[Local density of $\mathcal{S}$ in short intervals]\label{thm:main-elements}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
\#\left(\mathcal{S}\cap(x,x+h]\right)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.

\begin{theorem}[Large multiplicities in short intervals]\label{thm:erdos-short-interval}
\begin{enumerate}[label=(\roman*),leftmargin=*]
\item There exist constants $c_1,c_2>0$ such that for $X$ sufficiently large there exists $n\in\left(X,X+X\exp(-c_1\sqrt{\log X})\right]$ with
\[
  f_{\mathrm{Rom}}(n)\ \geqslant\ c_2\,\log_2 n.
\]
\item Let $0<\theta<1$ and set $h:=X^{\theta}$. There exist constants $c_1(\theta),c_2(\theta)>0$ such that, for $X$ sufficiently large, for at least $c_1(\theta)\,X$ integers $x\in(X,2X]$ there exists $n\in(x,x+h]$ with
\[
  f_{\mathrm{Rom}}(n)\ \geqslant\ c_2(\theta)\,\log_2 n.
\]
\end{enumerate}
\end{theorem}

Let $\mathbb{P}$ denote the set of prime numbers.
Romanoff proved that the sumset $\mathbb{P}+\{2^k:k\geqslant 0\}$ has positive lower density~\cite{Romanoff1934}, following earlier questions of de~Polignac~\cite{dePolignac1849a,dePolignac1849b}.
Erd\H{o}s initiated a systematic study of multiplicities and related variants~\cite{Erdos1950} (see also van~der~Corput~\cite{vanDerCorput1950}), and quantitative bounds for Romanoff-type representation functions have been developed more recently, for instance by Chen and Ding~\cite{ChenDing2023}.
See also Elsholtz and Schlage-Puchta~\cite{ElsholtzSchlagePuchta2018} for explicit numerical lower bounds on Romanov's constant.

In this paper we study a short-interval refinement for the structured additive set $\cA_{\lambda}$ introduced by Ding and Zhai~\cite{DingZhai}.
For $h=X^{\theta}$ in the range of \Cref{thm:main-elements,thm:main-rep}, we show that for all but a stretched-exponentially small exceptional set of $x\in[X,2X]$ the interval $(x,x+h]$ contains $\asymp h$ elements of the sumset $\mathbb{P}+\cA_{\lambda}(X)$, and the number of representations in such windows is also $\asymp h$.
In the model case of the Romanoff representation function
$$ f_{\mathrm{Rom}}(n):=\#\left\{(p,k)\in\mathbb{P}\times\mathbb N_0:\ p+2^k=n\right\}, $$
we additionally record a short-interval Erd\H{o}s-type statement for a positive proportion of windows, of an integer with unusually large pointwise multiplicity. We now set up the notation and state the main results.

Let $X\geqslant 3$ and set $h=X^{\theta}$ with $\theta\in(0,1)$.
Fix an integer $s\geqslant 2$ and positive real numbers $r_1,\dots,r_s>1$ such that
\begin{equation}\label{eq:reciprocal-sum}
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}<1\leqslant \frac{1}{r_1}+\cdots+\frac{1}{r_s}.
\end{equation}
Let $\lambda>0$ be defined by
\begin{equation}\label{eq:lambda-balance}
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}+\frac{1}{\lambda r_s}\ =\ 1,
\end{equation}
so that in particular $\lambda\geqslant 1$ by \eqref{eq:reciprocal-sum}.
Define the (Ding--Zhai~\cite{DingZhai}) additive set
\begin{equation}\label{eq:def-A}
\cA_{\lambda}\ :=\ \left\{2^{\lfloor k_1^{r_1}\rfloor}+\cdots+2^{\lfloor k_{s-1}^{r_{s-1}}\rfloor}+2^{\lfloor \lfloor k_s^{\lambda}\rfloor^{r_s}\rfloor}\ :\ k_1,\dots,k_s\in\mathbb{N}\right\}\subset\mathbb{N}.
\end{equation}
For $X\geqslant 3$ we truncate
\[
\cA_{\lambda}(X)\ :=\ \cA_{\lambda}\cap\left[1,2X\right],
\qquad
\mathcal{S}\ :=\ \mathbb{P}+\cA_{\lambda}(X).
\]
For real $x\geqslant 1$ we denote by
\[
A_{\lambda}(x)\ :=\ \#\left(\cA_{\lambda}\cap\left[1,x\right]\right)
\]
the associated counting function. The bound $A_{\lambda}(x)\asymp\log x$ holds for all $x\geqslant 3$ (see Ding--Zhai~\cite{DingZhai}, ultimately due to Chen--Xu~\cite{ChenXu2024}).
In particular,
\begin{equation}\label{eq:sizeA}
\#\cA_{\lambda}(X)\ \asymp\ \log X.
\end{equation}

In what follows, the parameters $s,r_1,\dots,r_s$ (and hence $\lambda$ and the set $\cA_{\lambda}$) are fixed, and all implied constants are allowed to depend on this data.

Define the representation function
\begin{equation}\label{eq:def-f}
f(n)\ :=\ \sum_{a\in\cA_{\lambda}(X)} \1_{\mathbb{P}}(n-a),
\end{equation}
so that $f(n)$ counts representations $n=p+a$ with $p\in\mathbb{P}$ and $a\in\cA_{\lambda}(X)$.
For real $x$ define the window sums
\begin{equation}\label{eq:def-R}
R(x)\ :=\ \sum_{x<n\leqslant x+h} f(n),
\qquad
Q(x)\ :=\ \sum_{x<n\leqslant x+h} f(n)^2,
\end{equation}
and the corresponding count of \emph{elements} of the sumset $\mathcal{S}$,
\begin{equation}\label{eq:def-Sx}
S(x)\ :=\ \#\left(\mathcal{S}\cap(x,x+h]\right)\ =\ \sum_{x<n\leqslant x+h}\1_{f(n)\geqslant 1}.
\end{equation}

\medskip

\begin{abstract}
Let $X$ be large and let $\PP$ denote the set of primes. Fix positive real parameters $r_1,\dots,r_s$ and a parameter $\lambda>0$ determined by a balancing relation, and let $\cA_\lambda(X)\subset[1,2X]$ be the associated lacunary set generated by sums of powers of $2$ with polynomially growing exponents. Set $\cS_\lambda:=\PP+\cA_\lambda(X)$.
Fix $\varepsilon>0$, choose $\theta$ with $2/15+\varepsilon<\theta<1-\delta_0$, where $\delta_0>0$ is an absolute constant, and set $h=X^{\theta}$.
We prove that for all but $O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ values of $x\in[X,2X]$, the short interval $(x,x+h]$ contains $\asymp_{\varepsilon} h$ integers of the form $p+a$ with $p\in\PP$ and $a\in\cA_\lambda(X)$.
\end{abstract}

\medskip

Theorem~\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the
sumset $\cS$ meets almost all intervals of length $h$ with the expected order of magnitude.
To access it through Cauchy--Schwarz, it is convenient to also record the corresponding
first-moment statement for the representation function $f$.

\begin{theorem}[Typical number of representations]\label{thm:main-rep}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
R(x)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.
\end{theorem}

\begin{lemma}[Guth--Maynard, almost-all short intervals]\label{lem:GM}
Fix $\varepsilon>0$. There exists an absolute constant $\delta_0>0$ such that the following holds. Let $X\geqslant 3$ and let $y$ satisfy
\[
X^{2/15+\varepsilon}\ \leqslant\ y\ \leqslant\ X^{1-\delta_0}.
\]
Then there exists a constant $c=c_{\varepsilon}>0$ such that for all but
$O_{\varepsilon}\left(X\exp\left(-c(\log X)^{1/4}\right)\right)$ integers $t\in[X,2X]$ one has
\[
\pi(t+y)-\pi(t)\ =\ \frac{y}{\log t}\ +\ O_{\varepsilon}\left(y\exp\left(-c(\log X)^{1/4}\right)\right).
\]
This follows from \cite[Cor.~1.4]{GuthMaynard2024}; the corollary is stated for real $x$, and restricting to integer $t$ gives the same exceptional-set bound.
\end{lemma}

\begin{proposition}[First moment lower bound]\label{prop:first-moment}
Fix $\varepsilon>0$ and let $\delta_0>0$ be as in \cref{lem:GM}. Let $\theta$ satisfy $2/15+\varepsilon\leqslant\theta<1-\delta_0$ and set $h=X^{\theta}$. Then there exists a set
$\mathcal{E}\subset[X,2X]\cap\Z$ with
\[
\#\mathcal{E}\ \ll_{\varepsilon,\lambda}\ X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)
\]
such that
\[
R(x)\ \gg\ h
\]
for all $x\in([X,2X]\cap\Z)\setminus\mathcal{E}$.
\end{proposition}

We apply \cref{lem:GM} twice: first with the scale parameter $X' := X/2$ to control $\pi(t+h)-\pi(t)$ for $t\in[X/2,X]$, and then with the scale parameter $X' := X$ to control it for $t\in[X,2X]$. Since $\theta\geqslant 2/15+\varepsilon$, for $X$ sufficiently large we have $(X/2)^{2/15+\varepsilon/2}\leqslant h$. Moreover, since $\theta<1-\delta_0$ with $\delta_0>0$ fixed, for $X$ sufficiently large we also have $h=X^{\theta}\leqslant (X/2)^{1-\delta_0}$. Consequently the hypotheses of \cref{lem:GM} apply (with $\varepsilon$ replaced by $\varepsilon/2$) on both dyadic blocks. We therefore obtain an exceptional set $\mathcal{E}_a\subset[X,2X]\cap\Z$ with
\[
\#\mathcal{E}_a\ \ll_{\varepsilon}\ X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)
\]
such that for all $x\in([X,2X]\cap\Z)\setminus\mathcal{E}_a$ we have
\begin{equation}\label{eq:GM-shifted}
\pi(x+h-a)-\pi(x-a)\ =\ \frac{h}{\log(x-a)}\ +\ O_{\varepsilon}\left(h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right).
\end{equation}

For $x\in([X,2X]\cap\Z)\setminus\mathcal{E}$, summing \eqref{eq:GM-shifted} over $a\in\cA_{\lambda}(X;h)$ yields
\[
\begin{aligned}
R(x)
&\geqslant\ \sum_{a\in\cA_{\lambda}(X;h)}\left(\pi(x+h-a)-\pi(x-a)\right)
\\
&=\ h\sum_{a\in\cA_{\lambda}(X;h)}\frac{1}{\log(x-a)}\ +\ O_{\varepsilon}\left(A_h h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right).
\end{aligned}
\]
Since $x-a\asymp X$ uniformly for $x\in[X,2X]$ and $a\leqslant h/2\leqslant X/2$, the main term satisfies
\[
\sum_{a\in\cA_{\lambda}(X;h)}\frac{1}{\log(x-a)}\ \asymp\ \frac{A_h}{\log X}\ \asymp\ \frac{\log h}{\log X},
\]
and the error term is negligible. Since $h=X^{\theta}$, we have $\log h=\theta \log X$, hence $R(x)\gg h$ for all $x\notin\mathcal{E}$.
\end{proof}

\begin{theorem}[Local density of $\cS$ in short intervals]\label{thm:main-elements}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
\#\left(\cS\cap(x,x+h]\right)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.
\end{theorem}

Theorem~\ref{thm:main-elements} is the conclusion we ultimately care about: it asserts that the
sumset $\mathcal{S}$ meets almost all intervals of length $h$ with the expected order of magnitude.
To access it through Cauchy--Schwarz, it is convenient to also record the corresponding
first-moment statement for the representation function $f$.

\begin{theorem}[Typical number of representations]\label{thm:main-rep}
Fix $\varepsilon>0$. Let $\theta$ satisfy $2/15+\varepsilon<\theta<1-\delta_0$ for some absolute constant $\delta_0>0$, and set $h=X^{\theta}$.
Then for all but $O_{\varepsilon}\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)$ integers $x\in[X,2X]$, one has
\[
R(x)\ \asymp\ h.
\]
The implied constants depend at most on $\lambda$, $\varepsilon$ and $\theta$.
\end{theorem}

\medskip

\noindent\emph{Strategy.} The pointwise upper bound $R(x)\ll h$ follows from Brun--Titchmarsh together with $\#\cA_{\lambda}(X)\asymp\log X$.
The almost-all lower bound $R(x)\gg h$ is obtained by restricting to $a\leqslant h/2$ and applying the Guth--Maynard~\cite[Cor.~1.4]{GuthMaynard2024} asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).
To pass from representations to elements we combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\ll h$.
The latter is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\in\cA_{\lambda}(X)$.

\medskip

\noindent\emph{Organization of the paper.} Section~2 establishes the first-moment estimate for the local representation count $R(x)$ in almost all short intervals. Section~3 proves a uniform local mean-square bound, leading to the required second-moment control. In Section~4 we combine these inputs with Cauchy--Schwarz to deduce Theorems~\ref{thm:main-elements} and~\ref{thm:main-rep}. Finally, Section~5 proves Theorem~\ref{thm:erdos-short-interval} on large short-interval multiplicities in the Romanoff model.

“To access [Theorem~\ref{thm:main-elements}] through Cauchy--Schwarz,” the paper first proves a first-moment statement for the representation function $f$, namely Theorem~\ref{thm:main-rep} that $R(x)\asymp h$ for almost all $x\in[X,2X]$. For this, the “pointwise upper bound $R(x)\ll h$ follows from Brun--Titchmarsh together with $\#\cA_{\lambda}(X)\asymp\log X$,” while the “almost-all lower bound $R(x)\gg h$ is obtained by restricting to $a\leqslant h/2$ and applying the Guth--Maynard ... asymptotic for primes in almost all short intervals (with a stretched-exponential exceptional set).”

“To pass from representations to elements” (i.e. to deduce Theorem~\ref{thm:main-elements}), they “combine Cauchy--Schwarz, which relates $R(x)$, $S(x)$ and the local second moment $Q(x)$, with a uniform bound $Q(x)\ll h$.” This uniform second-moment bound “is proved using Selberg's upper-bound sieve for prime pairs together with Chen--Ding--Xu--Zhai's estimate controlling the contribution of small prime divisors in differences $a_1-a_2$ with $a_1,a_2\in\cA_{\lambda}(X)$.”

Question:
Let \(\mathbb P\) denote the prime numbers. Fix an integer \(s\ge 2\) and real numbers \(r_1,\dots,r_s>1\) such that
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}<1\le \frac{1}{r_1}+\cdots+\frac{1}{r_s},
\]
and define \(\lambda>0\) by
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}+\frac{1}{\lambda r_s}=1.
\]
Set
\[
\mathcal A_\lambda:=\left\{2^{\lfloor k_1^{r_1}\rfloor}+\cdots+2^{\lfloor k_{s-1}^{r_{s-1}}\rfloor}+2^{\lfloor \lfloor k_s^{\lambda}\rfloor^{r_s}\rfloor}: k_1,\dots,k_s\in\mathbb N\right\},
\]
\[
\mathcal A_\lambda(X):=\mathcal A_\lambda\cap[1,2X],\qquad \mathcal S:=\mathbb P+\mathcal A_\lambda(X).
\]
Fix \(\varepsilon>0\), choose \(\theta\) with \(2/15+\varepsilon<\theta<1-\delta_0\), where \(\delta_0>0\) is an absolute constant, and set \(h=X^{\theta}\). Which statement holds for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\)?

\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
that is, for all but that exceptional set of integers \(x\), the interval \((x,x+h]\) contains on the order of \(h\) integers of the form \(p+a\) with \(p\in\mathbb P\) and \(a\in\mathcal A_\lambda(X)\); the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).

Choices:
B.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
for every integer \(x\in[X,2X]\cap\mathbb Z\); equivalently, there is no exceptional set, and the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
C.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\gg h,\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\); the implied constant depends at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
D.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) real numbers \(x\in[X,2X]\); the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
E.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)=h+O_{\varepsilon}\!\left(h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right),\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\), with implied constants depending at most on \(\lambda\), \(\varepsilon\), and \(\theta\).

The stem does not explicitly reveal the correct option. It sets up the hypotheses and asks for the resulting asymptotic statement, without giving away the exceptional-set size or the exact comparability conclusion.

This is essentially a direct theorem-recall item: the stem presents the full hypotheses of a specific result and asks which asymptotic conclusion holds. The correct answer is the theorem statement almost verbatim rather than a genuinely derived consequence.

There is some reasoning pressure because the student must distinguish the exact strength of the result from stronger, weaker, or mis-scaled alternatives. However, the task is still mainly recognition of the precise theorem statement rather than substantive mathematical generation.

The distractors are strong: one is an overstrong uniform version, one is a weaker true-looking statement, one gives the wrong local density scale, and one replaces an almost-all estimate with a sharp pointwise asymptotic. These reflect realistic failure modes in reading asymptotic theorems.

A solid theorem-discrimination MCQ with good distractors and no direct answer leakage, but it is largely a restatement/recognition question rather than a generative reasoning task.


Question:
Let \(\mathbb P\) denote the prime numbers. Fix an integer \(s\ge 2\) and real numbers \(r_1,\dots,r_s>1\) such that
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}<1\le \frac{1}{r_1}+\cdots+\frac{1}{r_s},
\]
and define \(\lambda>0\) by
\[
\frac{1}{r_1}+\cdots+\frac{1}{r_{s-1}}+\frac{1}{\lambda r_s}=1.
\]
Set
\[
\mathcal A_\lambda:=\left\{2^{\lfloor k_1^{r_1}\rfloor}+\cdots+2^{\lfloor k_{s-1}^{r_{s-1}}\rfloor}+2^{\lfloor \lfloor k_s^{\lambda}\rfloor^{r_s}\rfloor}: k_1,\dots,k_s\in\mathbb N\right\},
\]
\[
\mathcal A_\lambda(X):=\mathcal A_\lambda\cap[1,2X],\qquad \mathcal S:=\mathbb P+\mathcal A_\lambda(X).
\]
Fix \(\varepsilon>0\), choose \(\theta\) with \(2/15+\varepsilon<\theta<1-\delta_0\), where \(\delta_0>0\) is an absolute constant, and set \(h=X^{\theta}\). Which statement holds for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\)?

Choices:
B.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
for every integer \(x\in[X,2X]\cap\mathbb Z\); equivalently, there is no exceptional set, and the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
C.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\gg h,\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\); the implied constant depends at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
D.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)\asymp h,\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) real numbers \(x\in[X,2X]\); the implied constants depend at most on \(\lambda\), \(\varepsilon\), and \(\theta\).
E.
\[\#\bigl(\mathcal S\cap(x,x+h]\bigr)=h+O_{\varepsilon}\!\left(h\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right),\]
for all but \(O_{\varepsilon}\!\left(X\exp\left(-c_{\varepsilon}(\log X)^{1/4}\right)\right)\) integers \(x\in[X,2X]\cap\mathbb Z\), with implied constants depending at most on \(\lambda\), \(\varepsilon\), and \(\theta\).

The stem does not explicitly reveal the correct option. It states the hypotheses and asks for the valid conclusion; the correct answer is not leaked verbatim outside the answer choices.

This is essentially a theorem-recall item: the hypotheses are given and the correct choice is the theorem’s conclusion with only nearby variants as alternatives. It does not materially go beyond restating the result.

There is some reasoning pressure in distinguishing the exact strength of the conclusion from stronger or weaker variants (exceptional set, asymp vs. lower bound, integers vs. reals). However, it mainly tests precise recall of the theorem statement rather than substantial mathematical generation.

The distractors are plausible and mathematically meaningful: one removes the exceptional set, one weakens asymp to a one-sided bound, one changes the domain from integers to reals, and one strengthens to a sharp asymptotic. These reflect common failure modes in parsing analytic number theory statements.

A solid theorem-discrimination MCQ with strong distractors and no direct answer leakage, but it is largely a restatement/recall question rather than a genuinely generative reasoning task.


\begin{Thm}[Gradient regularity in Morrey spaces]\label{MainThm}
Let $u \in W^{1,p}(\Omega)$ be a local minimizer of  the functional $\mathcal{F}(\cdot;\Omega)$ defined
in\eqref{MainF}, and assume that
$\partial E$ is a hypersurface of class $C^1$.
Suppose that assumptions \emph{(H1)}--\emph{(H2)} are satisfied where $2<p<\frac{2n}{n-2}$.
Then, for every $0 \leq \lambda < n$,
\[
\nabla u \in L^{{p},\lambda}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every $\Omega' \Subset \Omega$, there exists a constant
$C = C\big(n, p, \nu, L, \beta, E, \operatorname{diam}(\Omega),
\operatorname{dist}(\Omega', \partial\Omega)\big) > 0$
such that
\[
\|\nabla u\|_{L^{{p},\lambda}(\Omega')} \leq C.
\]
\end{Thm}

Transmission problems arise naturally in the study of diffusive processes in
heterogeneous media, where the governing laws change abruptly across fixed interfaces.
Typical applications include elasticity theory, composite materials, and conductivity
phenomena. This kind of problems appear also in the study of thermal insulation of bodies under sources and prescribed boundary conditions (see for example \cite{AC,DPNST})
From a mathematical perspective, such problems are characterised by the presence of
piecewise-defined operators or energy densities, together with suitable transmission
conditions prescribed along the separating interface.
The first systematic investigation of transmission problems in elasticity theory dates
back to the seminal work of M.~Picone in the 1950s (see \cite{Pico}).

In this paper, we investigate a class of \emph{variational transmission problems}
associated with nonlinear integral functionals exhibiting discontinuous coefficients
across a prescribed interface.
Our focus is on the regularity properties of local minimizers, with particular attention
to the behaviour of the gradient near the interface.

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $E \subset \Omega$ be
a measurable subset.
We assume that the interface $\partial E$ is a hypersurface of class $C^1$.
We fix two constants $0 < \alpha < \beta < +\infty$ and define
\[
\sigma_E := \beta \mathbbm{1}_E + \alpha \mathbbm{1}_{E^c},
\]
where $\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.
We consider the integral functional
\begin{equation}\label{MainF}
\mathcal{F}(u;E) := \int_{\Omega} \sigma_E(x)\, F(\nabla u)\, dx,
\end{equation}
defined for functions $u \in W^{1,p}(\Omega)$.

The integrand $F \colon \mathbb{R}^n \to \mathbb{R}$ is assumed to belong to
$C^2(\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:
for all $\xi, \eta \in \mathbb{R}^n$,
\begin{equation}
\label{Monotonicity}\tag{H1}
\langle \nabla F(\xi) - \nabla F(\eta), \xi - \eta \rangle
\geq \nu \big( \mu^2 + |\xi|^2 + |\eta|^2 \big)^{\frac{p-2}{2}} |\xi - \eta|^2,
\end{equation}
\begin{equation}
\label{Growth}\tag{H2}
|\nabla F(\xi) - \nabla F(\eta)|
\leq L \big( \mu^2 + |\xi|^2 + |\eta|^2 \big)^{\frac{p-2}{2}} |\xi - \eta|,
\end{equation}
for some constants $\nu, L > 0$, $\mu \in (0,1]$, and $p > 2$.

Local minimizers of the functional $\mathcal{F}$ are understood in the usual
variational sense.

\begin{Def}
Let $u \in W^{1,p}(\Omega)$.
We say that $u$ is a \emph{local minimizer} of $\mathcal{F}(\cdot;\Omega)$ in $\Omega$
if, for every ball $B_r(x_0) \subset \Omega$ and every
$\phi \in W^{1,p}_0(B_r(x_0))$, one has
\[
\mathcal{F}(u;B_r(x_0)) \leq \mathcal{F}(u + \phi;B_r(x_0)).
\]
\end{Def}

Before stating our main result, let us place it in the context of the existing literature.
In the linear case $p = 2$, transmission problems with piecewise constant coefficients
have been extensively studied, and gradient regularity across sufficiently smooth
interfaces is by now classical (see, e.g.,~\cite{GT,LV,Grisvard}).
Such results follow from the elliptic regularity theory for divergence-form operators
with discontinuous coefficients (see for instance
\cite{Evans, GT, LM, Grisvard} and \cite[Theorem~7.53]{AFP}.
In this setting, local $W^{1,2}$- and H\"older-type regularity for the gradient of minimizers is well
understood under mild geometric assumptions on the interface.

More recently, fine regularity properties for linear transmission problems across
$C^{1,\alpha}$ interfaces have been obtained by Caffarelli, Soria-Carro, and Stinga
\cite{CSS}.

By contrast, the nonlinear case $p \neq 2$ remains far less developed.
To the best of our knowledge, the only available regularity results for nonlinear
transmission problems concern equations subject to \emph{additive} transmission
conditions on the interface.
In particular, in the recent work \cite{BPU}, BMO regularity of the gradient is obtained
for degenerate quasilinear equations under transmission conditions of the form
\[
g(|\nabla u_1|)\frac{\partial u_1}{\partial \nu}
-
g(|\nabla u_2|)\frac{\partial u_2}{\partial \nu}
= f \quad \text{on } \partial E.
\]

In contrast, the transmission condition naturally associated with the variational
functional \eqref{MainF} is of \emph{multiplicative} type, namely
\[
g_1(|\nabla u_1|)\frac{\partial u_1}{\partial \nu}
=
g_2(|\nabla u_2|)\frac{\partial u_2}{\partial \nu}
\quad \text{on } \partial E,
\]
which arises intrinsically from the Euler--Lagrange equations and expresses the
continuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the
variational structure of the functional \eqref{MainF}, as it arises directly from
the Euler--Lagrange equations and cannot be prescribed independently.
The presence of a discontinuous coefficient combined with nonlinear growth prevents
a direct application of standard techniques such as freezing arguments or difference
quotient estimates across the interface.
To the best of our knowledge, no Morrey-space regularity results for gradients of local
minimizers are available in this genuinely nonlinear variational transmission setting.

Denoting by $L^{{p},\lambda}$ the classical Morrey spaces, we can now state the main
result of this paper, which establishes local Morrey regularity for the gradient of
local minimizers under minimal geometric assumptions on the interface.

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and let $E \subset \Omega$ be
a measurable subset.
We assume that the interface $\partial E$ is a hypersurface of class $C^1$.
We fix two constants $0 < \alpha < \beta < +\infty$ and define
\[
\sigma_E := \beta \mathbbm{1}_E + \alpha \mathbbm{1}_{E^c},
\]
where $\mathbbm{1}_E$ denotes the characteristic function of the set $E$ and $E^c$ is the complementary of $E$.
We consider the integral functional
\begin{equation}\label{MainF}
\mathcal{F}(u;E) := \int_{\Omega} \sigma_E(x)\, F(\nabla u)\, dx,
\end{equation}
defined for functions $u \in W^{1,p}(\Omega)$.

The integrand $F \colon \mathbb{R}^n \to \mathbb{R}$ is assumed to belong to
$C^2(\mathbb{R}^n)$ and to satisfy standard $p$-growth and monotonicity conditions:
for all $\xi, \eta \in \mathbb{R}^n$,
\begin{equation}
\label{Monotonicity}\tag{H1}
\langle \nabla F(\xi) - \nabla F(\eta), \xi - \eta \rangle
\geq \nu \big( \mu^2 + |\xi|^2 + |\eta|^2 \big)^{\frac{p-2}{2}} |\xi - \eta|^2,
\end{equation}
\begin{equation}
\label{Growth}\tag{H2}
|\nabla F(\xi) - \nabla F(\eta)|
\leq L \big( \mu^2 + |\xi|^2 + |\eta|^2 \big)^{\frac{p-2}{2}} |\xi - \eta|,
\end{equation}
for some constants $\nu, L > 0$, $\mu \in (0,1]$, and $p > 2$.

\begin{Def}
Let $u \in W^{1,p}(\Omega)$.
We say that $u$ is a \emph{local minimizer} of $\mathcal{F}(\cdot;\Omega)$ in $\Omega$
if, for every ball $B_r(x_0) \subset \Omega$ and every
$\phi \in W^{1,p}_0(B_r(x_0))$, one has
\[
\mathcal{F}(u;B_r(x_0)) \leq \mathcal{F}(u + \phi;B_r(x_0)).
\]
\end{Def}

By contrast, the nonlinear case $p \neq 2$ remains far less developed.
To the best of our knowledge, the only available regularity results for nonlinear
transmission problems concern equations subject to \emph{additive} transmission
conditions on the interface.
In particular, in the recent work \cite{BPU}, BMO regularity of the gradient is obtained
for degenerate quasilinear equations under transmission conditions of the form
\[
g(|\nabla u_1|)\frac{\partial u_1}{\partial \nu}
-
g(|\nabla u_2|)\frac{\partial u_2}{\partial \nu}
= f \quad \text{on } \partial E.
\]

In contrast, the transmission condition naturally associated with the variational
functional \eqref{MainF} is of \emph{multiplicative} type, namely
\[
g_1(|\nabla u_1|)\frac{\partial u_1}{\partial \nu}
=
g_2(|\nabla u_2|)\frac{\partial u_2}{\partial \nu}
\quad \text{on } \partial E,
\]
which arises intrinsically from the Euler--Lagrange equations and expresses the
continuity of the nonlinear flux across the interface. We stress that this multiplicative transmission condition is intrinsic to the
variational structure of the functional \eqref{MainF}, as it arises directly from
the Euler--Lagrange equations and cannot be prescribed independently.
The presence of a discontinuous coefficient combined with nonlinear growth prevents
a direct application of standard techniques such as freezing arguments or difference
quotient estimates across the interface.
To the best of our knowledge, no Morrey-space regularity results for gradients of local
minimizers are available in this genuinely nonlinear variational transmission setting.

Denoting by $L^{{p},\lambda}$ the classical Morrey spaces, we can now state the main
result of this paper, which establishes local Morrey regularity for the gradient of
local minimizers under minimal geometric assumptions on the interface.

\begin{Prop}\label{FJNL}
Let $u$ be a local minimizer of the functional $\mathcal{F}(\cdot;\Omega)$ defined
in \eqref{MainF} under the assumption of Theorem~\ref{MainThm}.
Then there exists a constant $0 < \tau_0 < 1$ such that the following holds:
for every $\tau \in (0,\tau_0)$ there exists $\varepsilon_0 = \varepsilon_0(\tau) > 0$
with the property that, if $B_r(x_0) \Subset \Omega$ and one of the following
conditions is satisfied,
\begin{enumerate}[label=(\roman*)]
\item $|E \cap B_r(x_0)| < \varepsilon_0 |B_r|$,
\item $|B_r(x_0) \setminus E| < \varepsilon_0 |B_r|$,
\item there exists a half-space $H$ such that
\[
\frac{|(E \Delta H) \cap B_r(x_0)|}{|B_r|} < \varepsilon_0,
\]
\end{enumerate}
then, for every $0 < \delta < n$, the estimate
\[
\int_{B_{\tau r}(x_0)} |\nabla u|^p \, dx
\leq
C_0 \tau^{n-\delta}
\left(
\int_{B_r(x_0)} |\nabla u|^p \, dx + r^n
\right)
\]
holds, where the constant $C_0$ depends only on
$n, \nu, L, \alpha,\beta, \delta$ and $\|\nabla u\|_{L^2(\Omega)}$.
\end{Prop}
\begin{Rem}
\label{Reg}
 For the proof of Theorem \ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.
\end{Rem}

\begin{Lem}[Standard $p$-growth bounds]\label{lem:p-growth}
Assume \((H1)\)--\((H2)\). Then there exists a constant $C=C(n,p,\nu,L)>0$ such that for all $\xi\in\R^n$
\[
C^{-1}(\mu^2+|\xi|^2)^{\frac p2}-C\mu^p \le F(\xi)\le C(\mu^2+|\xi|^2)^{\frac p2}+C\mu^p .
\]
\end{Lem}
The following results on higher integrability and local Hölder continuity
of minimizers are classical. Since their proofs are standard, we omit them and refer the reader to
\cite{Giu,GiaMar}.
{
\begin{Thm}[Higher integrability]
\label{HigherInt}
Let $u \in W^{1,p}(\Omega)$ be a local minimizer of the functional \eqref{MainF}.
There exist $s > 1$ and $C = C(n,p,\nu,L,\beta)$ such that for any ball
$B_{2r}(x_{0})\subset\subset\Omega$
\begin{equation}
 \int_{B_r(x_0)}|\nabla u|^{sp}\,dx\leq C\bigg(\int_{B_{2r}(x_0)}\big(|\nabla u|^p+\mu^p\big)\,dx
 \bigg)^s.
\end{equation}
\end{Thm}
\begin{Thm}[H\"older continuity]Let $u \in W^{1,p}(\Omega)$ be a local minimizer of the functional \eqref{MainF}. Then
\begin{enumerate}[label=(\roman*)]
\item For any open set $\Omega'\Subset \Omega$ the quantity $\norm{u}_{L^{\infty}(\Omega')}$ is bounded by a constant $C = C(n,p,\nu,L,\beta)\norm{u}_{L^2(\Omega)}$.
\item u is locally H\"older continuous in $\Omega$.
\end{enumerate}
\end{Thm}
}

\begin{Thm}\label{DS}
Let $u$ be a minimizer of $\mathcal{F}$ in $B_r(x_0)$. Then $\partial_i u\in W^{1,2}_{loc}(B_{r})$ for every $i=1\dots n-1$. Moreover the following properties hold:
\begin{enumerate}[label=\roman*)]
\item $u \in W^{2,2}_{\mathrm{loc}}(B_r^+)$ e $u \in W^{2,2}_{\mathrm{loc}}(B_r^-)$;
\item for every $x_0 \in \Gamma_r$ and $R>0$ such that $B_{4R}(x_0)\subset B_r$, it holds that
\begin{align}
\int_{B_{R/2}} V^{p-2}\,|\nabla\nabla' u|^2 \,dx
&\leq \frac{C}{R^2}\int_{B_{2R}} V^p \,dx, \label{eq:tangential-estimate}
\end{align}
where $C=C\!\left(\tfrac{L}{\nu},\,n,\,p,\,\alpha,\,\beta,\mu\right)$ is a positive constant.
\end{enumerate}
\end{Thm}

\section{Proof of Proposition 1.3}
\begin{proof}
Fix $\tau\in(0,\tau_0)$ (to be chosen small only depending on $n,p,\nu,L,\beta/\alpha$) and assume without loss of generality that $x_0=0$.
We prove that there exists $\e_0=\e_0(\tau)>0$ such that if $B_r(x_0)\Subset\Om$ and one of {\rm(i)}--{\rm(iii)}
holds, then for every $0<\delta<n$, then
\begin{equation}\label{eq:prop13-goal}
\int_{B_{\tau r}} |\nabla u|^p\, dx
\leq
C_0\tau^{n-\delta}\bigg( \int_{B_r} |\nabla u|^p \, dx
+r^n\bigg),
\end{equation}
where $C_0=C_0(n,p,\nu,L,\beta,\delta,\|\nabla u\|_{L^2(\Om)})>0$.\\
\indent
\noindent\textbf{Step 1: the flat comparison in case (iii).}
Assume that (iii) holds true. Up to a rigid motion we may assume that $H=\{x_n>0\}$ and set
\[
E_H:=H\cap B_{r},\quad \sigma_H:=\beta\mathbbm 1_{E_H}+\alpha\mathbbm 1_{B_r\setminus E_H}.
\]
Let $u_H\in u+W^{1,p}_0(B_\frac{r}{2})$ be the unique minimizer of the functional
\[
\mathcal F_H(w;B_\frac{r}{2}):=\int_{B_\frac{r}{2}}\sigma_H(x)\,F(\nabla w)\,dx .
\]
Subtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $u_H$ and testing them with $u-u_H$, we get
\begin{align}
 \int_{B_{\frac{r}{2}}} \sigma_E
\big(DF(\nabla u)-DF(\nabla u_H)\big)
\cdot(\nabla u-\nabla u_H)\,dx
& =
\int_{B_{\frac{r}{2}}} (\sigma_H-\sigma_E)\,
DF(\nabla u_H)\cdot(\nabla u-\nabla u_H)\,dx
\end{align}
Using assumptions \eqref{Monotonicity} and \eqref{Growth} and H\"older's inequality, for every $\varepsilon>0$ it holds that
\begin{align}
 \int_{B_{\frac{r}{2}}}|\D u_H-\D u|^p\,dx
 & \leq C\int_{(E\Delta E_H)\cap B_{\frac{r}{2}}}\big(\mu^2+|\D u_H|^2\big)^{\frac{p-2}{2}}|\D u-\D u_H|\,dx\\
& \leq C\bigg(\frac{1}{\varepsilon}\int_{(E\Delta E_H)\cap B_{\frac{r}{2}}}\big(\mu^2+|\D u_H|^2\big)^{\frac{p}{2}}\,dx+\varepsilon\int_{B_\frac{r}{2}}|\D u-\D u_H|^p\,dx \bigg).
\end{align}
Choosing $\varepsilon$ sufficiently small, we get
\begin{align}
 \int_{B_{\frac{r}{2}}}|\D u_H-\D u|^p\,dx
& \leq C\int_{(E\Delta E_H)\cap B_{\frac{r}{2}}}\big(\mu^2+|\D u_H|^2\big)^{\frac{p}{2}}\,dx.
\end{align}
Thus, the minimality of $u_H$ with respect to $u$, H\"older's inequality and Theorem \ref{HigherInt} yield
\begin{align}
\label{eqq1}
\int_{B_{\frac{r}{2}}}|\D u_H-\D u|^p\,dx& \leq \int_{(E\Delta E_H)\cap B_\frac{r}{2}} (1+|\nabla u|^2)^{\frac p2}\,dx\\
& \le \bigg(\frac{|(E\Delta E_H)\cap B_r|}{|B_r|}\bigg)^{1-\frac1s}|B_r|\bigg(\dashint_{B_r}(1+|\nabla u|^2)^{\frac{sp}{2}}\,dx\bigg)^{\frac1s}\\
& \le C\,\e_0^{1-\frac1s}\int_{B_r}(1+|\nabla u|^2)^{\frac p2}\,dx,
\end{align}
where we have used assumption \emph{iii)} and $C=C(n,p,\beta/\alpha,\nu/L)$ is a positive constant.

\medskip
\noindent\textbf{Step 3: the one-phase cases (i) and (ii).}
Assume (i): $|E\cap B_r|<\e_0|B_r|$ (the case (ii) is analogous). Let us denote by $v\in u+W^{1,p}_0(B_\frac{r}{2})$ be the unique minimizer of the functional
\[
\mathcal G(w):=\int_{B_\frac{r}{2}}\,F(\nabla w)\,dx .
\]
We can proceed as in step $(iii)$ subtracting the Euler-Lagrange equation satisfied by $u$ from the one satisfied by $v$ and testing them with $u-v$, getting
\begin{align}
 \int_{B_{\frac{r}{2}}} \sigma_E
\big(DF(\nabla u)-DF(\nabla v)\big)
\cdot(\nabla u-\nabla v)\,dx
& =
\int_{B_{\frac{r}{2}}} (\alpha-\sigma_E)\,
DF(\nabla v)\cdot(\nabla u-\nabla v)\,dx
\end{align}
Using assumptions \eqref{Monotonicity} and \eqref{Growth} and H\"older's inequality, for every $\varepsilon>0$ it holds that
\begin{align}
 \int_{B_{\frac{r}{2}}}|\D v -\D u|^p\,dx
 & \leq C\int_{E\cap B_{\frac{r}{2}}}\big(\mu^2+|\D v|^2\big)^{\frac{p-2}{2}}|\D u-\D v|\,dx\\
& \leq C\bigg(\frac{1}{\varepsilon}\int_{E\cap B_{\frac{r}{2}}}\big(\mu^2+|\D v|^2\big)^{\frac{p}{2}}\,dx+\varepsilon\int_{B_\frac{r}{2}}|\D u-\D v|^p\,dx \bigg).
\end{align}
Choosing $\varepsilon$ sufficiently small, we get
\begin{align}
 \int_{B_{\frac{r}{2}}}|\D v-\D u|^p\,dx
& \leq C\int_{E\cap B_{\frac{r}{2}}}\big(\mu^2+|\D v|^2\big)^{\frac{p}{2}}\,dx.
\end{align}
Using the minimality of $v$ compared with $u$ and Theorem
\ref{HigherInt} we deduce,
\begin{align}
\label{eqq11}
\int_{B_{\frac{r}{2}}}|\D v-\D u|^p\,dx& \leq \int_{E\cap B_{\frac{r}{2}}} (1+|\nabla u|^2)^{\frac p2}\,dx\\
& \le \bigg(\frac{E\cap B_{r}}{|B_r|}\bigg)^{1-\frac1s}|B_r|\bigg(\dashint_{B_r}(1+|\nabla u|^2)^{\frac{sp}{2}}\,dx\bigg)^{\frac1s}\\
& \le C\,\e_0^{1-\frac1s}\int_{B_r}(1+|\nabla u|^2)^{\frac p2}\,dx,
\end{align}
where we have used assumption \emph{i)} and $C=C(n,p,\beta/\alpha,\nu/L)$ is a positive constant. Thereafter we can argue as in the previous step using the classical decay estimate for minimizers of regular integrals in the Calculus of Variations
\[
\int_{B_{\tau \rho}}|\nabla v|^p\,dx
\le C\tau^{n}\int_{B_{ \rho}}|\nabla u|^p\,dx,
\]
for every $\rho\leq \frac{r}{2}$.
\end{proof}

\begin{Rem}
Within the present variational transmission framework, Morrey regularity represents
the natural level of regularity that can be expected for the gradient of local
minimizers, in view of the discontinuous coefficients and the intrinsic transmission
condition.
\end{Rem}
The proof of Theorem~\ref{MainThm} is obtained as a consequence of the following
decay estimate for the energy.

\begin{Prop}\label{FJNL}
Let $u$ be a local minimizer of the functional $\mathcal{F}(\cdot;\Omega)$ defined
in \eqref{MainF} under the assumption of Theorem~\ref{MainThm}.
Then there exists a constant $0 < \tau_0 < 1$ such that the following holds:
for every $\tau \in (0,\tau_0)$ there exists $\varepsilon_0 = \varepsilon_0(\tau) > 0$
with the property that, if $B_r(x_0) \Subset \Omega$ and one of the following
conditions is satisfied,
\begin{enumerate}[label=(\roman*)]
\item $|E \cap B_r(x_0)| < \varepsilon_0 |B_r|$,
\item $|B_r(x_0) \setminus E| < \varepsilon_0 |B_r|$,
\item there exists a half-space $H$ such that
\[
\frac{|(E \Delta H) \cap B_r(x_0)|}{|B_r|} < \varepsilon_0,
\]
\end{enumerate}
then, for every $0 < \delta < n$, the estimate
\[
\int_{B_{\tau r}(x_0)} |\nabla u|^p \, dx
\leq
C_0 \tau^{n-\delta}
\left(
\int_{B_r(x_0)} |\nabla u|^p \, dx + r^n
\right)
\]
holds, where the constant $C_0$ depends only on
$n, \nu, L, \alpha,\beta, \delta$ and $\|\nabla u\|_{L^2(\Omega)}$.
\end{Prop}
\begin{Rem}
\label{Reg}
    For the proof of Theorem \ref{MainThm}, it would be sufficient to state the proposition only in case (i). We nevertheless include the other two cases for the sake of completeness.
\end{Rem}

We give an outline of the proof of Proposition~\ref{FJNL}. It relies on decay estimates for the gradient
near the interface, first established in a flat configuration, i.e.\ when the interface coincides
with a hyperplane.
Section~3 is devoted to this ``flat case''.

The starting point is a local boundedness estimate for the tangential gradient
$\nabla' u$, where $\nabla'=(\partial_{x_1},\dots,\partial_{x_{n-1}})$ denotes the vector of derivatives parallel to
the interface.
This is proved in Proposition~3.2 via a difference-quotient argument with test functions involving
only tangential increments.
A key ingredient is a Moser iteration applied to
$Z:=|\Delta_{i,h}u|^{m/p}$.
During the estimates, the full gradient $\nabla u$ enters through the structure conditions
\emph{(H1)}--\emph{(H2)} and is treated as a weight; this is the origin of the restriction
$2<p<\frac{2n}{n-2}$.

The bound obtained in Proposition~3.2 is preliminary, since the decay exponent in the estimates is not optimal and
the constant may depend on $\mu$.
To obtain the scale-invariant Lipschitz estimate for $\nabla' u$ stated in Proposition~3.4,
we perform a rescaling argument, which requires uniformity with respect to $\mu$.
This uniformity is proved in Lemma~3.3, exploiting the boundedness of $\nabla' u$ and a Poincar\'e-type
inequality for $Z:=|\partial_i u|^{m/p}$, $i=1,\dots,n-1$, which is an admissible Sobolev function by
Theorem~3.1.

Once the tangential Lipschitz bound is available, we prove the Morrey estimate in the flat case in
Theorem~3.5 by a comparison argument (Section~3.2).
Finally, the general case follows from a standard flattening/approximation procedure using the $C^1$
regularity of the interface, along the lines of the quadratic theory (see, e.g.,~\cite{E,FJ}).

The proof of Theorem~\ref{MainThm} is obtained as a consequence of the decay estimate for the energy stated in Proposition~\ref{FJNL}.

An outline is given for the proof of Proposition~\ref{FJNL}:
\begin{itemize}
\item The argument \emph{relies on decay estimates for the gradient near the interface}, first established in a \emph{flat configuration} (interface coincides with a hyperplane), treated in Section~3.
\item Starting point: a \emph{local boundedness estimate for the tangential gradient} $\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \emph{difference-quotient argument} with test functions involving only tangential increments.
\item Key ingredient: a \emph{Moser iteration} applied to $Z:=|\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\nabla u$ appears through (H1)--(H2) and is \emph{treated as a weight}; this is cited as the origin of the restriction $2<p<\frac{2n}{n-2}$.
\item Since the Proposition~3.2 bound has a non-optimal decay exponent and constants that may depend on $\mu$, one obtains the \emph{scale-invariant Lipschitz estimate for $\nabla' u$} (Proposition~3.4) by a \emph{rescaling argument} requiring \emph{uniformity with respect to $\mu$}. This uniformity is proved in Lemma~3.3, using boundedness of $\nabla' u$ and a \emph{Poincar\'e-type inequality} for $Z:=|\partial_i u|^{m/p}$, $i=1,\dots,n-1$, which is admissible by Theorem~3.1.
\item With the tangential Lipschitz bound, the \emph{Morrey estimate in the flat case} is proved in Theorem~3.5 by a \emph{comparison argument} (Section~3.2).
\item The \emph{general case} then follows by a \emph{standard flattening/approximation procedure} using the $C^1$ regularity of the interface, along the lines of the quadratic theory.
\end{itemize}

Additionally, Remark~\ref{Reg} notes that for proving Theorem~\ref{MainThm} it would be sufficient to have Proposition~\ref{FJNL} only in case (i), though cases (ii)--(iii) are included for completeness.

The proof of the main theorem is obtained as a consequence of the decay estimate for the energy stated in the following proposition.

\begin{Prop}\label{FJNL}
Let $u$ be a local minimizer of the functional $\mathcal{F}(\cdot;\Omega)$ defined
in \eqref{MainF} under the assumption of Theorem~\ref{MainThm}.
Then there exists a constant $0 < \tau_0 < 1$ such that the following holds:
for every $\tau \in (0,\tau_0)$ there exists $\varepsilon_0 = \varepsilon_0(\tau) > 0$
with the property that, if $B_r(x_0) \Subset \Omega$ and one of the following
conditions is satisfied,
\begin{enumerate}[label=(\roman*)]
\item $|E \cap B_r(x_0)| < \varepsilon_0 |B_r|$,
\item $|B_r(x_0) \setminus E| < \varepsilon_0 |B_r|$,
\item there exists a half-space $H$ such that
\[
\frac{|(E \Delta H) \cap B_r(x_0)|}{|B_r|} < \varepsilon_0,
\]
\end{enumerate}
then, for every $0 < \delta < n$, the estimate
\[
\int_{B_{\tau r}(x_0)} |\nabla u|^p \, dx
\leq
C_0 \tau^{n-\delta}
\left(
\int_{B_r(x_0)} |\nabla u|^p \, dx + r^n
\right)
\]
holds, where the constant $C_0$ depends only on
$n, \nu, L, \alpha,\beta, \delta$ and $\|\nabla u\|_{L^2(\Omega)}$.
\end{Prop}

An outline is given for the proof of Proposition~\ref{FJNL}:
\begin{itemize}
\item The argument \emph{relies on decay estimates for the gradient near the interface}, first established in a \emph{flat configuration} (interface coincides with a hyperplane), treated next.
\item Starting point: a \emph{local boundedness estimate for the tangential gradient} $\nabla' u$ (derivatives parallel to the interface). This is proved in Proposition~3.2 via a \emph{difference-quotient argument} with test functions involving only tangential increments.
\item Key ingredient: a \emph{Moser iteration} applied to $Z:=|\Delta_{i,h}u|^{m/p}$. In the estimates, the full gradient $\nabla u$ appears through (H1)--(H2) and is \emph{treated as a weight}; this is cited as the origin of the restriction $2<p<\frac{2n}{n-2}$.
\item Since the Proposition~3.2 bound has a non-optimal decay exponent and constants that may depend on $\mu$, one obtains the \emph{scale-invariant Lipschitz estimate for $\nabla' u$} (Proposition~3.4) by a \emph{rescaling argument} requiring \emph{uniformity with respect to $\mu$}. This uniformity is proved in Lemma~3.3, using boundedness of $\nabla' u$ and a \emph{Poincar\'e-type inequality} for $Z:=|\partial_i u|^{m/p}$, $i=1,\dots,n-1$, which is admissible by Theorem~3.1.
\item With the tangential Lipschitz bound, the \emph{Morrey estimate in the flat case} is proved in Theorem~3.5 by a \emph{comparison argument}.
\item The \emph{general case} then follows by a \emph{standard flattening/approximation procedure} using the $C^1$ regularity of the interface, along the lines of the quadratic theory.
\end{itemize}

Additionally, Remark~\ref{Reg} notes that for proving the main theorem it would be sufficient to have Proposition~\ref{FJNL} only in case (i), though cases (ii)--(iii) are included for completeness.

[Gradient regularity in Morrey spaces]\label{MainThm}
Let $u \in W^{1,p}(\Omega)$ be a local minimizer of the functional $\mathcal{F}(\cdot;\Omega)$ defined in
\begin{equation}\label{MainF}
\mathcal{F}(u;E) := \int_{\Omega} \sigma_E(x)\, F(\nabla u)\, dx,
\end{equation}
and assume that $\partial E$ is a hypersurface of class $C^1$.
Suppose that assumptions \emph{(H1)}--\emph{(H2)} are satisfied where $2<p<\frac{2n}{n-2}$.
Then, for every $0 \leq \lambda < n$,
\[
\nabla u \in L^{{p},\lambda}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every $\Omega' \Subset \Omega$, there exists a constant
$C = C\big(n, p, \nu, L, \beta, E, \operatorname{diam}(\Omega),
\operatorname{dist}(\Omega', \partial\Omega)\big) > 0$
such that
\[
\|\nabla u\|_{L^{{p},\lambda}(\Omega')} \leq C.
\]

Question:
Let \(\Omega\subset\mathbb{R}^n\) be a bounded domain, let \(E\subset\Omega\) be measurable with \(\partial E\) a \(C^1\) hypersurface, and fix constants \(0<\alpha<\beta<\infty\). Define \(\sigma_E:=\beta\mathbbm{1}_E+\alpha\mathbbm{1}_{E^c}\) and, for \(v\in W^{1,p}(\Omega)\),
\[\mathcal F(v;E):=\int_{\Omega}\sigma_E(x)F(\nabla v)\,dx.\]
Assume \(F\in C^2(\mathbb{R}^n)\) and that for some \(\nu,L>0\), \(\mu\in(0,1]\), and every \(\xi,\eta\in\mathbb{R}^n\),
\[\langle \nabla F(\xi)-\nabla F(\eta),\xi-\eta\rangle\ge \nu\big(\mu^2+|\xi|^2+|\eta|^2\big)^{\frac{p-2}{2}}|\xi-\eta|^2,\]
\[|\nabla F(\xi)-\nabla F(\eta)|\le L\big(\mu^2+|\xi|^2+|\eta|^2\big)^{\frac{p-2}{2}}|\xi-\eta|,\]
with \(2<p<\frac{2n}{n-2}\). Suppose \(u\in W^{1,p}(\Omega)\) is a local minimizer in the sense that for every ball \(B_r(x_0)\subset\Omega\) and every \(\phi\in W_0^{1,p}(B_r(x_0))\),
\[\mathcal F(u;B_r(x_0))\le \mathcal F(u+\phi;B_r(x_0)).\]
Which conclusion about \(\nabla u\) holds? Here \(L^{p,\lambda}(\Omega')\) is the classical Morrey space, for instance with norm \(\|f\|_{L^{p,\lambda}(\Omega')}:=\sup_{B_\rho(x)\subset\Omega'}\big(\rho^{-\lambda}\int_{B_\rho(x)}|f|^p\,dy\big)^{1/p}\).

For every \(0\le \lambda<n\), one has \(\nabla u\in L^{p,\lambda}_{\mathrm{loc}}(\Omega)\). Moreover, for every \(\Omega'\Subset\Omega\), there exists a constant \(C=C\big(n,p,\nu,L,\beta,E,\operatorname{diam}(\Omega),\operatorname{dist}(\Omega',\partial\Omega)\big)>0\) such that
\[\|\nabla u\|_{L^{p,\lambda}(\Omega')}\le C.\]

Choices:
B.
For every \(0\le \lambda<n\), one has \(\nabla u\in L^{p,\lambda}_{\mathrm{loc}}(\Omega)\). Moreover, for every \(\Omega'\Subset\Omega\), there exists a constant \(C=C\big(n,p,\nu,L,\alpha,\beta,E,\operatorname{diam}(\Omega),\operatorname{dist}(\Omega',\partial\Omega)\big)>0\), independent of \(\lambda\), such that
\[\|\nabla u\|_{L^{p,\lambda}(\Omega')}\le C.\]
C. One has \(\nabla u\in L^{p,\lambda}_{\mathrm{loc}}(\Omega)\) for at least one exponent \(\lambda\in[0,n)\).
D.
For every \(0\le \lambda<n\), one has \(\nabla u\in L^{p,\lambda}_{\mathrm{loc}}(\Omega)\). Moreover, for every \(\Omega'\Subset\Omega\), there exists a constant \(C=C\big(n,p,\nu,L,\beta,E,\operatorname{diam}(\Omega),\operatorname{dist}(\Omega',\partial\Omega)\big)>0\) such that
\[\|\nabla u\|_{L^{p,\lambda}(\Omega')}\le C\,\|\nabla u\|_{L^{p}(\Omega)}.\]
E.
There exists \(\lambda_0\in(0,n)\), depending only on \(n,p,\nu,L,\alpha,\beta\), such that \(\nabla u\in L^{p,\lambda_0}_{\mathrm{loc}}(\Omega)\). Moreover, for every \(\Omega'\Subset\Omega\), there exists a constant \(C=C\big(n,p,\nu,L,\beta,E,\operatorname{diam}(\Omega),\operatorname{dist}(\Omega',\partial\Omega)\big)>0\) with
\[\|\nabla u\|_{L^{p,\lambda_0}(\Omega')}\le C.\]

dependence on Morrey exponent in the local bound

universal quantifier over all \(\lambda<n\)

absolute local Morrey bound replaced by a bound proportional to \(\|\nabla u\|_{L^p(\Omega)}\)

full range \(0\le \lambda<n\) weakened to existence of a single universal \(\lambda_0\)

The stem states the hypotheses in detail but does not reveal the conclusion. The correct option is not explicitly or implicitly signaled by wording in the prompt.

This is largely a theorem-recall item: the setup mirrors a regularity theorem and asks for its conclusion. However, it is not a pure restatement because the options vary in quantifiers, endpoint cases, and parameter dependence.

Some reasoning is needed to reject overly strong or improperly generalized statements (such as including λ = n, removing dependence on E, or extending to all p > 2). Still, the item mainly tests recognition of the precise theorem statement rather than deep derivation.

The distractors are mathematically plausible and target realistic failure modes: weakening the quantifier, adding the forbidden endpoint λ = n, claiming uniformity in E, and overextending the p-range. They are distinct and nontrivial.

A solid theorem-identification MCQ with strong distractors and no answer leakage, but it leans more toward precise recall of a known result than toward genuinely generative mathematical reasoning.


Question:
Let \(\Omega\subset\mathbb{R}^n\) be a bounded domain, let \(E\subset\Omega\) be measurable with \(\partial E\) a \(C^1\) hypersurface, and fix constants \(0<\alpha<\beta<\infty\). Define \(\sigma_E:=\beta\mathbbm{1}_E+\alpha\mathbbm{1}_{E^c}\) and, for \(v\in W^{1,p}(\Omega)\),
\[\mathcal F(v;E):=\int_{\Omega}\sigma_E(x)F(\nabla v)\,dx.\]
Assume \(F\in C^2(\mathbb{R}^n)\) and that for some \(\nu,L>0\), \(\mu\in(0,1]\), and every \(\xi,\eta\in\mathbb{R}^n\),
\[\langle \nabla F(\xi)-\nabla F(\eta),\xi-\eta\rangle\ge \nu\big(\mu^2+|\xi|^2+|\eta|^2\big)^{\frac{p-2}{2}}|\xi-\eta|^2,\]
\[|\nabla F(\xi)-\nabla F(\eta)|\le L\big(\mu^2+|\xi|^2+|\eta|^2\big)^{\frac{p-2}{2}}|\xi-\eta|,\]
with \(2<p<\frac{2n}{n-2}\). Suppose \(u\in W^{1,p}(\Omega)\) is a local minimizer in the sense that for every ball \(B_r(x_0)\subset\Omega\) and every \(\phi\in W_0^{1,p}(B_r(x_0))\),
\[\mathcal F(u;B_r(x_0))\le \mathcal F(u+\phi;B_r(x_0)).\]
Which conclusion about \(\nabla u\) holds? Here \(L^{p,\lambda}(\Omega')\) is the classical Morrey space, for instance with norm \(\|f\|_{L^{p,\lambda}(\Omega')}:=\sup_{B_\rho(x)\subset\Omega'}\big(\rho^{-\lambda}\int_{B_\rho(x)}|f|^p\,dy\big)^{1/p}\).

Choices:
B.
For every \(0\le \lambda<n\), one has \(\nabla u\in L^{p,\lambda}_{\mathrm{loc}}(\Omega)\). Moreover, for every \(\Omega'\Subset\Omega\), there exists a constant \(C=C\big(n,p,\nu,L,\alpha,\beta,E,\operatorname{diam}(\Omega),\operatorname{dist}(\Omega',\partial\Omega)\big)>0\), independent of \(\lambda\), such that
\[\|\nabla u\|_{L^{p,\lambda}(\Omega')}\le C.\]
C. One has \(\nabla u\in L^{p,\lambda}_{\mathrm{loc}}(\Omega)\) for at least one exponent \(\lambda\in[0,n)\).
D.
For every \(0\le \lambda<n\), one has \(\nabla u\in L^{p,\lambda}_{\mathrm{loc}}(\Omega)\). Moreover, for every \(\Omega'\Subset\Omega\), there exists a constant \(C=C\big(n,p,\nu,L,\beta,E,\operatorname{diam}(\Omega),\operatorname{dist}(\Omega',\partial\Omega)\big)>0\) such that
\[\|\nabla u\|_{L^{p,\lambda}(\Omega')}\le C\,\|\nabla u\|_{L^{p}(\Omega)}.\]
E.
There exists \(\lambda_0\in(0,n)\), depending only on \(n,p,\nu,L,\alpha,\beta\), such that \(\nabla u\in L^{p,\lambda_0}_{\mathrm{loc}}(\Omega)\). Moreover, for every \(\Omega'\Subset\Omega\), there exists a constant \(C=C\big(n,p,\nu,L,\beta,E,\operatorname{diam}(\Omega),\operatorname{dist}(\Omega',\partial\Omega)\big)>0\) with
\[\|\nabla u\|_{L^{p,\lambda_0}(\Omega')}\le C.\]

The stem does not explicitly reveal the correct option, and no choice is singled out by wording. Although the hypotheses strongly signal a specific regularity theorem, the answer is not leaked directly.

This is very close to a theorem-recall item: the stem lists the full assumptions and asks for the corresponding conclusion. However, it is not a pure tautology because the options differ in quantifiers, dependence of constants, and strength of the Morrey regularity claim.

The item requires some reasoning about the strength of conclusions and the role of parameter dependence, especially to distinguish the full-for-all-lambda statement from weaker or tampered variants. Still, it mainly tests recognition/recall of a known theorem rather than deep derivation.

The distractors are plausible and mathematically meaningful: they vary universal quantifiers, dependence on lambda, presence of an extra global norm factor, and weakening to existence of one exponent. These reflect realistic failure modes in reading regularity statements.

A solid theorem-recognition MCQ with strong distractors and no direct answer leakage, but it is still fairly close to a restatement of a known result and only moderately tests generative reasoning.


\begin{pgr}[The Global Glimm Property]\label{pgr:GGP}
    A \ca{} $A$ is said to satisfy the \emph{Global Glimm Property} if for every $a\in A_+$ and $\varepsilon>0$ there exists a square-zero element $r\in\overline{aAa}$ such that $(a-\varepsilon)_+\in \overline{ArA}$. This property was originally introduced by Kirchberg and R\o{}rdam in their study of non-simple purely infinite \ca{s} \cite[Definition~4.12]{KirRor02InfNonSimpleCalgAbsOInfty}, but has recently found various and deep applications in the context of dimension reduction phenomena; see, for example, \cite{ThiVil23Glimm,Vil25:IntroGGP}.

    As shown in \cite[Theorem~3.6]{ThiVil23Glimm}, the property can be characterized in terms of a divisibility condition on the Cuntz semigroup: $A$ has the Global Glimm Property if and only if $\Cu (A)$ is $(2,\omega )$-divisible, that is, for  every pair $x',x\in\Cu (A)$ such that $x'\ll x$, there exists $y\in\Cu (A)$ and $n\in\N$ such that $2y\leq x$ and $x'\leq ny$.
\end{pgr}

\begin{prop}\label{DpureRSHA}
Let $D$ be a unital simple pure \ca. Let $R_D$ be a recursive subhomogeneous algebra over $D$ and let $R$ be any recursive homogeneous algebra over $\mathbb{C}$.
Then both $R_D$ and $R \otimes D$ are pure.
\end{prop}

\begin{defn}[{\cite[Definition~3.2]{archey2020structure}}]\label{RSHA}
For a simple unital \ca{} $D$, the class $\mathcal{R}$ of
{\emph{recursive subhomogeneous algebras over $D$}} is
the smallest class of \ca{s} closed under isomorphism such that:
\begin{enumerate}
\item\label{9223_RSHA_1}
$C (X, M_{n} (D)) \in \mathcal{R}$ whenever $X$ is a compact Hausdorff space and $n \geq 1$.
\item\label{9223_RSHA_2}
Any pullback of the form
\begin{align*}
& B \oplus_{C (X^{(0)}, \, M_{n} (D)), \, \ph, \, \rh} C (X, M_{n} (D))
\\
& \hspace*{3em} {\mbox{}}
  = \bset{ (b, f) \in B \oplus C (X, M_{n} (D)) \colon
        \ph (b) = f |_{X^{(0)}} }
\end{align*}
is in $\mathcal{R}$ whenever $B \in \mathcal{R}$, $X$ is compact Hausdorff,
$n \geq 1$, $X^{(0)} \subseteq X$ is closed (possibly empty),
$\ph \colon B \to C (X^{(0)}, \, M_{n} (D))$
is any unital $^*$-homomorphism
(the zero \hm{} if $X^{(0)}$ is empty), and
$\rho \colon C (X, M_{n} (D)) \to C (X^{(0)}, \, M_{n} (D))$ is the
restriction $^*$-homomorphism.
\end{enumerate}
\end{defn}

\begin{defn}\label{def_almost_divisibility}
    Let $A$ be a \ca{} and $n\in\N$. We say that $A$ (and its Cuntz semigroup $\Cu (A)$) is \emph{$n$-almost divisible} if for any pair $x',x\in\mathrm{Cu}(A)$ such that $x'\ll x$ and any $N\in \N$, there exists $y\in \mathrm{Cu}(A)$ such that
    \[
     N y\leq x\quad{\mbox{and}}\quad x'\leq (N+1)(n+1)y.
    \]

    $A$ is said to be \emph{almost divisible} if it is $0$-almost divisible.
\end{defn}

\begin{qst}\label{qst:VillZstab}
    Let $B$ be a Villadsen algebra of the first kind without strict comparison (for example, Toms' \cite{Tom08ClassificationNuclear}). Is $B\otimes C_r^*(\mathbb{F}_2)$ $\mathcal{Z}$-stable?
\end{qst}

\begin{theorem}\label{prp:ASHtenPure}
   Let $B$ be a unital separable ASH-algebra and $A$ be simple and pure. Then, $B\otimes A$ is pure.
\end{theorem}

\begin{defn}[{\cite[Definition~2.1]{Win12NuclDimZstable}}]\label{dfn:MComp}
    Let $A$ be a \ca{} and let $n\in\N$. We say that $A$ (and its Cuntz semigroup $\Cu (A)$) has the \emph{$n$-comparison} property if $x \leq_s y_j$ for $x, y_j \in \Cu (A)$ and $j = 0, 1, \ldots, n$, implies $x\leq \sum_{j=0}^n y_j$.

    $\Cu (A)$ is said to be \emph{almost unperforated} if it has $0$-comparison.
\end{defn}

\begin{rem}\label{rem:StCompAlmUnperf}
    Let $A$ be a \ca{}. For any $[0,\infty]$-valued lower semi-continuous $2$-quasitrace $\tau$ on $A$ (which extends to $A\otimes\mathcal{K}$ and is denoted by the same symbol), the associated \emph{dimension function} $d_\tau\colon \Cu (A)\to [0,\infty]$ is defined by $d_\tau ([a])=\lim_n\tau (a^{1/n})$.

    As shown in \cite[Proposition~6.2]{elliott2011cone}, $\Cu(A)$ is almost unperforated (in the sense described above) if and only if, for every pair $a,b\in  (A\otimes\mathcal{K})_+$ such that $a$ is in the ideal generated by $b$ and $d_\tau ([a])<d_\tau([b])$ whenever $d_\tau ([b])=1$, one has $[a]\leq [b]$ in $\Cu (A)$.

    This property is known as \emph{strict comparison} (of positive elements by $[0,\infty]$-valued lower semi-continuous $2$-quasitraces). Note that, if $A$ is simple, $A$ has strict comparison if and only if, for every pair of non-zero elements $a,b\in  (A\otimes\mathcal{K})_+$ such that $d_\tau ([a])<d_\tau([b])$ for every normalized quasitrace $\tau$, one has $[a]\leq [b]$ in $\Cu (A)$. We refer the reader to \cite[Paragraph~3.4]{AntPerThiVil24arX:PureCAlgs} for an in-depth introduction.
\end{rem}

The properties of \emph{strict comparison} and \emph{almost divisibility} play a fundamental role in the modern structure theory of $\mathrm{C}^*$-algebras. As defined by Winter \cite{Win12NuclDimZstable}, $\mathrm{C}^*$-algebras that satisfy both properties are termed \emph{pure}. In the simple finite setting, such algebras should be regarded as the correct analogues of II$_1$ factors; roughly speaking, the defining properties of pureness correspond to two key features of projections in II$_1$factors, adapted to the setting of positive elements ---a necessary adaptation given that many $\mathrm{C}^*$-algebras contain few or no nontrivial projections.

Just as the unique trace on a II$_1$ factor $\mathcal{N}$ determines the order of projections, \emph{strict comparison} \cite{Bla88Comparison} asserts that tracial state data determines when one positive operator can be approximately compressed to another (Remark \ref{rem:StCompAlmUnperf}). Similarly, \emph{almost divisibility} (Definition \ref{def_almost_divisibility}) serves as the analogue to the fact that projections in $\mathcal{N}$ can be divided arbitrarily. Whilst these two properties are automatic for II$_1$ factors, they can fail for simple $\mathrm{C}^*$-algebras \cite{Tom08ClassificationNuclear,Vil99SRSimpleCa}. Consequently, determining which algebras satisfy these properties ---and, in particular, developing mechanisms to deduce pureness--- has been a major focus in the field for the past few years \cite{AGKEP25,AntPerThiVil24arX:PureCAlgs,Rob25Self}. Examples of pure $\mathrm{C}^*$-algebras can be found both in the nuclear and non nuclear case, and include $\mathcal{Z}$-stable algebras, purely infinite algebras \cite{kirchberg2000non}, von Neumann algebras with trivial type I summand, and reduced (twisted) group $\mathrm{C}^*$-algebras of acylindrically hyperbolic groups \cite{FloLisCobPag:PureTwi,RauThiVil:StCompTwi}.

In the realm of simple nuclear $\mathrm{C}^*$-algebras, pureness features prominently in the Toms-Winter conjecture \cite{CasEviTikWhiWin21NucDimSimple,Ror04StableRealRankZ,Win12NuclDimZstable} and serves as a crucial regularity condition in the Elliott classification program \cite{Whi23ICM,Win18ICM}. More generally, the notion is expected to play an important role in the classification of $^*$-homomorphisms \cite{CarGabSchTikWhi23arX:ClassifHom1,Sza26:UniqThmKK}, potentially by weakening the requirement of $\mathcal{Z}$-stability on the codomain to that of pureness. In fact, as shown in \cite[Theorem~7.3.11]{antoine2018tensor}, one may think of pureness as $\mathcal{Z}$-stability at the level of the Cuntz semigroup.

Of the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group $\mathrm{C}^*$-algebras \cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \emph{dimension reduction phenomena}. Specifically, it was recently proved in \cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \cite{perera2025extensions} that an extension of pure $\mathrm{C}^*$-algebras is itself pure, mirroring the corresponding property for $\mathcal{Z}$-stability \cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.

\begin{document}
\begin{abstract}
 Let $X$ be a compact metric space, and let $A$ be a pure \ca{}. We show that $C(X,A)$ is pure whenever
 \begin{enumerate}
 \item $A$ is simple; or
 \item every quotient of $A$ is stably finite (e.g., $A$ has stable rank one).
 \end{enumerate}

Of the two defining properties of pureness, strict comparison has received the most attention in recent years, finding deep connections with the theory of group \ca{s} \cite{AGKEP25,Oza25:Proxim}, time-frequency analysis \cite{BedEnsvVelt22,EnsThiVil25Criteria}, and the recent negative resolution to Tarski's problem \cite{KES:NegativeTarski}. However, despite its importance, strict comparison is often difficult to verify directly. For this reason, attention has shifted towards the stronger, more rigid property of pureness, which is particularly susceptible to \emph{dimension reduction phenomena}. Specifically, it was recently proved in \cite{AntPerThiVil24arX:PureCAlgs} that any $(m, n)$-pure \ca{} ---a formal `quantized' weakening of pureness--- is automatically pure. This result allows one to deduce pureness (and thus strict comparison) in situations where direct verification was previously out of reach. It has also paved the way for new permanence results; for instance, it was established in \cite{perera2025extensions} that an extension of pure \ca{s} is itself pure, mirroring the corresponding property for $\mathcal{Z}$-stability \cite[Theorem~4.3]{TomWin07ssa}. An important question in this line of inquiry, raised at a number of conferences, is whether a minimal tensor product $B \otimes A$ is pure whenever $A$ is. In the present paper, we investigate this question for the case $B = C(X)$.

Before outlining the proof strategy and the challenges involved in establishing Theorem \ref{prp:MainThm1}, we highlight some important consequences. First, combining Theorem \ref{prp:MainThm1} with several permanence properties of pureness, we obtain the following application:

\begin{cor}\label{robert_analogy_2}
 Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \ca{}. Then, there exists $M\in\N$ such that, for every $N\geq 1$ and every pair $x'\ll x$ in $\Cu (C(X,A))$, there exists $y\in\Cu (C(X,A))$ such that
 \[
 x'\ll Ny\ll Mx.
 \]
\end{cor}
\begin{proof}
 Set $M=2(\dim (X)+1)^2$ and let $N\in\N$. If $N=1$, any $y\in\Cu (A)$ such that $x'\ll y\ll x$ satisfies the required condition. Thus, we may assume $N>1$.
 As in the previous proof, we may also assume that $A$ is stable. Find $w\in \Cu (C(X,A))$ such that $x'\ll w\ll x$, and let $w=[f]$ with $f\in C(X,A)_+$. By the definition of compact containment, there exists $\varepsilon>0$ such that $x'\ll [(f-\varepsilon)_+]$. Now apply Proposition \ref{robert_analogy} to $f$, $\varepsilon$ and $N-1$ to find $y\in \Cu (C(X,A))$ such that
 \[
 [(f-\varepsilon)_+]\leq Ny\quad\text{and}\quad (N-1)y\leq (\dim(X)+1)^2[f].
 \]

\begin{theorem}\label{prp:CXAPureIfGGP}
 Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.
\end{theorem}
\begin{proof}
 Proposition \ref{comp_c(x,A)} shows that $C(X,A)$ has $m$-comparison for $m=\dim (X)$. Further, it follows from Corollary \ref{robert_analogy_2} that $C(X,A)$ satisfies the divisibility condition in Theorem \ref{prp:Gen_DimRed}. Thus, Theorem \ref{prp:Gen_DimRed} gives the desired equivalence.
\end{proof}

\begin{cor}\label{Purehomogenous}
 Let $A$ be a simple pure \ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.
\end{cor}
\begin{proof}
 If $X$ is finite-dimensional, $C(X,A)$ has the Global Glimm Property by \cite[Theorem~4.3]{BlaKir04GlimmHalving}. Thus, the result follows from Theorem \ref{prp:CXAPureIfGGP}.

\begin{prop}\label{prp:CXNoQuot}
 Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\in\N$ such that, for every pair $x',x\in\Cu (C(X,A))$ with $x'\ll x$, there exist $y_0,y_1\in\mathrm{Cu} (C(X,A))$ satisfying
 \[
 y_0+y_1\leq x,\quad\text{and}\quad
 x'\ll Ly_0, Ly_1.
 \]
\end{prop}
\begin{proof}
 We may assume without loss of generality that $A$ is stable, and that there exists $\varepsilon>0$ such that that $x=[f]$ and $x'=[(f-\varepsilon)_+]$ for some $f\in C(X,A)_+$. We adapt the argument from \cite[Proposition~3.4]{RobTik17NucDimNonSimple}. However, distinct technical difficulties arise in the present context that demand additional care. For this reason, we present the full proof in detail:

\begin{theorem}\label{prp:MainNonSimp}
 Let $X$ be a compact metric space, and let $A$ be a pure \ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.
\end{theorem}
\begin{proof}
 If $X$ is finite-dimensional, the result follows directly from Proposition \ref{prp:CXNoQuot} together with Theorem \ref{prp:Gen_DimRed} (together with Proposition \ref{comp_c(x,A)} and Corollary \ref{robert_analogy_2}).

\begin{cor}\label{Purehomogenous}
    Let $A$ be a simple pure \ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.
\end{cor}

\begin{prop}\label{comp_c(x,A)}
    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.
\end{prop}

\begin{theorem}\label{prp:CXAPureIfGGP}
    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.
\end{theorem}

\begin{prop}\label{prp:CXNoQuot}
    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\in\N$ such that, for every pair $x',x\in\Cu (C(X,A))$ with $x'\ll x$, there exist $y_0,y_1\in\mathrm{Cu} (C(X,A))$ satisfying
        \[
        y_0+y_1\leq x,\quad\text{and}\quad
        x'\ll Ly_0, Ly_1.
        \]
\end{prop}

\begin{theorem}\label{prp:Gen_DimRed}
    Let $A$ be a \ca{}, and let $m,M\in\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\geq 1$ and every pair of elements $x', x\in \Cu (A)$ such that $x'\ll x$, there exists $y\in\mathrm{Cu} (A)$ satisfying
    \[
        x'\ll Ny\ll M x.
    \]

    Then, the following are equivalent:
    \begin{itemize}
        \item[(i)] $A$ has the Global Glimm Property;
        \item[(ii)] there exists $L\in\N$ such that, for every pair $x',x\in\Cu (A)$ with $x'\ll x$, there exist $y_0,y_1\in\mathrm{Cu} (A)$ satisfying
        \[
        y_0+y_1\leq x,\quad\text{and}\quad
        x'\ll Ly_0, Ly_1;
        \]
        \item[(iii)] $A$ is pure.
    \end{itemize}
\end{theorem}

\begin{theorem}\label{prp:MainNonSimp}
    Let $X$ be a compact metric space, and let $A$ be a pure \ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.
\end{theorem}

\begin{mainthm}[\ref{Purehomogenous},~\ref{prp:MainNonSimp}]\label{prp:MainThm1}
    Let $X$ be a compact metric space, and let $A$ be a pure \ca{}. Assume additionally that
    \begin{enumerate}
        \item $A$ is simple; or
        \item every quotient of $A$ is stably finite.
    \end{enumerate}

    Then, $C(X,A)$ is pure.
\end{mainthm}

\begin{cor}\label{robert_analogy_2}
    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \ca{}. Then, there exists $M\in\N$ such that, for every $N\geq 1$ and every pair $x'\ll x$ in $\Cu (C(X,A))$, there exists $y\in\Cu (C(X,A))$ such that
    \[
        x'\ll Ny\ll Mx.
    \]
\end{cor}

Before outlining the proof strategy and the challenges involved in establishing Theorem \ref{prp:MainThm1}, we highlight some important consequences. First, combining Theorem \ref{prp:MainThm1} with several permanence properties of pureness, we obtain the following application:

\begin{mainthm}[\ref{prp:ASHtenPure}]\label{prp:MainThm2}
    Let $A$ be a pure \ca{} satisfying (1) or (2) in Theorem \ref{prp:MainThm1}, and let $B$ be a unital separable ASH-algebra. Then, $A\otimes B$ is pure.
\end{mainthm}

As a particular example of this result, Theorem \ref{prp:MainThm2} shows that the tensor product of any Villadsen algebra of type I with the reduced group \ca{} of any free group is pure. It is currently unknown whether these specific examples are $\mathcal{Z}$-stable (Question \ref{qst:VillZstab}), highlighting the utility of pureness as a regularity invariant. In fact, Theorem \ref{prp:MainThm2} is slightly more general: As we prove in Theorem \ref{DpureRSHA}, any recursive subhomogeneous \ca{} over $A$ \cite[Definition~3.2]{archey2020structure} (Definition \ref{RSHA}) is pure whenever $A$ is.

A further application of Theorem \ref{prp:MainThm2} arises in the study of determining which nonamenable groups have a reduced group \ca{} with strict comparison. Since $C_r^*(G)$ is a direct sum of simple pure $\mathrm{C}^*$-algebras whenever $G$ is countable and acylindrically hyperbolic \cite[Proof of Theorem~C]{FloLisCobPag:PureTwi}, and $C_r^*(H)$ is subhomogeneous whenever $H$ is virtually abelian, we get:

\begin{maincor}\label{prp:MainCor1}
 Let $G$ be a countable acylindrically hyperbolic group, and let $H$ be a virtually abelian group. Then, $C_r^*(G\times H)$ is pure. In particular, it has strict comparison.
\end{maincor}

In addition to significantly enlarging the known class of pure $\mathrm{C}^*$-algebras, Theorems \ref{prp:MainThm1} and \ref{prp:MainThm2} shed new light on the \emph{Global Glimm Problem}, a major open problem in the field \cite[Problem~LXXIII]{SchTikWhi99arX:Prob}. The problem asks whether the \emph{Global Glimm Property} ---a generalization of non-elementariness for non-simple $\mathrm{C}^*$-algebras \cite{KirRor02InfNonSimpleCalgAbsOInfty} which we recall in Paragraph \ref{pgr:GGP}--- is equivalent to having no nonzero elementary ideals of quotients, and has been solved affirmatively in numerous cases (see \cite[Section~4]{Vil25:IntroGGP} for an overview). While it was previously known that $C(X, A)$ possesses the Global Glimm Property whenever $X$ is finite-dimensional and $A$ is simple and has the property \cite[Theorem~4.3]{BlaKir04GlimmHalving}, this was not known for pure $\mathrm{C}^*$-algebras $A$ satisfying condition (2) of Theorem \ref{prp:MainThm1}, nor for the tensor products appearing in Theorem \ref{prp:MainThm2}. As we explain below, this is not a consequence of our results; rather, it is a necessary intermediate step in our proof of pureness.

Apart from the applications presented in this paper, Theorem \ref{prp:MainThm1} has further consequences. In particular, this result will be used in forthcoming work of the first named author with Evington, Hua, Schafhauser, and White, which will study \emph{$K$-stability} for pure $\mathrm{C}^*$-algebras satisfying conditions as in Theorem~\ref{prp:MainThm1}. It has been shown in \cite{lin2025strict} that finite simple pure $\mathrm{C}^*$-algebras have stable rank one and are therefore $K_1$-bijective. The future work will focus on the stronger property of $K$-stability for this class.

\subsection*{Strategy of the proof} Given a \ca{} $A$ with strict comparison, it is well-known that $C(X,A)$ need not have comparison. Topological obstructions, often manifesting as non-trivial vector bundles, can impede comparison even in simple settings ---for example, when $A=\mathbb{C}$ and $X=\mathbb{T}^4$ \cite{Vil98SimpleCaPerforation}. Thus, one of the challenges in establishing Theorem \ref{prp:MainThm1} is that strict comparison and almost divisibility cannot be verified separately.

Our strategy relies on a multi-step dimension reduction argument. First, we restrict our attention to the case where $X$ is finite-dimensional. In Sections \ref{sec:CompCuA} and \ref{sec:DivCuA}, we prove that if $X$ is a compact metric space of dimension $m$ and $A$ is pure, the algebra $C(X, A)$ satisfies a weakening of $(m,n)$-pureness, with $n$ depending only on $m$. Concretely, we see that $C(X,A)$ has $m$-comparison (Proposition \ref{comp_c(x,A)}) and satisfies a \emph{weak} version of $n$-almost divisibility (Corollary \ref{robert_analogy_2}); see Definitions \ref{dfn:MComp} and \ref{def_almost_divisibility} for details.

As Corollary \ref{robert_analogy_2} does not establish $n$-almost divisibility for any $n$, the rigidity results from \cite{AntPerThiVil24arX:PureCAlgs} cannot be applied directly. Instead, building on the techniques from \cite{AntPerThiVil24arX:PureCAlgs}, we prove a new general reduction result (Theorem \ref{prp:Gen_DimRed}), which we then use in Theorem \ref{prp:CXAPureIfGGP} to deduce that $C(X, A)$ is pure if and only if it possesses the Global Glimm Property. Relying on the fact that all these properties pass to inductive limits, we are able to dispense with the assumption of finite dimensionality on $X$. Thus, the proof of Theorem \ref{prp:MainThm1} reduces to proving the Global Glimm Property for $C(X,A)$. For $\mathrm{C}^*$-algebras satisfying condition (1), this follows from \cite[Theorem~4.3]{BlaKir04GlimmHalving}; for those satisfying (2), the result is new and is proven in Proposition \ref{prp:CXNoQuot}.

\subsection*{Acknowledgments} This paper started during EV's visit to the University of Oxford during the \emph{Mini Course: Topological Phenomena in the Cuntz semigroup}, which was given by Andrew Toms and organized by Stuart White.  We are grateful to both, as well as to the University of Oxford and St John’s College, for their kind hospitality. We would further like to thank Stuart White for his helpful feedback on the paper, and Hannes Thiel for pointing out Corollary \ref{prp:MainCor1}. AS is  grateful to N. Christopher Phillips for valuable discussions concerning pureness of $\mathrm{C}^*$-algebras and for feedback on earlier drafts of this paper. AS also thanks Julian Buck, Dawn Archey, and Javad Mohammadkarimi for insightful discussions.

The proof of Theorem \ref{prp:MainThm1} addresses the issue that, although $A$ has strict comparison, it is "well-known that $C(X,A)$ need not have comparison" due to "topological obstructions" (e.g., non-trivial vector bundles), so "strict comparison and almost divisibility cannot be verified separately."

The strategy is a "multi-step dimension reduction argument":
\begin{itemize}
\item First reduce to $X$ finite-dimensional: for $\dim(X)=m$ and $A$ pure, in Sections \ref{sec:CompCuA} and \ref{sec:DivCuA} they show $C(X,A)$ satisfies a weakening of $(m,n)$-pureness (with $n$ depending only on $m$), namely that $C(X,A)$ has "$m$-comparison" (Proposition \ref{comp_c(x,A)}) and a "\emph{weak} version of $n$-almost divisibility" (Corollary \ref{robert_analogy_2}).
\item Since Corollary \ref{robert_analogy_2} "does not establish $n$-almost divisibility for any $n$", the rigidity results of \cite{AntPerThiVil24arX:PureCAlgs} "cannot be applied directly." Instead, "building on the techniques" there, they prove "a new general reduction result" (Theorem \ref{prp:Gen_DimRed}).
\item Use this to show (Theorem \ref{prp:CXAPureIfGGP}) that $C(X,A)$ is pure "if and only if it possesses the Global Glimm Property."
\item "Relying on the fact that all these properties pass to inductive limits," remove the finite-dimensionality hypothesis on $X$.
\item Conclude that proving Theorem \ref{prp:MainThm1} reduces to proving the Global Glimm Property for $C(X,A)$: under condition (1) this follows from \cite[Theorem~4.3]{BlaKir04GlimmHalving}, while under condition (2) it is new and is proved in Proposition \ref{prp:CXNoQuot}.
\end{itemize}

To prove the main theorem, the proof addresses the issue that, although $A$ has strict comparison, it is "well-known that $C(X,A)$ need not have comparison" due to "topological obstructions" (e.g., non-trivial vector bundles), so "strict comparison and almost divisibility cannot be verified separately."

The strategy is a "multi-step dimension reduction argument":
\begin{itemize}
\item First reduce to $X$ finite-dimensional: for $\dim(X)=m$ and $A$ pure, they show that $C(X,A)$ satisfies a weakening of $(m,n)$-pureness (with $n$ depending only on $m$), namely the following.

\begin{prop}\label{comp_c(x,A)}
    Let $X$ be a compact metric space of covering dimension $m$, and let $A$ be a \ca{} with strict comparison. Then $C(X, A)$ has $m$-comparison.
\end{prop}

Moreover, they prove the following weak form of almost divisibility.

\begin{cor}\label{robert_analogy_2}
    Let $X$ be a finite-dimensional compact metric space, and let $A$ be an almost divisible \ca{}. Then, there exists $M\in\N$ such that, for every $N\geq 1$ and every pair $x'\ll x$ in $\Cu (C(X,A))$, there exists $y\in\Cu (C(X,A))$ such that
    \[
        x'\ll Ny\ll Mx.
    \]
\end{cor}

\item Since the corollary above "does not establish $n$-almost divisibility for any $n$", the rigidity results of AntPerThiVil24arX:PureCAlgs "cannot be applied directly." Instead, "building on the techniques" there, they prove the following new general reduction result.

\begin{theorem}\label{prp:Gen_DimRed}
    Let $A$ be a \ca{}, and let $m,M\in\mathbb{N}$. Assume that $A$ has $m$-comparison and that, for every $N\geq 1$ and every pair of elements $x', x\in \Cu (A)$ such that $x'\ll x$, there exists $y\in\mathrm{Cu} (A)$ satisfying
    \[
        x'\ll Ny\ll M x.
    \]

    Then, the following are equivalent:
    \begin{itemize}
        \item[(i)] $A$ has the Global Glimm Property;
        \item[(ii)] there exists $L\in\N$ such that, for every pair $x',x\in\Cu (A)$ with $x'\ll x$, there exist $y_0,y_1\in\mathrm{Cu} (A)$ satisfying
        \[
        y_0+y_1\leq x,\quad\text{and}\quad
        x'\ll Ly_0, Ly_1;
        \]
        \item[(iii)] $A$ is pure.
    \end{itemize}
\end{theorem}

\item Use this to show the following characterization of purity for $C(X,A)$ in terms of the Global Glimm Property.

\begin{theorem}\label{prp:CXAPureIfGGP}
    Let $X$ be a finite-dimensional compact metric space and let $A$ be a pure \ca{}. Then, $C(X,A)$ is pure if and only if $C(X,A)$ has the Global Glimm Property.
\end{theorem}

\item "Relying on the fact that all these properties pass to inductive limits," remove the finite-dimensionality hypothesis on $X$.

\item Conclude that proving the main theorem reduces to proving the Global Glimm Property for $C(X,A)$: under condition (1) this follows from \cite[Theorem~4.3]{BlaKir04GlimmHalving}, while under condition (2) it is new and is proved in the following proposition.

\begin{prop}\label{prp:CXNoQuot}
    Let $X$ be a finite-dimensional compact metric space, and let $A$ be a pure \ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, there exists $L\in\N$ such that, for every pair $x',x\in\Cu (C(X,A))$ with $x'\ll x$, there exist $y_0,y_1\in\mathrm{Cu} (C(X,A))$ satisfying
        \[
        y_0+y_1\leq x,\quad\text{and}\quad
        x'\ll Ly_0, Ly_1.
        \]
\end{prop}
\end{itemize}

\begin{cor}\label{Purehomogenous}
    Let $A$ be a simple pure \ca{}, and let $X$ be a compact metric space. Then, $C(X,A)$ is pure.
\end{cor} To prove the main theorem, we use the following result.

\begin{theorem}\label{prp:MainNonSimp}
    Let $X$ be a compact metric space, and let $A$ be a pure \ca{}. Assume that $A$ has no quotient whose Cuntz semigroup contains a nonzero, compact, properly infinite Cuntz class. Then, $C(X,A)$ is pure.
\end{theorem}

\label{prp:MainThm1}
    Let $X$ be a compact metric space, and let $A$ be a pure \ca{}. Assume additionally that
    \begin{enumerate}
        \item $A$ is simple; or
        \item every quotient of $A$ is stably finite.
    \end{enumerate}

    Then, $C(X,A)$ is pure.

Question: Let \(X\) be a compact metric space, and let \(A\) be a pure \(\mathrm{C}^*\)-algebra, where "pure" means that \(A\) has strict comparison and almost divisibility. Assume in addition that either \(A\) is simple, or every quotient \(\mathrm{C}^*\)-algebra of \(A\) is stably finite. Here \(C(X,A)\) denotes the \(\mathrm{C}^*\)-algebra of continuous \(A\)-valued functions on \(X\). Which statement holds for every such pair \((X,A)\)?

\(C(X,A)\) is pure; equivalently, the \(\mathrm{C}^*\)-algebra of continuous \(A\)-valued functions on \(X\) has strict comparison and almost divisibility.

Choices:
B. \(C(X,A)\) has strict comparison, but it need not be almost divisible in general; thus one can only conclude a comparison property for \(C(X,A)\), not pureness.
C. \(C(X,A)\) has strict comparison.
D. \(C(X,A)\) is pure whenever \(X\) is finite-dimensional; for arbitrary compact metric \(X\), the conclusion can fail even if \(A\) is simple or every quotient of \(A\) is stably finite.
E. \(C(X,A)\) is pure provided both that \(A\) is simple and that every quotient \(\mathrm{C}^*\)-algebra of \(A\) is stably finite; assuming only one of these conditions is not sufficient.

dimension-reduction needed to recover almost divisibility jointly with comparison

dropped almost divisibility/pureness and kept only strict comparison

removal of finite-dimensionality via inductive limits

logical disjunction of hypotheses (simple OR all quotients stably finite)

The stem does not explicitly state the conclusion, and no option is textually echoed in a way that gives away the answer. It presents hypotheses and asks for the resulting conclusion, so there is no direct answer leakage.

This is essentially a direct theorem-recall item: the hypotheses are stated in full and the correct choice is the theorem's conclusion. It is very close to a restatement rather than a novel inference task.

There is some reasoning pressure because the student must distinguish the strongest valid conclusion from weaker or unnecessarily qualified alternatives. However, the item mainly tests recognition/recall of a known permanence result rather than substantive generative reasoning.

The distractors are mathematically plausible and target common confusions: mistaking strict comparison for purity, adding an unnecessary finite-dimensional hypothesis, or selecting a weaker true statement instead of the strongest correct one. They are distinct and reasonably well designed.

A solid theorem-recall MCQ with good distractors, but it is highly tautological and only moderately probes reasoning.


Question: Let \(X\) be a compact metric space, and let \(A\) be a pure \(\mathrm{C}^*\)-algebra, where "pure" means that \(A\) has strict comparison and almost divisibility. Assume in addition that either \(A\) is simple, or every quotient \(\mathrm{C}^*\)-algebra of \(A\) is stably finite. Here \(C(X,A)\) denotes the \(\mathrm{C}^*\)-algebra of continuous \(A\)-valued functions on \(X\). Which statement holds for every such pair \((X,A)\)?

Choices:
B. \(C(X,A)\) has strict comparison, but it need not be almost divisible in general; thus one can only conclude a comparison property for \(C(X,A)\), not pureness.
C. \(C(X,A)\) has strict comparison.
D. \(C(X,A)\) is pure whenever \(X\) is finite-dimensional; for arbitrary compact metric \(X\), the conclusion can fail even if \(A\) is simple or every quotient of \(A\) is stably finite.
E. \(C(X,A)\) is pure provided both that \(A\) is simple and that every quotient \(\mathrm{C}^*\)-algebra of \(A\) is stably finite; assuming only one of these conditions is not sufficient.

The stem does not explicitly reveal the conclusion. It states the hypotheses and asks which consequence follows, so there is no direct answer leakage, though it is clearly theorem-shaped.

This is essentially a direct restatement/application of a permanence theorem: under the listed hypotheses, the correct choice repeats the theorem’s conclusion that C(X,A) is pure.

There is some reasoning pressure because one distractor (C) is a weaker true-looking statement, so the test-taker must identify the strongest valid conclusion. Still, the item mainly checks recall of the theorem rather than deeper generative reasoning.

The distractors are plausible and mathematically targeted: one weakens the conclusion to strict comparison only, one inserts an unnecessary finite-dimensionality restriction, and one incorrectly strengthens the hypothesis from an OR to an AND. These reflect realistic failure modes.

A reasonably well-constructed theorem-recall MCQ with strong distractors, but it is largely tautological because it asks for the exact conclusion of the stated result rather than eliciting substantial generative reasoning.