\begin{corollary}\label{DG3}
The following holds when $T$ is sufficiently large and
$
\sup_{\tb_1 } |\psi| \leq \ve\ll 1
$
is sufficiently small. In the coordinates $T^{1/2} \zeta_1,\ldots T^{1/2} \zeta_m, z_{m+1},\ldots z_n$, the Calabi-Yau metric $\omega_T+ \ddbar \psi_T$ satisfies
\[
T \|  \ddbar \psi_T \|_{ C^k(\tb_{T^{-1/2}}(y), T\omega_T^{\rm ref}) }\leq C(k) (\ve+ T^{-\alpha/2}) \ll 1.
\]
\end{corollary}

We are interested in the behavior of Ricci-flat K\"ahler metrics on compact Calabi-Yau manifolds whose complex structure degenerates. This is a much studied problem in the literature, see e.g. the survey \cite{To} and references therein. The general setting is the following: we are given a proper flat holomorphic map $\pi:\mathfrak{X}\to\mathbb{D}\subset\mathbb{C}$, from a normal $(n+1)$-fold, with a relative polatization $L\to\mathfrak{X}$, such that $X:=\pi^{-1}(\mathbb{D}^*)$ is smooth with $K_X\cong\mathcal{O}_X$ and $\pi$ is a submersion over $\mathbb{D}^*$. The smooth fibers $X_t$ for $t\in\mathbb{D}^*$ are thus compact Calabi-Yau $n$-folds, and by Yau \cite{Ya} they carry the Ricci-flat K\"ahler metric $\omega_{\rm CY,t}\in c_1(L|_{X_t})$. These metrics have total volume independent of $t$, but their coarse geometry is controlled by an integer $m\in\{0,\dots,n\}$ which is the real dimension of the essential skeleton $\mathrm{Sk}(\mathfrak{X})$ of the family \cite{MN,NX} (namely the dual intersection complex of the central fiber of a relatively minimal model of a semistable reduction of $\mathfrak{X}$). Indeed, in our previous work \cite{LT} it is shown that when $m>0$ we have the uniform equivalence
\begin{equation}
\mathrm{diam}(X_t,\omega_{{\rm CY},t})\sim |\log|t||^{\frac{1}{2}},
\end{equation}
while when $m=0$ the earlier works \cite{Ta,To2} proved that
\begin{equation}
\mathrm{diam}(X_t,\omega_{{\rm CY},t})\sim 1.
\end{equation}

The case $m=0$ is by now well-understood, and in this case the Ricci-flat manifolds $(X_t,\omega_{{\rm CY},t})$ converge in the Gromov-Hausdorff topology to a singular Calabi-Yau metric on a Calabi-Yau variety with klt singularities \cite{DS,EGZ,RZ}. On the other hand, when $m>0$ to obtain a nontrivial limit, one has to look at the rescaled Ricci-flat manifolds
$(X_t,|\log|t||^{-1}\omega_{{\rm CY},t})$, and the main question is then to understand the limit of these metrics. These rescaled metrics are now volume-collapsed, and their Gromov-Hausdorff limit is expected to be homeomorphic to the real $m$-dimensional simplicial complex $\mathrm{Sk}(\mathfrak{X})$, see e.g. \cite[Conjecture 4.6]{To}.

Recent progress \cite{Li,Li2,Li3,Li4,Li5} (see also \cite{AH,HJMM}) has shown that the K\"ahler potentials of the rescaled metrics converge in a suitable $C^0$-hybrid topology to a potential function on the associated Berkovich space, which is completely determined by a potential function on $\mathrm{Sk}(\mathfrak{X})$ solving a non-Archimedean Monge-Amp\`ere equation. The case when $m=n$ is known as a large complex structure limit, and is the subject of the Strominger-Yau-Zaslow conjecture \cite{SYZ}. In this case, by leveraging the small perturbation theorem of Savin \cite{Sa} in elliptic PDE theory, it was shown that on the ``generic region'' of $X_t$, which fills up almost the entire volume, the convergence of the potentials is actually in the smooth topology.

The remaining cases when $0<m<n$ are known as ``intermediate complex structure limits". A large class of examples come from transverse complete intersections inside a given Fano manifold $M$ with $-K_M= dL$ for some ample line bundle $L$,
\[
X_t= \{ F_0 F_1\ldots F_m+ tF=0\} \subset M,
\]
where the sections $F_i\in H^0(M, d_i L)$ and $F\in H^0(M, dL)$ define smooth and transverse divisors, and $d=\sum_{i=0}^m d_i$. Our goal is to improve the $C^0$-potential theoretic convergence to metric convergence. The non-Archimedean potential defines a family of ansatz metrics $\omega_t$ in the class $c_1(L)$, on some large open subset $U_t\subset X_t$ whose percentage of Calabi-Yau measure tends to one. We call $U_t$ the ``generic region", and we allow for $U_t$ to be slighted shrunken in the arguments.

The main result is the following, which is the analog for intermediate complex structure limits of the aforementioned result for large complex structure limits:

The main result is the following, which is the analog for intermediate complex structure limits of the aforementioned result for large complex structure limits:

\begin{rmk}
In fact, we prove $C^\infty$ estimates for $\omega_{{\rm CY},t}$ with respect to a local Euclidean metric after stretching the base coordinates and the $T^m$-fibers by $|\log|t||^{\frac{1}{2}},$ see Corollary \ref{DG3} for a precise statement.
\end{rmk}

\subsection{Ricci-flat metrics}
For $t\in\mathbb{D}^*$, let $\omega_{{\rm CY},t}$ be the unique Ricci-flat K\"ahler metric on $X_t$ in the class $c_1(L)$. Their volume form can be described as follows. Given a trivializing section $\Omega$ of $K_{\mathfrak{X}/\mathbb{D}}\cong\mathcal{O}_{\mathfrak{X}}$ (cf. \cite[Proposition 4.1]{PS}), up to multiplying $\Omega$ by a suitable integer power of $t$, we may assume without loss that the vanishing orders of $\Omega$ along the $E_i$'s are nonnegative, and their minimum equals zero. Writing $\Omega_t=\Omega|_{X_t}$ for $t\neq 0$, the smooth positive volume form $i^{n^2}\Omega_t\wedge\ov{\Omega_t}$ on $X_t$ has total mass asymptotic to
\begin{equation}
\int_{X_t}i^{n^2}\Omega_t\wedge\ov{\Omega_t}\sim |\log|t||^m,
\end{equation}
as $t\to 0$, see \cite{BJ}. The Ricci-flat metrics $\omega_{{\rm CY},t}$ thus have volume form equal to
\begin{equation}
\frac{\omega_{{\rm CY},t}^n}{(L^n)}=
\frac{i^{n^2}\Omega_t\wedge\ov{\Omega_t}}{\int_{X_t}i^{n^2}\Omega_t\wedge\ov{\Omega_t}}=:\mu_t,
\end{equation}
where $\int_{X_t}\omega_{{\rm CY},t}^n=\int_{X_t}c_1(L)^n=:(L^n)$ is independent of $t$. We will refer to this as the Calabi-Yau measure.

Let $\omega_M$ be a K\"ahler metric on the ambient Fano manifold, whose restriction to $Y= \cap_0^m E_j$ agrees with the Calabi-Yau metric $\omega_Y$ on $Y$ in the class $c_1(L)$.
The ansatz reference metrics are
$$\omega_t=\omega_M+\ddbar\vp_t,$$
where
$$\vp_t=|\log|t||\, u\left(-\frac{\log|F_0|_{h^{d_0}}}{|\log|t||},\dots,-\frac{\log|F_m|_{h^{d_m}}}{|\log|t||}\right),$$
and we can write
$$\omega_{{\rm CY},t}=\omega_t+|\log|t||\ddbar\psi_t,$$
with the normalization $\min_{X_t}\psi_t=0$.
The main result in \cite[Proposition 6.13]{Li2} is that
\begin{equation}\label{c0}
\|\psi_t\|_{L^\infty(X_t)}\to 0,\quad t\to 0.
\end{equation}
Using the function $u$, \cite[Corollary 4.6]{Li2} defines open subsets $\Delta_k\subset\Delta, 0\leq k\leq m,$ whose union has full measure in $\Delta$, and $u$ is smooth on each $\Delta_k$ by \cite[Lemma 4.7]{Li2}.
Futhermore, \cite[Lemma 4.8]{Li2} shows that there is an open neighborhood $U_k\supset \Delta_k$ where the function $u$ \emph{does not depend on the coordinate} $x_k$, and as a function on $\Delta_k$, the function $u$ satisfies the real Monge-Amp\`ere PDE
\begin{equation}
 \det(D^2 u) \left(1+\sum_{i=0}^m d_i D_i u\right)^{n-m}= c_0.
\end{equation}
Variants of this PDE have also featured in recent works on complete noncompact Calabi-Yau manifolds \cite{CL,CTY}.
We emphasize that because $u$ does not depend on $x_k$ on $U_k$, it can be locally regarded as a function of $n$ variables. The Hessian $D^2 u$ is taken with respect to these $n$ variables, and $D_k u=0$.

Near any point in the generic region, we can choose a chart containing it over which $L$ is trivialized, with local coordinates $(z_1,\dots,z_n)$ on $X_t$, where $z_i$ is identified with $F_i$ using the trivialization for $1\leq i\leq m$, while $(z_{m+1}, \dots, z_n)$ are local coordinates on $Y$.
Each fiber of $\pi_t$ is a $T^m$-bundle over $Y$, and it will be convenient to ``unwrap'' the torus fibers. We thus call
\begin{equation}
\zeta_i=-|\log|t||^{-1} \log z_i ,\quad 1\leq i\leq m.
\end{equation}
The rescaled ansatz metric $|\log|t||^{-1}\omega_t$ is uniformly equivalent to
\begin{equation}
\omega^{\rm ref}_t:=\sum_{i=1}^m \sqrt{-1} d\zeta_i\wedge d\ov{\zeta_i}+|\log|t||^{-1}\sum_{i=m+1}^n \sqrt{-1} dz_i\wedge d\ov{z_i},
\end{equation}
and furthermore, $|\log|t||\omega^{\rm ref}_t$ is $C^\infty$-uniformly equivalent to the standard Euclidean metric in the rescaled coordinates
$|\log|t||^{\frac{1}{2}}\zeta_1,\dots,|\log|t||^{\frac{1}{2}}\zeta_m,$ $z_{m+1},\dots, z_n$. Recall that
$\omega_{{\rm CY},t}=\omega_t+|\log|t||\ddbar\psi_t.$ Our goal is to show
\[
\| \omega_{CY,t}
- \omega_t\|_{C^0(U_t)}\to 0,\quad t\to 0.
\]

Given any point $x_0\in D_k,$ choose $\hat{x}_0\in \pi_T^{-1}(x_0)$ with $\psi(\hat{x}_0)=\underline{\psi}(x_0)$, and fix a coordinate chart near $\hat{x}_0$ over which $L$ is trivialized, with local coordinates $(z_1,\dots,z_n)$ where $z_i$ is identified with $F_i$ using the trivialization for $1\leq i\leq m$, while $(z_{m+1}, \dots, z_n)$ are local coordinates on $Y$.
Each fiber of $\pi_T$ is a $T^m$-bundle over $Y$, and it will be convenient to ``unwrap'' the torus fibers. We thus call
\begin{equation}
\zeta_i=-T^{-1} \log z_i ,\quad 1\leq i\leq m,
\end{equation}
and observe that
\begin{equation}\label{appr}
\mathrm{Re}\zeta_i = x_i+O\left(T^{-1}\right).
\end{equation}
while $\theta_i=\mathrm{Im}\zeta_i$ are $2\pi T^{-1}$-periodic coordinates. We write
\[
(x_1,\dots,x_m,\theta_1,\dots,\theta_m, z_{m+1},\dots,z_n)=:(x,\theta,z).
\]
The ansatz metric $\omega_T$ is uniformly equivalent to
\begin{equation}
\omega^{\rm ref}_T:=\sum_{i=1}^m \sqrt{-1} d\zeta_i\wedge d\ov{\zeta_i}+T^{-1}\sum_{i=m+1}^n \sqrt{-1} dz_i\wedge d\ov{z_i},
\end{equation}
while its volume form is uniformly equivalent to
\[
T^{m-n}\omega_M^{n-m}\wedge id\zeta_1\wedge d\ov{\zeta_1}\wedge\cdots\wedge id\zeta_m\wedge d\ov{\zeta_m}
\]
The fibers of $\pi_T$ exhibit two different length scales as $T\to +
\infty$: the length scale of the Calabi-Yau manifold $Y$ is $T^{-\frac{1}{2}}$ while the length scale of the $T^m$-fibers is much smaller, of order $T^{-1}$.

Finally, we sketch how to modify this to prove the main Theorem \ref{main}. We recall that we are allowed to slightly shrink the ``generic region" $U_t$ in the argument, as long as the normalized Calabi-Yau measure of its complement is smaller than any prescribed $\delta>0$. We can find compact subsets $K_0,\ldots K_m$ properly contained inside the interior of $\Delta_0,\ldots \Delta_m$, such that its complement has Lebesgue measure $\ll \delta$. Then $r_0= \text{dist}(K_k, \partial \Delta_k )>0$ is a small number independent of $t$, and we can define $U_t$ as above. By the main result of \cite{Li2}, the $C^0$-norm $\| \psi \|_{C^0(X_t)} \to 0$ as $t\to 0$, so $\| \psi \|_{C^0(X_t)}$ is far smaller than the constant $\ve_0$, and we can run the De Giorgi type blow up argument, to prove Corollary \ref{DG2}. We can then follow the proof of
Corollary \ref{DG3}, noting that the change of the Calabi-Yau volume form from the model to the geometric case only results in $O(e^{-cT})$-error, which is negligible. The conclusion is that up to slightly shrinking the compact sets $K_0,\ldots K_m$, then for any $y\in K_k$ ($k=0,1,\ldots m$),
\[
T \| \ddbar \psi_T \|_{ C^k(\tb_{T^{-1/2}}(y), T\omega_T^{\rm ref}) }\leq C(k) ( \| \psi\|_{C^0} + T^{-\alpha/2}) \to 0,\quad t\to 0 .
\]
In particular, we have the metric convergence
\[
\lim_{t\to 0} \| \omega_{CY,t}- \omega_t \|_{C^0(\tb_{T^{-1/2}}(y), \omega_t)} =0.
\]
But the region on $X_t$ not covered by these $\tb_{T^{-1/2}}(y)$ has normalized Calabi-Yau measure $<\delta$, so we conclude the metric convergence on the generic region.

\begin{rmk}
In fact, we prove $C^\infty$ estimates for $\omega_{{\rm CY},t}$ with respect to a local Euclidean metric after stretching the base coordinates and the $T^m$-fibers by $|\log|t||^{\frac{1}{2}},$ see Corollary \ref{DG3} for a precise statement.
\end{rmk}

\begin{rmk}
For a specific intermediate complex structure limit family with $m=1$, this result was known thanks to the work of Sun-Zhang \cite{SZ} who in that case constructed the Calabi-Yau metrics $\omega_{{\rm CY},t}$ by a delicate gluing process.
\end{rmk}

The idea to prove this result is to adapt the proof of Savin's small perturbation theorem to our collapsing setup. One fundamental difference between the case $m=n$ of large complex structure limits in \cite{Li} and our case $0<m<n$ of intermediate complex structure limits, is that in the first case the generic region of $X_t$ is modeled on a $T^n$-fibration over a ball in $\mathbb{R}^n$, while in our case the model is a $Z$-fibration over a ball in $\mathbb{R}^m$, where $Z$ is a compact manifold which is itself a  $T^m$-bundle over a compact Calabi-Yau $(n-m)$-fold $Y$.  In the case $m=n$ one can then ``unwrap'' the $T^n$-fibers, and directly apply Savin's small perturbation theorem to bootstrap uniform convergence to smooth convergence. In our case $0<m<n$ this idea fails, as one can still unwrap the $T^m$-fibers but not in general the Calabi-Yau manifold $Y$, which is still collapsing (albeit at a slower rate: the torus fibers have size $|\log|t||^{-1}$ while $Y$ has size $|\log|t||^{-\frac{1}{2}}$). Our idea is then to adapt the proof of Savin's theorem in our setting, and prove the analogous main steps (a Harnack inequality and a De Giorgi type iteration), but only in a  truncated sense: these estimates will hold only up to the scale $|\log|t||^{-1}$.

To prove these, we observe that the fiberwise maximum and minimum of the potential are respectively a subsolution and supersolution of the limiting real Monge-Amp\`ere type equation. We apply ``half'' of Savin's results downstairs to these two functions, and working upstairs after unravelling the $T^m$-fibers, we are able to modify Savin's methods to close the proof of Harnack. In the De Giorgi iteration part, the approximating quadratic polynomials are now pulled back from the base, and arguing by contradiction, an important point is that the fiberwise maximum and minimum coincide in the limit. In the end we obtain a truncated H\"older bound, which is only valid up to the scale of the torus fibers. However, this bound is precisely what is needed to apply Savin's original theorem after rescaling the fibers, and thus conclude the proof.

\begin{rmk}
In \cite{Li, Li3, Li4}, the first-named author used the higher order estimates on the generic region for the Ricci-flat metrics under a large complex structure limit, together with an implicit function theorem argument, to construct special Lagrangian torus fibrations on these generic regions. It seems likely that using our main theorem and some implicit function theorem argument, one could then construct coisotropic fibrations on the generic region of our intermediate complex structure limits $X_t$. We leave this question to the interested reader.
\end{rmk}

\begin{rmk}
The techniques of this paper can also be applied in the setting of \cite{HT}, where one has a family of Ricci-flat K\"ahler metrics on a fixed Calabi-Yau manifold, which admits a holomorphic fibration over a lower-dimensional base, and the metrics are shrinking in the fiber directions (which can be an arbitrary Calabi-Yau manifold). However, the resulting higher order estimates one would obtain in this way will not recover the results in \cite{HT} (which give in particular $C^\infty$ estimates for the collapsing Ricci-flat metrics), but only the weaker estimates proved in \cite[Theorem 1.1]{TZ} (which give $C^\infty$ estimates for the Ricci-flat metrics after stretching out the base directions and rescaling).
\end{rmk}

\noindent
{\bf Acknowledgments.} The first-named author was supported by a Royal Society University Research Fellowship. The second-named author was partially supported by NSF grant DMS-2404599.

The idea to prove Theorem~\ref{main} is to “adapt the proof of Savin's small perturbation theorem to our collapsing setup.” Unlike the case $m=n$ (large complex structure limits) where one can “unwrap” the $T^n$-fibers and “directly apply Savin's small perturbation theorem to bootstrap uniform convergence to smooth convergence,” in the intermediate case $0<m<n$ this fails because one can unwrap the $T^m$-fibers “but not in general the Calabi-Yau manifold $Y$,” which is still collapsing (torus fibers of size $|\log|t||^{-1}$ and $Y$ of size $|\log|t||^{-\frac{1}{2}}$). The approach is to “prove the analogous main steps (a Harnack inequality and a De Giorgi type iteration), but only in a truncated sense,” namely “only up to the scale $|\log|t||^{-1}$.

To do this, they “observe that the fiberwise maximum and minimum of the potential are respectively a subsolution and supersolution of the limiting real Monge-Amp\`ere type equation.” They apply “\"half\" of Savin's results downstairs to these two functions,” and then “working upstairs after unravelling the $T^m$-fibers,” they “modify Savin's methods to close the proof of Harnack.” For the De Giorgi iteration, “the approximating quadratic polynomials are now pulled back from the base,” and in a contradiction argument “an important point is that the fiberwise maximum and minimum coincide in the limit.” This yields “a truncated H\"older bound, which is only valid up to the scale of the torus fibers,” but “this bound is precisely what is needed to apply Savin's original theorem after rescaling the fibers, and thus conclude the proof.”

The idea to prove the main theorem is to “adapt the proof of Savin's small perturbation theorem to our collapsing setup.” Unlike the case $m=n$ (large complex structure limits) where one can “unwrap” the $T^n$-fibers and “directly apply Savin's small perturbation theorem to bootstrap uniform convergence to smooth convergence,” in the intermediate case $0<m<n$ this fails because one can unwrap the $T^m$-fibers “but not in general the Calabi-Yau manifold $Y$,” which is still collapsing (torus fibers of size $|\log|t||^{-1}$ and $Y$ of size $|\log|t||^{-\frac{1}{2}}$). The approach is to “prove the analogous main steps (a Harnack inequality and a De Giorgi type iteration), but only in a truncated sense,” namely “only up to the scale $|\log|t||^{-1}$.

To do this, they “observe that the fiberwise maximum and minimum of the potential are respectively a subsolution and supersolution of the limiting real Monge-Amp\`ere type equation.” They apply “\"half\" of Savin's results downstairs to these two functions,” and then “working upstairs after unravelling the $T^m$-fibers,” they “modify Savin's methods to close the proof of Harnack.” For the De Giorgi iteration, “the approximating quadratic polynomials are now pulled back from the base,” and in a contradiction argument “an important point is that the fiberwise maximum and minimum coincide in the limit.” This yields “a truncated H\"older bound, which is only valid up to the scale of the torus fibers,” but “this bound is precisely what is needed to apply Savin's original theorem after rescaling the fibers, and thus conclude the proof.”

\label{main}
On the generic region of $X_t$,  the
Calabi-Yau metric $\omega_{{\rm CY},t}$ converges in $C^0$-sense to the ansatz metric $\omega_t$ as $t\to 0$, namely
\[
\lim_{t\to 0} \|  \omega_{{\rm CY},t}- \omega_t    \|_{C^0(U_t, \omega_t)} =0.
\],

Question: Let \(\pi:\mathfrak{X}\to\mathbb{D}\) be a degeneration whose smooth fibers \(X_t\) for \(t\in\mathbb{D}^*\) are compact Calabi--Yau manifolds, and let \(\omega_{{\rm CY},t}\in c_1(L|_{X_t})\) be the unique Ricci-flat K\"ahler metric on \(X_t\). Suppose that, for each \(t\neq 0\), a paper-specific ansatz metric \(\omega_t\) is defined on a large open subset \(U_t\subset X_t\), called the generic region, with the normalized Calabi--Yau measure of \(X_t\setminus U_t\) tending to zero as \(t\to 0\). Here \(\|\cdot\|_{C^0(U_t,\omega_t)}\) denotes the sup norm on \(U_t\) measured using \(\omega_t\). As \(t\to 0\), which limiting statement holds on the generic region?

The Calabi--Yau metrics converge to the ansatz metrics in \(C^0\) on the generic region: \[\lim_{t\to 0}\big\|\omega_{{\rm CY},t}-\omega_t\big\|_{C^0(U_t,\omega_t)}=0.\]

Choices:
B. After rescaling by the collapsing factor, the Calabi--Yau metrics converge uniformly to the ansatz metrics on the generic region: \[\lim_{t\to 0}\big\||\log|t||^{-1}\omega_{{\rm CY},t}-\omega_t\big\|_{C^0(U_t,\omega_t)}=0.\]
C. On the generic region, the difference between the Calabi--Yau metrics and the ansatz metrics becomes uniformly bounded in the \(\omega_t\)-sup norm as \(t\to 0\): \[\sup_{0<|t|\ll 1}\big\|\omega_{{\rm CY},t}-\omega_t\big\|_{C^0(U_t,\omega_t)}<\infty.\]
D. The Calabi--Yau metrics converge to the ansatz metrics in \(C^0\) on every compact subset of \(X_t\), namely for each compact \(K\subset X_t\) one has \[\lim_{t\to 0}\big\|\omega_{{\rm CY},t}-\omega_t\big\|_{C^0(K,\omega_t)}=0.\]
E. The Calabi--Yau metrics converge to the ansatz metrics in \(C^\infty\) on the generic region: for every \(k\ge 0\), \[\lim_{t\to 0}\big\|\omega_{{\rm CY},t}-\omega_t\big\|_{C^k(U_t,\omega_t)}=0.\]

The stem does not literally state the correct conclusion, but it strongly cues it by introducing the generic region and the specific \(C^0(U_t,\omega_t)\) norm that appears verbatim in the correct choice.

This is largely a theorem-recall item: it asks which limiting statement holds under the given setup, with one option essentially matching the target result. The alternatives prevent it from being a pure tautology, but it is still close to a restatement.

Some reasoning is needed to reject tempting variants such as unnecessary rescaling, weaker boundedness, stronger \(C^\infty\) convergence, or an overextended domain. However, the item mainly tests recognition of the precise theorem statement rather than deeper derivation.

The distractors are mathematically plausible and target common errors: incorrect normalization, replacing convergence by boundedness, extending from the generic region to all compact subsets, and overclaiming higher-regularity convergence.

A reasonably well-constructed theorem-discrimination MCQ with strong distractors, but it is still close to theorem restatement and offers only moderate generative reasoning.


\begin{equation}\label{k cond S}
    |k_{i}-\mu_{K}|\leq C_{g} \textrm{ and } k_{1}+k_{2}+k_{3}\equiv M\md{g-1},
\end{equation}

\begin{theorem}\label{Main theorem 1}
   For any $g\geq 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3.
\end{theorem}

A central question in additive number theory is to establish whether a given set of integers $\mathcal{S}$ is an \emph{asymptotic basis} for the integers, that is, to determine whether there exists a natural number $k$ such that any sufficiently large integer can be written as the sum of $k$ elements of $\mathcal{S}$. Here, $k$ denotes the \textit{order} of the basis.

Famously, Lagrange's theorem gives that the squares are a basis of order 4, and Waring's problem, solved by Hilbert, shows that $k$\textsuperscript{th } powers are also an additive basis. Some interesting variants of Waring's problem consider $k$\textsuperscript{th } powers of integers which have restrictions on their digits in some base. For example, Pfeiffer and Thuswaldner \cite{pfeiffer2007waring} show that the $k$\textsuperscript{th } powers of integers with certain congruence conditions on their sums of digits in different bases is an asymptotic basis. More recently, Green \cite{green2025waring} established that, given any two digits which are coprime, the integers whose base-$g$ expansions consists of only these digits satisfy Waring's problem. Further references for additive bases coming from sets of integers with digit restrictions are given in the introduction of \cite{sanna2021additive}.

A \emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \textit{Harshad numbers}. It is shown in \cite{de2003counting}, and independently in \cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\eta_{g}x/\log x$ for some constant $\eta_g>0$.

It is conjectured that the set of base-$g$ Niven numbers is an asymptotic basis of order 2. In \cite{sanna2021additive} Sanna established, conditionally upon a certain generalisation of the Riemann Hypothesis, that the set of base-$g$ Niven numbers is an asymptotic basis with order growing linearly in $g$. Our result is the following unconditional statement.

A \emph{base-$g$ Niven number} is a natural number that is divisible by its base-$g$ sum of digits. Such integers are also referred to as \textit{Harshad numbers}. It is shown in \cite{de2003counting}, and independently in \cite{mauduit2005distribution}, that the number of base-$g$ Niven numbers less than $x$ is asymptotically $\eta_{g}x/\log x$ for some constant $\eta_g>0$.

Therefore $|\mathcal{E}_{0}|\geq |\mathcal{E}_{y}|\geq R$. Recall the definition of the tuple $\mathbf{a}$ from \cref{a tuple defn}; as $x=0$ we have that \[\mathbf{a}=(|\mathcal{E}_{0}|,0,\ldots,0,|\mathcal{E}_{y}|,0,\ldots,0 ).\]
Our aim is to bound $\Psi(\mathbf{a};0)$ using \cref{psi 0 bound} with $m=R$ and $t=y$. To satisfy the assumptions of \cref{psi 0 bound} we need $R\ll_{C} |\mathcal{E}_{0}|^{1/4}$, which follows from the fact that $|\mathcal{E}_{0}|\geq (K-|\mathcal{D}|)/2\geq (K-LR)/2$. As $LR\leq CK^{1/4}$ by assumption, we have that $|\mathcal{E}_{0}|\gg_{C} K$, and certainly $R\leq CK^{1/4}$, giving $R\ll_{C} |\mathcal{E}_{0}|^{1/4} $. Applying \cref{psi 0 bound} gives the bound
\begin{equation*}\label{psi cancellation}
 \Psi(\mathbf{a};0)\ll_{C,g} g^{-R}+R^{2}K^{-3/2}.
\end{equation*}
This gives \cref{main term cancellation psi}, using that $R\geq \tfrac{3}{2}\log_{g}K$.
\end{proof}
\subsection{Decoupling the averages over $\xbd$ and $\xbee$}\label{section decoupling}
In this section we prove the decoupling result, \cref{decoupling lemma}. This lemma allows us to replace the condition on the digits of $X$, $s(\xbd)+s(\xbee)=\xi$, with the condition $s(\xbee)=0$, even as the digits in $\xbd$ vary.
\begin{proof}[Proof of \cref{decoupling lemma}] Let $(g-1)\theta$ have centred base-$g$ expansion given by \cref{theta digit exp}. Dividing through by $(g-1)$ in this expansion gives
\[\theta=\frac{1}{g-1}\sum_{j=1}^{w}\varepsilon_{n_{j}}g^{-n_{j}}+\frac{\eta}{g-1},\]
where $|\eta|<g^{-K}$ and $w\coloneq w_{K}((g-1)\theta)$. Thus we can separate the contribution to $e(\xb\theta)$ from each non-zero digit as follows,
\begin{equation}\label{decoupling proof eq 1}
 e(\xb\theta)=e\Big(\xb\frac{\eta}{g-1}\Big)\prod_{j=1}^{w} e\Big(\xb \frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big).
\end{equation}
We have that
\[\Big|\xb_{[0,n_{j}-R-1]}\frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big|\leq g^{-R} \textrm{ and } \xb_{[n_{j},K-1]}\frac{\varepsilon_{n_{j}}}{g^{n_{j}}}\equiv X_{[n_{j},K-1]}\varepsilon_{n_{j}}\md{g-1}, \]
where the latter statement uses \cref{trans inv}. Thus the contribution to $e(X\theta)$ from the $n_{j}$\sth digit of $(g-1)\theta$ is
 \begin{align}
 e\Big(\xb\frac{\varepsilon_{j}}{(g-1)g^{n_{j}}}\Big)=e\Big(\xb_{[n_{j}-R,n_{j}-1]}\frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big )\ru{\varepsilon_{n_{j}}\xb_{[n_{j},K-1]}}+O(g^{-R})\label{ni contribution}.
 \end{align}
The $\eta$ term gives a similar contribution: let $u\coloneq n_{w_{K}((g-1)\theta)+1}-R$, then
 \begin{equation}\label{eta contribution}
 e(\xb\eta/(g-1))=e(\xb_{[u,K-1]}\eta/(g-1))+O(g^{-R}).
 \end{equation}
 If $u\geq K$, then the interval $[u,K-1]\cap \{0,\ldots, K-1\}$ is empty and $e(X\eta/(g-1))=1+O(g^{-R})$. From \cref{decoupling proof eq 1}, \cref{ni contribution} and \cref{eta contribution}, we have
\begin{align}
 \E_{\xb}e(\xb\theta)\ind_{\sx{\xb}=\xi}&=\E_{\xb}e\Big(\xb\frac{\eta}{g-1}\Big)\prod_{i=1}^{w}e\Big (\xb\frac{\varepsilon_{n_{i}}}{(g-1)g^{n_{i}}}\Big)\ind_{\sx{\xb}=\xi}\nonumber\\
 &=\E_{\xb}e\Big(\xb_{[u,K-1]}\frac{\eta}{g-1}\Big)\prod_{j=1}^{w}e\Big (\xb_{[n_{j}-R,n_{j}-1]}\frac{\varepsilon_{n_{j}}}{(g-1)g^{n_{j}}}\Big)\nonumber\\&\hspace{12em}\times \ru{\xb_{[n_{j},K-1]}\varepsilon_{n_{j}}}\ind_{\sx{\xb}=\xi}+O_{C,g}(L g^{-R}).\label{decoupling eg 2}\end{align}
 Here, we have used that $L\leq CK^{1/4}$ and $R\geq \tfrac{3}{2}\log_{g}K$ to obtain this error term.
Let $\mathcal{E}\coloneq \{0,\ldots, K-1\}\setminus \mathcal{D}$, and note that $\mathcal{E} $ is partitioned into $\mathcal{E}= \mathcal{E}_{0}\cup \ldots \cup \mathcal{E}_{g-2}$ for $\mathcal{E}_{r}$ defined by \cref{E set defn}. With this notation, we can rewrite \cref{decoupling eg 2} as
\begin{equation}\label{decoupling eq 2.5}\E_{X}e(X\theta)\ind_{s(X)=\xi}=\E_{\xbd}e(\xbd\theta)\E_{\xbee}\prod_{j=1}^{w}\ru{\xb_{[n_{j},K-1]\cap \mathcal{E}}\varepsilon_{j}}\ind_{\sx{\xb}=\xi}+O_{C,g}(L g^{-R}).\end{equation}
As the terms $n_{j}$ index precisely the non-zero digits of $(g-1)\theta$, each random variable $X_{u}$ for $u\in \mathcal{E}$ occurs in the above product once for each non-zero digit with index $n_{j}\leq u$, weighted by the value of that non-zero digit, $\varepsilon_{n_{j}}$. Rearranging the above product gives
 \[\prod_{j=1}^{w}\ru{\xb_{[n_{j},K-1]\cap \mathcal{E}}\varepsilon_{n_{j}}}=\prod_{u\in \mathcal{E}}\ru{X_{u}g^{u}\sum_{r\leq u}\varepsilon_{r}}=\prod_{v=1}^{g-2}\ru{v\xb_{\mathcal{E}_{v}}},\]
where the final equality follows from the definition of the sets $\mathcal{E}_{v}$ given by \cref{E set defn}. Hence from \cref{decoupling eq 2.5} and the above equation, $\E_{X}e(X\theta)\ind_{s(X)=\xi}$ equals
 \begin{equation}\label{decoupling eq 3}
 \E_{\xbd}e(\xbd\theta) \E_{\xbee}\ru{\xb_{\mathcal{E}_{1}}+2\xb_{\mathcal{E}_{2}}+\ldots+(g-2)\xb_{\mathcal{E}_{g-2}}}\ind_{\sx{\xbee}=\xi-\sx{\xbd}}+O_{C,g}(L g^{-R}).
 \end{equation}
Let $\mathbf{a}_{0}\coloneq(|\mathcal{E}_{0}|,\ldots,|\mathcal{E}_{g-2}|)$. From \cref{Psi expectation form} the average over $\xbee$ equals
\begin{align*}\label{decoupling psi for a0}
 \E_{\xbee}\ru{\xb_{\mathcal{E}_{1}}+2\xb_{\mathcal{E}_{2}}+\ldots+(g-2)\xb_{\mathcal{E}_{g-2}}}\ind_{\sx{\xbee}=\xi-\sx{\xbd}}=\Psi(\mathbf{a}_{0};\xi-\sx{\xbd}),
\end{align*}
and thus \cref{decoupling eq 3} equals
\[\E_{\xbd}e(\xbd\theta) \Psi(\mathbf{a}_{0};\xi-\sx{\xbd})+O_{C,g}(L g^{-R}).\]
Our aim is to remove the dependence on $\xbd$ from the term $ \Psi(\mathbf{a}_{0};\xi-\sx{\xbd})$ by applying \cref{psi lemma}. However, in order to apply \cref{psi lemma} with the tuple $\mathbf{a}_{0}$, we require that the first coordinate of $\mathbf{a}_{0}$, $|\mathcal{E}_{0}|$, satisfies $|\mathcal{E}_{0}|\geq |\mathcal{E}_{j}|$ for all $1\leq j \leq g-2$, which may not be the case. To circumvent this issue, we exploit the fact that multiplying \cref{decoupling eq 3} through by factors of $\ru{X}$ increases the coefficients of each $X_{\mathcal{E}_{r}}$, essentially allowing us to cycle the coordinates of $\mathbf{a}_{0}$. To do this, we use the following relation. As $\xb\equiv \sx{\xb} \md{g-1}$, we have $\E_{X}\ru{\xb}\ind_{\sx{\xb}=\xi}=\ru{\xi}$, and thus for any integer $m$,
 \begin{equation}\label{decoupling eq 5}
 \E_{\xb}e(\xb\theta)\ind_{\sx{\xb}}=\ru{m\xi}\E_{\xb}e(\xb\theta)\ru{-m\xb }\ind_{\sx{\xb}=\xi}.
 \end{equation}
Let $x$ be defined as in the statement of \cref{decoupling lemma}, that is, $x\in\{0,\ldots, g-2\}$ is such that $\mathcal{E}_{x}$ is maximal among the $|\mathcal{E}_{r}|$, and let $\mathbf{a}$ be as defined by \cref{a tuple defn},
\[ \mathbf{a}\coloneq(|\mathcal{E}_{x}|,|\mathcal{E}_{x+1}|,\ldots, |\mathcal{E}_{g-2}|,|\mathcal{E}_{0}|,|\mathcal{E}_{1}|,\ldots, |\mathcal{E}_{x-1}|).\]
From \cref{decoupling eq 5}, multiplying \cref{decoupling eq 3} through by $\ru{-x\xbee}$ gives
\begin{align}
 \E_{\xb}e(X\theta)\ind_{s(X)=\xi}&=\ru{x\xi}\E_{X}e(X\theta)\ru{-xX}\ind_{s(X)=\xi}\nonumber\\
 &=\ru{x\xi}\E_{\xbd}e\Big (\xbd\Big (\theta-\frac{x}{g-1}\Big )\Big )\E_{\xbee}\prod_{j=0}^{g-2}\ru{(j-x)\xb_{\mathcal{E}_{j}}}\ind_{\sx{\xbee}=\xi-\sx{\xbd}}\nonumber\\
 &=\ru{x\xi}\E_{\xbd}e\Big (\xbd\Big (\theta-\frac{x}{g-1}\Big )\Big )\Psi(\mathbf{a};\xi-\sx{\xbd}),\label{decoupling eq 4}
\end{align}
using \cref{Psi expectation form} and the definition of $\mathbf{a}$ for the final equality. As $|\mathcal{E}_{x}|$ is maximal among the sets $\mathcal{E}_{r}$, the tuple $\mathbf{a}$ fulfils the requirement that its first coordinate is the largest.

We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum. By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers. To state these results more precisely, we need the following notation.

Let $s_{g}(n)$ denote the base-$g$ digit sum of $n$. Let $K\geqslant 1$ be a sufficiently large integer, and let $M\in (g^{K-1},g^{K}]$. For $k\in \N$, let
\[S_{g}(k)\coloneq \{n<g^{K}\ensuremath{:} s_{g}(n)=k\} \textrm{ and }\mathcal{N}_{g}(k)\coloneq \{n\in S(k) \ensuremath{:} k\mid n\}.\]
Thus $\mathcal{N}_{g}(k)$ is the set of Niven numbers in $S_{g}(k)$, that is, Niven numbers of a certain size with fixed digit sum. Let \begin{equation*}
    \mu_{K}\coloneq \frac{(g-1)K}{2},
\end{equation*}
which is the average digit sum of $n<g^{K}$. For a fixed choice of $k_{1},k_{2},k_{3}\in \N$, let $S_{i}\coloneq S_{g}(k_{i})$ for $i=1,2,3$, and let $\mathcal{N}_{i}\coloneq \mathcal{N}_{g}(k_{i})$ for $i=1,2,3$. Suppose that $k_{1},k_{2},k_{3}\in \N$ are such that
\begin{equation}\label{k cond S}
    |k_{i}-\mu_{K}|\leqslant C_{g} \textrm{ and } k_{1}+k_{2}+k_{3}\equiv M\md{g-1},
\end{equation}
where $C_g := g(g-1)\prod_{p\leqslant 10g^{2}}p$, and the product is over primes.  The specific choice of constant here is unimportant, and any sufficiently large value would do, but this value is large enough to ensure that $k_{i}$ with further desired properties exist. Let $r_{S_{1}+S_{2}+S_{3}}(M)$ be the number of representations of $M=s_{1}+s_{2}+s_{3}$, for $s_{i}\in S_{i}$. Our main result is the following theorem, from which we can later deduce the corresponding result for $\mathcal{N}_{i}\subset S_{i}$.
\begin{theorem}\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\geqslant 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \cref{k cond S}. Then
    \[r_{S_{1}+S_{2}+S_{3}}(M)=\frac{(g-1)M^{2}}{2(2\pi\sigma^{2}K)^{3/2}}(1+O_{g}((\log K)^{4}K^{-1/4})),\]
where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}
Let $r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)$ be the number of representations of $M=n_{1}+n_{2}+n_{3}$, where $n_{i}\in \mathcal{N}_{i}$ for $i=1,2,3$. In order to relate the quantity $r_{S_{1}+S_{2}+S_{3}}(M)$ to $r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)$, we require some further conditions on the choice of $k_{1},k_{2},k_{3}$. Throughout, $(a,b)$ denotes the greatest common divisor of $a$ and $b$. Suppose that $k_{1},k_{2},k_{3}\in\N$ also satisfy, in addition to \cref{k cond S},
\begin{align}\label{k cond N}
  (k_{i},k_{j})=1 \textrm{ and }(k_{i},g)=1 \textrm{ for }i,j=1,2,3 \textrm{ and } i\neq j.
 \end{align}
Let
\[c_{g}(k_{1},k_{2},k_{3})\coloneq \frac{4}{(g-1)^{2}}\prod_{i=1}^{3}(g-1,k_{i}).\]
Then we have the following theorem, from which \cref{Main theorem 1} is a corollary after showing that such a choice of $k_{1},k_{2}$ and $k_{3}$ exist.
\begin{theorem}\label{Main theorem N count}
Let $g,K$ and $M$ be integers such that $g\geqslant 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \cref{k cond S} and \cref{k cond N}, then
\[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\]
In particular,
 \[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\frac{M^{2}}{(2\pi\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\log K)^{4}K^{-19/4}),\]
 where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}
Our methods can be adapted to show the analogous versions of \cref{Main theorem 1}, \cref{Main theorem S count} and \cref{Main theorem N count} for base 2, however there is one main technical difference. For readability, we do not give the details here, but comment on the necessary changes in \cref{Section: minor arc}.
\subsection{Notation}\label{notation}
Throughout, we consider $g$ to be fixed. Let $\Vert x \Vert_{\R/\Z}$ denote the distance of a real number $x$ to the nearest integer. We write $e(x)$ for $\exp(2\pi ix)$, and $e_{g-1}(x)$ for $e(x/(g-1))$. We require the quantity $\ell$, defined to be
\begin{equation}\label{l choice}
    \ell\coloneqq \lceil 384g^{3}\log K\rceil.
\end{equation}
Throughout, $\sigma=\sqrt{(g^{2}-1)/12}.$
\stoptoc

To prove Theorem~\ref{Main theorem 1}, the introduction explains the following strategy.

1. "We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum." Concretely, for $M\in (g^{K-1},g^{K}]$ and digit sums $k_i$ satisfying $|k_i-\mu_K|\le C_g$ and $k_1+k_2+k_3\equiv M\pmod{g-1}$, one studies
\[r_{S_1+S_2+S_3}(M)=\#\{(s_1,s_2,s_3)\in S_g(k_1)\times S_g(k_2)\times S_g(k_3): s_1+s_2+s_3=M\},\]
with the goal of obtaining an asymptotic formula (Theorem \ref{Main theorem S count}).

2. "By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers." This is implemented by relating the above count to
\[r_{\mathcal N_1+\mathcal N_2+\mathcal N_3}(M)=\#\{(n_1,n_2,n_3)\in \mathcal N_g(k_1)\times\mathcal N_g(k_2)\times\mathcal N_g(k_3): n_1+n_2+n_3=M\}.\]
Under additional coprimality conditions on the digit sums, namely $(k_i,k_j)=1$ for $i\ne j$ and $(k_i,g)=1$, one obtains a comparison
\[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_g(M^{2}K^{-29/6})\]
(Theorem \ref{Main theorem N count}), hence a positive proportion of the $S_1+S_2+S_3$ representations contribute to Niven-summand representations.

3. Finally, the introduction states that Theorem~\ref{Main theorem 1} follows because Theorem \ref{Main theorem N count} implies it "after showing that such a choice of $k_1,k_2$ and $k_3$ exist" (i.e., satisfying the near-average digit-sum congruence condition and the extra coprimality conditions).

To prove the main theorem, the introduction explains the following strategy.

1. "We first count, via the circle method, the number of representations of a sufficiently large integer $M$ as the sum of three integers with a near-average digit sum." Concretely, for $M\in (g^{K-1},g^{K}]$ and digit sums $k_i$ satisfying $|k_i-\mu_K|\le C_g$ and $k_1+k_2+k_3\equiv M\pmod{g-1}$, one studies
\[r_{S_1+S_2+S_3}(M)=\#\{(s_1,s_2,s_3)\in S_g(k_1)\times S_g(k_2)\times S_g(k_3): s_1+s_2+s_3=M\},\]
with the goal of obtaining an asymptotic formula. We first prove the following theorem.

\begin{theorem}\label{Main theorem S count} Let $g,K$ and $M$ be integers such that $g\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ satisfy \cref{k cond S}. Then
    \[r_{S_{1}+S_{2}+S_{3}}(M)=\frac{(g-1)M^{2}}{2(2\pi\sigma^{2}K)^{3/2}}(1+O_{g}((\log K)^{4}K^{-1/4})),\]
where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}

2. "By showing that a proportion of such representations have each summand being a Niven number, we provide a lower bound for the number of representations of $M$ as the sum of three Niven numbers." This is implemented by relating the above count to
\[r_{\mathcal N_1+\mathcal N_2+\mathcal N_3}(M)=\#\{(n_1,n_2,n_3)\in \mathcal N_g(k_1)\times\mathcal N_g(k_2)\times\mathcal N_g(k_3): n_1+n_2+n_3=M\}.\]
Under additional coprimality conditions on the digit sums, namely $(k_i,k_j)=1$ for $i\ne j$ and $(k_i,g)=1$, one obtains the following comparison theorem.

\begin{theorem}\label{Main theorem N count}
Let $g,K$ and $M$ be integers such that $g\geq 3$, $K$ is sufficiently large in terms of $g$, and $M\in (g^{K-1},g^{K}]$. Suppose that $k_{1},k_{2},k_{3}$ fulfil \cref{k cond S} and \cref{k cond N}, then
\[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=\frac{(g-1)^{2}c_{g}(k_{1},k_{2},k_{3})}{4k_{1}k_{2}k_{3}}r_{S_{1}+S_{2}+S_{3}}(M)+O_{g}(M^{2}K^{-29/6}).\]
In particular,
 \[r_{\mathcal{N}_{1}+\mathcal{N}_{2}+\mathcal{N}_{3}}(M)=c_{g}(k_{1},k_{2},k_{3})\frac{M^{2}}{(2\pi\sigma^{2})^{3/2}K^{9/2}}+O_{g}(M^{2}(\log K)^{4}K^{-19/4}),\]
 where $\sigma^{2}=(g^{2}-1)/12$.
\end{theorem}

Hence a positive proportion of the $S_1+S_2+S_3$ representations contribute to Niven-summand representations.

3. Finally, the introduction states that the main theorem follows because the preceding theorem implies it after showing that such a choice of $k_1,k_2$ and $k_3$ exist (i.e., satisfying the near-average digit-sum congruence condition and the extra coprimality conditions).

\label{Main theorem 1}
For any $g\geqslant 3$, the set of base-$g$ Niven numbers is an asymptotic basis of order 3. This completes the proof of the main theorem.

Question: Fix an integer base $g\ge 3$, and let $\mathcal S_g$ be the set of base-$g$ Niven numbers, i.e. natural numbers $n$ such that $n$ is divisible by the sum of its digits in base $g$. Recall that a set $\mathcal S\subseteq \mathbb N$ is an asymptotic basis of order $3$ if there exists $N$ such that every integer $m\ge N$ can be written as $m=s_1+s_2+s_3$ with $s_1,s_2,s_3\in \mathcal S$. Under these assumptions, which asymptotic additive statement holds for $\mathcal S_g$?

For every integer $g\ge 3$, the set $\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis of order $3$; equivalently, for each such $g$ there exists $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of three base-$g$ Niven numbers.

Choices:
B. For every integer $g\ge 3$, the set $\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis of order $2$; equivalently, for each such $g$ there exists $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of two base-$g$ Niven numbers.
C. For every integer $g\ge 3$, the set $\mathcal S_g$ of base-$g$ Niven numbers is an asymptotic basis; equivalently, for each such $g$ there exists an integer $h_g\ge 1$ and $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of $h_g$ base-$g$ Niven numbers.
D. There exists an absolute integer $N$ such that for every base $g\ge 3$ and every integer $m\ge N$, one can write $m=s_1+s_2+s_3$ with $s_1,s_2,s_3\in \mathcal S_g$; that is, the base-$g$ Niven numbers form an asymptotic basis of order $3$ uniformly in $g$.
E. For every integer $g\ge 3$, there exists $N_g$ such that every integer $m\ge N_g$ can be expressed as the sum of three base-$g$ Niven numbers, and moreover each summand may be chosen to have the same base-$g$ digit sum.

three-summand representation count upgraded to two summands

fixed order $3$ dropped to existence of some order depending on $g$

threshold dependence on $g$ replaced by uniform absolute threshold

near-average digit-sum auxiliary choice misread as equal digit sums for the three Niven summands

The stem does not state the theorem's conclusion; it only defines base-g Niven numbers and what 'asymptotic basis of order 3' means. The correct choice is not explicitly leaked, though the terminology in the stem naturally frames the target result.

The item is close to a theorem-recall question: the correct answer essentially restates the claimed result in the same language as the stem. However, the presence of stronger, weaker, and quantifier-altered alternatives prevents it from being a pure tautology.

Some reasoning is needed to distinguish the exact statement from nearby variants such as order 2, existence of some order, or uniformity in g. Still, the item mainly tests recognition of the theorem's precise formulation rather than substantial mathematical generation or derivation.

The distractors are strong and mathematically meaningful: one is a stronger false claim (order 2), one is a weaker true statement, one alters quantifier dependence, and one adds an implausible extra constraint. These reflect common failure modes in reading theorem statements.

A solid theorem-statement discrimination MCQ with good distractors and little answer leakage, but it leans more toward precise recall of a result than genuine generative reasoning.


\begin{remark}\label{rem: compute v(F(gamma))}
    Define the \textbf{depth} of $\mathfrak{s}$ by $d_\mathfrak{s} \coloneqq \min_{\alpha, \alpha'} \val(\alpha - \alpha')$.  When $|{\mathfrak{s}}| > 1$ and $\gamma_{\mathfrak{s}} \in X_{\mathfrak{s}}$, we have that $\val(\gamma_{\mathfrak{s}} - \alpha_j) = d_{\mathfrak{s}}$, and hence $\val(F(\gamma_{\mathfrak{s}})) = |\mathfrak{s}| d_{\mathfrak{s}} +c_{\mathfrak{s}}$.
\end{remark}

\begin{thm}\label{thm: main thm}
    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Then, $\mathcal{D}(C/K)$ is not cofinite if and only if $\val(F(0)) \not \equiv 0 \pmod q$, and for every Galois-invariant cluster $\mathfrak{s}$ of roots of $F(x)$,
    \begin{enumerate}[label=$(\roman*)$]
        \item $|\mathfrak{s}| \equiv 0 \pmod q$ and $c_{\mathfrak{s}} \not \equiv 0 \pmod{q}$, and
        \item if $\gamma_{\mathfrak{s}} \in X_{\mathfrak{s}}$, then $\val(F(\gamma_{\mathfrak{s}})) \not \equiv 0 \pmod{q}$.
    \end{enumerate}
    When these conditions are satisfied, we have that
    \begin{align*}
        \mathcal{D}(C/K) \subseteq q\mathbb{N} \cup \bigcup_{\substack{ \mathfrak{s} \, \text{\rm not Galois} \\ \text{\rm invariant}}} |\mathcal{O}(\mathfrak{s})| \mathbb{N}.
    \end{align*}
\end{thm}

\begin{remark}\label{rem: dokchitser}
The code of \cite{Regular-Models} applied to \Cref{ex: computed degree set} gives the following diagram.

$$
\begin{tikzpicture}[xscale=0.8,yscale=0.7,
  l1/.style={shorten >=-1.3em,shorten <=-0.5em,thick},
  l2/.style={shorten >=-0.3em,shorten <=-0.3em},
  lfnt/.style={font=\tiny},
  rightl/.style={right=-3pt,lfnt},
  mainl/.style={scale=0.8,above left=-0.17em and -1.5em},
  facel/.style={scale=0.5,blue,below right=-0.5pt and 6pt},
  redbull/.style={red,label={[red,scale=0.6,above=-0.17]#1}}]
\draw[l1] (0.00,0.00)--(2.96,0.00) node[mainl] {6} node[facel] {$F_1$};
\node[redbull=a] at (0.00,0.00) {$\bullet$};
\node[redbull=b] at (0.50,0.00) {$\bullet$};
\node[redbull=c] at (1.00,0.00) {$\bullet$};
\draw[l2] (1.50,0.00)--node[rightl] {3} (1.50,0.66);
\draw[l2] (2.30,0.00)--node[rightl] {3} (2.30,0.66);
\end{tikzpicture}
$$

Note that this method is unable to fully compute the special fiber, and thus is insufficient for computing the degree set in this case.  Using the main result of \cite{Degrees-on-Varieties} and the fact that \Cref{ex: computed degree set} will have the same degree set over $\mathbb{Q}_7^\text{unr}$, we see that every component of the special fiber must have multiplicity divisible by 2 or 3.

\end{remark}

Given a variety $V/K$ over a field $K$, one would like to determine $V(K)$, the set of $K$-rational points. If it happens that $V(K)$ is empty, one would then like to determine for which extensions $L/K$ is $V(L)$ nonempty. Explicitly, define the degree set of $V/K$ by
\begin{align*}
 \mathcal{D}(V/K) \coloneqq \{\deg_K (P) \mid P \in V, \, P \text{ closed} \}
\end{align*}
where $\deg_K P$ denotes the degree of the residue field \textbf{k}$(P)$ over $K$, and the index of $V/K$ by $\ind(V/K) \coloneqq \gcd \mathcal{D}(V)$. We would like to determine the degree set of $V/K$.

When $K$ is a discretely valued Henselian field, determining the index and the degree set is possible. Gabber, Liu, and Lorenzini have shown that that the index over $K$ depends only on data related to the special fiber \cite{index-special-fiber}*{Theorem 8.2}, and Creutz and Viray showed that the degree set can be computed explicitly from the special fiber of a strict normal crossings model \cite{Degrees-on-Varieties}*{Theorem 1.1}. Thus, provided knowledge of the special fiber, one can compute the degree set. For a large class of curves with affine model $f(x,y) = 0$, \cite{Regular-Models}*{Theorem 1.1} gives an effective method, and for hyperelliptic curves with potential semistable reduction, \cite{models-of-hyperelliptic-curves}*{Theorem 1.2} describes the special fiber in terms of cluster diagrams. The special fiber, however, is difficult to compute in general.

Creutz and Viray give examples of curves $C/K$ with index $1$ having degree set that excludes infinitely many integers. These degree sets do not arise over finite fields or global fields. Indeed, over finitely generated fields, the degree set contains all sufficiently large multiples of the index (e.g. \cite{new-points}*{Theorem 7.5}). We say that $\mathcal{D}(C/K)$ is \textbf{not cofinite} if the complement of $\mathcal{D}(C/K)$ in $\ind(C/K)\mathbb{N}$ is not finite. We investigate when this behavior occurs for curves $C/K$ which are ``superelliptic" of prime degree (curves with a cyclic cover $C \to \mathbb{P}^1$ of prime degree $q$). Over fields of characteristic not $q$, superelliptic curves are birational to an affine plane curve of the form $y^q = F(x)$. When $q$ is 2, the curve is hyperelliptic.

Following \cite{cluster-diagrams}, we give an answer in terms of clusters. A \textbf{cluster} is a subset of the roots of $F(x)$ contained in an open disk. As the absolute Galois group $\text{Gal}(K^{\text{sep}}/K)$ acts on the roots of $F(x)$, there is a natural action of $\text{Gal}(K^{\text{sep}}/K)$ on clusters (see \Cref{defn: cluster}). Let $\mathcal{O}(\mathfrak{s})$ denote the orbit of a cluster $\mathfrak{s}$ under $\text{Gal}(K^{\text{sep}}/K)$, let $c_\mathfrak{s}$ be the integer defined in \Cref{lem: valuation of polynomial}, and let $\gamma_\mathfrak{s} \in K$ be defined as in \Cref{lem: shifting argument}. Note that $c_\mathfrak{s}$ and $\gamma_\mathfrak{s}$ are computable by hand from the roots of $F(x)$, see \Cref{ex: computed degree set}.

\begin{definition}\label{defn: cluster}
    A \textbf{cluster} is a nonempty subset $\mathfrak{s} \subseteq \{\alpha_i\}$ of the form $\mathfrak{s} = D \cap \{\alpha_i\}$ for some disk $D_{z,\, d} = \{x \in \overline{K}\mid |x - z|\leq d\}$ for some $z \in \overline{K}$ and $d \in \mathbb{R}$.  Let $\mathcal{O}(\mathfrak{s})$ denote the orbit of $\mathfrak{s}$ as a set under $\text{Gal}(K^{\text{\rm sep}}/K)$, and let $\mathfrak{s}^c$ denote the complement of $\mathfrak{s}$ in $\{\alpha_i\}$.
\end{definition}

\begin{example}\label{ex: computed degree set}
    Let $\alpha = 7^{\frac{1}{2}} + 7^{\frac{3}{5}}$, $\beta = 2\cdot7^{\frac{1}{2}} + 7^{\frac{3}{5}}$, and $\gamma = 3 \cdot 7^{\frac{1}{2}} + 7^{\frac{3}{5}}$ as elements of $\overline{\mathbb{Q}_7}$, and let $f_1(x)$, $f_2(x)$, and $f_3(x)$ be the minimal polynomials of $\alpha$, $\beta$, and $\gamma$ respectively over $\mathbb{Q}_7$. Let $F(x) = 7 f_1(x)f_2(x)f_3(x)$, and let $C/\mathbb{Q}_7$ be the curve given by $y^3 = F(x)$. Then, we have
    \begin{align*}
        \mathcal{D}(C/\mathbb{Q}_7) = 3\mathbb{N} \cup 2\left( \{8,11,13,14\} \cup \mathbb{N}_{>15}\right) \cup 10\mathbb{N} \subseteq 3\mathbb{N} \cup 2 \mathbb{N}.
    \end{align*}
\end{example}

\begin{lem}\label{lem: shifting argument}
    Suppose that the residue field of $K$ is algebraically closed and that the roots of $F(x)$ are tamely ramified. Let $\mathfrak{s}$ be a Galois-invariant cluster. There exists $\gamma_{\mathfrak{s}} \in K$ and $\alpha_{\min} \in {\mathfrak{s}}$ such that ${\mathfrak{s}} = \{ \alpha \mid \val(\alpha - \gamma_{\mathfrak{s}}) \geq \val(\alpha_{\min} - \gamma_{\mathfrak{s}}) \}$.
\end{lem}

\begin{lem}\label{lem: valuation of polynomial}
    Fix a cluster $\mathfrak{s}$. There exists $c_{\mathfrak{s}} \in \mathbb{Q}$ such that for all $x_0 \in X_{\mathfrak{s}}$ and any $\alpha \in {\mathfrak{s}}$,
    $$
    \val(F(x_0)) = |\mathfrak{s}|\cdot\val(x_0 - \alpha) + c_{\mathfrak{s}},
    $$
    where $c_{\mathfrak{s}}$ is given by
    \begin{align*}
        c_{\mathfrak{s}} &= \val(a_d) + \sum_{\beta \in {\mathfrak{s}^c}} \val(\alpha - \beta)
    \end{align*}
    and all expressions above are independent of the choice of $\alpha \in {\mathfrak{s}}$.
\end{lem}

Following \cite{cluster-diagrams}, we give an answer in terms of clusters. A \textbf{cluster} is a subset of the roots of $F(x)$ contained in an open disk. As the absolute Galois group $\text{Gal}(K^{\text{sep}}/K)$ acts on the roots of $F(x)$, there is a natural action of $\text{Gal}(K^{\text{sep}}/K)$ on clusters (see \Cref{defn: cluster}). Let $\mathcal{O}(\mathfrak{s})$ denote the orbit of a cluster $\mathfrak{s}$ under $\text{Gal}(K^{\text{sep}}/K)$, let $c_\mathfrak{s}$ be the integer defined in \Cref{lem: valuation of polynomial}, and let $\gamma_\mathfrak{s} \in K$ be defined as in \Cref{lem: shifting argument}. Note that $c_\mathfrak{s}$ and $\gamma_\mathfrak{s}$ are computable by hand from the roots of $F(x)$, see \Cref{ex: computed degree set}.

The two main tools used in the proof of \Cref{thm: main thm} are \Cref{lem: valuation of polynomial} and \Cref{lem: no points not zero mod q}. \Cref{lem: valuation of polynomial} provides a formula to compute $\val(F(x_0))$ as a function of $x_0 \in \overline{K}$ and the nearest cluster, and may be of independent interest. \Cref{lem: no points not zero mod q} shows that there is an obstruction arising from the valuation to points of arbitrary degree.

\begin{lem}\label{lem: cS integer}
 Suppose that the residue field of $K$ is algebraically closed and that the roots of $F(x)$ are tamely ramified. Let $\mathfrak{s}$ be a Galois-invariant cluster. Then $c_{\mathfrak{s}} \in \mathbb{Z}$.
\end{lem}
\begin{proof}
 Note that we can apply a rational change of coordinates: if $\gamma \in K$, consider $F(x + \gamma)$. The set ${\mathfrak{s}}_\gamma=\{ \alpha - \gamma \mid \alpha \in {\mathfrak{s}}\}$ is a Galois-invariant cluster for $F(x+ \gamma)$ and $c_{{\mathfrak{s}}_\gamma} = c_{\mathfrak{s}}$. By \Cref{lem: shifting argument}, we can assume that there exists $\alpha_\text{min}$ such that ${\mathfrak{s}} = \{ \alpha \mid \val(\alpha) \geq \val(\alpha_\text{min})\}$.

\begin{lem}\label{lem: no points not zero mod q}
 Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\mathfrak{s}$. Let $\gamma_{\mathfrak{s}}$ be constructed from $\mathfrak{s}$ as in \Cref{lem: shifting argument}. Let $r$ be a natural number not divisible by $q$. If there exists $P \in U(\overline{K})$ of degree $r$ with $x(P) \in X_{\mathfrak{s}}$, then one of the following holds:
 \begin{enumerate}[label=$(\roman*)$]
 \item\label{condition: bS not 0 mod q first lemma} $|\mathfrak{s}| \not \equiv 0 \pmod{q}$, or
 \item\label{condition: bS and cS both 0 mod q first lemma} $|\mathfrak{s}| \equiv 0 \pmod q$ and $c_{\mathfrak{s}} \equiv 0 \pmod{q}$, or
 \item\label{condition: constant term 0 mod q first lemma} $0 \in X_{\mathfrak{s}}$ and $\val(F(0)) \equiv 0 \pmod{q}$, or
 \item\label{condition: F(gamma) mod q first lemma} $\gamma_{\mathfrak{s}} \in X_{\mathfrak{s}}$ and $\val(F(\gamma_{\mathfrak{s}})) \equiv 0 \pmod{q}$.
 \end{enumerate}
\end{lem}

\begin{lem}\label{lem: congruence conditions imply N>r}
 Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\mathfrak{s}$. If one of the following holds:
 \begin{enumerate}[label=$(\roman*)$]
 \item\label{condition: bS not 0 mod q} $|\mathfrak{s}| \not \equiv 0 \pmod{q}$, or
 \item\label{condition: bS and cS both 0 mod q} $|\mathfrak{s}| \equiv 0 \pmod q$ and $c_{\mathfrak{s}} \equiv 0 \pmod{q}$, or
 \item\label{condition: constant term 0 mod q} $0 \in X_{\mathfrak{s}}$ and $\val(F(0)) \equiv 0 \pmod{q}$, or
 \item\label{condition: gamma term 0 mod q} $\gamma_{\mathfrak{s}} \in X_{\mathfrak{s}}$ and $\val(F(\gamma_{\mathfrak{s}})) \equiv 0 \pmod{q}$,
 \end{enumerate}
 then there exists $r_0 \in \mathbb{N}$ such that for all $r \geq r_0$ with $\gcd(r,q) = 1$, there exists a point $P \in U(\overline{K})$ of degree $r$ with $x(P) \in X_{\mathfrak{s}}$.
\end{lem}
\begin{proof}
 Following the argument in the proof of case \ref{case: close to root} of \Cref{lem: no points not zero mod q}, without loss of generality we can take $\gamma_{\mathfrak{s}} = 0$, so condition \ref{condition: gamma term 0 mod q} reduces to condition \ref{condition: constant term 0 mod q}.
 Suppose that condition \ref{condition: constant term 0 mod q} holds, so that $0 \in X_{\mathfrak{s}}$ and hence ${\mathfrak{s}}$ is the set of roots of maximal valuation. We let $r_0 = 1$ and construct points of all degrees $r \geq r_0$. Let $N$ be an integer greater than $\max\{\val(\alpha)\}$, and let $x_0 = \pi^{N + \frac{1}{r}}$. By construction, $x_0 \in X_{\mathfrak{s}}$, and hence by \Cref{lem: valuation of polynomial}, for any $\alpha \in \mathfrak{s}$,
 \begin{align*}
 \val(F(x_0)) = |\mathfrak{s}| \val(x_0 - \alpha) + c_{\mathfrak{s}} = |\mathfrak{s}|\val(\alpha) + c_{\mathfrak{s}} = \val(F(0)) \equiv 0 \pmod q.
 \end{align*}
 Thus, $(x_0,\sqrt[q]{F(x_0)})$ is a $\overline{K}$-point of degree $r$ over $K$.

\begin{proof}
 We begin with the case where $q$ is odd. Define $m_k = \{ n_k \mid n_k >1\}$, and reorder so that both $n_k$ and $m_k$ are weakly increasing. We construct a superelliptic curve $C/K$ with equation $y^q = F(x)$ such that
 \begin{align*}
 \mathcal{D}(C/K) = q\mathbb{N} \cup \bigcup_{n_k> 1} n_k \mathbb{N} = q\mathbb{N} \cup \bigcup_{k} m_k \mathbb{N}.
 \end{align*}
 Define $c \coloneqq D - q\sum_{k}m_k$ and let $a$ be relatively prime to $c$, congruent to $1 \pmod q$, and less than $-c$. As $D$ is a multiple of $q$, so is $c$. Define $F(x)$ by
 \begin{align}\label{eq: definition of F(x)}
 F(x) \coloneqq \pi (x^c - \pi^a)\prod_{n_k>1} \prod_{i=1}^q (x^{n_k} - \pi)
 \end{align}
 and consider the curve $C/K$ defined by $y^q = F(x)$. By \Cref{lem: contains qN} and \Cref{lem: degree of root}, we have that $q\mathbb{N} \cup \bigcup_{n_k > 1} n_k \mathbb{N} \subseteq \mathcal{D}(C/K)$. We show that $\mathcal{D}(C/K) \subseteq q\mathbb{N} \cup \bigcup_{n_k > 1} n_k \mathbb{N}$. Let $P \in U(\overline{K})$ and let $\mathfrak{s}$ be such that $x(P) \in X_{\mathfrak{s}}$. By \Cref{lem: types of S}, $X_{\mathfrak{s}} \subseteq B_{|\alpha|}(\alpha)$ or there exists $\alpha_\text{min}$ such that ${\mathfrak{s}}$ indexes the set $\{ \alpha \mid \val(\alpha) \geq \val(\alpha_\text{min}) \}$.

Suppose $|\mathfrak{s}| = 5$. We detail only the case where $\mathfrak{s} = \{7^{\frac{1}{2}} + \zeta_5^i7^{\frac{3}{5}}\}_{i = 1}^{5}$, as the other cases are similar. We have that $c_{\mathfrak{s}} = \frac{27}{2}$. Fix $\alpha \in \mathfrak{s}$.
 By \Cref{lem: types of S}, we have that $X_{\mathfrak{s}} \subseteq B_{|\alpha|}(\alpha)$, and applying \Cref{lem: tamely ramified generalized krasners}, we find that for all $\eta \in B_{|\alpha|}(\alpha)$, $K(7^{\frac{1}{2}}) \subseteq K(\eta)$, so that if $x(P) \in X_{\mathfrak{s}}$, then $K(7^{\frac{1}{2}}) \subseteq \textbf{k}(x(P))$. We extend to $K(7^{\frac{1}{2}})$ and renormalize the valuation, remembering that we have divided the degree of our point by 2.
 Following \Cref{lem: congruence conditions imply N>r} over $K(7^{\frac{1}{2}})$, we have that $\gamma_\mathfrak{s} = 7^{\frac{1}{2}}$ and $(M, N) = (1, \frac{6}{5})$. If $\val(x(P) - \gamma_\mathfrak{s}) = \frac{a}{r}$, then
 \begin{align*}
 \val(y(P)) = \frac{5}{3} \cdot \frac{a}{r} + \frac{27}{3} = \frac{5a + 27r}{3r}
 \end{align*}
 and so 3 divides the denominator of $\val(y(P))$ unless $5a + 27r \equiv 0 \pmod{3}$, which occurs if and only if $a \equiv 0 \pmod{3}$. We search for $r \in \mathbb{N}$ that produce $\frac{a}{r} \in (1, \frac{6}{5})$ such that $a \equiv 0 \pmod 3$. For $r$ large enough, by \Cref{lem: sol in interval}, this condition is always satisfied, and a computer finds the condition is satisfied for $8$, $11$, $13$, $14$, and all integers greater than $15$.
\end{proof}

The two main tools used in the proof of \Cref{thm: main thm} are \Cref{lem: valuation of polynomial} and \Cref{lem: no points not zero mod q}.  \Cref{lem: valuation of polynomial} provides a formula to compute $v(F(x_0))$ as a function of $x_0 \in \overline{K}$ and the nearest cluster, and may be of independent interest.  \Cref{lem: no points not zero mod q} shows that there is an obstruction arising from the valuation to points of arbitrary degree.

The results of \cite{Degrees-on-Varieties} show that \Cref{thm: main thm} must relate to the multiplicities of the components of the special fiber of a regular model with strict normal crossings. Following the construction of \cite{Regular-Models}, for many Galois invariant clusters $\mathfrak{s}$ (specifically, for those that fall in case \ref{case: annulus type} of \Cref{lem: types of S}), there exists a $v$-edge $L_\mathfrak{s}$, and that $\delta_{L_\mathfrak{s}}$ equals 1 if and only if the conditions of \Cref{thm: main thm} are not satisfied. When $\delta_{L_\mathfrak{s}}$ equals 1, there are intersecting copies of $\mathbb{P}^1$ with coprime multiplicity, and thus \cite{Degrees-on-Varieties}*{Theorem 1.1} shows that for some $r$, $\mathbb{N}_{>r} \subseteq \mathcal{D}(C/K)$. We compute the degrees of all points reducing to the $\mathbb{P}^1$'s arising from $L_\mathfrak{s}$ simultaneously in \Cref{lem: congruence conditions imply N>r}. When $C/K$ is not $\Delta_v$-regular, the methods of \cite{Regular-Models} do not give a regular model, see \Cref{rem: dokchitser}. As in \Cref{ex: computed degree set}, we can often compute the degree set even when $C/K$ is not $\Delta_v$-regular.

Restricting briefly to the case of hyperelliptic curves, \cite{models-of-hyperelliptic-curves}*{Theorem 1.2} shows that, if the curve has potential semistable reduction (which is equivalent to every root of $F(x)$ being tamely ramified), the special fiber can be computed from a \textit{cluster diagram} (see \cite{original-cluster-diagrams} or the overview \cite{cluster-diagrams-users-guide}) and thus the degree set depends only on the cluster diagram.  Using an additional quantity $c_{\mathfrak{s}}$, both $|\mathfrak{s}|$ and $v(F(\gamma_{\mathfrak{s}}))$ can be computed from the cluster diagram (see \Cref{rem: compute v(F(gamma))}). We do not know how $c_\mathfrak{s}$ relates to the cluster diagram.

\begin{thm}\label{thm: nice generate examples theorem}
    Let $K$ be a discretely valued Henselian field with algebraically closed residue field of characteristic $p$. Let $q$ be a prime not equal to $p$, and let $n_1, \ldots, n_\ell$ be positive integers. Then, there exists a superelliptic curve $C/K$ such that
    \begin{align*}
        \mathcal{D}(C/K) = q\mathbb{N} \cup n_1\mathbb{N} \cup \ldots \cup n_\ell \mathbb{N}.
    \end{align*}
\end{thm}

As a special case of \Cref{thm: nice generate examples theorem}, we find that for every pair of odd primes $p$ and $q$, there exists a curve $C/K$ such that $\mathcal{D}(C/K) = p \mathbb{N} \cup q\mathbb{N}$. Thus, the density of $\mathcal{D}(C/K)$ in $\ind(C/K)\mathbb{N}$ can be arbitrarily small.

The authors would like to thank Bianca Viray, Paul Fili, and Carlos Rivera. The first author was supported in part by NSF grant DGE-2140004.

The post-theorem introduction says that "[t]he two main tools used in the proof of \Cref{thm: main thm} are \Cref{lem: valuation of polynomial} and \Cref{lem: no points not zero mod q}." The role of these is: \Cref{lem: valuation of polynomial} "provides a formula to compute $v(F(x_0))$ as a function of $x_0 \in \overline{K}$ and the nearest cluster," while \Cref{lem: no points not zero mod q} "shows that there is an obstruction arising from the valuation to points of arbitrary degree."

It further explains that, via \cite{Degrees-on-Varieties}, \Cref{thm: main thm} is tied to "the multiplicities of the components of the special fiber of a regular model with strict normal crossings." Using the construction of \cite{Regular-Models}, for many Galois-invariant clusters $\mathfrak{s}$ (those in "case \ref{case: annulus type} of \Cref{lem: types of S}"), there is a $v$-edge $L_\mathfrak{s}$ such that "$\delta_{L_\mathfrak{s}}$ equals 1 if and only if the conditions of \Cref{thm: main thm} are not satisfied." When "$\delta_{L_\mathfrak{s}}$ equals 1," one gets "intersecting copies of $\mathbb{P}^1$ with coprime multiplicity," and then \cite{Degrees-on-Varieties}*{Theorem 1.1} implies that "for some $r$, $\mathbb{N}_{>r} \subseteq \mathcal{D}(C/K)$." It also notes that the paper computes "the degrees of all points reducing to the $\mathbb{P}^1$'s arising from $L_\mathfrak{s}$ simultaneously in \Cref{lem: congruence conditions imply N>r}."

The post-theorem introduction says that "[t]he two main tools used in the proof of the main theorem are \begin{lem}\label{lem: valuation of polynomial}
    Fix a cluster $\mathfrak{s}$. There exists $c_{\mathfrak{s}} \in \mathbb{Q}$ such that for all $x_0 \in X_{\mathfrak{s}}$ and any $\alpha \in {\mathfrak{s}}$,
    $$
    \val(F(x_0)) = |\mathfrak{s}|\cdot\val(x_0 - \alpha) + c_{\mathfrak{s}},
    $$
    where $c_{\mathfrak{s}}$ is given by
    \begin{align*}
        c_{\mathfrak{s}} &= \val(a_d) + \sum_{\beta \in {\mathfrak{s}^c}} \val(\alpha - \beta)
    \end{align*}
    and all expressions above are independent of the choice of $\alpha \in {\mathfrak{s}}$.
\end{lem} and \begin{lem}\label{lem: no points not zero mod q}
    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\mathfrak{s}$. Let $\gamma_{\mathfrak{s}}$ be constructed from $\mathfrak{s}$ as in \Cref{lem: shifting argument}. Let $r$ be a natural number not divisible by $q$. If there exists $P \in U(\overline{K})$ of degree $r$ with $x(P) \in X_{\mathfrak{s}}$, then one of the following holds:
    \begin{enumerate}[label=$(\roman*)$]
        \item\label{condition: bS not 0 mod q first lemma} $|\mathfrak{s}| \not \equiv 0 \pmod{q}$, or
        \item\label{condition: bS and cS both 0 mod q first lemma} $|\mathfrak{s}| \equiv 0 \pmod q$ and $c_{\mathfrak{s}} \equiv 0 \pmod{q}$, or
        \item\label{condition: constant term 0 mod q first lemma} $0 \in X_{\mathfrak{s}}$ and $\val(F(0)) \equiv 0 \pmod{q}$, or
        \item\label{condition: F(gamma) mod q first lemma} $\gamma_{\mathfrak{s}} \in X_{\mathfrak{s}}$ and $\val(F(\gamma_{\mathfrak{s}})) \equiv 0 \pmod{q}$.
    \end{enumerate}
\end{lem}." The role of these is: the first lemma "provides a formula to compute $v(F(x_0))$ as a function of $x_0 \in \overline{K}$ and the nearest cluster," while the second lemma "shows that there is an obstruction arising from the valuation to points of arbitrary degree."

It further explains that, via \cite{Degrees-on-Varieties}, the main theorem is tied to "the multiplicities of the components of the special fiber of a regular model with strict normal crossings." Using the construction of \cite{Regular-Models}, for many Galois-invariant clusters $\mathfrak{s}$ (those in case \ref{case: annulus type} of the following lemma), there is a $v$-edge $L_\mathfrak{s}$ such that "$\delta_{L_\mathfrak{s}}$ equals 1 if and only if the conditions of the main theorem are not satisfied." We first recall the relevant dichotomy for clusters:

\begin{lem}\label{lem: types of S}
    Let $\mathfrak{s}$ be a cluster. Then, at least one of the following is true:
    \begin{enumerate}[label=$(\alph*)$]
        \item\label{case: annulus type} there exists some root $\alpha_{\min}$ of $\mathfrak{s}$ such that $\mathfrak{s} = \{ \alpha \mid \val(\alpha) \geq \val(\alpha_{\min})\}$, or
        \item\label{case: ball type} all roots of $\mathfrak{s}$ are contained in $B_{|\alpha|}(\alpha)$ for any $\alpha \in \mathfrak{s}$, and further $X_{\mathfrak{s}} \subseteq B_{|\alpha|}(\alpha)$.
    \end{enumerate}
\end{lem}

When "$\delta_{L_\mathfrak{s}}$ equals 1," one gets "intersecting copies of $\mathbb{P}^1$ with coprime multiplicity," and then \cite{Degrees-on-Varieties}*{Theorem 1.1} implies that "for some $r$, $\mathbb{N}_{>r} \subseteq \mathcal{D}(C/K)$." It also notes that the paper computes "the degrees of all points reducing to the $\mathbb{P}^1$'s arising from $L_\mathfrak{s}$ simultaneously in \begin{lem}\label{lem: congruence conditions imply N>r}
    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Fix a Galois-invariant cluster $\mathfrak{s}$. If one of the following holds:
    \begin{enumerate}[label=$(\roman*)$]
        \item\label{condition: bS not 0 mod q} $|\mathfrak{s}| \not \equiv 0 \pmod{q}$, or
        \item\label{condition: bS and cS both 0 mod q} $|\mathfrak{s}| \equiv 0 \pmod q$ and $c_{\mathfrak{s}} \equiv 0 \pmod{q}$, or
        \item\label{condition: constant term 0 mod q} $0 \in X_{\mathfrak{s}}$ and $\val(F(0)) \equiv 0 \pmod{q}$, or
        \item\label{condition: gamma term 0 mod q} $\gamma_{\mathfrak{s}} \in X_{\mathfrak{s}}$ and $\val(F(\gamma_{\mathfrak{s}})) \equiv 0 \pmod{q}$,
    \end{enumerate}
    then there exists $r_0 \in \mathbb{N}$ such that for all $r \geq r_0$ with $\gcd(r,q) = 1$, there exists a point $P \in U(\overline{K})$ of degree $r$ with $x(P) \in X_{\mathfrak{s}}$.
\end{lem}."

\label{thm: main thm}
    Suppose that the residue field of $K$ is algebraically closed and that every root of $F(x)$ is tamely ramified. Then, $\mathcal{D}(C/K)$ is not cofinite if and only if $v(F(0)) \not \equiv 0 \pmod q$, and for every Galois-invariant cluster $\mathfrak{s}$ of roots of $F(x)$,
    \begin{enumerate}[label=$(\roman*)$]
        \item $|\mathfrak{s}| \equiv 0 \pmod q$ and $c_{\mathfrak{s}} \not \equiv 0 \pmod{q}$, and
        \item if $\gamma_{\mathfrak{s}} \in X_{\mathfrak{s}}$, then $v(F(\gamma_{\mathfrak{s}})) \not \equiv 0 \pmod{q}$.
    \end{enumerate}
    When these conditions are satisfied, we have that
    \begin{align*}
        \mathcal{D}(C/K) \subseteq q\mathbb{N} \cup \bigcup_{\substack{ \mathfrak{s} \, \text{\rm not Galois} \\ \text{\rm invariant}}} |\mathcal{O}(\mathfrak{s})| \mathbb{N}.
    \end{align*},

Biconditional or Equivalence
Implication

Question:
Let $C/K$ be the curve $y^q=F(x)$, and let
\[
\mathcal{D}(C/K)=\{\deg_K(P)\mid P\in C\text{ is a closed point}\}.
\]
Write $\operatorname{ind}(C/K)=\gcd\mathcal{D}(C/K)$, and say that $\mathcal{D}(C/K)$ is not cofinite if its complement in $\operatorname{ind}(C/K)\mathbb N$ is not finite. A cluster of roots of $F(x)$ is a nonempty subset of the roots contained in a disk $D_{z,d}=\{x\in \overline K:|x-z|\le d\}$; $\mathcal O(\mathfrak s)$ denotes its Galois orbit. For a cluster $\mathfrak s$, let $c_{\mathfrak s}$ be the quantity characterized by
\[
v(F(x_0))=|\mathfrak s|\,v(x_0-\alpha)+c_{\mathfrak s}
\]
for $x_0\in X_{\mathfrak s}$ and $\alpha\in\mathfrak s$, where $X_{\mathfrak s}$ is the associated region on which this formula holds. For a Galois-invariant cluster $\mathfrak s$, let $\gamma_{\mathfrak s}\in K$ be such that
\[
\mathfrak s=\{\alpha\mid v(\alpha-\gamma_{\mathfrak s})\ge v(\alpha_{\min}-\gamma_{\mathfrak s})\}
\]
for some $\alpha_{\min}\in\mathfrak s$. Suppose the residue field of $K$ is algebraically closed and every root of $F(x)$ is tamely ramified. Which statement is equivalent to $\mathcal D(C/K)$ being not cofinite?

One has $v(F(0))\not\equiv 0\pmod q$, and for every Galois-invariant cluster $\mathfrak s$ of roots of $F(x)$, (i) $|\mathfrak s|\equiv 0\pmod q$ and $c_{\mathfrak s}\not\equiv 0\pmod q$, and (ii) if $\gamma_{\mathfrak s}\in X_{\mathfrak s}$, then $v(F(\gamma_{\mathfrak s}))\not\equiv 0\pmod q$; moreover, in this case,
\[
\mathcal D(C/K)\subseteq q\mathbb N\cup \bigcup_{\substack{\mathfrak s\text{ not Galois}\\ \text{invariant}}}|\mathcal O(\mathfrak s)|\,\mathbb N.
\]

Choices:
B.
One has $v(F(0))\not\equiv 0\pmod q$, and for every cluster $\mathfrak s$ of roots of $F(x)$, (i) $|\mathfrak s|\equiv 0\pmod q$ and $c_{\mathfrak s}\not\equiv 0\pmod q$, and (ii) if $\gamma_{\mathfrak s}\in X_{\mathfrak s}$, then $v(F(\gamma_{\mathfrak s}))\not\equiv 0\pmod q$; moreover, in this case,
\[
\mathcal D(C/K)\subseteq q\mathbb N\cup \bigcup_{\mathfrak s}|\mathcal O(\mathfrak s)|\,\mathbb N.
\]
C. One has $v(F(0))\not\equiv 0\pmod q$, and for every Galois-invariant cluster $\mathfrak s$ of roots of $F(x)$, (i) $|\mathfrak s|\equiv 0\pmod q$ and $c_{\mathfrak s}\not\equiv 0\pmod q$, and (ii) if $\gamma_{\mathfrak s}\in X_{\mathfrak s}$, then $v(F(\gamma_{\mathfrak s}))\not\equiv 0\pmod q$.
D.
One has $v(F(0))\not\equiv 0\pmod q$, and there exists a Galois-invariant cluster $\mathfrak s$ of roots of $F(x)$ such that (i) $|\mathfrak s|\equiv 0\pmod q$ and $c_{\mathfrak s}\not\equiv 0\pmod q$, and (ii) if $\gamma_{\mathfrak s}\in X_{\mathfrak s}$, then $v(F(\gamma_{\mathfrak s}))\not\equiv 0\pmod q$; moreover, in this case,
\[
\mathcal D(C/K)\subseteq q\mathbb N\cup \bigcup_{\substack{\mathfrak s\text{ not Galois}\\ \text{invariant}}}|\mathcal O(\mathfrak s)|\,\mathbb N.
\]
E.
One has $v(F(0))\equiv 0\pmod q$, and for every Galois-invariant cluster $\mathfrak s$ of roots of $F(x)$, at least one of the following holds: (i) $|\mathfrak s|\equiv 0\pmod q$ and $c_{\mathfrak s}\not\equiv 0\pmod q$, or (ii) if $\gamma_{\mathfrak s}\in X_{\mathfrak s}$, then $v(F(\gamma_{\mathfrak s}))\not\equiv 0\pmod q$; moreover, in this case,
\[
\mathcal D(C/K)\subseteq q\mathbb N\cup \bigcup_{\substack{\mathfrak s\text{ not Galois}\\ \text{invariant}}}|\mathcal O(\mathfrak s)|\,\mathbb N.
\]

Galois-invariant scope of the cluster conditions

dropped the concluding containment for $\mathcal D(C/K)$

universal quantification over all Galois-invariant clusters

obstruction congruences from the no-points lemma, especially the sign of $v(F(0))$ mod $q$ and conjunction of cluster conditions

The stem does not reveal the correct option; it asks for an equivalent characterization and provides technical setup without singling out choice A. There is no explicit answer leakage beyond the general fact that one option should match a theorem statement.

This is essentially a theorem-recall item: the task is to identify the exact statement equivalent to non-cofiniteness. It is very close to selecting the correctly restated theorem rather than deriving a conclusion from novel data.

Some reasoning is needed to compare subtle changes in scope, quantifiers, and side conditions across the options, but the main demand is recognition of the precise theorem statement rather than substantial mathematical generation or synthesis.

The distractors are strong: they vary universal vs. existential quantification, Galois-invariant vs. all clusters, conjunction vs. disjunction, and omission of the concluding containment. These are plausible mathematical failure modes and are meaningfully distinct.

Technically well-constructed with strong distractors and little answer leakage, but it mainly tests precise theorem recall/statement matching rather than genuine generative reasoning.


\label{thm-characterization}
Let $(M, \varphi)$ be a diffuse separable $\mathrm{W}^*$-probability space. The following assertions are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $(M, \varphi)$ is selfless.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed for some nontrivial $\mathrm{W}^*$-probability space $(N, \psi)$.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\operatorname{ L}(\mathbb{Z}), \tau_\mathbb{Z})$ is existentially closed, where $\tau_\mathbb{Z}$ is the canonical trace on $\operatorname{ L}(\mathbb{Z})$.

\item $\operatorname{B}(M, \varphi) = \mathbb{C} 1$.
\end{enumerate}

\begin{letterthm} \label{thm-characterization}
Let $(M, \varphi)$ be a diffuse separable $\mathrm{W}^*$-probability space. The following assertions are equivalent:
\begin{enumerate}[{\rm (i)}]
\item $(M, \varphi)$ is selfless.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed for some nontrivial $\mathrm{W}^*$-probability space $(N, \psi)$.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z)$ is existentially closed, where $\tau_\Z$ is the canonical trace on $\rL(\Z)$.

\item $\rB(M, \varphi) = \C 1$.
\end{enumerate}
\end{letterthm}

In \cite{Ro23}, Robert introduced a new class of C$^*$-probability spaces, which he called \emph{selfless}, characterized by the existence of a copy of themselves in their ultrapower that is freely independent from the diagonal copy (thus being ``free from themselves"). This property quickly attracted the attention of numerous researchers as it implies many important regularity properties and is satisfied by a large class of examples (see \cite{AGKEP24, HKER25, RTV25, Vi25, Oz25}).

In this short note, we introduce a parallel notion of selfless W$^*$-probability space and we relate this notion to Connes'\! bicentralizer problem.

A W$^*$-{\em probability space} is a pair $(M, \varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\varphi \in M_\ast$. For W$^*$-probability spaces $(M, \varphi)$ and $(N, \psi)$, we say that $(M, \varphi) \subset (N, \psi)$ is an {\em inclusion} of W$^*$-probability spaces if $M \subset N$ and if there exists a faithful normal conditional expectation $\operatorname{ E} : N \to M$ such that $\varphi\circ \operatorname{ E} = \psi$. In that case, $\operatorname{ E} : N \to M$ is the unique faithful normal conditional expectation such that $\varphi \circ \operatorname{ E} = \psi$.

Following \cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \varphi) \subset (N, \psi)$ is {\em existentially closed} if there exists a nonprincipal ultrafilter $\mathcal U$ on some directed set $I$ such that $(M, \varphi) \subset (N, \psi) \subset (M, \varphi)^{\mathcal U}$, where $(M, \varphi) \subset (M, \varphi)^{\mathcal U}$ is the diagonal inclusion. Note that if  $N$ is separable (i.e.\! $N$ has separable predual), then $\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\mathbb{N}$.

Adapting \cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \varphi)$ is {\em selfless} if the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (M, \varphi)$ is existentially closed.

Popa's seminal work \cite{Po95} shows that $(M, \tau)$ is selfless for any separable type ${\rm II_1}$ factor $M$ endowed with its canonical trace $\tau$. Houdayer--Isono \cite{HI14} extended Popa's result by showing that $(M, \varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\varphi \in M_\ast$ for which $(M_\varphi)' \cap M = \C1$.

In this note, we show that a diffuse separable W$^*$-probability space $(M, \varphi)$ is selfless if and only if it has a trivial \emph{bicentralizer}. Recall from \cite{Co80, Ha85}, that the {\em bicentralizer} $\operatorname{B}(M, \varphi)$ of a W$^*$-probability space $(M, \varphi)$ is the set of all elements $x \in M$ that satisfy the following condition:
\begin{verse}
\noindent
For every $\varepsilon > 0$, there exists $\delta > 0$ such that for every $u \in \mathscr U(M)$, if $\|u \varphi - \varphi u\| < \delta$, then $\|u x - x u\|_\varphi < \varepsilon$.
\end{verse}

We then obtain the following characterization.

A W$^*$-{\em probability space} is a pair $(M, \varphi)$ that consists of a von Neumann algebra $M$ endowed with a faithful normal state $\varphi \in M_\ast$. For W$^*$-probability spaces $(M, \varphi)$ and $(N, \psi)$, we say that $(M, \varphi) \subset (N, \psi)$ is an {\em inclusion} of W$^*$-probability spaces if $M \subset N$ and if there exists a faithful normal conditional expectation $\operatorname{ E} : N \to M$ such that $\varphi\circ \operatorname{ E} = \psi$. In that case, $\operatorname{ E} : N \to M$ is the unique faithful normal conditional expectation such that $\varphi \circ \operatorname{ E} = \psi$.

Following \cite{GH21}, we say that an inclusion of W$^*$-probability spaces $(M, \varphi) \subset (N, \psi)$ is {\em existentially closed} if there exists a nonprincipal ultrafilter $\mathcal U$ on some directed set $I$ such that $(M, \varphi) \subset (N, \psi) \subset (M, \varphi)^{\mathcal U}$, where $(M, \varphi) \subset (M, \varphi)^{\mathcal U}$ is the diagonal inclusion. Note that if $N$ is separable (i.e.\! $N$ has separable predual), then $\mathcal U$ can be chosen to be a nonprincipal ultrafilter on $\mathbb{N}$.

Adapting \cite[Definition 2.1]{Ro23} to the von Neumann algebraic realm, we say that a W$^*$-probability space $(M, \varphi)$ is {\em selfless} if the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (M, \varphi)$ is existentially closed.

Popa's seminal work \cite{Po95} shows that $(M, \tau)$ is selfless for any separable type ${\rm II_1}$ factor $M$ endowed with its canonical trace $\tau$. Houdayer--Isono \cite{HI14} extended Popa's result by showing that $(M, \varphi)$ is selfless for any separable factor $M$ endowed with a faithful normal state $\varphi \in M_\ast$ for which $(M_\varphi)' \cap M = \C1$.

In this note, we show that a diffuse separable W$^*$-probability space $(M, \varphi)$ is selfless if and only if it has a trivial \emph{bicentralizer}. Recall from \cite{Co80, Ha85}, that the {\em bicentralizer} $\operatorname{B}(M, \varphi)$ of a W$^*$-probability space $(M, \varphi)$ is the set of all elements $x \in M$ that satisfy the following condition:
\begin{verse}
\noindent
For every $\varepsilon > 0$, there exists $\delta > 0$ such that for every $u \in \mathscr U(M)$, if $\|u \varphi - \varphi u\| < \delta$, then $\|u x - x u\|_\varphi < \varepsilon$.
\end{verse}

We then obtain the following characterization.

We then obtain the following characterization.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed for some nontrivial $\mathrm{W}^*$-probability space $(N, \psi)$.

\item The first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z)$ is existentially closed, where $\tau_\Z$ is the canonical trace on $\rL(\Z)$.

We point out that the first three conditions above are analogous to the ones appearing in \cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.

Observe that if $\rB(M, \varphi) = \C1$, then $M$ must be a factor. Moreover, according to \cite{Ok21}, one and exactly one of the following assertions hold:
\begin{itemize}
\item $(M, \varphi)$ is a tracial factor of type ${\rm I}_n$ for $n \in \N^*$ or of type ${\rm II_1}$.
\item There exists $\lambda \in (0, 1)$ such that $(M, \varphi)$ is a type ${\rm III_\lambda}$ factor endowed with its $\frac{2\pi}{|\log(\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\varphi)' \cap M = \C 1$.
\item $M$ is a type ${\rm III_1}$ factor. In that case, using \cite[Corollary 1.5]{Ha85}, we further have $\rB(M, \psi) = \C1$ for \emph{every} faithful normal state $\psi \in M_\ast$.
\end{itemize}

\begin{lettercor}\label{cor}
Let $M$ be a separable type ${\rm III_1}$ factor satisfying Connes'\! bicentralizer conjecture. Then for {\em every} faithful normal state $\varphi \in M_\ast$, the $\mathrm{W}^*$-probability space $(M, \varphi)$ is selfless.
\end{lettercor}

$(\rm ii) \Rightarrow (\rm iii)$ Let $(N, \psi)$ be a nontrivial W$^*$-probability space such that the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (N, \psi)$ is existentially closed. We may assume that $N$ is separable. Choose a nonprincipal ultrafilter $\mathcal U$ on $\N$ such that we have $(M, \varphi) \subset (M, \varphi) \ast (N, \psi) \subset (M, \varphi)^{\mathcal U}$, where $(M, \varphi) \subset (M, \varphi)^{\mathcal U}$ is the diagonal inclusion. There are two cases to consider.

Secondly, assume that $N_\psi \neq \C 1$. Upon replacing $(N, \psi)$ by $(N_\psi, \psi)$, we may assume that $(N, \psi)$ is tracial. Reasoning as is the first case, we obtain
\begin{align*}
(M, \varphi) &\subset (M, \varphi) \ast (N, \psi)^{\ast 2} \\
&= \left( (M, \varphi) \ast (N, \psi) \right ) \ast (N, \psi) \\
&\subset (M, \varphi)^{\mathcal U} \ast (N, \psi) \\
&\subset (M, \varphi)^{\mathcal U} \ast (N, \psi)^{\mathcal U} \\
& \subset (M^{\mathcal U}, \varphi^{\mathcal U})^{\mathcal U} = (M, \varphi)^{\mathcal W}.
\end{align*}
By \cite[Lemma 2.5]{Ro23}, there exists $n \in \N$ large enough so that the iterated free product $(N, \psi)^{\ast n}$ is diffuse and so $(\rL(\Z), \tau_{\Z}) \subset (N, \psi)^{\ast n}$. Upon iterating $n$ times the ultraproduct construction and replacing $\mathcal U$ by the appropriate ultrafilter $\mathcal W = \mathcal U^{\otimes n}$, we obtain
$$(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z) \subset (M, \varphi) \ast (N, \psi)^{\ast n} \subset (M, \varphi)^{\mathcal W},$$ where $(M, \varphi) \subset (M, \varphi)^{\mathcal W}$ is the diagonal inclusion. Therefore, the first factor inclusion $(M, \varphi) \subset (M, \varphi) \ast (\rL(\Z), \tau_\Z)$ is existentially closed.

We point out that the first three conditions above are analogous to the ones appearing in \cite[Theorem 2.6]{Ro23} for C$^*$-probability spaces.

The bicentralizer $\operatorname{B}(M, \varphi) \subset M$ is a von Neumann subalgebra such that $\operatorname{B}(M, \varphi) \subset (M_\varphi)' \cap M$ (see \cite[Proposition 1.3]{Ha85}). Thus Theorem \ref{thm-characterization} strengthens Houdayer--Isono's result \cite[Theorem A]{HI14}. In fact, the proof of Theorem \ref{thm-characterization} relies on \cite[Theorem A]{HI14} combined with a diagonal argument.

Observe that if $\operatorname{B}(M, \varphi) = \C1$, then $M$ must be a factor. Moreover, according to \cite{Ok21}, one and exactly one of the following assertions hold:
\begin{itemize}
\item $(M, \varphi)$ is a tracial factor of type ${\rm I}_n$ for $n \in \mathbb{N}^*$ or of type ${\rm II_1}$.
\item There exists $\lambda \in (0, 1)$ such that $(M, \varphi)$ is a type ${\rm III_\lambda}$ factor endowed with its $\frac{2\pi}{|\log(\lambda)|}$-periodic faithful normal state. In that case, we have $(M_\varphi)' \cap M = \mathbb{C} 1$.
\item $M$ is a type ${\rm III_1}$ factor. In that case, using \cite[Corollary 1.5]{Ha85}, we further have $\operatorname{B}(M, \psi) = \C1$ for \emph{every} faithful normal state $\psi \in M_\ast$.
\end{itemize}

In particular, Theorem \ref{thm-characterization} implies that no W$^*$-probability space of type ${\rm II_\infty}$ or type ${\rm III_0}$ can be selfless (this also follows by combining \cite[Theorem 3.5]{GH21} and \cite[Theorem 4.1]{Ue11}).

On the other hand, a famous conjecture of Connes, known as Connes'\! bicentralizer problem, claims that $\operatorname{B}(M,\varphi)=\mathbb{C} 1$ for \emph{every} type ${\rm III_1}$ factor $M$ and every faithful normal state $\varphi \in M_*$.  This conjecture has been verified for several families of type ${\rm III_1}$ factors such as amenable factors \cite{Ha85}, factors with a Cartan subalgebra, free products \cite{HU15}, semisolid factors \cite{HI15}, $q$-deformed Araki--Woods factors \cite{HI20, Bi24} and tensor products of type ${\rm III_1}$ factors \cite{Ma25}.

\begin{lettercor}\label{cor}
Let $M$ be a separable type ${\rm III_1}$ factor satisfying Connes'\! bicentralizer conjecture. Then for {\em every} faithful normal state $\varphi \in M_\ast$, the $\mathrm{W}^*$-probability space $(M, \varphi)$ is selfless.
\end{lettercor}

The post-theorem introduction explicitly says: ``the proof of Theorem \ref{thm-characterization} relies on \cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.

The post-theorem introduction explicitly says: ``the proof of Theorem \label{thm-characterization} relies on \cite[Theorem A]{HI14} combined with a diagonal argument.'' No further steps or outline are provided there.

Biconditional or Equivalence
Classification or Bijection

Question: Let $(M,\varphi)$ be a diffuse separable $\mathrm{W}^*$-probability space, i.e. a von Neumann algebra $M$ with separable predual and a faithful normal state $\varphi$. Say that an inclusion $(M,\varphi)\subset (N,\psi)$ is existentially closed if there exists a nonprincipal ultrafilter $\mathcal U$ such that $(M,\varphi)\subset (N,\psi)\subset (M,\varphi)^{\mathcal U}$, where $(M,\varphi)\subset (M,\varphi)^{\mathcal U}$ is the diagonal inclusion. Say that $(M,\varphi)$ is selfless if the first-factor inclusion $(M,\varphi)\subset (M,\varphi)\ast (M,\varphi)$ is existentially closed. Also, the bicentralizer $\operatorname{B}(M,\varphi)$ is the set of all $x\in M$ such that for every $\varepsilon>0$ there exists $\delta>0$ with the property that for every unitary $u\in\mathscr U(M)$, if $\|u\varphi-\varphi u\|<\delta$, then $\|ux-xu\|_\varphi<\varepsilon$. Which of the following statements is equivalent to both (1) $(M,\varphi)$ being selfless and (2) the existence of some nontrivial $\mathrm{W}^*$-probability space $(N,\psi)$ such that the first-factor inclusion $(M,\varphi)\subset (M,\varphi)\ast (N,\psi)$ is existentially closed?

The bicentralizer of $(M,\varphi)$ is trivial; equivalently, $\operatorname{B}(M,\varphi)=\mathbb C1$.

Choices:
B. The first-factor inclusion $(M,\varphi)\subset (M,\varphi)\ast (\operatorname{L}(\mathbb Z),\tau_{\mathbb Z})$ is existentially closed for every nonprincipal ultrafilter $\mathcal U$, in the sense that one has $(M,\varphi)\subset (M,\varphi)\ast (\operatorname{L}(\mathbb Z),\tau_{\mathbb Z})\subset (M,\varphi)^{\mathcal U}$ via the diagonal inclusion.
C. The bicentralizer of $(M,\varphi)$ is trivial on the center; equivalently, $\operatorname{B}(M,\varphi)\cap \mathcal Z(M)=\mathbb C1$.
D. For every nontrivial $\mathrm{W}^*$-probability space $(N,\psi)$, the first-factor inclusion $(M,\varphi)\subset (M,\varphi)\ast (N,\psi)$ is existentially closed.
E. The first-factor inclusion $(M,\varphi)\subset (M,\varphi)\ast (\operatorname{L}(\mathbb Z),\tau_{\mathbb Z})$ is an inclusion of $\mathrm{W}^*$-probability spaces and admits the canonical faithful normal conditional expectation onto the first factor.

existential-ultrafilter is existential not uniform in ultrafilter

full triviality of bicentralizer dropped to central part only

existence of some nontrivial free product partner strengthened to all partners

existentially closed replaced by mere inclusion with conditional expectation

The stem does not state the correct answer outright, but it explicitly introduces the bicentralizer and then asks for an equivalent condition. That makes the bicentralizer-based options especially salient and gives a noticeable hint toward A/C.

This is close to theorem recall: it asks for a statement equivalent to two listed conditions, and the correct choice is essentially the third equivalent formulation. Still, the presence of nearby strengthenings/weakenings means it is not a pure verbatim restatement.

There is some reasoning pressure because the distractors are logical variants: weaker (C), stronger (D), overly uniform/effective (B), and irrelevant weakening (E). However, the item mainly tests recognition of a known equivalence rather than deep derivation.

The distractors are mathematically plausible and distinct. They reflect common failure modes: replacing equivalence by a weaker central-only statement, strengthening 'some' to 'every', adding unnecessary ultrafilter uniformity, or confusing existential closedness with the existence of a conditional expectation.

A reasonably well-constructed advanced MCQ with strong distractors, but it leans heavily on theorem recognition and gives some topical hinting by foregrounding the bicentralizer in the stem.