\begin{corollary}\label{cor:CompPresAF}
Every c.e. presentation of a unital AF algebra is computable.
\end{corollary}

\begin{proposition}\label{prop:PreimCeOpen}
Suppose $\boldA^\#$ and $\boldB^\#$ are c.e. presentations of \cstar-algebras, and let
$U$ be a c.e. open set of $\boldB^\#$.  If $f$ is a computable map from
$\boldA^\#$ to $\boldB^\#$, then $f^{-1}[U]$ is a c.e. open set of $\boldA^\#$.
Moreover, it is possible to compute an $\boldA^\#$ index of $f^{-1}[U]$ from a $\boldB^\#$ index
of $U$ and an $(\boldA^\#, \boldB^\#)$ index of $f$.
\end{proposition}

Suppose $\boldA^\#$ is a c.e. presentation of a stably finite unital \cstar-algebra.
In \cite{UHFPaper}, it is shown that $\unit_\boldA$ is a computable point of
$\boldA^\#$.
We say that $e \in \N$ is a \emph{unital index of $\boldA^\#$} if
$e$ is a code of a pair $(e_0, e_1)$ that consists of an index of $\boldA^\#$ and
an $\boldA^\#$ index of $\unit_\boldA$.  Our motivation for introducing such indexes
is that even for the class of stably finite unital \cstar-algebras, an $\boldA^\#$-index of $\unit_\boldA$ may not be computable from an index of
$\boldA^\#$; see \cite{mcnicholl2024evaluative}.

We now discuss prior developments regarding the computability of $K$-theory.

\subsubsection{$K$-theory}\label{subsubsec:BackCompKThy}

In \cite{UHFPaper}, it is shown that there is a \emph{computable} functor from the
category of c.e. presentations of unital \cstar-algebras (where the morphisms are
computable unital $\star$-morphisms) to the category of c.e. presentations of abelian groups.
This functor is denoted $K_0$.
The functor is computable in the sense that it is possible to compute an index of $K_0(\boldA^\#)$ from
an index of $\boldA^\#$.  and from an $(\boldA^\#, \boldB^\#)$ index of a computable $\star$-homomorphism $\phi$, it is possible to compute a $(K_0(\boldA^\#), K_0(\boldB^\#))$-index of $K_0(\phi)$.

The application of computable $K$-theory to UHF algebras relied heavily on the
concept of computable weak stability, which will also support our proof of the Effective Glimm Lemma.
Hence, it is our next topic for review.

\subsubsection{Computable weak stability}

The concept of computable weak stability was introduced by Fox, Goldbring, and Hart \cite
{FoxGoldbringHart.2024+} based on the corresponding notion of weak stability in continuous
model theory, which was introduced in \cite{FarahEtAll.2021}.
In \cite{UHFPaper}, we presented computable weak stability in a way that illuminated
its connection with c.e. closedness.
For the sake of referencing some of its details, we recall this definition now.

\begin{definition}\label{def:CompWklyStbl}
Suppose that $\vec{x} = (x_1, \ldots, x_N)$ is a tuple of variables, $C_1, \ldots, C_N \in \N$,
and $p_1(\vec{x}), \ldots, p_M(\vec{x})$ are rational $*$-polynomials.  Consider the set of relations
$$\mathcal{R} = \{p_i(\vec{x})=0 \ : \ i=1,\ldots,M\}\cup\{\norm{x_j} \leq C_j \ : \ j=1,\ldots,N\}$$.
\begin{enumerate}
    \item For a \cstar-algebra $\boldB$ and $w_1, \ldots, w_N \in \boldB$,
    let $\mathcal{R}^{\boldB}$ denote the quantity
    $$
    \max(\{\norm{p_j(\vec{w})}\ :\ j \in \{1, \ldots, M\}\} \cup \{\norm{w_j}- C_j\ : j \in \{1, \ldots, N\}\}$$
    and write $\boldB \models \mathcal{R}(\vec{w})$ if
    $p_j(\vec{w}) = 0$ for all $j \in \{1, \ldots, M\}$ and $\norm{w_j} \leq C_j$ for all $j \in \{1, \ldots, N\}$.

    \item A function $g : \N \rightarrow \N$ is a \emph{modulus of weak stability} for
    $\mathcal{R}$ provided that, for every $k \in \N$, every \cstar-algebra $\boldB$,
    and every $w_1, \ldots, w_N \in \boldB$, if $\mathcal{R}^\boldB(\vec{w}) < 2^{-g(k)}$,
    then there exists $z_1, \ldots, z_N \in \boldB$ so that
    $\max_j \norm{w_j - z_j} < 2^{-k}$ and $\mathbf{B} \models \mathcal{R}(\vec{z})$.

    \item We say that $\mathcal{R}$ is \emph{weakly stable} if it has a modulus of weak stability and \emph{computably weakly stable} if it has a computable modulus of weak stability.
\end{enumerate}
\end{definition}

In \cite[Theorem 1.12]{UHFPaper} we showed that computably weakly stable relations define c.e. closed sets in c.e. presentations of \cstar-algebras.

We have now completed our description of relevant prior developments in
classical and computable mathematics.  We now attend to some purely prelminary matters
in these realms which will support our main efforts later.

\section{Preliminaries from classical mathematics}\label{sec:PrlmClscl}

\subsection{Ordered abelian groups}

\subsubsection{Inductive limits}

Our goal here is to arrive at a more concrete understanding of inductive limits
of ordered abelian groups which will be valuable when developing their computability
theory. We begin with abelian groups.

\begin{lemma}\label{lm:IndLimAbl}
Suppose $(G_s, \phi_s)_{s \in \N}$ is an inductive sequence of abelian groups and let
$(G, (\nu_s)_{s \in \N})$ be an inductive limit of $(G_s, \phi_s)_{s \in \N}$.  Then:
\begin{enumerate}
    \item $G = \bigcup_{s \in \N} \ran(\nu_s)$.\label{lm:IndLimAbl:ran}

    \item For each $s \in \N$, $\ker(\nu_s) = \bigcup_{k \in \N} \ker(\phi_{s, s+k})$.\label{lm:IndLimAbl:ker}
\end{enumerate}
\end{lemma}

\begin{proof}[Proof sketch]
We first review a standard construction of
an inductive limit of $(G_s, \phi_s)_{s \in \N}$. For each $s \in \N$, let $\rho_s : G_s \rightarrow \prod_n G_n$ be the homomorphism defined by
setting $\rho_s(a) = (\phi_{s,n}(a))_{n \in \N}$.
When $f,g \in \prod_n G_n$, write $f \sim g$ if there exists $k \in \N$
so that $f(k+n) = g(k + n)$ for all $n \in \N$.  Thus, $\sim$ is a congruence relation on
$\prod_n G_n$.  Let $\pi : \prod_n G_n \rightarrow \prod_n G_n/ \sim$ denote the canonical epimorphism,
and set $\mu_s = \pi \circ \rho_s$ for all $s \in \N$.

Define $H$ to be $\bigcup_{s \in \N} \ran(\mu_s)$.  By construction,
$\ran(\mu_s) \subseteq \ran(\mu_{s+1})$.  Hence, $H$ is an abelian group.
It is well-known, and easy to verify, that $(H, (\mu_n)_{n \in \N})$ is
an inductive limit of $(G_s, \phi_s)_{s \in \N}$.  Furthermore,
for each $s \in \N$, $\ker(\mu_s) = \bigcup_{k \in \N} \ker(\phi_{s,s+k})$.

Let $\lambda$ be the reduction of $(G, (\nu_s)_{s \in \N})$ to
$(H, (\mu_s)_{s \in \N})$.
By Proposition \ref{prop:IndLimUni}, $\lambda$ is an isomorphism.
Since $(H, (\mu_s)_{s \in \N})$ satisfies (\ref{lm:IndLimAbl:ran}) and
(\ref{lm:IndLimAbl:ker}), it follows that $(G, (\nu_s)_{s \in \N})$ does as well.
\end{proof}

We now proceed to describe inductive limits of ordered abelian groups.

\begin{lemma}\label{lm:IndLimOrdAbl}
Suppose $(\calG_s, \phi_s)_{s \in \N}$ is an inductive sequence of ordered abelian groups.
Assume $(\calG, (\nu_s)_{s \in \N})$ is an inductive upper limit of $(\calG_s, \phi_s)_{s \in \N}$.
Then $(\calG, (\nu_s)_{s \in \N})$ is an inductive limit of $(\calG_s, \phi_s)_{s \in \N}$
if and only if it satisfies the following:
\begin{enumerate}
    \item $\calG^+ = \bigcup_{s \in \N} \nu_s[\calG_s^+]$.\label{lm:IndLimOrdGrp:ran}

    \item For each $s \in \N$, $\ker(\nu_s) = \bigcup_{k \in \N} \ker(\phi_{s,s+k})$. \label{lm:IndLimOrdGrp:ker}
\end{enumerate}
\end{lemma}

\begin{proof}
Assume $\calG = (G, P)$ and $\calG_s = (G_s, P_s)$ for all $s \in \N$.
Let $(H, (\mu_s)_{s \in \N})$ be an inductive limit of
$(G_s, \phi_s)_{s \in \N}$, and let $\lambda$ be the reduction of
$(H, (\mu_s)_{s \in \N})$ to $(G, (\nu_s))_{s \in \N}$.
Let $P' = \bigcup_{s \in \N} \mu_s[P_s]$ and set $\calH = (H, P')$.
By \cite[Proposition 6.2.6]{Rordam.Larsen.Laustsen.2000},
$(\calH, (\mu_s)_{s \in \N})$ is an inductive limit of $(\calG_s, \phi_s)_{s \in \N}$.
It also follows that $\lambda$ is positive and so
$\lambda$ is the reduction of $(\calH, (\mu_s)_{s \in \N})$ to $(\calG, (\nu_s)_{s \in \N})$.
Furthermore, by construction, $\calH^+ = \bigcup_s \mu_s[\calG^+_s]$, and by
Lemma \ref{lm:IndLimAbl}, $\ker(\mu_s) = \bigcup_{k \in \N} \ker(\phi_{s,s+k})$ for all $s \in \N$.

Suppose $(\calG, (\nu_s)_{s \in \N})$ is an inductive limit of
$(\calG_s, \phi_s)_{s \in \N}$.
Then $\lambda$ is an isomorphism.  Since $(\calH, (\mu_s)_{s \in \N})$ satisfies
(\ref{lm:IndLimOrdGrp:ran}) and (\ref{lm:IndLimOrdGrp:ker}),
it follows that $(\calG, (\nu_s)_{s \in \N})$ does as well.

Finally, suppose $(\calG, (\nu_s)_{s \in \N})$ satisfies (\ref{lm:IndLimOrdGrp:ran}) and (\ref{lm:IndLimOrdGrp:ker}).  It follows from (\ref{lm:IndLimOrdGrp:ran}) that $\lambda$
is surjective.  Let $h \in \ker(\lambda)$.  Then, there exists $s$ and $g \in \calG_s$ so
that $\mu_s(g) = h$.  Hence, $g \in \ker(\nu_s)$, and so there exists $k \in \N$
so that $g \in \ker(\phi_{s,s+k})$.
Thus, $h = \mu_{s+k}(\phi_{s,s+k}(g)) = \zerovec_G$, and so $\lambda$ is an isomorphism.
Hence, $(\calG, (\nu_s)_{s \in \N})$ is an inductive limit of $(\calG_s, \phi_s)_{s \in \N}$.
\end{proof}

\subsubsection{Unital dimension groups}

Unital dimension groups are central to the $K$-theory of unital AF algebras, and
accordingly we elaborate on our definition of certificate of dimensionality
as follows.

\begin{definition}\label{def:UntlCertDim}
Suppose $(\calG, u)$ is a unital dimension group.  We say that $(n_s, u_s \nu_s, \phi_s)_{s \in \N}$
is a \emph{unital certificate of dimensionality} of $(\calG, u)$ if $u_s$ is an order unit of
$\Z^{n_s}$ and if
$((\calG, u), (\nu_s)_{s \in \N})$ is an inductive limit of
$((\Z^{n_s}, u_s), \phi_s)_{s \in \N}$ in the category of unital ordered abelian groups.
\end{definition}

These certificates will form the backbone of our proof of Main Theorem \ref{mthm:ClsCePresDimGrp::Untl}.

\subsection{\cstar-algebras}

\subsubsection{Inductive limits}

As in the previous section, we give a concrete description of
inductive limits of \cstar-algebras.

\begin{lemma}\label{lm:IndLimAlg}
Suppose $(\boldA_s, \phi_s)_{s \in \N}$ is an inductive sequence of \cstar-algebras,
and let $(\boldA, (\nu_s)_{s \in \N})$ be an inductive upper limit of
$(\boldA_s, \phi_s)_{s \in \N}$.  Then, $(\boldA, (\nu_s)_{s \in \N})$ is an inductive limit of
$(\boldA_s, \phi_s)_{s \in \N}$ if and only if it satisfies the following:
\begin{enumerate}
    \item $\boldA = \overline{\bigcup_{s \in \N} \ran(\nu_s)}$.\label{lm:IndLimAlg::ran}

    \item For every $s \in \boldA_s$, $\norm{\nu_s(a)} = \lim_k \norm{\phi_{s, s + k}(a)}$.\label{lm:IndLimAlg::norm}

    \item For every $s \in \N$, $\ker(\nu_s) = \{a \in A_s\ : \lim_k \norm{\phi_{s,s+k}(a)} = 0\}$.\label{lm:IndLimAlg::ker}
\end{enumerate}
\end{lemma}

\begin{proof}
The forward direction is \cite[Proposition 6.2.4]{Rordam.Larsen.Laustsen.2000}.

Suppose (\ref{lm:IndLimAlg::ran}), (\ref{lm:IndLimAlg::norm}), and (\ref{lm:IndLimAlg::ker}) are satisfied.
Let $(\boldB, (\mu_s)_{s \in \N})$ be an inductive upper limit of
$(\boldA_s, \phi_s)_{s \in \N}$.

We first demonstrate that $\ker(\nu_s) \subseteq \ker(\mu_s)$.
Let $a \in \ker(\nu_s)$.
Then, by (\ref{lm:IndLimAlg::norm}), $\lim_k \norm{\phi_{s, s+k}(a)} = 0$.
Since every $\star$-homomorphism is $1$-Lipschitz,
we have $\lim_k \norm{\mu_{s + k}(\phi_{s, s+k}(a))} = 0$.  However, since
$(\boldB, (\mu_s)_{s \in \N})$ is an inductive upper limit of $(\boldA_s, \phi_s)_{s \in \N}$,
it follows that $\mu_{s+k}(\phi_{s,s+k}(a)) = \mu_s(a)$ and so $a \in \ker(\mu_s)$.

It now follows that if $\nu_s(a_0) = \nu_{s + k}(a_1)$, then
$\mu_s(a_0) = \mu_{s+k}(a_1)$.

We now construct a $\star$-homomorphism $\lambda'$ from $\bigcup_{s \in \N} \ran(\nu_s)$ to $\boldB$.
To begin, let $a \in \ran(\nu_s)$, and choose any $a' \in \boldA_s$ so that $\nu_s(a') = a$.
Define $\lambda'(a)$ to be $\mu_s(a')$.  It follows from what has just been shown that
$\lambda'$ is well defined. Since each $\nu_s$ and each $\mu_s$ is a $\star$-homomorphism,
it also follows that $\lambda'$ is a $\star$-homomorphism.

We now claim that $\lambda'$ is $1$-Lipschtiz.  Let $a \in \ran(\nu_s)$, and suppose
$\nu_s(a') = a$.  By (\ref{lm:IndLimAlg::norm}), $\norm{\nu_s(a')} = \lim_k \norm{\phi_{s, s+k}(a')}$.
Since each $\mu_{s+k}$ is $1$-Lipschitz, it follows that
$\norm{\nu_s(a')} \geq \lim_k \norm{\mu_{s + k}(\phi_{s, s+k}(a'))}$. But,
$\mu_{s + k}(\phi_{s, s + k}(a')) = \mu_s(a') = \lambda'(a)$.

It now follows that $\lambda'$ has a unique continuous extension $\lambda$ to
$\boldA$.  Furthermore, $\lambda$ is a $\star$-homomorphism and
by construction $\lambda \circ \nu_s = \mu_s$ for all $s \in \N$.

Finally, it directly follows from the definitions that any two reductions of
$(\boldA, (\nu_s)_{s \in \N})$ to $(\boldB, (\mu_s)_{s \in \N})$ agree on
$\bigcup_{s \in \N} \ran(\nu_s)$ and hence must be identical.
\end{proof}

An immediate consequence of Lemma \ref{lm:IndLimAlg} is the following connection
 between AF certificates and inductive limits.

\begin{corollary}\label{cor:IndLimAFCert}
If $(F_j,\psi_j)_{j \in \N}$ is an AF certificate of $\boldA$,
then $(\boldA, (\psi_j)_{j \in \N})$ is an inductive limit of
$(\bigoplus_{n \in F_j} M_n(\C), (\psi_{j + 1}^{-1} \circ \psi_j)_{j \in \N})$.
\end{corollary}

\subsubsection{Matricial systems}

Our only item here is to present some terminology which will support
our proof of the Effective Glimm Lemma.

\begin{definition}\label{def:CompMat}
Suppose $\boldA$ is a \cstar-algebra and let $n_1, \ldots, n_m$ be positive integers.
An array $(e^s_{i,j})_{s,i,j}$ is a \emph{type-$(n_1, \ldots, n_m)$ matricial system of $\boldA$}
if it satisfies the following:
\begin{enumerate}
    \item For each $s \in \{1, \ldots, m\}$,
    $(e^s_{i,j})_{i,j}$ is an $n_s \times n_s$ system of matrix units of $\boldA$.

    \item When $s,s' \in \{1, \ldots, m\}$ are distinct, then $e^s_{i,j} e^{s'}_{i',j'} = 0$
    for all $i,j \in \{1, \ldots, n_s\}$ and all $i',j' \in \{1, \ldots, n_{s'}\}$.
\end{enumerate}
If, in addition, $\boldA$ is unital and $\sum_{s,i} e^s_{i,i} = \unit_\boldA$, then we say that the system $(e^s_{i,j})_{s,i,j}$ is
\emph{unital}.
\end{definition}

If $\boldF$ is a \cstar-algebra that is generated by a matricial system $(e^s_{i,j})_{s,i,j}$,
then
we call $(e^s_{i,j})_{s,i,j}$ a \emph{matricial generating system for $\boldF$}.
Thus a \cstar-algebra is finite-dimensional if and only if it has a matricial generating system.

\section{Preliminaries from effective mathematics}\label{sec:PrlmEff}

Herein, we present our frameworks for the computability of inductive limits
in a computably indexed category and for the computability of ordered
abelian groups.

\subsection{Inductive limits in computably indexed categories}

We now present a computable picture of the material discussed in \ref{subsec:BackIndLimCat}.
Thus, we again assume $\calC$ is a category with a zero object.
However, we assume we have fixed an \emph{indexing} of $\calC$.  This consists of
an assignment of natural numbers to the objects of $\calC$.
The number assigned to an object is referred to as one of its indexes.
Each object has at least one index.  A number may not index more the one object.
Each morphism is indexed by at least one triple of natural numbers.
If $(e_0, e_1,e)$ indexes a morphism $\gamma$, then $e_0$ must index its
domain and $e_1$ must index its co-domain.

We also assume this indexing is \emph{computable} in the sense that there is
an algorithm that given an index $(e_0,e_1,e)$ of $\phi : A \rightarrow B$ and
an index $(e_1, e_2,e')$ of $\psi : B \rightarrow C$
computes an index of $\psi \circ \phi$ of the form $(e_0, e_2, e'')$.
Furthermore, we require that from an index $e$ of an object $A$ it is possible to
compute an index of its identity morphism.

Now that the computable indexing is fixed, our remaining definitions are
straightforward modifications of those in Section \ref{subsec:BackIndLimCat}.
Fix an inductive sequence $(A_n, \phi_n)_{n \in \N}$ of $\calC$.
We say that $(A_n, \phi_n)_{n \in \N}$ is \emph{computable} if there is an algorithm
that given $n \in \N$ computes indexes of $A_n$ and $\phi_n$.
Now suppose
$(A, (\nu_n)_{n \in \N})$ is an inductive upper limit of $(A_n, \phi_n)_{n \in \N}$,
and assume it is possible to compute an index of $\nu_n$ from $n$.
We say that an upper inductive limit $(A, (\nu_n)_{n \in \N})$
of $(A_n, \phi_n)_{n \in \N}$ is \emph{computable} with index $e \in \N$ if $e$ is the code of a pair $(e_0, e_1)$ so that
$e_0$ is an index of $A$ and $e_1$ is an index of an algorithm that computes an index of
$\nu_n$ from $n$.
We say that $(A, (\nu_n)_{n \in \N})$ is \emph{a computable inductive limit} if
it is a computable inductive upper limit and if it is possible to compute from an index of a computable inductive upper limit
$(B, (\mu_n)_{n \in \N})$ of $(A_n, \phi_n)_{n \in \N}$ an index of the reduction from $(A, (\nu_n)_{n \in \N})$
to $(B, (\mu_n)_{n \in \N})$.

\subsection{Abelian groups}

\begin{lemma}\label{lm:IndLimPresAbl}
Suppose $(G, (\nu_s)_{s \in \N})$ is an inductive limit of
$(G_s, \phi_s)_{s \in \N}$ in the category of
abelian groups.
\begin{enumerate}
    \item If $(G_s^\#, \phi_s)_{s\in \N}$ is an inductive sequence in the category
    of c.e. presentations of abelian groups, and if
    $(G^\#, (\nu_s)_{s \in \N})$ is a computable inductive upper limit
    of this sequence, then $(G^\#, (\nu_s)_{s \in \N})$ is
    a computable inductive limit of $(G^\#, (\nu_s)_{s \in \N})$ in the category
    of c.e. presentations of abelian groups.

    \item If $(G_s^\#, \phi_s)_{s\in \N}$ is an inductive sequence in the category
    of computable presentations of abelian groups, and if
    $(G^\#, (\nu_s)_{s \in \N})$ is a computable inductive upper limit
    of this sequence, then $(G^\#, (\nu_s)_{s \in \N})$ is
    a computable inductive limit of $(G^\#, (\nu_s)_{s \in \N})$ in the category
    of computable presentations of abelian groups.
\end{enumerate}
\end{lemma}

\begin{proof}
Suppose $(G_s^\#, \phi_s)_{s\in \N}$ is an inductive sequence in the category
    of c.e. presentations of abelian groups, and assume
    $(G^\#, (\nu_s)_{s \in \N})$ is a computable inductive upper limit
    of this sequence.
    Let $(H^\#, (\mu_s)_{s \in \N}$ be a computable inductive upper limit of
    $(G_s^\#, \phi_s)_{s \in \N}$ in the category of c.e. presentations of abelian groups.
    Thus, $(H, (\mu_s)_{s \in \N})$ is an inductive upper limit of
    $(G_s, \phi_s)_{s \in \N}$ in the category of abelian groups.
    Hence, there is a unique reduction $\lambda$ of
$(G, (\nu_s)_{s \in \N})$ to $(H, (\mu_s)_{s \in \N})$.

It only remains to show that $\lambda$ is a computable map from $G^\#$ to $H^\#$.
Suppose $w$ is a $G^\#$ label of $g \in G$.  Then, there exists $s_0 \in \N$ so that
$g \in \ran(\nu_{s_0})$ and so there is a $\ran(\nu_{s_0})^\#$ label $w'$ of $g$.
Since the inclusion map is a computable map from $\ran(\nu_s)^\#$ to
$G^\#$ uniformly in $s$, $s_0$ and $w'$ can be found by a search procedure.
Then, by a search procedure, we can compute a $G_{s_0}^\#$ label $w''$ of a
$g' \in G_{s_0}$ so that $\nu_{s_0}(g') = g$ (since $\nu_s$ is a computable map
from $G_s^\#$ to $G^\#$ uniformly in $s$).  Since $\lambda(g) = \mu_{s_0}(g')$,
it follows that we can now compute an $H^\#$ label of $\lambda(g)$ from $w$.

The proof of the second part is almost identical.
\end{proof}

\begin{proposition}\label{prop:CompIndLimPresAbl}
\begin{enumerate}
    \item Every computable inductive sequence of c.e. presentations of abelian
    groups has a computable inductive limit.

    \item Every computable inductive sequence of computable presentations of abelian
    groups has a computable inductive limit.
\end{enumerate}
\end{proposition}

\begin{corollary}\label{cor:NotCompCatAF}

\

\begin{enumerate}
    \item The AF algebra $C(\omega +1 )$ is not computably categorical.

    \item There is a unital dimension group that has two c.e. presentations that are
    not computably isomorphic.

    \item There exist two computably presentable Bratteli diagrams that are
    equivalent but not computably equivalent.
\end{enumerate}
\end{corollary}

\begin{theorem}[Classification of finite-dimensional algebras]\label{thm:ClsFinDimAlg}
A \cstar-algebra $\boldA$ is finite-dimensional if and only if
there is a finite multiset $F$ of positive integers so that
$\boldA$ is $\star$-isomorphic to $\bigoplus_{n \in F} M_n(\C)$.
\end{theorem}

Suppose $F$ is a finite multiset of positive integers and set $\boldB = \bigoplus_{n \in F} M_n(\C)$.
When $\pi$ is the projection map associated with a summand
of $\boldB$, we associate $\pi$ with an injection map $\iota_\pi :\ran(\pi) \rightarrow \boldB$
so that $\pi \circ \iota_\pi = \Id_{\ran(\pi)}$.
In addition, we ensure that $\pi \circ \iota_{\pi'} = \zerovec$ if $\pi \neq \pi'$.

We now discuss the $K$-theory of finite-dimensional algebras.
Let $k = $ the cardinality of $F$.
Then, by the properties of the $K_0$ functor (see, e.g., \cite[Section 3.2]{Rordam.Larsen.Laustsen.2000}), it is possible to construct from the isomorphisms and order units  described in
Section \ref{subsubsec:BackMtrx} an order unit
$v_F$ for $\Z^k$ and
a canonical isomorphism $\zeta_F$ from $\uK{\boldB}$ to $(\Z^k, v_F)$.
Moreover, this construction provides a
canonical bijective mapping $\xi_F$ from the projection maps of the
summands of $\boldB$ to the generators of $\Z^k$.

Suppose $F'$ is a finite multiset of positive integers and set $\boldC = \bigoplus_{n \in F'} M_n(\C)$.
Let $k' = $ the cardinality of $F'$.
Suppose $\phi$ is a $\star$-homomorphism from $\boldB$ to $\boldC$.
The properties of the $K_0$ functor allow one to construct, from the maps
$\eta_{m,n}$ described in Section \ref{subsubsec:BackMtrx},
a canonical homomorphism $\eta_{F,F'}(\phi)$ from the ordered group $\Z^k$ to
$\Z^{k'}$
so that $\zeta_{F'} \circ \phi  = \eta_{F,F'}(\phi)\circ\zeta_F$.
In addition, the construction of $\eta_{F,F'}$ ensures that whenever
$\pi$ is the projection map for a summand of $\boldB$ and $\pi'$ is a projection
map for a summand of $\boldC$,
$\pi_{\xi_{F'}(\pi')}(\eta_{F,F'}(\phi)(\xi_F(\pi))) = $
the multiplicity of $\pi' \circ \phi \circ \iota_\pi$.
It follows form the discussion in Section \ref{subsubsec:BackMtrx}
that $\eta_{F, F'}(\phi)(v_F) \leq v_{F'}$ and so
$K_0(\phi)(K_0(\unit_\boldB)) \leq K_0(\unit_\boldC)$.
Conversely, if $\gamma : (K_0(\boldB), K_0(\boldB)^+) \rightarrow (K_0(\boldC), K_0(\boldC)^+)$
is a homomorphism and if $\gamma(K_0(\unit_\boldB)) \leq K_0(\unit_\boldC)$, then
there is a $\star$-homomorphism $\delta$ from $\boldA$ to $\boldB$ so
that $K_0(\delta) = \gamma$.

\subsubsection{AF algebras}\label{subsubsec:BackAF}
We finally arrive at a formal treatment of AF algebras.
\begin{definition}\label{def:AFCert}
Suppose $\boldA$ is a \cstar-algebra.
 An \emph{AF certificate} of $\boldA$ is a sequence
$(F_j, \psi_j)_{j \in \N}$ that satisfies the following:
\begin{enumerate}
    \item Each $F_j$ is a finite multiset of positive integers.

    \item For each $j \in \N$, $\psi_j$ is a unital $\star$-embedding of $\bigoplus_{n \in F_j} M_n(\C)$
    into $\boldA$.

    \item For all $j$, $\ran(\psi_j) \subseteq \ran(\psi_{j+1})$.

    \item $\boldA = \overline{\bigcup_{j \in \N} \ran(\psi_j)}$.
\end{enumerate}
\end{definition}

A \cstar-algebra is an \emph{AF algebra} if it has an AF certificate.
We remark that this definition is not, but nevertheless is equivalent to,
the standard definition of an AF algebra as an inductive limit of
finite-dimensional algebras.  Our choice of this definition is motivated by forthcoming
computabilty considerations.

We can now state Elliott's Theorem formally, a proof of which can be found in
\cite{Rordam.Larsen.Laustsen.2000}.

\begin{theorem}[Elliott's Theorem]\label{thm:Elliott}
Suppose $\boldA$ and $\boldB$ are AF algebras.  If
$\uK{\boldA}$ and
$\uK{\boldB}$ are isomorphic,
then $\boldA$ and $\boldB$ are  $\star$-isomorphic.
Moreover, if $\alpha$ is an isomorphism from $\uK{\boldA}$
to $\uK{\boldB}$,
then there is a unital $\star$-isomorphism $\phi : \boldA \rightarrow \boldB$
so that $K_0(\phi) = \alpha$.
\end{theorem}

\begin{maintheorem}\label{mthm:CompAFCert} If a unital AF algebra has a c.e. presentation, then it also has a computable AF certificate.
\end{maintheorem}

\begin{theorem}[Effective Glimm Lemma]\label{thm:EffGlimm}
There is a computable function $\Delta:\mathbb{N}^2\to \mathbb{N}$
so that for every c.e.-closed $\star$-subalgebra $\boldB$ of $\boldA^\#$,
 all $n,k \in \N$, and every c.e.-closed $\star$-subalgebra $\boldF$ of $\boldA^\#$ for which $\unit_\boldA \in \boldF$,
if $\dim(\boldF) = n$, and if $B(\boldB; 2^{-\Delta(n,k)})$ contains a matricial generating
system of $\boldF$, then there is a computable unitary $u$ of $\boldA^\#$ so that
$\norm{u - \unit_\boldA} < 2^{-k}$ and $u^*\boldF u \subseteq \boldB$.
Furthermore, an $\boldA^\#$ index of $u$ can be computed from an index of $\boldA^\#$ and $\boldA^\#$ indexes of $\boldF$ and $\boldB$.
\end{theorem}

\label{sec:Intro}
In our previous work \cite{UHFPaper} with R. Miller, we initiated the effective study of $K$-theory for $\mathrm{C}^*$-algebras. More precisely, we established a computable functor which associated, to each computably enumerable (c.e.) presentation $\mathbf{A}^\#$ of a $\mathrm{C}^*$-algebra $\mathbf{A}$, a c.e. presentation $K_0(\mathbf{A}^\#)$ of its $K_0$ group $K_0(\mathbf{A})$. In addition, this functor maps
computable $\ast$-homomorphisms to computable group homomorphisms. These statements are
made precise in Section \ref{subsubsec:BackCompKThy} below.
A similar such computable functor was also defined for $K_1$ as well. As a proof of concept, the special case of the $K_0$ functor restricted to \emph{uniformly hyperfinite (UHF)} algebras was completely analyzed.

In this paper, we broaden our scope and consider our computable $K_0$ functor restricted to the class of \emph{approximately finite-dimensional (AF)} algebras, which are the $\mathrm{C}^*$-algebras isomorphic to those obtained as inductive limits of finite-dimensional $\mathrm{C}^*$-algebras. Every AF algebra $\mathbf{A}$ has $K_1(\mathbf{A}) = 0$, so the $K$-theory for AF algebras is entirely focused on $K_0$. Since AF algebras are stably finite, their $K_0$ group has a natural ordering which renders it a (partially) ordered abelian group.
In addition, for a c.e. presentation $\mathbf{A}^\#$, the positive cone of this group,
which we denote $K_0(\mathbf{A})^+$, is a c.e. set of the presentation $K_0(\mathbf{A}^\#)$;
accordingly, we refer to $(K_0(\mathbf{A}^\#), K_0(\mathbf{A})^+)$ as the \emph{ordered presented group
of $\mathbf{A}^\#$}.
The ordered abelian groups that arise as ordered $K_0$ groups of AF algebras are known as \emph{dimension groups}.
By the Effros-Handelman-Shen Theorem, these are precisely the ordered abelian groups isomorphic to an inductive limit of \emph{simplicial groups}, that is, groups of the form $\mathbb{Z}^n$ with their canonical ordering. By distinguishing the image $K_0(\unit_\mathbf{A})$ of the unit of $\mathbf{A}$ in $K_0(\mathbf{A})$, one views $K_0(\mathbf{A})$ as a \emph{unital (or scaled) group}, that is, as an ordered abelian group with a distinguished order unit. Elliott's classification theorem \cite{Elliott.1976} states that AF algebras are characterized, up to isomorphism, by their unital ordered $K_0$ groups. In fact, the $K_0$ functor yields an equivalence of categories between the category of AF algebras, with approximate unitary equivalence classes of $*$-algebra homomorphisms as morphisms, and the category of unital dimension groups, with order-unit preserving positive group homomorphisms as morphisms.

While $K_0$ groups provide an algebraic representation of AF algebras, Bratteli diagrams provide a combinatorial method of representing AF algebras, namely as
infinite multi-graphs arranged in a sequence of levels.
Labeled Bratteli diagrams assign numbers to the vertexes, and every AF algebra is completely described
by a labeled Bratteli diagram.

Our first overarching goal is to classify the c.e. presentations of AF algebras.
To this end, we introduce the concept of an \emph{AF certificate} (defined in Section \ref{subsubsec:BackAF}
below). These essentially describe an AF algebra as a specific inductive limit of finite dimensional
algebras. Our first main result is the following.

\section{Introduction}\label{sec:Intro}
In our previous work \cite{UHFPaper} with R. Miller, we initiated the effective study of $K$-theory for \cstar-algebras. More precisely, we established a computable functor which associated, to each computably enumerable (c.e.) presentation $\boldA^\#$ of a \cstar-algebra $\boldA$, a c.e. presentation $K_0(\boldA^\#)$ of its $K_0$ group $K_0(\boldA)$. In addition, this functor maps
computable $\star$-homomorphisms to computable group homomorphisms. These statements are
made precise in Section \ref{subsubsec:BackCompKThy} below.
A similar such computable functor was also defined for $K_1$ as well. As a proof of concept, the special case of the $K_0$ functor restricted to \emph{uniformly hyperfinite (UHF)} algebras was completely analyzed.

Our first overarching goal is to classify the c.e. presentations of AF algebras.
To this end, we introduce the concept of an \emph{AF certificate} (defined in Section \ref{subsubsec:BackAF}
below). These essentially describe an AF algebra as a specific inductive limit of finite dimensional
algebras. Our first main result is the following.

In classical mathematics, when an AF algebra is considered, a representation of it
as an inductive limit of finite-dimensional algebras is assumed. However, a c.e.
presentation of an AF algebra, which merely provides information about the
norm on a certain dense set, does not in and of itself readily provide such a description.
Rather, from the information provided by a c.e. presentation of an AF algebra $\boldA$, one must algorithmically discern the finite-dimensional $\star$-subalgebras of $\boldA$ (which are not the same
as the finitely generated subalgebras). Locating these $\star$-subalgebras turns out to be no
mean feat, and as an intermediate step we go to great lengths to first prove an effective
version of Glimm's Lemma (Theorem \ref{thm:EffGlimm} below). We note that our proof of Main Theorem \ref{mthm:CompAFCert} is uniform in that it provides an algorithm for producing AF certificates
from c.e. presentations of AF algebras.

\item If $u$ is an order unit of $\calG$,
 then there is a c.e. presentation $\boldA^\#$ of a
 unital AF algebra so that the presented unital ordered $K_0$ group of $\boldA^\#$
is computably isomorphic to $(\calG, u)^\#$.\label{mthm:ClsCePresDimGrp::Untl}
\end{enumerate}
\end{maintheorem}

\begin{maintheorem}[Classification of computably presentable unital AF algebras]\label{mthm:ClsCompPrsntAF}
Suppose $\boldA$ is a unital AF algebra. Then, the following are equivalent.
\begin{enumerate}
 \item $\boldA$ has a c.e. presentation. \label{mthm:ClsCompPrsntAF::CePres}

\item There is a computable inductive sequence of finite dimensional algebras whose inductive limit is $\star$-isomorphic
 to $\boldA$ and so that each bonding map is unital.\label{mthm:ClsCompPrsntAF::CompIndSeqFinDimAlg}

\begin{maintheorem}\label{mthm:CompIso}
Suppose $\boldA^\#$ and $\boldB^\#$ are c.e. presentations of unital AF algebras.
Then, the following are equivalent.
\begin{enumerate}
 \item $\boldA^\#$ is computably $\star$-isomorphic to $\boldB^\#$.\label{mthm:CompIso::Pres}

\begin{definition}\label{def:CompAFCert}
Suppose $(F_j, \psi_j)_{j \in \N}$ is an AF certificate for $\boldA$, and assume $\boldA^\#$ is a
presentation of $\boldA$. We say that $(F_j, \psi_j)_{j \in \N}$ is a \emph{computable AF certificate of $\boldA^\#$} if $(F_j)_{j \in \N}$ is computable and if each $\psi_j$ is a computable map
from $\bigoplus_{n \in F_j}M_n(\C)$
 to $\boldA^\#$.
\end{definition}

In classical mathematics, when an AF algebra is considered, a representation of it
as an inductive limit of finite-dimensional algebras is assumed.  However, a c.e.
presentation of an AF algebra, which merely provides information about the
norm on a certain dense set, does not in and of itself readily provide such a description.
Rather, from the information provided by a c.e. presentation of an AF algebra $\mathbf{A}$, one must algorithmically discern the finite-dimensional $\ast$-subalgebras of $\mathbf{A}$ (which are not the same
as the finitely generated subalgebras).  Locating these $\ast$-subalgebras turns out to be no
mean feat, and as an intermediate step we go to great lengths to first prove an effective
version of Glimm's Lemma (Theorem \ref{thm:EffGlimm} below).  We note that our proof of Main Theorem \ref{mthm:CompAFCert} is uniform in that it provides an algorithm for producing AF certificates
from c.e. presentations of AF algebras.

Our second main result, which will also contribute to our first overarching goal,
is an effective version of the Effros-Handelman-Shen Theorem referred to above.

\begin{maintheorem}\label{mthm:ClsCePresDimGrp} Suppose $\mathcal{G}^\#$ is a c.e. presentation of a dimension group.
\begin{enumerate}
    \item There is a c.e. presentation $\mathbf{A}^\#$ of an AF algebra so that
the presented ordered $K_0$ group of $\mathbf{A}^\#$
is computably isomorphic to $\mathcal{G}^\#$.\label{mthm:ClsCePresDimGrp::NotUntl}

    \item If $u$ is an order unit of $\mathcal{G}$,
    then there is a c.e. presentation $\mathbf{A}^\#$ of a
    unital AF algebra so that the presented unital ordered $K_0$ group of $\mathbf{A}^\#$
is computably isomorphic to $(\mathcal{G}, u)^\#$.\label{mthm:ClsCePresDimGrp::Untl}
\end{enumerate}
\end{maintheorem}

Again, the proof is uniform.  With these two results in hand, we then
achieve our goal of classifying the c.e. presentations of unital AF algebras.

\begin{maintheorem}[Classification of computably presentable unital AF algebras]\label{mthm:ClsCompPrsntAF}
Suppose $\mathbf{A}$ is a unital AF algebra. Then, the following are equivalent.
\begin{enumerate}
    \item $\mathbf{A}$ has a c.e. presentation. \label{mthm:ClsCompPrsntAF::CePres}

    \item There is a computable inductive sequence     of finite dimensional algebras whose inductive limit is $\ast$-isomorphic
    to $\mathbf{A}$ and so that each     bonding map is unital.\label{mthm:ClsCompPrsntAF::CompIndSeqFinDimAlg}

    \item $\mathbf{A}$ is computably presentatable. \label{mthm:ClsCompPrsntAF::CompPres}

    \item The unital ordered $K_0$ group of $\mathbf{A}$ is computably presentatable. \label{mthm:ClsCompPrsntAF::CompPresUntlOrdAbl}

    \item $\mathbf{A}$ has a computably presentable labeled Bratteli diagram.\label{mthm:ClsCompPrsntAF::CompBrat}
\end{enumerate}
\end{maintheorem}

Once again, the proof is uniform.  Our next overarching goal is to strengthen this
classification result by showing that computably isomorphic objects
map to computably isomorphic objects.  This is the content of the following.

\begin{maintheorem}\label{mthm:CompIso}
Suppose $\mathbf{A}^\#$ and $\mathbf{B}^\#$ are c.e. presentations of unital AF algebras.
Then, the following are equivalent.
\begin{enumerate}
    \item $\mathbf{A}^\#$ is computably $\ast$-isomorphic to $\mathbf{B}^\#$.\label{mthm:CompIso::Pres}

    \item The presented unital ordered $K_0$ groups of $\mathbf{A}^\#$ and $\mathbf{B}^\#$ are
    computably isomorphic.\label{mthm:CompIso::Grp}

    \item Every Bratteli diagram of a computable AF certificate of
    $\mathbf{A}^\#$ is computably equivalent to
    every Brattelii diagram of a computable AF certificate of $\mathbf{B}^\#$. \label{mthm:CompIso::Brat}
\end{enumerate}
\end{maintheorem}

Not surprisingly, the proof is uniform.  Our results so far suggest that
it should be possible to demonstrate an effective version of Elliott's
equivalence of categories, and in fact it is:

\begin{maintheorem}\label{mthm:CompEqvCat}
The following categories are computably equivalent:
\begin{enumerate}
    \item The category of c.e. presentations of unital AF algebras.

    \item The category of c.e. presentations of unital dimension groups.
\end{enumerate}
\end{maintheorem}

We formally define the terminology in this theorem in Section \ref{sec:CompEqvCat} below; for now
let us say that computable equivalence means the required functors and natural isomorphisms
can be computed.  The proof is very much based on the uniformity of the proofs of the prior
theorems.

Our main theorems have several interesting consequences.  For one,
every c.e. presentation of a unital AF algebra is computable (Corollary \ref{cor:CompPresAF}).
Also, in contrast to UHF algebras, there is an AF algebra that is not computably categorical
(Corollary \ref{cor:NotCompCatAF}).  We can use this observation to prove
similar statements for dimension groups and Bratteli diagrams.
Finally, we use our results to determine the complexity of the index set and isomorphism
problems for AF and UHF algebras.

This paper brings together material from several areas of mathematics: computability theory,
category theory, group theory, and the theory of $\mathrm{C}^*$-algebras.
In order to make the presentation fairly self-contained and navigable, we have chosen
the following organization.  We first cover background material from these areas; this consists
of information that is already in the literature but which we attempt to organize and summarize
so that the reader need not constantly consult an array of other sources.
We divide this coverage into two components: the background from classical mathematics
(Section \ref{sec:BackClscl}) and from effective mathematics (Section \ref{sec:BackEff}).
The latter presents a picture of the
computability theory of the topics covered in the former with a parallel organization.
We then attend to the development of preliminary matters from classical and effective mathematics
(Sections \ref{sec:PrlmClscl} and \ref{sec:PrlmEff}).
The Effective Glimm Lemma is proven in Section \ref{sec:EffGlimm}, and our main theorems are proven in Sections \ref{sec:CompAFCert} through \ref{sec:CompEqvCat}.
Section \ref{sec:IndxIso} contains our results on index sets

From a c.e. presentation of an AF algebra $\mathbf{A}$ (which “merely provides information about the norm on a certain dense set”), one must “algorithmically discern the finite-dimensional $\ast$-subalgebras of $\mathbf{A}$ (which are not the same as the finitely generated subalgebras).” As an intermediate step toward proving Main Theorem~\ref{mthm:CompAFCert}, the authors “go to great lengths to first prove an effective version of Glimm's Lemma (Theorem \ref{thm:EffGlimm} below)” in order to help locate these $\ast$-subalgebras. The proof of Main Theorem~\ref{mthm:CompAFCert} is stated to be “uniform in that it provides an algorithm for producing AF certificates from c.e. presentations of AF algebras.”

Question: Let \(\mathbf A\) be a unital approximately finite-dimensional (AF) \(C^*\)-algebra, and suppose \(\mathbf A\) has a computably enumerable (c.e.) presentation. An AF certificate for \(\mathbf A\) is data \((F_j,\psi_j)_{j\in\mathbb N}\) describing \(\mathbf A\) as a specific inductive limit of finite-dimensional algebras, where each \(\psi_j\) is a map from \(\bigoplus_{n\in F_j} M_n(\mathbb C)\) into \(\mathbf A\). A computable AF certificate of the given presentation \(\mathbf A^\#\) means such an AF certificate for which the sequence \((F_j)_{j\in\mathbb N}\) is computable and each \(\psi_j\) is computable as a map into \(\mathbf A^\#\). Under these assumptions, which statement holds?

There exists a computable AF certificate for the c.e. presentation of \(\mathbf A\); equivalently, every unital AF algebra with a c.e. presentation also has a computable AF certificate.

Choices:
B. There exists a single algorithm which, given any c.e. presentation of a unital AF algebra, computes a computable AF certificate and moreover computes each map \(\psi_j\) uniformly from the presentation with no additional information.
C. There exists an AF certificate for the c.e. presentation of \(\mathbf A\); equivalently, every unital AF algebra with a c.e. presentation has some AF certificate.
D. There exists a computable sequence \((F_j)_{j\in\mathbb N}\) of finite-dimensional building blocks for \(\mathbf A\), but the maps \(\psi_j\colon \bigoplus_{n\in F_j} M_n(\mathbb C)\to \mathbf A^\#\) need not be computable.
E. There exists a computable AF certificate for the c.e. presentation of \(\mathbf A\), and in addition the finite-dimensional \(^*\)-subalgebras appearing in the certificate can be effectively recovered as the finitely generated \(^*\)-subalgebras of \(\mathbf A^\#\).

dropped computability of the certificate

computability of each embedding map \(\psi_j\)

identifying finite-dimensional \(^*\)-subalgebras with finitely generated \(^*\)-subalgebras

The stem defines the relevant notions but does not explicitly reveal that a computable AF certificate exists. The correct answer is not directly stated in the prompt.

Choice A is essentially the theorem statement being tested, with only minimal reformulation. The item mainly asks for recognition of the exact claimed conclusion.

Some reasoning is needed to distinguish the exact theorem from stronger uniform-effectivity claims and weaker existence claims, but the task is still largely theorem recall rather than genuine derivation.

The distractors are plausible and mathematically meaningful: one is a stronger uniformity claim, one is a weaker true-but-insufficient statement, one confuses computability of objects versus maps, and one adds an unjustified extra effectiveness claim.

A technically well-constructed theorem-recognition MCQ with strong distractors, but it is close to a direct restatement of the target result and only moderately tests generative reasoning.


Define $H_N=\{(x,y)\in \mathbb{R}^2: xy\ge N, x,y>0\}$ which is an unbounded convex set and $N$ is a (large) positive integer. Its {\sl integer convex hull} or {\sl integer hull} for short is the convex hull of the integer (or lattice) points in $H_N$:
\[
I(H_N)=\conv(\mathbb{Z}^2 \cap H_N).
\]
This is bounded by the two halflines $\{(x,1): x\ge 1\}$ and $\{(1,y): y\ge 1\}$ and a convex lattice chain connecting $(1,N)$ with $(N,1)$. Our target is to determine the order of magnitude of the number of vertices of $I(H_N)$. The traditional notation for the number of vertices of a polytope $P\subset \mathbb{R}^d$ is $f_0(P)$, so we are interested in $f_0(I(H_N))$.

\medskip
Motivation for this question comes from integer programming and, more generally, from the theory of geometry of numbers. The determination or estimation of $f_0(I(H_N))$ is considered in a recent paper by Alc\'antara et al. \cite{Santos}. They establish among other results that
\begin{equation}\label{eq:Sant}
N^{1/3} \ll f_0(I(H_N)) \ll N^{1/3} \log N,
\end{equation}
and ask what the right order of magnitude of $f_0(I(H_N))$ is. Here and throughout the paper the notation $f(N)\ll g(N)$ means that there is a constant $C>0$ such that $0<f(N)<Cg(N)$ for every $N \in \mathbb{N}$. The notation $f(N)\gg g(N)$ is analogous. We will also use the $O(.)$ notation. Our main result establishes the order of magnitude of $f_0(I(H_N))$.

\medskip
Motivation for this question comes from integer programming and, more generally, from the theory of geometry of numbers. The determination or estimation of $f_0(I(H_N))$ is considered in a recent paper by Alc\'antara et al. \cite{Santos}. They establish among other results that
\begin{equation}\label{eq:Sant}
N^{1/3} \ll f_0(I(H_N)) \ll N^{1/3} \log N,
\end{equation}
and ask what the right order of magnitude of $f_0(I(H_N))$ is. Here and throughout the paper the notation $f(N)\ll g(N)$ means that there is a constant $C>0$ such that $0<f(N)<Cg(N)$ for every $N \in \NN$. The notation $f(N)\gg g(N)$ is analogous. We will also use the $O(.)$ notation. Our main result establishes the order of magnitude of $f_0(I(H_N))$.

A similar question is considered in our earlier paper \cite{BalBar}. Writing $B^2$ for the Euclidean unit disk centered at the origin the set $P_r=I(rB^2)$ is a convex lattice polytope, the integer hull of $rB^2$. It is shown in \cite{BalBar} that
\begin{equation}\label{eq:BalB}
 0.33 r^{2/3} < f_0(P_r) < 5.55 r^{2/3}.
\end{equation}
Most likely the limit $r^{-2/3}f_0(P_r)$ exists. Balog and Deshoullier \cite{BalD}
proved that the average of $r^{-2/3}f_0(P_r)$ in the interval $[R,R+M]$ tends to an explicit constant 3.453.. as $R \to \infty$ and $M \to\infty$ (for instance $M=\log R$). The upper bound in (\ref{eq:BalB}) is easier and follows from Andrews theorem (Theorem~\ref{th:Andr} below). The lower bound is more difficult and the method of its proof will be used here with suitable modifications. But the case of hyperbola is more involved because the curvature changes from $N^{-1/2}$ to $N^{-2}$ while it is constant for the disk.

\medskip
We observe that $I(H_N)$ has few, about $2N^{1/3}$ vertices outside a square $[1,N_2]^2$, where $N_2$ is about $N^{2/3}$, because we can actually list these vertices. They are
\[
(x,y) = (k,\lceil\frac Nk\rceil),\quad 1\leq k\leq N^{1/3},
\]
and by symmetry
\[
(x,y) = (\lceil\frac Nk\rceil,k),\quad 1\leq k\leq N^{1/3}.
\]
Indeed, the integer point $(1,N)$ is a vertex and $(k,\lceil\frac Nk\rceil)$ is certainly a vertex whenever $(k-1,\lceil\frac N{k-1}\rceil)$ is a vertex and
\[
\frac N{k+1} > 2\lceil\frac Nk\rceil - \lceil\frac N{k-1}\rceil.
\]
This inequality follows if $k\leq N^{1/3}$. We mention that the above list of vertices is practically the lower bound of Alc\'antara et al. ~\cite{Santos}. For later use we point out that with
\begin{equation}\label{eq:N_12}
 N_1=\lfloor N^{1/3}\rfloor =N^{1/3}+O(1),\quad N_2=\lceil \frac N{N_1} \rceil=N^{2/3}+O(N^{1/3}),
\end{equation}
the points $(N_1,N_2)$ and $(N_2,N_1)$ are vertices of $I(H_N)$.
Introducing the notation $Q_N=H_N\cap [1,N_2]^2$ another form of Theorem~\ref{th:main} is the following.

\begin{thm}\label{th:main2}
\[
N^{1/3} \log N \ll f_0(I(Q_N)) \ll N^{1/3} \log N.
\]
\end{thm}

\begin{thm}\label{th:area}
\[ N^{1/3} \log N \ll A_N \ll N^{1/3} \log N.\]
\end{thm}

\medskip
Our Theorem~\ref{th:main} claims that a positive proportion of the integer points in the above hyperbolic strip are actually vertices of $I(H_N)$. For technical reasons, we change to a somewhat narrower strip. In this way, we can miss a few vertices, but we can afford to get a lower bound.
Set $\Delta=\frac12 N^{1/3}$, and write $H^1$ and $H^2$ for the hyperbolas with equations $xy=N$ and $xy=N+\Delta$, respectively, and finally $S$ for the strip between $H^1$ and $H^2$. The above argument shows that
\begin{equation}\label{eq:delta}
|S \cap \ZZ^2| = \Delta \log N+O(N^{1/3}).
\end{equation}

\medskip
The number of non-vertices associated with the fixed primitive vector $(a,b)$ is then at most $M_1+M_2\le 4\sqrt{\Delta/ab}$. Suppose $(a_i,b_i)\in \ZZ^2$ for $i=1,\ldots,R$ is the set of primitive vectors (with distinct directions) that take part in the relations $z\to v$ with $z \in NV$ and $v \in V$. Then $R\le 2|V|$ as one vertex is used with at most two vectors $(a_i,b_i).$ At the same time
\begin{equation}\label{eq:3root}
|NV| \le \sum_{i=1}^R 4\sqrt{\frac{\Delta}{a_ib_i}}=4\sqrt{\Delta}\sum_{i=1}^R \frac1{\sqrt{a_ib_i}}.
\end{equation}
The last sum is the largest when we choose $R$ different primitive vectors with $a_ib_i$ as small as possible, so with $a_ib_i\le w$ for the least possible $w\in \NN$. From {\bf Fact ~\ref{fa:divisors}} this $w$ satisfies $F(w-1)<R\le F(w)$ and then
\[
R=\frac1{\zeta(2)}w\log w+O(w).
\]
This $w$ satisfies
\[ w\ll \frac R{\log R}\ll \frac {|V|}{\log |V|}. \]

Let $G_j$ be the set of inner normals $p$ with $x_p \in I_j$. We are going to show that $ \Area \left(\bigcup_{p\in G_j}\right) C^p \ll N^{1/3}$ for every $j$. This will complete the proof as there are $O(\log N)$ such intervals. For a fixed $p \in G_j$, the cap $C^p$ has parameters $x_p,h_x,r_x$ and $\rho_x$. The advantage of the intervals $I_j$ is that the radius of curvature $r_x\asymp \frac {x^3}N$ on $I_j$ is almost constant, and so is the slope $-\frac N{x^2}$ of the tangent to $H^1$ at $x$:
\[
2^{-3j}N \ll r_x \ll 2^{-3j+3}N \mbox{ and } -2^{2j}N^{-1/3} \le -\frac N{x^2} \le -2^{2j-2}N^{-1/3}
\]
We set $R_j:= 2^{-3j}N$ and define $\Phi_j\in (0,\pi/4]$ via $\tan \Phi_j:= 2^{2j}N^{-1/3}$. So for all $x\in I_j$
\begin{equation}\label{eq:bounded}
x\asymp 2^{-j}N^{2/3}\mbox{ and } r_x\asymp R_j \mbox{ and } \frac N{x^2}\asymp \tan \Phi_j \mbox{ and }|I_j|=2^{-j}N^{2/3}.
\end{equation}
We distinguish three cases.

\begin{equation}\label{eq:BalB}
 0.33 r^{2/3} < f_0(P_r) < 5.55 r^{2/3}.
\end{equation}

\begin{thm}\label{th:Andr} For a lattice polytope $P \subset \RR^d$ with $\vol P>0$
\[
f_0(P) \ll (\vol P)^{\frac {d-1}{d+1}}
\]
with the implied constant depending only on $d$.
\end{thm}

\begin{thm}\label{th:main}
\[
N^{1/3} \log N \ll f_0(I(H_N)) \ll N^{1/3} \log N.
\]
\end{thm}

A similar question is considered in our earlier paper \cite{BalBar}. Writing $B^2$ for the Euclidean unit disk centered at the origin the set $P_r=I(rB^2)$ is a convex lattice polytope, the integer hull of $rB^2$. It is shown in \cite{BalBar} that
\begin{equation}\label{eq:BalB}
 0.33 r^{2/3} < f_0(P_r) < 5.55 r^{2/3}.
\end{equation}
Most likely the limit $r^{-2/3}f_0(P_r)$ exists. Balog and Deshoullier \cite{BalD}
proved that the average of $r^{-2/3}f_0(P_r)$ in the interval $[R,R+M]$ tends to an explicit constant 3.453.. as $R \to \infty$ and $M \to\infty$ (for instance $M=\log R$). The upper bound in (\ref{eq:BalB}) is easier and follows from Andrews theorem (Theorem~\ref{th:Andr} below). The lower bound is more difficult and the method of its proof will be used here with suitable modifications. But the case of hyperbola is more involved because the curvature changes from $N^{-1/2}$ to $N^{-2}$ while it is constant for the disk.

\medskip
We observe that $I(H_N)$ has few, about $2N^{1/3}$ vertices outside a square $[1,N_2]^2$, where $N_2$ is about $N^{2/3}$, because we can actually list these vertices. They are
\[
(x,y) = (k,\lceil\frac Nk\rceil),\quad 1\leq k\leq N^{1/3},
\]
and by symmetry
\[
(x,y) = (\lceil\frac Nk\rceil,k),\quad 1\leq k\leq N^{1/3}.
\]
Indeed, the integer point $(1,N)$ is a vertex and $(k,\lceil\frac Nk\rceil)$ is certainly a vertex whenever $(k-1,\lceil\frac N{k-1}\rceil)$ is a vertex and
\[
\frac N{k+1} > 2\lceil\frac Nk\rceil - \lceil\frac N{k-1}\rceil.
\]
This inequality follows if $k\leq N^{1/3}$. We mention that the above list of vertices is practically the lower bound of Alc\'antara et al. ~\cite{Santos}. For later use we point out that with
\begin{equation}\label{eq:N_12}
    N_1=\lfloor N^{1/3}\rfloor =N^{1/3}+O(1),\quad N_2=\lceil \frac N{N_1} \rceil=N^{2/3}+O(N^{1/3}),
\end{equation}
the points $(N_1,N_2)$ and $(N_2,N_1)$ are vertices of $I(H_N)$.
Introducing the notation $Q_N=H_N\cap [1,N_2]^2$ another form of Theorem~\ref{th:main} is the following.

\begin{thm}\label{th:main2}
\[
N^{1/3} \log N \ll f_0(I(Q_N)) \ll N^{1/3} \log N.
\]
\end{thm}

Our second main result is about how well $I(H_N)$ approximates $H_N$ when the approximation is measured by the area $H_N$ missed by $I(H_N)$. But this area is infinite because $H_N \setminus I(H_N)$ contains a large part of the set $\{(x,y)\in \mathbb{R}^2: x\in [0,\infty), y\in [0,1)\}$. So it is better to consider $(H_N \setminus I(H_N))\cap [1,N]^2$. The area of this set is of order $N$; see the remark at the end of Section~\ref{sec:prepa}. So we rather work with $Q_N$ again and define
\[
A_N=\Area(Q_N \setminus I(Q_N)).
\]

\begin{thm}\label{th:area}
\[ N^{1/3} \log N \ll A_N \ll N^{1/3} \log N.\]
\end{thm}

\bigskip

The post-theorem introduction indicates how the proof of Theorem~\ref{th:main} will proceed only at a high level: the authors note that for the disk case the upper bound “follows from Andrews theorem (Theorem~\ref{th:Andr} below)”, while “the lower bound is more difficult and the method of its proof will be used here with suitable modifications.” They then explain an initial structural step for the hyperbola case: “we observe that $I(H_N)$ has few, about $2N^{1/3}$ vertices outside a square $[1,N_2]^2$, where $N_2$ is about $N^{2/3}$, because we can actually list these vertices,” namely $(x,y)=(k,\lceil N/k\rceil)$ and by symmetry $(\lceil N/k\rceil,k)$ for $1\le k\le N^{1/3}$. They justify these are vertices via an inductive condition and an inequality that “follows if $k\le N^{1/3}$,” and record that with $N_1=\lfloor N^{1/3}\rfloor$ and $N_2=\lceil N/N_1\rceil$ the points $(N_1,N_2)$ and $(N_2,N_1)$ are vertices. Finally, they reformulate Theorem~\ref{th:main} as the equivalent statement for $Q_N=H_N\cap[1,N_2]^2$ (Theorem~\ref{th:main2}), indicating the argument reduces to bounding $f_0(I(Q_N))$. They caution that “the case of hyperbola is more involved because the curvature changes from $N^{-1/2}$ to $N^{-2}$ while it is constant for the disk.”

Question:
Let
\[
H_N=\{(x,y)\in\mathbb{R}^2: xy\ge N,\ x,y>0\},
\]
where $N$ is a large positive integer. Its integer hull is
\[
I(H_N)=\operatorname{conv}(\mathbb{Z}^2\cap H_N),
\]
and $f_0(P)$ denotes the number of vertices of a polytope $P$. Using the notation $f(N)\ll g(N)$ to mean that $f(N)\le Cg(N)$ for some absolute constant $C>0$, which quantitative estimate holds for the number of vertices of $I(H_N)$?

\[
N^{1/3}\log N\ll f_0\bigl(I(H_N)\bigr)\ll N^{1/3}\log N,
\]
so $f_0(I(H_N))$ has order of magnitude $N^{1/3}\log N$.

Choices:
B.
\[
N^{1/3}\ll f_0\bigl(I(H_N)\bigr)\ll N^{1/3},
\]
so $f_0(I(H_N))$ has order of magnitude $N^{1/3}$.
C.
\[
N^{1/3}\ll f_0\bigl(I(H_N)\bigr)\ll N^{1/3}\log N.
\]
In particular, the number of vertices grows between constant multiples of $N^{1/3}$ and $N^{1/3}\log N$.
D.
\[
N^{1/3}(\log N)^2\ll f_0\bigl(I(H_N)\bigr)\ll N^{1/3}(\log N)^2,
\]
so $f_0(I(H_N))$ has order of magnitude $N^{1/3}(\log N)^2$.
E.
\[
N^{1/3}\log N\ll f_0\bigl(I(H_N)\bigr)\ll N^{2/3}\log N,
\]
so one only gets a lower bound of order $N^{1/3}\log N$ together with an upper bound of order $N^{2/3}\log N$.

The stem does not explicitly reveal the correct asymptotic estimate, nor does it contain strong hints favoring the matching upper/lower bound with a log factor.

This is essentially a theorem-recall question: it asks for the quantitative estimate for a specific object rather than requiring derivation from intermediate facts. The nearby alternatives make it slightly less tautological, but it remains close to a restatement of a known result.

There is some reasoning pressure in comparing closely related asymptotic forms and deciding whether the sharp order includes a log factor, but the item mainly tests recall of the exact result. Also, option C is a weaker statement that is still true, which weakens the demand for precise generative reasoning.

The distractors are mathematically plausible and target common asymptotic mistakes (missing log, extra log, wrong exponent). However, one distractor (C) appears to be a weaker true statement, making the options non-exclusive and reducing distractor quality in a single-correct MCQ.

Reasonably well-formed and free of answer leakage, with plausible asymptotic distractors, but it is close to theorem restatement and suffers from ambiguity because a weaker true option is included alongside the intended sharp answer.


An \emph{additively idempotent semiring} (abbreviated as ai-semiring)
is an algebraic structure $(S, +, \cdot)$ equipped with two binary operations $+$ and $\cdot$ such that:
\begin{itemize}
\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;
\item the multiplicative reduct $(S, \cdot)$ is a semigroup;
\item the distributive laws hold:
\[
(x+y)z \approx xz + yz, \qquad x(y+z) \approx xy + xz.
\]
\end{itemize}
This class of algebras includes well-known examples such as the Kleene semiring of regular languages~\cite{con},
the max-plus algebra~\cite{aei}, the power semiring of a semigroup~\cite{dgv24},
the endomorphism semiring of a semilattice~\cite{dgv25},
the semiring of all binary relations on a set \cite{dolinka2009}, and distributive lattices~\cite{burris1981}.
These and other similar algebras have found significant applications in several branches of mathematics,
such as algebraic geometry~\cite{cc}, tropical geometry~\cite{ms}, information science~\cite{gl},
theoretical computer science~\cite{go}, and soft constraint solving and programming~\cite{bis2004}.

Let $S$ be an ai-semiring. Define a binary relation $\leq$ on $S$ by
\[
a\leq b\Leftrightarrow a+b=b.
\]
Then $\leq$ is a partial order on $S$, and $(S, \leq)$ becomes an upper semilattice in which
the supremum of any elements $a$ and $b$ is $a+b$.
Consequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.
Therefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \leq)$.
Moreover, one can easily verify that this semilattice order is compatible with multiplication.
For this reason, an ai-semiring is often called a \emph{semilattice-ordered semigroup}.

Let $n\geq 1$ be an integer,
let $S$ be an ai-semiring, and let $\mathbf{M}_n(S)$ denote the set of all $n \times n$ matrices over $S$.
Then $\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,
\[
A+B = [a_{ij}+b_{ij}]_{n\times n}, \quad A \cdot B = \left[ \sum_{k=1}^n a_{ik} b_{kj} \right]_{n\times n}
\]
for all $A=[a_{ij}]_{n\times n}, B=[b_{ij}]_{n\times n}\in \mathbf{M}_n(S)$.
It is obvious that $S$ is isomorphic to $\mathbf{M}_1(S)$.
For $n\geq 2$, $S$ can be embedded into $\mathbf{M}_n(S)$,
since the mapping
\[
\varphi\colon S \rightarrow \mathbf{M}_n(S),\quad a \mapsto [a]_{n\times n},
\]
where $[a]_{n\times n}$ denotes the constant matrix with all entries equal to $a$,
is an embedding mapping.
Moreover, if $S$ has an additive identity that is also a multiplicative zero,
then we can easily obtain the following infinite strict ascending chain
\[
S\hookrightarrow \mathbf{M}_2(S)\hookrightarrow \mathbf{M}_{3}(S)\hookrightarrow  \mathbf{M}_{4}(S)\hookrightarrow\cdots.
\]

Let $\mathcal{V}$ be a class of algebras.
Then $\mathcal{V}$ is called a \emph{variety} if it is closed under
taking subalgebras, homomorphic images, and arbitrary direct products.
By the celebrated Birkhoff's theorem \cite{birkhoff1935}, $\mathcal{V}$ is a variety if and only if
it is an \emph{equational class}, that is, the class of all algebras satisfying some set of identities.

A variety is \emph{finitely based} if it can be defined by a finite set of identities;
otherwise, it is \emph{nonfinitely based}.
An algebra $A$ is finitely based or nonfinitely based if the variety $\mathsf{V}(A)$
it generates is finitely based or not.
The \emph{finite basis problem} for a class of algebras,
one of the central problems in universal algebra,
concerns the classification of
its members with respect to the property of being finitely based.

A variety is \emph{locally finite} if each of its finitely generated members is finite.
A locally finite variety is \emph{inherently nonfinitely based}
if it is not contained in any finitely based locally finite variety.
Since every finite algebra generates a locally finite variety,
we say that a finite algebra $A$ is inherently nonfinitely based if
the variety $\mathsf{V}(A)$ is inherently nonfinitely based.
It follows immediately that every finite algebra whose variety contains
an inherently nonfinitely based algebra is nonfinitely based.
Consequently, every inherently nonfinitely based algebra must be nonfinitely based.

Over the past two decades,
the finite basis problem for ai-semirings has attracted considerable attention,
resulting in substantial
progress~\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.
In particular, Dolinka~\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.
Pastijn et al.~\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \approx x$ is finitely based;
Ren et al.~\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \approx x$.
Shao and Ren~\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.
Subsequently,
Jackson et al.~\cite{jrz} and Zhao et al.~\cite{zrc}
provided a complete classification of all three-element ai-semirings with respect to the finite basis property.
Most recently,
Dolinka, Gusev and Volkov~\cite{dgv25, gv2501} settled the finite basis problem for
the endomorphism semirings of finite semilattices.

To the best of our knowledge,
the study of the finite basis problem for matrix semirings was initiated by
Dolinka~\cite{dolinka2009}, who considered the problem for $\mathbf{M}_n(D_2)$ for $n\geqslant 2$,
where $D_2$ denotes the two-element distributive lattice.
Gusev and Volkov~\cite{gv2510, volkov2024} are currently investigating
the same problem for upper triangular matrix semirings over $D_2$.
Motivated by these works,
we undertake a systematic classification of matrix semirings  $\mathbf{M}_n(S)$,
where $S$ ranges over all two-element ai-semirings.
The main result of this paper provides the following complete characterization.

An \emph{additively idempotent semiring} (abbreviated as ai-semiring)
is an algebraic structure $(S, +, \cdot)$ equipped with two binary operations $+$ and $\cdot$ such that:
\begin{itemize}
\item the additive reduct $(S, +)$ is a commutative idempotent semigroup;
\item the multiplicative reduct $(S, \cdot)$ is a semigroup;
\item the distributive laws hold:
\[
(x+y)z \approx xz + yz, \qquad x(y+z) \approx xy + xz.
\]
\end{itemize}
This class of algebras includes well-known examples such as the Kleene semiring of regular languages~\cite{con},
the max-plus algebra~\cite{aei}, the power semiring of a semigroup~\cite{dgv24},
the endomorphism semiring of a semilattice~\cite{dgv25},
the semiring of all binary relations on a set \cite{dolinka2009}, and distributive lattices~\cite{burris1981}.
These and other similar algebras have found significant applications in several branches of mathematics,
such as algebraic geometry~\cite{cc}, tropical geometry~\cite{ms}, information science~\cite{gl},
theoretical computer science~\cite{go}, and soft constraint solving and programming~\cite{bis2004}.

Let $S$ be an ai-semiring. Define a binary relation $\leq$ on $S$ by
\[
a\leq b\Leftrightarrow a+b=b.
\]
Then $\leq$ is a partial order on $S$, and $(S, \leq)$ becomes an upper semilattice in which
the supremum of any elements $a$ and $b$ is $a+b$.
Consequently, the additive reduct $(S, +)$ is uniquely determined by this semilattice order.
Therefore, it is often convenient to represent the addition operation using the Hasse diagram of $(S, \leq)$.
Moreover, one can easily verify that this semilattice order is compatible with multiplication.
For this reason, an ai-semiring is often called a \emph{semilattice-ordered semigroup}.

Let $n\geq 1$ be an integer,
let $S$ be an ai-semiring, and let $\mathbf{M}_n(S)$ denote the set of all $n \times n$ matrices over $S$.
Then $\mathbf{M}_n(S)$ forms an ai-semiring under the usual matrix addition and multiplication, that is,
\[
A+B = [a_{ij}+b_{ij}]_{n\times n}, \quad A \cdot B = \left[ \sum_{k=1}^n a_{ik} b_{kj} \right]_{n\times n}
\]
for all $A=[a_{ij}]_{n\times n}, B=[b_{ij}]_{n\times n}\in \mathbf{M}_n(S)$.
It is obvious that $S$ is isomorphic to $\mathbf{M}_1(S)$.
For $n\geq 2$, $S$ can be embedded into $\mathbf{M}_n(S)$,
since the mapping
\[
\varphi\colon S \rightarrow \mathbf{M}_n(S),\quad a \mapsto [a]_{n\times n},
\]
where $[a]_{n\times n}$ denotes the constant matrix with all entries equal to $a$,
is an embedding mapping.
Moreover, if $S$ has an additive identity that is also a multiplicative zero,
then we can easily obtain the following infinite strict ascending chain
\[
S\hookrightarrow \mathbf{M}_2(S)\hookrightarrow \mathbf{M}_{3}(S)\hookrightarrow \mathbf{M}_{4}(S)\hookrightarrow\cdots.
\]

Let $\mathcal{V}$ be a class of algebras.
Then $\mathcal{V}$ is called a \emph{variety} if it is closed under
taking subalgebras, homomorphic images, and arbitrary direct products.
By the celebrated Birkhoff's theorem \cite{birkhoff1935}, $\mathcal{V}$ is a variety if and only if
it is an \emph{equational class}, that is, the class of all algebras satisfying some set of identities.

Over the past two decades,
the finite basis problem for ai-semirings has attracted considerable attention,
resulting in substantial
progress~\cite{dol07, dolinka2009, dgv24, dgv25, gpz05, gv23, gv2501, gv2510, jrz, pas05, rz16, rzw, sr, volkov2024, yr25, yrzs, zrc}.
In particular, Dolinka~\cite{dol07} constructed the first example of a nonfinitely based finite ai-semiring.
Pastijn et al.~\cite{gpz05, pas05} proved that every ai-semiring satisfying the identity $x^2 \approx x$ is finitely based;
Ren et al.~\cite{rz16, rzw} later extended this to ai-semirings satisfying the identity $x^3 \approx x$.
Shao and Ren~\cite{sr} established that every algebra in the variety generated by all two-element ai-semirings is finitely based.
Subsequently,
Jackson et al.~\cite{jrz} and Zhao et al.~\cite{zrc}
provided a complete classification of all three-element ai-semirings with respect to the finite basis property.
Most recently,
Dolinka, Gusev and Volkov~\cite{dgv25, gv2501} settled the finite basis problem for
the endomorphism semirings of finite semilattices.

To the best of our knowledge,
the study of the finite basis problem for matrix semirings was initiated by
Dolinka~\cite{dolinka2009}, who considered the problem for $\mathbf{M}_n(D_2)$ for $n\geqslant 2$,
where $D_2$ denotes the two-element distributive lattice.
Gusev and Volkov~\cite{gv2510, volkov2024} are currently investigating
the same problem for upper triangular matrix semirings over $D_2$.
Motivated by these works,
we undertake a systematic classification of matrix semirings $\mathbf{M}_n(S)$,
where $S$ ranges over all two-element ai-semirings.
The main result of this paper provides the following complete characterization.

\begin{abstract}
We provide a complete classification of matrix semirings $\mathbf{M}_n(S)$ over two-element additively idempotent semirings $S$
with respect to the finite basis property.
Our main theorem shows that for every integer $n \geq 2$,
the semiring $\mathbf{M}_n(S)$ is finitely based
if and only if $S$ is distinct from a distributive lattice.
\end{abstract}

To the best of our knowledge,
the study of the finite basis problem for matrix semirings was initiated by
Dolinka~\cite{dolinka2009}, who considered the problem for $\mathbf{M}_n(D_2)$ for $n\ge 2$,
where $D_2$ denotes the two-element distributive lattice.
Gusev and Volkov~\cite{gv2510, volkov2024} are currently investigating
the same problem for upper triangular matrix semirings over $D_2$.
Motivated by these works,
we undertake a systematic classification of matrix semirings $\mathbf{M}_n(S)$,
where $S$ ranges over all two-element ai-semirings.
The main result of this paper provides the following complete characterization.

For the proof,
we first note that the case where $S$ is a distributive lattice follows from Dolinka's work.
Specifically,
\cite[Theorem B]{dolinka2009} states that the semiring $\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based.
Since $\mathcal{R}_2 $ is isomorphic to $\mathbf{M}_2(D_2)$,
it follows immediately that $\mathbf{M}_2(D_2)$ inherits this property.
Furthermore, because $\mathbf{M}_n(D_2)$ contains a copy of $\mathbf{M}_2(D_2)$ for all $n \geq 2$,
the same property extends to $\mathbf{M}_n(D_2)$; this result is explicitly stated in \cite[Corollary 6.2]{dolinka2009}.
(We note that $D_2$ is also denoted $\mathbb{B}_2$ and called the two-element Boolean semiring; our notation follows \cite{sr}.)

Therefore, to establish Theorem~\ref{main}, it remains to prove the converse:
that the matrix semiring $\mathbf{M}_n(S)$ is finitely based
for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n \geq 2$.
The necessary preliminaries are collected in Section 2, and the proof is carried out in Sections 3 and 4.

\begin{pro}\label{pro21}
Let $n\geq 2$ be an integer. Then
$\mathsf{V}(\mathbf{M}_n(L_2))$ is the ai-semiring variety determined by the identities \eqref{eq:T22} and
\begin{align}
&xy\approx xz. \label{f1}
\end{align}
\end{pro}
\begin{proof}
We first show that $\mathbf{M}_n(L_2)$ satisfies identities \eqref{eq:T22} and \eqref{f1}.
Let $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$ be arbitrary matrices in $\mathbf{M}_n(L_2)$, where all entries belong to $L_2$.
For any indices $1 \leq i, j \leq n$, we have
\[
(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \stackrel{\eqrefbasis{eq:L2}}=
\sum_{k=1}^{n} a_{ik} \stackrel{\eqrefbasis{eq:L2}}= \sum_{k=1}^{n} a_{ik}c_{kj} = (AC)_{ij},
\]
and
\begin{align*}
(A^2)_{ij} = \sum_{k=1}^{n} a_{ik}a_{kj}
&= \left( \sum_{k=1}^{n} a_{ik}a_{kj} \right) + a_{ij}a_{jj} \\
&\stackrel{\eqrefbasis{eq:L2}}= \left( \sum_{k=1}^{n} a_{ik}a_{kj} \right) + a_{ij} \\
&= (A^2)_{ij} + A_{ij} = (A^2 + A)_{ij}.
\end{align*}
This shows that $AB = AC$ and $A^2 = A^2 + A$.
Hence $\mathbf{M}_n(L_2)$ satisfies the identities \eqref{eq:T22} and \eqref{f1}.

\begin{pro}\label{pro22}
Let $n\geq 2$ be an integer. Then
$\mathsf{V}(\mathbf{M}_n(R_2))$ coincides with $\mathsf{V}(S_{56})$, properly contains $\mathsf{V}(R_2)$,
and is the ai-semiring variety determined by the identities \eqref{eq:T22} and
\begin{align}
&xy\approx zy. \label{f3}
\end{align}
\end{pro}
\begin{proof}
Observe that $R_2$ and $L_2$ have dual multiplications.
The remaining argument parallels those of Proposition~\ref{pro21} and Remark~\ref{rem1}.
\end{proof}

\begin{pro}\label{pro233}
Let $n\geq 1$ be an integer. Then
$\mathsf{V}(\mathbf{M}_n(N_2))$ is the ai-semiring variety determined by the identities \eqref{eq:N21} and \eqref{eq:N22}.
\end{pro}
\begin{proof}
Let $A = [a_{ij}]$, $B = [b_{ij}]$, $C = [c_{ij}]$, $D = [d_{ij}]$ be arbitrary matrices in $\mathbf{M}_n(N_2)$.
For any $1 \leq i, j \leq n$, we have
\[
(AB)_{ij} = \sum_{k=1}^{n} a_{ik}b_{kj} \stackrel{\eqrefbasis{eq:N21}}= \sum_{k=1}^{n} c_{ik}d_{kj} = (CD)_{ij},
\]
and
\begin{align*}
(A + A^2)_{ij}
&= A_{ij} + (A^2)_{ij}=a_{ij} + \left(\sum_{k=1}^n a_{ik}a_{kj}\right)\\
&\stackrel{\eqrefbasis{eq:N21}}= a_{ij} + a_{ij}a_{ij}
\stackrel{\eqrefbasis{eq:N22}}= a_{ij}= A_{ij}.
\end{align*}
This shows that $AB = CD$ and $A=A+A^2$.
Hence $\mathbf{M}_n(N_2)$ satisfies both identities \eqref{eq:N21} and \eqref{eq:N22}.

\section{Conclusion}
We have provided a complete classification of
matrix semirings $\mathbf{M}_n(S)$ over two-element ai-semirings $S$
with respect to the finite basis property.
From Remark~\ref{rem1} and \cite[Figure 1]{yr25},
the variety $\mathsf{V}(\mathbf{M}_n(L_2))$ has precisely $5$ subvarieties;
they form a distributive lattice and are all finitely based.
The same conclusion holds for the variety $\mathsf{V}(\mathbf{M}_n(R_2))$.
By \cite[Theorem 1]{polin} together with Propositions~\ref{pro233} and \ref{pro23},
the varieties $\mathsf{V}(\mathbf{M}_n(N_2))$ and $\mathsf{V}(\mathbf{M}_n(T_2))$
are both minimal nontrivial varieties.
In contrast, the subvariety lattice of the variety $\mathsf{V}(\mathbf{M}_n(M_2))$
remains to be explicitly described.

\begin{thm}\label{main}
Let $S$ be a two-element ai-semiring, and let $n \geq 2$ be an integer.
Then the matrix semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.
\end{thm}

For the proof of Theorem~\ref{main}: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \cite[Theorem B]{dolinka2009} shows that the semiring $\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\mathcal{R}_2\cong \mathbf{M}_2(D_2)$, it follows that $\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\mathbf{M}_n(D_2)$ contains a copy of $\mathbf{M}_2(D_2)$ for all $n\ge2$, the same property extends to $\mathbf{M}_n(D_2)$ (explicitly \cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish Theorem~\ref{main} it remains to prove the converse: $\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\ge2$. The preliminaries are in Section~2 and the proof is carried out in Sections~3 and~4.

For the proof of the main theorem: (1) In the case where $S$ is a distributive lattice, the result follows from Dolinka's work: \cite[Theorem B]{dolinka2009} shows that the semiring $\mathcal{R}_2$ of all binary relations on a two-element set is inherently nonfinitely based. Since $\mathcal{R}_2\cong \mathbf{M}_2(D_2)$, it follows that $\mathbf{M}_2(D_2)$ is inherently nonfinitely based; and because $\mathbf{M}_n(D_2)$ contains a copy of $\mathbf{M}_2(D_2)$ for all $n\ge2$, the same property extends to $\mathbf{M}_n(D_2)$ (explicitly \cite[Corollary 6.2]{dolinka2009}). (2) Thus, to establish the main theorem it remains to prove the converse: $\mathbf{M}_n(S)$ is finitely based for every two-element ai-semiring $S$ distinct from a distributive lattice and every $n\ge2$. The preliminaries are given next, and the proof is then carried out subsequently.

Question: An additively idempotent semiring (ai-semiring) is an algebra $(S,+,\cdot)$ such that $(S,+)$ is a commutative idempotent semigroup, $(S,\cdot)$ is a semigroup, and multiplication distributes over addition on both sides. Let $S$ be a two-element ai-semiring, and let $n\ge 2$ be an integer. Write $\mathbf{M}_n(S)$ for the ai-semiring of all $n\times n$ matrices over $S$ with the usual matrix addition and multiplication. An algebra is called finitely based if the variety it generates is definable by finitely many identities. Which statement holds for every such $S$ and $n$?

The matrix semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ is not a distributive lattice.

Choices:
B. The matrix semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ is a distributive lattice.
C. If $S$ is not a distributive lattice, then the matrix semiring $\mathbf{M}_n(S)$ is finitely based.
D. For every two-element ai-semiring $S$, the matrix semiring $\mathbf{M}_n(S)$ is finitely based for all sufficiently large integers $n\ge 2$.
E. The matrix semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ itself is finitely based.

dropped the converse implication excluding distributive lattices

uniformity in n from the embedding of $\mathbf{M}_2(D_2)$ into all $\mathbf{M}_n(D_2)$ for $n\ge2$

replacement of the matrix-semiring criterion by a criterion on the base semiring

The stem does not explicitly reveal the correct criterion. It asks for an equivalent characterization, and the correct answer is not implied by the wording alone.

The item is close to a theorem-recall question: the correct option is essentially the theorem statement itself. However, the presence of nearby stronger, weaker, and boundary-case variants makes it slightly more than a pure restatement.

Some reasoning is needed to reject the distractors by checking sufficiency vs. equivalence, the range restriction on n, and uniformity claims. Still, the question mainly tests precise recall of a characterization rather than substantial generative mathematical reasoning.

The distractors are plausible and well targeted: one weakens the biconditional, one overextends the range of n, one adds an unjustified uniform finite-basis claim, and one substitutes an unrelated finite-basis criterion on S itself.

A solid MCQ with no answer leakage and strong distractors, but it is primarily theorem recognition rather than a deeply generative reasoning task.


Question: An additively idempotent semiring (ai-semiring) is an algebra $(S,+,\cdot)$ such that $(S,+)$ is a commutative idempotent semigroup, $(S,\cdot)$ is a semigroup, and multiplication distributes over addition on both sides. Let $S$ be a two-element ai-semiring, and let $n\ge 2$ be an integer. Write $\mathbf{M}_n(S)$ for the ai-semiring of all $n\times n$ matrices over $S$ with the usual matrix addition and multiplication. An algebra is called finitely based if the variety it generates is definable by finitely many identities. Which statement holds for every such $S$ and $n$?

Choices:
B. The matrix semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ is a distributive lattice.
C. If $S$ is not a distributive lattice, then the matrix semiring $\mathbf{M}_n(S)$ is finitely based.
D. For every two-element ai-semiring $S$, the matrix semiring $\mathbf{M}_n(S)$ is finitely based for all sufficiently large integers $n\ge 2$.
E. The matrix semiring $\mathbf{M}_n(S)$ is finitely based if and only if $S$ itself is finitely based.

The stem defines the objects and asks for a universal characterization, but it does not reveal or strongly hint at the correct criterion. The key discriminator involving distributive lattices appears only in the answer choices.

The correct option appears to be essentially the theorem statement itself: a precise if-and-only-if characterization of when \(\mathbf{M}_n(S)\) is finitely based. The item mainly asks for recognition of the exact theorem rather than deriving a new consequence.

There is some reasoning pressure because the student must distinguish the exact biconditional from a weaker true statement and several plausible overgeneralizations. However, the question primarily tests theorem recall/recognition rather than substantial generative mathematical reasoning.

The distractors are plausible and well targeted: one reverses the distributive-lattice condition, one gives a weaker one-way implication, one overgeneralizes in \(n\), and one confuses properties of the matrix semiring with those of the base semiring. These reflect realistic logical and structural failure modes.

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


[Intermittency] \label{prop:intermittency}For any $p\geq 1$ integer, we have
\begin{align*}
   n \exp\left((p+\frac{2p(p-1)}{n+2})\tau \right) \leq  \mathbb{E} |F_\tau|^{2p}\leq n^{p} \exp\left((p+\frac{2p(p-1)}{n+2})\tau \right),
\end{align*}
where $|F|^2:={\rm tr} F^*F$ denotes the square of the Frobenius norm.

\begin{align}
    \E|F_\tau|^{2p} \sim_{n,p}\footnotemark \left(\E^p |F_\tau|^2\right)^{1+\frac{2(p-1)}{n+2}}. \label{2pthmoment}
\end{align}

\begin{proposition}[Intermittency] \label{prop:intermittency}For any $p\geq 1$ integer, we have
\begin{align*}
   n \exp\left((p+\frac{2p(p-1)}{n+2})\tau \right) \leq  \E |F_\tau|^{2p}\leq n^{p} \exp\left((p+\frac{2p(p-1)}{n+2})\tau \right),
\end{align*}
where $|F|^2:=\tr F^*F$ denotes the square of the Frobenius norm.
\end{proposition}

\noindent
We are interested in the process $\{F_\tau\}_{\tau \geq 0}$ that solves the Stratonovich stochastic differential equation (SDE)
\begin{align}\label{F}
 dF=F_\tau\circ dB, \quad F_{\tau=0}=id ,
\end{align}
where $\{B_\tau\}_{\tau \geq 0}$ is a Brownian motion on ${\frak{sl}}(n):=\{{\rm tr} B=0\}$ that satisfies the following three assumptions:
\begin{enumerate}[label=(B\arabic*)]
 \item \label{brownian1} $OB O^{-1}=_{\text{law}}B$ for all $O\in \textbf{O}(n):=\{O^*O=id\}$,
 \item \label{brownian2} $\mathbb E[B_{\tau} B_\tau] =0$, and
 \item\label{brownian3} $\mathbb E[B_\tau^* B_\tau]=\tau \text{id}$.
 \end{enumerate}
Since the chain rule holds for Stratonovich SDEs, $F\in \textbf{SL}(n):=\{\text{det} F=1\}$. In \cite{MOW25}, it is proved that for $n\geq 2$, there is a unique Brownian motion on $\frak{sl}(n)$ that satisfies the Assumption \ref{brownian1}-\ref{brownian3}. We are interested in exploring the intermittent behavior of $F$ in \eqref{F}. Our first result is on the moments:

\begin{align}\label{F}
    dF=F_\tau\circ dB, \quad F_{\tau=0}=id ,
\end{align}

\begin{enumerate}[label=(B\arabic*)]
    \item \label{brownian1} $OB O^{-1}=_{\text{law}}B$ for all $O\in \textbf{O}(n):=\{O^*O=id\}$,
    \item \label{brownian2} $\bbE[B_{\tau} B_\tau] =0$, and
    \item\label{brownian3} $\bbE[B_\tau^* B_\tau]=\tau \text{id}$.
    \end{enumerate}

\begin{enumerate}[label=(B\arabic*)]
    \item \label{brownian1} $OB O^{-1}=_{\text{law}}B$ for all $O\in \textbf{O}(n):=\{O^*O=id\}$,
    \item \label{brownian2} $\bbE[B_{\tau} B_\tau] =0$, and
    \item\label{brownian3} $\bbE[B_\tau^* B_\tau]=\tau \text{id}$.
    \end{enumerate}

\end{abstract}
\maketitle

\medskip \noindent
The fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\ref{prop:intermittency} generalizes \cite[Lemma 2]{MOW25} for $n=2$ to general $n\ge 2$.

\medskip \noindent
While Proposition~\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\nabla u$. This is because we can capture the proximity of $\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.
 \begin{proposition}[Non-tightness] \label{prop:nontightness}
 For $\tau \gg_n 1$\footnote{Here $\tau \gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\tau$ with $\tau \geq C(n)$.}, we have \begin{align}\label{prop02}
 \E | F_{ \tau } |^2 I \Big( { \textstyle \frac{ 1 }{ n } } | F_{ \tau } |^2 \leq { \textstyle \frac{ 1 }{ e } } \big( { \textstyle \E \frac{ 1 }{ n } } | F_{ \tau } |^2 \big)^{ \frac{ n + 4 }{ n + 2 } } \Big)
 \leq \Big( \frac{ 1 }{ 2 } + \frac{ C(n) }{ \sqrt{\tau} } \Big) \E| F_{ \tau } |^2 .
 \end{align}
 where $I(A)$ is the indicator function of the event $A$.
\end{proposition}
\begin{remark}\label{cor:nontightness} The information in Proposition~\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \cite[Theorem 3 \& (118)]{MOW25}
 \begin{align*}
\E |F_\tau|^{2p} \gtrsim_{n,p} \bbE^{1+\frac{n+4}{n+2}(p-1)} |F_\tau|^{2},
\end{align*}
which is weaker than \eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\frac{2p(p-1)}{n+2}$ around $p=1$.
\end{remark}

\subsection{Proof of Proposition~\ref{prop:intermittency}}
Recall from Lemma~\ref{lem:odes}, we have for $p\geq 1$ integer
\begin{align*}
 \frac{ d\bbE[\tr G^p]}{d\tau} =(p+\frac{p(p-1)(n-2)}{\alpha_n})\bbE \tr G_\tau^p +\frac{pn}{\alpha_n}\sum_{j=1}^{p-1}\bbE \tr G_\tau^j \tr G_\tau^{p-j}
\end{align*}
 with initial condition $\tr G_{\tau=0}=\tr id=n$. Using $\tr G^j \tr G^{p-j} \geq \tr G^p$ for $0\leq j\leq p$, we can bound the second term
 \begin{align*}
 \sum_{j=1}^{p-1}\bbE \tr G_\tau^j \tr G_\tau^{p-j} \geq (p-1) \bbE \tr G_\tau^p
\end{align*}
which leads to the differential inequality
\begin{align*}
 \frac{ d\bbE\tr G^p}{d\tau} \geq (p+\frac{p(p-1)(2n-2)}{\alpha_n}) \bbE\tr G_\tau^p
\end{align*}
where $\alpha_n$ is as defined \eqref{alpha}. Hence we get
\begin{align*}
 \bbE\tr G_\tau^p \geq n e^{(p+\frac{2p(p-1)}{n+2})\tau} .
\end{align*}
On the other hand, recall again from Lemma~\ref{lem:odes}, $\bbE \tr^p G_\tau$ satisfies
\begin{align*}
 \frac{ d\bbE\tr^p G}{d\tau}& =(p-\frac{2p(p-1)}{\alpha_n}) \bbE\tr^p G_\tau +\frac{2p(p-1)n}{\alpha_n} \bbE\tr^{p-2} G_\tau\tr G_\tau^2
\end{align*}
with initial condition $\bbE\tr^p G_{\tau=0}=n^p$. This time using $\tr G^2\leq \tr^2 G$, we get the differential inequality
\begin{align*}
 \frac{d\bbE \tr^p G}{d\tau}& \leq (p+\frac{p(p-1)(2n-2)}{\alpha_n}) \bbE\tr^p G_\tau
\end{align*}
which leads to
\begin{align*}
 \bbE\tr^p G_\tau \leq n^pe^{(p+\frac{2p(p-1)}{n+2})\tau} .
\end{align*}
Hence we obtain \eqref{lpestimates} which is a restatement of Proposition~\ref{prop:intermittency}.
\qed

For parameters $ \tau^* \geq 0 $ and $ \hat{\sigma}^* $ that we will choose later we consider the terminal condition
\begin{align}\label{tc}
 \hat{\zeta}( \tau^*,\hat{\sigma})\begin{cases}
 = 1, \text{ if } \hat{\sigma} \leq \hat{\sigma}^*,\\
 \in [0,1] \text{ for all } \hat{\sigma},\\
 =0, \text{ if }\hat{\sigma}\geq \hat{\sigma}^*+1,
 \end{cases}
 \text{ with } \left| \frac{\partial^2 \hat{\zeta}( \tau^* ,\hat{\sigma}) }{\partial \hat{\sigma}^2}\right| \lesssim 1
 \end{align}
 for the equation \eqref{zetahat2} in Lemma~\ref{lem:zeta}.
It is readily seen from \eqref{tc} that we have
 \begin{align}
\bbE \hat R_{ \tau^* } \hat{\zeta}(\tau^*, \hat R_{ \tau^* }) \stackrel{ (\ref{tc}) }{ \geq } \bbE \hat R_{ \tau^* } I( \hat R_{ \tau^* } \leq \hat{R}^*)
\quad \text{provided} \quad \hat\sigma^* =: \ln \hat R^{ * }. \label{ononehand}
\end{align}
Our goal is to choose $ \hat\sigma^* $ large enough so that at initial time $ \tau = 0 $ it holds
\begin{align}\hat{\zeta}( \tau=0, \hat\sigma = 0 ) \leq \frac{1}{2} , \label{initial}
\end{align}
so that Lemma~\ref{lem:zeta} implies
\begin{align}
 \bbE \hat R_{ \tau^* } I( \hat R_{ \tau^* } \leq \hat{R}^*) - \frac{ 1 }{ 2 }
 \stackrel{ (\ref{ononehand}) }{ \leq } \bbE \hat{R}_{ \tau^* } \hat{\zeta}( \tau^*, \hat R_{ \tau^* }) - \frac{ 1 }{ 2 }
 \stackrel{ \eqref{zetahatineq} \& (\ref{initial}) }{ \lesssim_{ n } } \int_0^{\tau^*}d \tau\, e^{ - ( 1 - \frac{ \lambda_2 }{ 2 }) \tau} \sup_{\hat \sigma}\left|\frac{\partial \hat \zeta}{\partial\hat \sigma}+ \frac{\partial^2 \hat \zeta}{\partial\hat \sigma^2} \right| . \label{zetahatineqb}
\end{align}

\noindent
To conclude the proof, it is left to show that the integral in \eqref{zetahatineqb} is small. To this end, note that using the estimates for the terminal data
\begin{align*}
 \sup_{\hat\sigma } \left|\frac{\partial \hat{\zeta}( \tau^*,\hat{\sigma})}{\partial {\hat\sigma}}\right|+\left|\frac{\partial^2 \hat{\zeta}(\tau^*,\hat{\sigma})}{\partial^2 {\hat \sigma } }\right|
 \lesssim 1
 \quad \text{and} \quad
 \int d{\hat{\sigma}}\left|\frac{\partial \hat{\zeta}( \tau^*,\hat{\sigma})}{\partial {\hat\sigma}}\right|
 \lesssim 1
\end{align*}
we learn (by convolution with the heat kernel) that
\begin{align*}
 \sup_{ \hat \sigma } \left|\frac{\partial \hat{\zeta} ( \tau^* , \hat\sigma ) }{\partial \hat{\sigma}}\right| \lesssim_{ n } \frac{1}{ \sqrt{1+ \tau^*- \tau}}, \hskip 10pt \sup_{ \hat\sigma } \left|\frac{\partial^2 \hat{\zeta} ( \tau^* , \hat\sigma ) }{\partial \hat{\sigma}^2}\right|
 \lesssim_{ n } \frac{1}{ 1+ \tau^*- \tau} .
\end{align*}
Hence, the integral term in \eqref{zetahatineqb} is bounded by
\begin{align*} \int_0^{\tau*}d\tau e^{ - ( 1 - \frac{ \lambda_2 }{ 2 } ) \tau} \sup_{\hat \sigma}\left|\frac{\partial \hat \zeta}{\partial\hat \sigma}+ \frac{\partial^2 \hat \zeta}{\partial\hat \sigma^2} \right| \lesssim_{ n } \int_0^{\tau^*} d\tau e^{-\frac{4}{n-1} \tau } \frac{1}{\sqrt{1+\tau^*-\tau}} \lesssim_{ n } \frac{ 1 }{\sqrt{\tau^*}},
\end{align*}
and with some constant $ C = C ( n ) $ estimate \eqref{zetahatineqb} turns into
\begin{align}\label{ontheotherhand}
 \bbE \hat R_{ \tau^* } I( \hat R_{ \tau^* } \leq \hat{R}^*) \leq \frac{ 1 }{ 2 } + \frac{ C(n) }{ \sqrt{ \tau^* } }.
\end{align}
Finally, undoing the change of variables we have
\begin{align*}
 \hat R_{ \tau^* }
 \stackrel{ \eqref{Rhat} }{ = } e^{ - \tau^* } { \textstyle \frac{ 1 }{ n } } R_{ \tau^* }
 \stackrel{ \ref{F3} , (\ref{R}) \& (\ref{:G}) }{ = } \frac{ | F_{ \tau^* } |^2 }{ \E | F_{ \tau^* } |^2 } ,
 \quad
 \hat R^{ * }
 \stackrel{ (\ref{eqnHatRStar}) }{ \leq } e^{ \frac{ 2 }{ n + 2 } \tau^* - 1 }
 \stackrel{\ref{F3} }{ = } \frac{ 1 }{ e } \Big( \frac{ \E | F_{ \tau^* } |^2 }{ n } \Big)^{ \frac{ 2 }{ n + 2 } } ,
\end{align*}
so that since $ \frac{ 2 }{ n + 2 } + 1 = \frac{ n + 4 }{ n + 2 } $
\begin{align*}
 \hat R_{ \tau^* } I( \hat R_{ \tau^* } \leq \hat{R}^*)
 = \frac{ | F_{ \tau^* } |^2 }{ \E | F_{ \tau^* } |^2 } I \Big( { \textstyle \frac{ 1 }{ n } } | F_{ \tau^* } |^2 \leq { \textstyle \frac{ 1 }{ e } } \big( { \textstyle \frac{ \E | F_{ \tau^* } |^2 }{ n } } \big)^{ \frac{ n + 4 }{ n + 2 } } \Big) .
\end{align*}
Thus \eqref{ontheotherhand} turns into \eqref{prop02}. \qed

\noindent
Proposition~\ref{prop:intermittency} implies that
\begin{align}
    \mathbb{E}|F_\tau|^{2p} \sim_{n,p}\footnotemark \left(\mathbb{E}^p |F_\tau|^2\right)^{1+\frac{2(p-1)}{n+2}}. \label{2pthmoment}
\end{align}
\footnotetext{Here $A \sim_{n,p} B$ means that there exist constants $c=c(n,p)$ and $C=C(n,p)$ depending only on $n$ and $p$ such that $cB\leq A\leq CB$.}

\medskip \noindent
The fact that $2p$-th moment scales with a rate much larger than the $p$-th power of the second moment amounts to strongly non-Gaussian and intermittent behavior. Proposition~\ref{prop:intermittency} generalizes \cite[Lemma 2]{MOW25} for $n=2$ to general $n\ge 2$.

\medskip \noindent
The interest in studying the geometric Brownian motion comes from its
intriguing relation to the drift-diffusion process
\begin{align}
    dX=b(X_t)dt+\sqrt{2}dW \label{dd}
\end{align}
with $b$ being a divergence-free and time-independent vector field. One is interested in the case where $b$ is sampled from a stationary and isotropic Gaussian ensemble with the scaling
 \begin{align}
   \label{b} b(\mu \cdot)=\frac{1}{\mu}b \text{ in law for all } \mu>0,
\end{align}
in which convection and diffusion balance at every scale. After implementing an ultraviolet cutoff, w.l.o.g.\ at scale 1, $b$ is fixed up to a single constant which we describe in terms of $\mathbb{E}|b|^2=\epsilon^2 \frac{ n }{ 4 } $, where $\epsilon \ll 1$ can be interpreted as the P\'{e}clet number. It is now well-known that the mean-square displacement displays super-diffusive behavior $\frac{1}{2t}\mathbb E|X_t|^2 \approx \lambda(t)$ for $t \gg 1$, where
\begin{align*}
\lambda(t):=\sqrt{1+\frac{\epsilon^2}{2} \ln(1+t)}.
\end{align*}
This behavior was rigorously established with increasing precision in \cite{TV12, CHT22, CMOW22,ABK24, MOW25}.

\medskip \noindent
The relationship between the geometric Brownian motion process $F$ in \eqref{F} and the drift-diffusion process in \eqref{dd} was discovered in \cite{MOW25} on the level of the expected position, $u(x,t)$, (w.r.t. $W_t$) of the process $\{X_t\}_{t\geq 0}$ starting from $X_{t=0}=x$. More precisely, one obtains $\nabla u(0,t) \approx F_{\tau(T)}$ in law on average over $t\in (0,T)$, where $$\tau(T):=\ln \lambda(T), $$ see \cite[Theorem 2 \& (19)]{MOW25}.

\medskip \noindent
Our result shows that the intermittency of the drift-diffusion equation depends less on topology (stream lines of the divergence-free $b$ are closed iff $n=2$) and more on geometry ($\textbf{SL}(n)$ has less curvature as $n$ increases).

\medskip \noindent
While Proposition~\ref{prop:intermittency} clearly expresses intermittency at the level of $F$, we are not able to transfer this higher-moment information to $\nabla u$. This is because we can capture the proximity of $\nabla u$ and $F$ only on the second moments. Hence we need to capture the shadow of the intermittency on the level of the second moment. In fact, it comes as a non-tightness result in the second moment in the sense that the extreme tails contain a substantial fraction of the second moment.
 \begin{proposition}[Non-tightness] \label{prop:nontightness}
    For $\tau \gg_n 1$\footnote{Here $\tau \gg_n 1$ means that there is a possibly large constant $C(n)$ depending only on $n$ such that the statement is true for any $\tau$ with $\tau \geq C(n)$.}, we have 	\begin{align}\label{prop02}
		\mathbb{E} | F_{ \tau } |^2 I \Big( { \textstyle \frac{ 1 }{ n } } | F_{ \tau } |^2 \leq { \textstyle \frac{ 1 }{ e  } } \big( { \textstyle \mathbb{E} \frac{ 1 }{ n } } | F_{ \tau } |^2 \big)^{ \frac{ n + 4 }{ n + 2 } } \Big)
		\leq \Big( \frac{ 1 }{ 2 } + \frac{ C(n) }{ \sqrt{\tau} } \Big) \mathbb{E}| F_{ \tau }  |^2  .
	\end{align}
 where $I(A)$ is the indicator function of the event $A$.
\end{proposition}
\begin{remark}\label{cor:nontightness} The information in Proposition~\ref{prop:nontightness} can easily be upgraded to the lower bound as is done in \cite[Theorem 3 \& (118)]{MOW25}
 \begin{align*}
\mathbb{E}  |F_\tau|^{2p} \gtrsim_{n,p}  \mathbb E^{1+\frac{n+4}{n+2}(p-1)}  |F_\tau|^{2},
\end{align*}
which is weaker than \eqref{2pthmoment} but not completely unrelated. This exponent is precisely the linearization of the quadratic exponent $p+\frac{2p(p-1)}{n+2}$ around $p=1$.
\end{remark}

\noindent
The results in Propositions~\ref{prop:intermittency} and ~\ref{prop:nontightness} were previously proven in the case $n=2$ (see \cite[Lemma 2, Lemma 4]{MOW25}). The argument relied on the fact that the law of $R=\frac{1}{2}{\rm tr} F^*F$ can be identified as the Itô evolution \begin{align*}
    dR= R_{\tau} d\tau +\sqrt{R_\tau^2-1} dw, \quad R_{\tau =0}=1,
\end{align*}
where $\{w_\tau\}_{\tau\geq 0}$ is a one-dimensional Brownian motion. In any dimension $n$, as a consequence of the $\textbf{O}(n)$-invariance, it is not difficult to see that one obtains an evolution for the $(n-1)$ quantities ${\rm tr} F^*F, {\rm tr} (F^*F)^2,\dots ,{\rm tr} (F^*F)^{n-1}$. However, we are unable to leverage this evolution in order to characterize the evolution of $R$. In this paper, we take a different route: Relying on the observation that the quantities $\mathbb{E} {\rm tr} (F^*F)^2 $ and $\mathbb{E}{\rm tr}^2 F^*F $ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly, we learn that $\mathbb E{\rm tr} (F^*F)^2$ can be well approximated by $\mathbb E {\rm tr}^2 F^* F$. This in turn allows us to approximately close the equation for $ R $ in dimensions $ n > 2 $.

\medskip
 \noindent
Note that $|F_\tau|^2$ is close to a stochastic exponential, which is consistent with Proposition~\ref{prop:intermittency} and \ref{prop:nontightness}, namely $\ln |F_\tau|^2$ behaves like a Gaussian random variable with mean $(1-\frac{2}{n+2})\tau$ and variance $\frac{4\tau}{n+2}$.

\subsection*{Acknowledgment}
We thank Anna Wienhard and Corentin Le Bars for helpful discussions concerning random walks on groups. We thank Peter Morfe and Ofer Zeitouni for helpful discussions.

For $n=2$, the previously used argument (\cite[Lemma 2, Lemma 4]{MOW25}) identified the law of $R:=\frac{1}{2}{\rm tr}\,F^*F$ via the one-dimensional It\^o SDE
\[
 dR=R_{\tau}\,d\tau+\sqrt{R_{\tau}^2-1}\,dw,\qquad R_{\tau=0}=1,
\]
which yields the needed moment information.

In general dimension $n$, by $\mathbf O(n)$-invariance “one obtains an evolution for the $(n-1)$ quantities ${\rm tr}F^*F,\,{\rm tr}(F^*F)^2,\dots,{\rm tr}(F^*F)^{n-1}$,” but the authors state they are “unable to leverage this evolution in order to characterize the evolution of $R$.”

Instead, they “take a different route”: they use that “$\mathbb E\,{\rm tr}(F^*F)^2$ and $\mathbb E\,{\rm tr}^2 F^*F$ satisfy a linear system of ODEs which is closed in any dimension $n$ and can be solved explicitly.” From this explicit solution, they “learn that $\mathbb E\,{\rm tr}(F^*F)^2$ can be well approximated by $\mathbb E\,{\rm tr}^2 F^*F$,” which “in turn allows us to approximately close the equation for $R$ in dimensions $n>2$.”,

Question:
Let $n\ge 2$, let $\mathfrak{sl}(n)=\{A:\operatorname{tr}A=0\}$, and let $\{B_\tau\}_{\tau\ge 0}$ be a Brownian motion on $\mathfrak{sl}(n)$ such that for every $O\in \mathbf O(n)=\{O^*O=\mathrm{id}\}$ one has $OBO^{-1}=_{\mathrm{law}}B$, and moreover $\mathbb E[B_\tau B_\tau]=0$ and $\mathbb E[B_\tau^*B_\tau]=\tau\,\mathrm{id}$. Let $\{F_\tau\}_{\tau\ge 0}$ solve the Stratonovich SDE
\[
 dF = F_\tau\circ dB,\qquad F_{\tau=0}=\mathrm{id}.
\]
If $|F|^2:=\operatorname{tr}(F^*F)$ denotes the square of the Frobenius norm, which statement holds for every integer $p\ge 1$ and every $\tau\ge 0$?

\[
 n\exp\!\left(\left(p+\frac{2p(p-1)}{n+2}\right)\tau\right)
 \le \mathbb E|F_\tau|^{2p}
 \le n^p\exp\!\left(\left(p+\frac{2p(p-1)}{n+2}\right)\tau\right).
\]

Choices:
B.
\[
 n\exp\!\left(\left(p+\frac{2p(p-1)}{n}\right)\tau\right)
 \le \mathbb E|F_\tau|^{2p}
 \le n^p\exp\!\left(\left(p+\frac{2p(p-1)}{n}\right)\tau\right).
\]
C.
\[
 \mathbb E|F_\tau|^{2p}
 \le n^p\exp\!\left(\left(p+\frac{2p(p-1)}{n+2}\right)\tau\right).
\]
D.
\[
 \text{For each integer }p\ge 1\text{ there exists }C_p\in[1,\infty)\text{ such that for all }\tau\ge 0,
\quad C_p^{-1}\exp\!\left(\left(p+\frac{2p(p-1)}{n+2}\right)\tau\right)
 \le \mathbb E|F_\tau|^{2p}
 \le C_p\exp\!\left(\left(p+\frac{2p(p-1)}{n+2}\right)\tau\right).
\]
E.
\[
 n\exp\!\left(\left(p+\frac{2p(p-1)}{n+2}\right)\tau\right)
 \le \mathbb E|F_\tau|^{2p}
 \le n^p\exp\!\left(\left(p+\frac{2p(p-1)}{n+2}\right)\tau\right)
\]
for every real number \(p\ge 1\).

explicit sharp prefactors n and n^p replaced by non-explicit constants depending on p

range of p extended from integers to all reals

The stem does not state the target estimate or uniquely reveal the correct option. It only provides the setup and asks which bound holds.

This is very close to a direct theorem-recall item: the task is essentially to recognize the exact stated estimate among near-variants, rather than derive a new conclusion from the data.

There is some reasoning pressure because the options differ in sharp constants, exponent structure, and strength of conclusion, including a weaker true statement. But the item mainly tests precise recall/recognition of the theorem rather than substantial generative mathematical reasoning.

The distractors are plausible and mathematically targeted: one perturbs the denominator, one gives a weaker true statement, one alters quantifier dependence, and one inflates the upper exponential rate. These reflect realistic failure modes.

Well-constructed in terms of plausible distractors and lack of leakage, but it is primarily a theorem-recognition question and thus only moderately tests genuine reasoning.