.

Definition 3.4. Let R be a ring, let (T, F) be a torsion pair in R - Mod. Then a non-zero module B is torsionfree, almost torsion if:
(i) B ∈ F
(ii) Every proper quotient of B is in T.
(iii) For every short exact sequence

0 → B → F → C → 0

with F ∈ F, we have that C ∈ F.

Proposition 3.5 ([2, Theorem 3.6]). A module B is torsionfree, almost torsion with respect to a torsion pair (T, F) if and only if it is a simple object in the heart H which is torsion with respect to (F, T[−1]).

To obtain the desired connection we will make use of the following:

Lemma 3.6. Let (t, f) be a torsion pair in Λ–mod and let {Fi, (αij)} be a direct system of non-zero maps between f–simple modules. Then F = lim→{Fi, (αij)} is lim→ f–simple.

Proof. First, we notice that the direct limit is non-zero, as any non-zero map between f–simple objects is a monomorphism. In particular, each of the canonical maps Fi → F must be a monomorphism.

Now we have to show that every proper quotient of F is contained in the torsion class lim→ t.

Using the fact that torsion classes are closed under quotients, this task can be reduced to proving that every maximal quotient of F is a torsion module.

So let S be a simple module with some non-zero map f : S → F. As S is a finite-dimensional module, any map from S to F can be factored trough the direct system {Fi, (αij)}. This yields a direct system of short exact sequences:

0 → S  →^{fi}  Fi → coker(fi) → 0

where each coker(fi) is contained in t as Fi is f–simple.
The direct limit of this system of sequences is

0 → S  →^f  F → coker(f) → 0

so that coker(f) is in lim→ t. This shows that every maximal quotient of F is torsion. □

While lim→f–simple might be infinite-dimensional, it easy to observe that they are finitely generated as objects of the heart. So we can obtain a second useful lemma:

Lemma 3.7. Let F be a lim→f–simple module, then there exists a torsionfree, almost torsion module B such that F is isomorphic to a submodule of B.

Proof. Let F be lim→f–simple and consider any non-trivial finite-dimensional submodule M of F. This results in a short exact sequence

0 → M → F → N → 0

where N must be a torsion module, by simplicity. But then by Lemma 3.3, the inclusion M → F is an epimorphism, when considered as a map in the heart of the HRS-tilt at (lim→ t, lim→ f). So F is the epimorphic image of a finitely presented object, thus it is finitely generated.

As the heart is a locally coherent Grothendieck category, we know that every finitely generated object has a simple quotient, thus we have an epimorphism F → S. In particular, as F is an element of the torsion class lim→f, S will be a torsion simple.

Now by Proposition 3.5, this means that S as a module is torsionfree, almost torsion for the pair (lim→ t, lim→ f).

We therefore have a non-zero map F → S which must be a monomorphism, by simplicity of F. □

We need a last technical lemma:

Lemma 3.8. Let G be a non-empty connected graph, then G contains a spanning tree (i.e. an acyclic connected subgraph containing all the vertices).

Proof. This is a well-known consequence of Zorn's lemma. We give the proof for completeness. Let $\mathcal{T}$ be the collection of all the subtrees of $G$ which we can order by inclusion. Notice that $\mathcal{T}$ is non-empty as every singleton gives a subtree of $G$ and $G$ itself is non-empty. It is easy to notice that every ascending chain of elements of $\mathcal{T}$ has an upper bound, this is just the union of the elements of the chain.

Then by Zorn's lemma the set $\mathcal{T}$ contains some maximal element $T$. I claim that $T$ must be a spanning tree.

It is a tree by definition, so assume there is some vertex $v$ of $G$ such that $v \notin T$. If $T$ were empty, then clearly $\{v\}$ would be a tree containing $T$ contradicting maximality. Thus we may assume that $T$ contains some element $t$. As $G$ is connected there exists a path starting at $t = v_0 \to v_1 \to \cdots \to v_n = v$, we may assume that this is the shortest path connecting an element of $T$ to $v$. In particular, $v_1$ is not in $T$ and we can enlarge $T$ to a new subtree $T'$ containing the vertex $v_1$ and the edge $v_0 \to v_1$.

Then $T \subsetneq T'$ giving a contradiction. Thus $T$ must be a spanning tree.

We can now prove our main result:

Proposition 3.9. Let $f$ be a join-irreducible torsionfree class in $\Lambda\text{−mod}$. Then there exists a brick $B$ in $\Lambda\text{−Mod}$ which is torsionfree, almost torsion for $\varinjlim f$ such that $f = \mathbf{F}(B)\cap \Lambda\text{−mod}$.

Proof. By Proposition 2.9 the poset of $f$−simples is directed. We show that the graph $G$ underlying the Hasse quiver of this poset is connected.

Let $F_1$ and $F_2$ be two isomorphism classes of $f$−simples. Then by directedness there exists a third isoclass $F_3$ with $F_1 \leq F_3$ and $F_2 \leq F_3$. As all the isoclasses involved correspond to finite-dimensional modules, the poset of objects smaller than $F_3$ has finite height, therefore we have finite paths in the Hasse quiver connecting $F_1$ and $F_2$ to $F_3$.

This proves that the graph $G$ is connected.

Now by Lemma 3.8 we can find a spanning tree $T$ in $G$, and consequently a directed acyclic subquiver $T'$ of the Hasse quiver containing all the vertices. Using the axiom of choice we assign to each vertex $v_i$ of $T'$ an $f$−simple module $V_i$ in the isoclass $v_i$ and successively to each edge $v_i \to v_j $ a non-zero map $V_i \to V_j$ (such a map exists by the definition of the order on $f$−simples).

This gives a directed system of $f$−simples consisting of monomorphisms (we don't have any compatibility condition to check on the maps as the underlying graph is a tree), therefore we can use Lemma 3.6 to obtain a $\varinjlim f$−simple $F$ with the property that every $f$−simple is isomorphic to a submodule of $F$.

But now, Lemma 3.7 tells us that $F$ is a submodule of some torsionfree, almost torsion module $B$ of the class $\varinjlim f$.

So we obtain that $f = \mathbf{F}(B)\cap \Lambda\text{−mod}$ by Theorem 2.5. $\square$

[4] W. Crawley-Boevey; Locally finitely presented additive categories; Communications in Algebra,
22(5), 1641–1674 (1994).
[5] L. Demonet, O. Iyama, G. Jasso; τ-tilting finite algebras, bricks, and g-vectors, Int. Math. Res.
Not. IMRN (2019) no. 3, 852–892.
[6] L. Demonet, O. Iyama, N. Reading, I. Reiten, H. Thomas; Lattice theory of torsion classes: Beyond
τ-tilting theory, Trans. Amer. Math. Soc. 10, 542–612, (April 2023).
[7] H. Enomoto; Monobrick, a uniform approach to torsion-free classes and wide subcategories, Adv.
Math., 393 (2021).
[8] C. Parra, M. Saorín; Direct limits in the heart of a t-structure: the case of a torsion pair, Journal
of Pure and Applied Algebra, 219(9):4117–4143 (2015).
[9] C. Parra, M. Saorín; Addendum to Direct limits in the heart of a t-structure: the case of a torsion
pair, J. Pure Appl. Algebra 219 (9).
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(2019).
[11] F. Sentieri; A brick version of a theorem of Auslander, Nagoya Math. J., 249, 88–106 (March 2023).


[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]
Mu-Te Lau*†, Chin-Yi Cheng*†, Cheng-Hua Lu*‡, Chia-Hsu Chuang*†,
Yi-Hsiang Kuo†, Hsiang-Chun Yang†, Chien-Tung Kuo†, Hsin-Yu Chen†, Chen-Ying Tung†, Cheng-En Tsai†,
Guan-Hao Chen†, Leng-Kai Lin†, Ching-Huan Wang†, Tzu-Hsu Wang†, Chung-Yang Ric Huang†

†National Taiwan University, R.O.C., ‡Stanford University, U.S.
josh.lau2011@gmail.com, chin-yi.cheng@utexas.edu, cyhuang@ntu.edu.tw

_Abstract—In this paper, we introduce Qsyn, a novel quantum circuit synthesis (QCS) framework designed to facilitate the research, development, testing, and experimentation of QCS algorithms and tools. Our framework is more developer-friendly than other modern QCS frameworks in three aspects: (1) Qsyn provides a comprehensive command-line interface that enables developers to design various testing scenarios with ease and conduct flexible experiments on their algorithms. This feature significantly streamlines the development process, making it more efficient and user-friendly. (2) Qsyn offers detailed access to multiple data representations at different abstraction levels of quantum circuits. This capability allows developers to optimize their algorithms extensively, gaining deeper insights and control over the structure and behavior of quantum circuits. By understanding the intricacies of circuit design, developers can achieve higher levels of optimization and performance in their algorithms. (3) Qsyn implements a rigorous development flow and environment to help developers maintain high-quality standards using modern software engineering practices, including robust quality assurance measures like regression testing, continuous integration and continuous delivery (CI/CD) pipelines, and code linting. We demonstrate Qsyn’s superior performance through fair comparisons with PyZX [1], highlighting its efficiency and optimization capabilities. By providing a unified and user-friendly development environment, Qsyn empowers researchers and developers to prototype, implement, and evaluate their QCS algorithms effectively._

_Index Terms—Quantum Computing, Quantum Systems Software, Quantum Software Engineering_

To push the advancement of the quantum circuit synthesis (QCS) algorithms in the future, we need a framework that offers a friendly environment for more QCS algorithm/tool developers to easily contribute their advanced ideas and conduct thorough experiments. As a result, we have open-sourced Qsyn, a developer-friendly QCS framework, to aid further development in this field. Our main contributions are:

With Qsyn, developers can easily accelerate the implementation and evaluation of new QCS algorithms by leveraging the provided data structures and development environments.

Qsyn embodies end-to-end synthesis by invoking functionalities for each synthesis stage as commands. It is designed with extensibility in mind, leveraging a data-oriented approach that focuses on robust data management and manipulation capabilities. Below, we introduce the core functionalities in Qsyn (https://github.com/DVLab-NTU/qsyn) for QCS.

For ZX-based methods, Qsyn has implementation advantages over PyZX. Qsyn revises the original algorithms in two aspects: (1) adopt a more compact ZX-diagram representation for MCT gates and an early-stopping strategy to avoid producing redundant graph complexity [11] (2) improves the conversion routine from ZX-diagrams to circuits. Specifically, the "gadget removal" strategy during conversion was improved to extract fewer CZ gates, contributing to a more compacted quantum circuit.

On the other hand, tableau, another data structure in Qsyn, represents Pauli rotation and phase polynomial which is used in QCS approaches. This representation unification eliminates

This work is supported by the National Science and Technology Council R.O.C., Project No.: NSTC 113-2119-M-002-024.

frequent data conversions required when switching between these two synthesis paradigms.

Qsyn offers a variety of capabilities that facilitate users’ expanding of Qsyn for their own algorithms and applications. The extension is initiated with the design of the usage flow based on Qsyn’s command-line interface (CLI), which processes user inputs and handles command execution. Then with the key capabilities described below, Qsyn provides a robust and adaptable framework for quantum circuit synthesis:

Qsyn offers an extensible and highly expressive quantum circuit model. In addition to fundamental gate types, it allows developers to freely add new types of quantum gates. This capability enables developers and researchers to combine a wide range of quantum operations, simplifying the management of these complex tasks.

Developers can integrate Qsyn with external quantum synthesis tools, enhancing workflow flexibility through employing scripts. This shell interoperability becomes particularly useful when comparing optimization results between Qsyn and other synthesis tools.

One of Qsyn’s main missions is to assist developers in creating and assessing new synthesis algorithms. This is made possible by its powerful and easy-to-maintain argument parser along with its useful CLI utilities.

We compare Qsyn’s efficiency and optimization power with PyZX [1]. In Table I, we performed gate-level synthesis to examine the run time and memory usage of Qsyn against PyZX.

Compared to PyZX, Qsyn tended to optimize circuits with shorter program runtimes and lower memory usage. This distinction was particularly notable in circuits such as gf2^128, hwb12, and urf4, where PyZX encounters TLE and/or MLE (Time/Memory Limit Exceeded). For circuit statistics, Qsyn achieved either smaller or equal RZ-counts and, on average, shallower circuits. This is attributed to Qsyn’s integration of ZX-calculus and Tableau combined methods, further compressing rotation counts.

COMPARISON WITH PYZX [1] ON THE GATE-LEVEL SYNTHESIS STAGE
TIME UNIT: s, MEMORY UNIT: MiB

This work is supported by the National Science and Technology Council, R.O.C., Project No.: NSTC 113-2119-M-002-024.