# Property (T) in k-gonal random groups

MurphyKate Montee  Department of Mathematics and Statistics, Carleton College
<EMAIL_ADDRESS>
(Date: March 2021)

###### Abstract.

The $k$-gonal models of random groups are defined as the quotients of free
groups on $n$ generators by cyclically reduced words of length $k$. As $k$
tends to infinity, this model approaches the Gromov density model. In this
paper we show that for any fixed $d_{0}\in(0,1)$, if positive $k$-gonal random
groups satisfy Property (T) with overwhelming probability for densities
$d>d_{0}$, then so do $nk$-gonal random groups, for any $n\in\mathbb{N}$. In
particular, this shows that for densities above 1/3, groups in $3k$-gonal
models satisfy Property (T) with probability 1 as $n$ approaches infinity.

##  1\. Introduction

The study of random groups is one way to approach questions of the form ‘Does
a typical group have property P?’ To answer this question, one needs to choose
a model of random groups, and then calculate the probability that property P
is satisfied. Generally speaking we are interested in the asymptotic behavior
of this probability as some constraint goes to infinity.

There are many models of random groups (an excellent introduction to this
topic is [Oll05]), the most studied of which is the Gromov density model,
first introduced by Gromov in [Gro93]. In this model, a group presentation is
chosen uniformly at random by fixing the number of generators, and choosing
cyclically reduced relators of a certain length. The number of relators chosen
is controlled by a fixed constant called the density. One then analyzes the
probability of a property being satisfied as both the length and the number of
relators goes to infinity. This probability generally tends to 0 or 1, and in
many important cases there is a sharp threshold $d_{0}$, so that for densities
above $d_{0}$ the probability is 1, and below $d_{0}$ the probability is 0 (or
vice versa). Some examples of this phase shift behavior include properties
such as hyperbolicity [Gro93, Oll04], small cancellation [Gro93], and
satisfying Dehn’s algorithm [Alo+91].

It is known that there is a sharp threshold for Kazhdan’s Property (T);
however its precise value is unknown. Property (T) was introduced by Kazhdan
in [Kaž67] as a tool to study lattices in Lie groups, but turned out to be
important in several areas of mathematics and computer science, especially in
the study of expander graphs (for more information, see [Lub94]). In [Żuk03],
Żuk stated and sketched a proof that for $d>1/3$, random groups in the Gromov
density model satisfy, with overwhelming probability, Property (T). The proof
was completed in [KK13].

Similarly, we do not yet know a sharp threshold for cubulation of random
groups (i.e. admitting a cocompact action on a CAT(0) cube complex without a
global fixed point). In [MP15], Mackay-Przytycki built on the work of
Ollivier-Wise [OW11] to show that at densities $d<5/24$, Gromov random groups
are, with probability 1, cubulated. This bound was improved to $d<3/14$ in
[Mon] using a generalization of the methods in [MP15].

For infinite hyperbolic groups, being cubulated and having Property (T) are
mutually exclusive, so together these results show that the sharp threshold at
which Gromov random groups fail to satisfy Property (T) must be between 1/3
and 3/14.

In order to prove his theorem, Żuk introduced a new model of random groups,
called the _triangular model_. This was extended to the _square model_ in
[Odr19], and in [AR20] this was further generalized this to a class of models,
here called the _$k$ -gonal models_. Roughly speaking, as $k$ approaches
infinity the behavior of $k$-gonal random groups should reflect the behavior
of Gromov random groups. Odrzygóźdź showed that in the $6$-gonal model,
$d=1/3$ is the sharp threshold at which the probability of a group satisfying
Property (T) switches from 1 to 0. Figure 1 illustrates what is currently
known about values of the sharp threshold for Property (T) in the $k$-gonal
models.

Figure 1. Possible values of the sharp threshold for Property (T) in the
$k$-gonal models. Red indicates (some of the) results from this paper.

In this paper, we show that show that any upper bound on the sharp threshold
for Property (T) in the positive $k$-gonal model gives an upper bound in the
$jk$-gonal model for all $j\in\mathbb{N}$. More precisely, we prove the
following theorem:

######  Theorem 1.1.

Let $d_{0}$ be some fixed value in $(0,1)$, and suppose that with overwhelming
probability a random group at density $d>d_{0}$ in $\mathcal{M}^{+}_{k}(n,d)$
satisfies Property (T). Then with overwhelming probability, for any
$d^{\prime}\in(d_{0},d)$ and for any $j\in\mathbb{N}$, a random group in
$\mathcal{M}_{jk}(n,d^{\prime})$ or $\mathcal{M}^{+}_{jk}(n,d^{\prime})$
satisfies Property (T).

In particular, this improves the known upper bounds in the $3j$-gonal models
from $1/2$ to $1/3$. It also shows that to find upper bounds for the sharp
threshold of Property (T), it suffices to investigate the _positive_ $k$-gonal
models. Further, this provides evidence supporting the following conjecture:

######  Conjecture 1.2.

Let $d_{(T)}(k)$ denote the sharp threshold for Property (T) in the $k$-gonal
model. Then $\lim_{k\to\infty}d_{(T)}(k)$ exists, and is equal to the sharp
threshold of the Gromov model.

### Acknowledgements

I would like to thank Danny Calegari and Tomasz Odrzygóźdź for valuable
comments and feedback. This research was partially supported by a Clare Boothe
Luce Professorship in Mathematics from the Henry Luce Foundation.

##  2\. Background

We begin by defining the $k$-gonal model of random groups.

######  Definition 2.1 ([AR20]).

Fix a set $S$ of cardinality $n$. The _$k$ -gonal model of random groups_,
denoted $\mathcal{M}_{k}(n,d),$ is the set of group presentations $\langle
S|R\rangle$ such that $R$ is a set of $(2n-1)^{kd}$ cyclically reduced words
on $S\cup S^{-1}$ of length $k$, along with the uniform probability measure.
We say that a $k$-gonal random group at density $d$ satisfies a property P
_with overwhelming probability_ , abbreviated w.o.p, if

$\lim_{n\to\infty}\mathbb{P}(\mbox{the group defined by a presentation in
}\mathcal{M}_{k}(n,d)\mbox{ satisfies P})=1.$

A related family of models of random groups restricts possible relators to
positive words, i.e. words on $S$ but not on $S^{-1}$. These are called the
positive $k$-gonal models of random groups and are denoted
$\mathcal{M}^{+}_{k}(n,d).$

We now define (Kazhdan’s) Property (T) for discrete, finitely generated
groups. The notation is suggestive - the idea of Property (T) is that the
trivial representation of the group must be isolated in some metric. More
formally, we can define this property via (almost) invariant vectors: Let
$\Gamma$ be a finite generated group with finitely generating set $S$. If
$\mathcal{H}$ is a Hilbert space and $\pi:\Gamma\to U(\mathcal{H})$ is a
unitary representation of $\Gamma$ on $\mathcal{H}$, then $\pi$ has _almost
invariant vectors_ if, for every $\varepsilon>0$, there exists some
$u_{\varepsilon}\in\mathcal{H}$ such that for every $s\in S$,
$\|\pi(s)u_{\varepsilon}-u_{\varepsilon}\|<\varepsilon\|u_{\varepsilon}\|.$ A
vector $u\in\mathcal{H}$ is _invariant_ if $\pi(g)(u)=u$ for every
$g\in\Gamma$.

######  Definition 2.2 ([Kaž67]).

A finitely generated group $\Gamma$ has _Property (T)_ if, for every
$\mathcal{H}$ and $\pi$, the following holds: if $\pi$ has almost invariant
vectors, then $\pi$ has a nonzero invariant vector.

The first results towards demonstrating Property (T) in random groups were
first stated in [Żuk03], and fully proved in [KK13]. The proof involved
passing between several different models of random groups and analyzing
spectral properties of random graphs.

######  Theorem 2.3 ([KK13] Theorem 3.14).

A random group in $\mathcal{M}^{+}_{3}(n,d^{\prime})$ with density
$d^{\prime}>1/3$ satisfies Property (T) with overwhelming probability.

In fact, they also showed that this is true in $\mathcal{M}_{3}(n,d)$ and in
the Gromov model of random groups (at the same density).

######  Remark 2.4.

Property (T) is preserved by surjections and finite extensions (see [KK13]
Remark 2.10 and [BHV08] Theorem 1.7.1). In other words, if $H$ is a group with
Property (T) and $H$ surjects onto $H^{\prime}$, then $H^{\prime}$ has
Property (T); and if $H\stackrel{{\scriptstyle f.i.}}{{\leq}}H^{\prime\prime}$
is a sub-group of finite index, then $H^{\prime\prime}$ has Property (T).

The tempting argument is to say if $G\in\mathcal{M}_{jk}(n,d)$ then $G=\langle
S|R\rangle$ and any relator in $R$ can be interpreted as a word of length $k$
over the alphabet of words of length $j$ on $S\cup S^{-1}$, denoted $|W_{j}|$.
Then $G$ is in $\mathcal{M}_{k}(|W_{j}|,d).$ However, as is pointed out in
[KK13], problems arise around the requirement that relators be reduced. In
particular, a word of length $kj$ is a word of length $k$ in the alphabet of
words of length $j$; however, a reduced word of length $k$ in the alphabet of
words of length $j$ might not be reduced as a word of length $jk$. So we must
first consider an intermediate group, $G^{+}$, whose relators are positive
words on $S$. This removes the reduction problem, and we can safely interpret
this group as a random group in $\mathcal{M}_{k}^{+}(n,d)$.

The method of proof closely follows the argument used to prove Theorem 4.3 in
[Odr19], which shows that groups in the $6$-gonal model w.o.p. satisfy
Property (T). In particular, to prove that a random group
$G\in\mathcal{M}_{jk}(n,d)$ satisfies Property (T), we construct two groups
$G^{+}$ and $\Gamma$ so that

$\Gamma\stackrel{{\scriptstyle\phi}}{{\twoheadrightarrow}}\phi(\Gamma)\stackrel{{\scriptstyle
f.i.}}{{\leq}}G^{+}\twoheadrightarrow G,$

and then show that $\Gamma$ satisfies Property (T).

There are two parts to this proof. The first is showing that $\phi(\Gamma)$ is
finite index in $G^{+}$. We find a finite set of coset representatives to
prove this explicitly, generalizing the method used in [Odr19]. The second is
in showing that the group $\Gamma$ can be interpreted as a typical
representative of random groups in $\mathcal{M}^{+}_{k}(m,d^{\prime})$ for
$d^{\prime}>1/3.$ For this part of the argument, we mimic the method developed
in the proof of Theorem A in [KK13].

##  3\. Property (T) in jk-Gonal Models

Fix a generating set $S$ of cardinality $n$. Let $W^{+}$ be the set of
positive words on $S$ and let $W_{j}$ be the words of length $j$ on $S$. For
any set $R$ of words on $S$, let $R^{+}=R\cap W^{+}$. Then for any group
$G=\langle S|R\rangle$ in $\mathcal{M}_{jk}(n,d)$, there is a group
$G^{+}=\langle S|R^{+}\rangle$ that surjects onto $G$.

Since $G$ is in the $jk$-gonal model, every word in $R$ and $R^{+}$ has length
$jk$. There is a natural injective map $\phi:W^{+}_{j}\to F(S)$, where $F(S)$
denotes the free group on $S$, given by the labeling on each word in
$W^{+}_{j}$. Define $R^{+}_{k}=\phi^{-1}(R^{+}),$ and notice that every word
in $R^{+}_{k}$ has length $k$ over $W_{j}^{+}$. Let $\Gamma=\langle
W_{j}^{+}|R^{+}_{k}\rangle$. Then $\phi$ induces a homomorphism, also denoted
$\phi$, mapping $\Gamma\to G^{+}$.

Note also that $|R^{+}_{k}|=|\phi(R^{+}_{k})|=|R^{+}|,$ so if
$G^{+}\in\mathcal{M}^{+}_{jk}(n,d^{\prime}),$ then
$\Gamma\in\mathcal{M}^{+}_{k}(|W^{+}_{j}|,d^{\prime})$. Furthermore, this map
takes the uniform probability distribution on
$\mathcal{M}^{+}_{jk}(n,d^{\prime})$ to the uniform probability distribution
on $\mathcal{M}^{+}_{k}(|W^{+}_{j}|,d^{\prime})$, because $\phi$ induces a 1-1
map on possible sets of relators.

We will show that $\phi(\Gamma)$ is a finite index subgroup of $G^{+}$ by
finding an explicit set of coset representatives of $\phi(\Gamma)$ in $G^{+}$.
We note that an element of $G^{+}$ is in $\phi(\Gamma)$ if and only if it is
expressible as a string of words of length $k$, all of which are positive or
the inverse thereof. We do this by inserting trivial words that allow us to
group positive and negative letters into subwords of length $j$.

In the following proof, for any word $w$, $|w|_{j}$ denotes the word length of
$w$ modulo $j$.

######  Lemma 3.1.

$\phi(\Gamma)$ is a finite index subgroup of $G^{+}$.

###### Proof.

Let $w$ be a word on $S\cup S^{-1}$. Without loss of generality, assume that
the first letter of $w$ is in $S^{-1}$. Then $w$ can be written as

$w=A_{\ell}a_{\ell}\cdots A_{2}a_{2}A_{1}a_{1},$

where $\\{a_{i}\\}$ are positive words on $S$, $\\{A_{i}\\}$ are negative
words on $S$, and $a_{1}$ might be empty.

Let $x$ be a positive word of length $j-|a_{1}|_{j}$ and let $y$ be a positive
word of length $j-|A_{1}x^{-1}|_{j}$. Then we have

$w=A_{\ell}a_{\ell}\cdots A_{2}a_{2}yy^{-1}A_{1}x^{-1}xa_{1}.$

Note that $xa$ is a positive word with length divisible by $j$, and
$y^{-1}A_{1}x^{-1}$ is a negative word with length divisible by $j$. Therefore
$w=A_{\ell}a_{\ell}\cdots A_{2}a^{\prime}_{2}u,$ where $u\in\phi(\Gamma)$. By
induction, we see that $w\in A^{\prime}_{\ell}\phi(\Gamma)$ where
$A^{\prime}_{\ell}$ is a positive word on $S$. Then $A^{\prime}_{\ell}=az,$
where $j$ divides $|z|$ and $|a|<j$, so $z\in\phi(\Gamma)$. Therefore $w=azv$,
where $zv\in\phi(\Gamma)$ and $|a|<j$.

Thus the (finite) set of words of length $<j$ contains a transversal of
$\phi(\Gamma)$ in $G^{+}$, and $\phi(\Gamma)$ has finite index in $G^{+}$. ∎

We are now ready to prove Theorem 1.1:

See 1.1

###### Proof.

We first show that a random group $G^{+}$ in $\mathcal{M}^{+}_{kj}(n,d)$ has
Property (T). Let $m=|W^{+}_{j}|$. By the construction above, given such a
$G^{+}$ we can find a group $\Gamma$ in $\mathcal{M}^{+}_{k}(m,d)$ and a map
$\phi:\Gamma toG^{+}$. The distribution induced on $\mathcal{M}^{+}_{k}(m,d)$
in this way is uniform and by Lemma 3.1 $\phi(\Gamma)$ is a finite index
subgroup of $G^{+}$. Since $\Gamma$ satisfies Property (T) with overwhelming
probability, by Remark 2.4, so does $G^{+}$.

Now let $G=\langle S|R\rangle$ be a group in $\mathcal{M}_{kj}(n,d).$ Note
that $|W^{+}_{j}|>\frac{1}{2^{j}}|W_{j}|.$ Therefore, the expected number of
elements of $R^{+}$ is larger than $\frac{1}{2^{j}}|R|.$ Furthermore, for any
$d^{\prime}<d$ there is some $N_{d^{\prime}}$ so that for $n>N_{d^{\prime}},$
we have

$|R^{+}|>\frac{1}{2^{j}}|R|=\frac{1}{2^{j}}(2n-1)^{kjd}>(2n-1)^{kjd^{\prime}}.$

Fix $d^{\prime}\in(d_{0},d)$. If we enumerate the elements of $W^{+}_{j}$ and
take $R^{\prime}$ to be the set of the first $(2n-1)^{kjd^{\prime}}$ elements
of $W^{+}_{j}$ which are in $R$, we can find a group $G^{+}\langle
S|R^{\prime}\rangle\in\mathcal{M}^{+}_{kj}(n,d^{\prime})$ that surjects onto
$G$. This gives a uniform distribution on
$\mathcal{M}^{+}_{kj}(n,d^{\prime})$, so w.o.p $G^{+}$ satisfies Property (T),
and by Remark 2.4 so does $G$. ∎

By Theorem 3.14 in [KK13], with overwhelming probability random groups in
$\mathcal{M}^{+}_{3}(n,d)$ satisfy Property (T) at density $d>1/3$. By
applying Theorem 1.1 we get the following result.

######  Corollary 3.2.

With overwhelming probability, a random group at density $d>1/3$ in
$\mathcal{M}_{3k}(n,d)$ or $\mathcal{M}^{+}_{3k}(n,d)$ satisfies Property (T).

## References

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$\mathbb{KC}^{L}_{SD}$.

The question then is: can we approximate a semiconcept of $\mathbb{K}$ by
definable semiconcepts of $\mathbb{KC}_{SD}$? For that purpose, the lower and
upper approximations of a semiconcept are defined. We split the set
$\mathfrak{H}(\mathbb{K})$ as
$\mathfrak{H}(\mathbb{K}):=(\mathfrak{H}(\mathbb{K})_{\sqcap}\setminus\mathcal{B}(\mathbb{K}))\cup(\mathfrak{H}(\mathbb{K})_{\sqcup}\setminus\mathcal{B}(\mathbb{K}))\cup\mathcal{B}(\mathbb{K})$.

######  Definition 26.

  * (a)

Let $(A,B)\in\mathcal{B}(\mathbb{K})$. The lower approximation and upper
approximation of $(A,B)$ are $(A,B)$ itself.

  * (b)

Let
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcap}\setminus\mathcal{B}(\mathbb{K})$.
The lower approximation and upper approximation of $(A,B)$ are defined as
$\underline{(A,B)}:=(\underline{A}_{E_{1}},(\underline{A}_{E_{1}})^{\prime})$
and $\overline{(A,B)}:=(\overline{A}^{E_{1}},(\overline{A}^{E_{1}})^{\prime})$
respectively.

  * (c)

Let
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcup}\setminus\mathcal{B}(\mathbb{K})$.
The lower approximation and upper approximation of $(A,B)$ are defined as
$\underline{(A,B)}:=((\overline{B}^{E_{2}})^{\prime},\overline{B}^{E_{2}})$
and
$\overline{(A,B)}:=((\underline{B}_{E_{2}})^{\prime},\underline{B}_{E_{2}})$
respectively.

######  Observation 1.

Recall the operators $f_{R}$ and $f_{S}$ defined in Section 2.3.

  1. 1.

For
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcap}\setminus\mathcal{B}(\mathbb{K})$,
$\underline{(A,B)}=f_{E_{1}}((A,B))$,
$\overline{(A,B)}=f^{\delta}_{E_{1}}((A,B))$.

  2. 2.

For
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcup}\setminus\mathcal{B}(\mathbb{K})$,
$\underline{(A,B)}=f_{E_{2}}((A,B))$,
$\overline{(A,B)}=f^{\delta}_{E_{2}}((A,B))$.

######  Example 3.

In Example 2 considered above, the lower and upper approximations of the
semiconcept $(\\{Leech,$ $Bream,Dog\\},\\{a,g\\})$ are
$(\\{Leech,Bream\\},\\{a,b,g\\})$ and $(\\{Leech,Bream,Cat,Dog\\},\\{a,g\\})$
respectively.

######  Proposition 22.

Let $A\subseteq G$ and $B\subseteq M$.

  1. 1.

$\underline{(\underline{A}_{E_{1}})^{\prime}}_{E_{2}}=(\underline{A}_{E_{1}})^{\prime}$
and
$\overline{(\overline{A}^{E_{1}})^{\prime}}^{E_{2}}=(\overline{A}^{E_{1}})^{\prime}$.

  2. 2.

$\underline{(\underline{B}_{E_{2}})^{\prime}}_{E_{1}}=(\underline{B}_{E_{2}})^{\prime}$
and
$\overline{(\overline{B}^{E_{2}})^{\prime}}^{E_{1}}=(\overline{B}^{E_{2}})^{\prime}$.

###### Proof.

1\. By Proposition 6(v),
$\underline{(\underline{A}_{E_{1}})^{\prime}}_{E_{2}}\subseteq(\underline{A}_{E_{1}})^{\prime}$.
Let $m\in(\underline{A}_{E_{1}})^{\prime}$. Then $gIm$ for all
$g\in\underline{A}_{E_{1}}$, which implies that $gIm$ for all $g\in G$ such
that $E_{1}(g)\subseteq A$. Let $m_{1}\in E_{2}(m)$, which implies that $gIm$
if and only if $gIm_{1}$ for all $g\in G$. So $gIm_{1}$ for all $g\in G$,
$E_{1}(g)\subseteq A$, which implies that $gIm_{1}$ for all
$g\in\underline{A}_{E_{1}}$. So $m_{1}\in(\underline{A}_{E_{1}})^{\prime}$,
which implies that $E_{2}(m)\subseteq(\underline{A}_{E_{1}})^{\prime}$. So
$m\in\underline{(\underline{A}_{E_{1}})^{\prime}}_{E_{2}}$, whence
$(\underline{A}_{E_{1}})^{\prime}\subseteq\underline{(\underline{A}_{E_{1}})^{\prime}}_{E_{2}}$.
So
$\underline{(\underline{A}_{E_{1}})^{\prime}}_{E_{2}}=(\underline{A}_{E_{1}})^{\prime}$.
By Proposition 6(v),
$(\overline{A}^{E_{1}})^{\prime}\subseteq\overline{(\overline{A}^{E_{1}})^{\prime}}^{E_{2}}$.
Let $m\in\overline{(\overline{A}^{E_{1}})^{\prime}}^{E_{2}}$. Then
$E_{2}(m)\cap(\overline{A}^{E_{1}})^{\prime}\neq\emptyset$, which implies that
there exists $m_{1}\in(\overline{A}^{E_{1}})^{\prime}$ such that $m_{1}\in
E_{2}(m)$. Now for all $g\in\overline{A}^{E_{1}}$, $gIm_{1}$, which implies
that for all $g\in\overline{A}^{E_{1}}$, $gIm$ , as $m_{1}\in E_{2}(m)$. So
$m\in(\overline{A}^{E_{1}})^{\prime}$, whence
$\overline{(\overline{A}^{E_{1}})^{\prime}}^{E_{2}}\subseteq(\overline{A}^{E_{1}})^{\prime}$.
So
$\overline{(\overline{A}^{E_{1}})^{\prime}}^{E_{2}}=(\overline{A}^{E_{1}})^{\prime}$.
2\. The proof is similar to that of 1. ∎

######  Corollary 8.

  1. 1.

The lower and upper approximations of semiconcepts of $\mathbb{K}$ are
definable semiconcepts of $\mathbb{KC}_{SD}$.

  2. 2.

If $(A,B)$ is a definable semiconcept of $\mathbb{KC}_{SD}$ then
$\underline{(A,B)}=(A,B)$ and $\overline{(A,B)}=(A,B)$.

  3. 3.

For $(A,B)\in\mathfrak{H}(\mathbb{K})$,
$\underline{(\underline{(A,B)})}=\underline{(A,B)}$ and
$\overline{(\overline{(A,B)})}=\overline{(A,B)}$.

  4. 4.

A left semiconcept $(A,B)$ of $\mathbb{K}$ is definable if and only if $A$ is
a category in $(G,E_{1})$.

  5. 5.

A right semiconcept $(A,B)$ of $\mathbb{K}$ is definable if and only if $B$ is
a category in $(M,E_{2})$.

###### Proof.

1\. The proof follows from Propositions 21, 22 and 6(vii and viii).

2\. Let $(A,B)$ be a definable semiconcept. If
$(A,B)\in\mathcal{B}(\mathbb{K})$, the proof follows from Definition 26(a).

Let
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcap}\setminus\mathcal{B}(\mathbb{K})$.
Then
$\underline{(A,B)}=(\underline{A}_{E_{1}},(\underline{A}_{E_{1}})^{\prime})=(A,A^{\prime})$,
as $A$ is a category in $(G,E_{1})$. So
$\underline{(A,B)}=(A,A^{\prime})=(A,B)$. Similarly, one gets the result for
the case when
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcup}\setminus\mathcal{B}(\mathbb{K})$.
Similar to the above proof, we can show that $\overline{(A,B)}=(A,B)$.

3\. The proof follows from 1 and 2.

4\. Let $(A,B)$ be a left semiconcept, so that $(A,B)=(A,A^{\prime})$. If $A$
is a category then by Proposition 22(1) it follows that $A^{\prime}$ is a
category. So $(A,B)$ is definable semiconcept. The other direction follows
from the definition of definability of semiconcepts.

5\. The proof is similar to that of 4. ∎

Now, we prove sequents in MPDBL5 that will yield fundamental properties of the
approximations of semiconcepts, as will be seen in Proposition 24 below.

######  Proposition 23.

Let $\gamma\in\mathfrak{F}_{1}$. Then the following sequents are derivable in
MPDBL5.

$\begin{array}[]{ll}(1)~{}\blacklozenge\gamma\vdash\gamma\sqcup\gamma&(2)~{}\gamma\sqcap\gamma\vdash\lozenge\gamma\\\
\end{array}$

###### Proof.

We give the proof for (1). The proof of (2) is similar.

(1)

$\blacklozenge\gamma\vdash\gamma\sqcup\gamma\lrcorner\blacksquare\lrcorner\gamma\vdash\gamma\sqcup\gamma~{}~{}(R4)(R3)^{\prime}~{}~{}\lrcorner\blacksquare\lrcorner\gamma\vdash\lrcorner\lrcorner\gamma~{}~{}\lrcorner\lrcorner\gamma\vdash\gamma\sqcup\gamma~{}~{}(\mbox{Theorem
\ref{thempdbl}(10b))}\lrcorner\gamma\vdash\blacksquare\lrcorner\gamma~{}~{}(19b)$

∎

######  Proposition 24.

For any $(A,B)\in\mathfrak{H}(\mathbb{K})$,
$\underline{(A,B)}\sqsubseteq(A,B)$ and $(A,B)\sqsubseteq\overline{(A,B)}$.

###### Proof.

Let $(A,B)\in\mathfrak{H}(\mathbb{K})$. If $(A,B)\in\mathcal{B}(\mathbb{K})$,
the results are trivially true.

Case I: Let
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcap}\setminus\mathcal{B}(\mathbb{K})$.
We consider a model $\mathbb{M}:=(\mathbb{KC}_{SD},v)$ and $p\in\textbf{OV}$
such that $v(p)=(A,B)$. As $\square p\vdash p$ is a valid sequent of MPDBL5,
$v(\square p)\sqsubseteq v(p)$, by Proposition 16. Using Proposition 15 and
Definition 26(b), $v(\Box
p)=(\underline{A}_{E_{1}},(\underline{A}_{E_{1}})^{\prime})=\underline{v(p)}$.
So we get $\underline{v(p)}\sqsubseteq v(p)$, which implies that
$\underline{(A,B)}\sqsubseteq(A,B)$. Case II: Let
$(A,B)\in\mathfrak{H}(\mathbb{K})_{\sqcup}\setminus\mathcal{B}(\mathbb{K})$.
We consider a model $\mathbb{M}:=(\mathbb{KC}_{SD},v)$ and $P\in\textbf{PV}$
such that $v(P)=(A,B)$. By Proposition 23(1), $\blacklozenge P\vdash P\sqcup
P$. $v(\blacklozenge P)\sqsubseteq v(P\sqcup P)$, by Proposition 16. Using
Proposition 15 and Definition 26(c), $v(\blacklozenge
P)=((\overline{B}^{E_{2}})^{\prime},\overline{B}^{E_{2}})=\underline{v(P)}$.
Hence $\underline{v(P)}\sqsubseteq v(P\sqcup P)=v(P)\sqcup v(P)=v(P)$, as
$v(P)\in\mathfrak{H}(\mathbb{K})_{\sqcup}$ which implies that
$\underline{(A,B)}\sqsubseteq(A,B)$.

Similar to the above derivations, using the valid sequents
$P\vdash\blacksquare P$ and $p\sqcap p\vdash\lozenge p$ of MPDBL5 (the last
from Proposition 23(2), we get the property $(A,B)\sqsubseteq\overline{(A,B)}$
in $\mathfrak{H}(\mathbb{K})$. ∎

######  Observation 2.

From the proof of Proposition 24, we have the following.

  1. 1.

For a formula $\alpha$, if $v(\alpha)$ is a left semiconcept of $\mathbb{K}$
then $v(\square\alpha)$ is its lower approximation and $v(\lozenge\alpha)$ is
its upper approximation. Moreover, the valid sequents
$\square\alpha\vdash\alpha$ and $\alpha\sqcap\alpha\vdash\lozenge\alpha$
respectively represent the properties that the lower approximation of a left
semiconcept lies ``below'' the left semiconcept, while its upper approximation
lies ``above" it.

  2. 2.

On the other hand, if $v(\alpha)$ is a right semiconcept of $\mathbb{K}$ then
$v(\blacklozenge\alpha)$ is its lower approximation and
$v(\blacksquare\alpha)$ is its upper approximation. The valid sequents
$\blacklozenge\alpha\vdash\alpha\sqcup\alpha$ and
$\alpha\vdash\blacksquare\alpha$ translate into properties of approximations
of right semiconcepts: the lower approximation of a right semiconcept lies
below it, while its upper approximation lies above it.

Let $\underline{(A,B)}=(X,Y)$. By Proposition 22, it follows that the extent
$X$ and intent $Y$ are categories in $(G,E_{1})$ and $(M,E_{2})$,
respectively. By Proposition 24, we have $X\subseteq A$ and $B\subseteq Y$,
which implies that $X\subseteq\underline{A}\subseteq A$ and
$\overline{B}\subseteq Y$. From $X\subseteq\underline{A}\subseteq A$, we can
say that the objects in the extent $X$ are certainly classified as objects in
the extent $A$ of the concept $(A,B)$. Now $\overline{B}\subseteq Y$ implies
that $Y^{c}\subseteq\underline{(B^{c})}$. So we can say that properties in
$Y^{c}$ are certainly classified as properties in the intent $B^{C}$ of
$\lrcorner(A,B)$. Similarly, we have $\underline{A}\subseteq
ext(\overline{(A,B)})$ and $int(\overline{(A,B)})\subseteq\overline{B}$. So
the properties in $int(\overline{(A,B)})$ are the properties that possibly
belong to the intent $B$ of the concept $(A,B)$. Now $\underline{A}\subseteq
ext(\overline{(A,B)})$ implies that
$ext(\overline{(A,B)})^{c}\subseteq\overline{(A^{c})}$. So the objects in the
set $ext(\overline{(A,B)})^{c}$ are the objects that possibly belong to the
extent $A^{C}$ of $\neg(A,B)$.

We end the section by making a comparison with the work in [29]. Consider a
semiconcept $(A,A^{\prime})$ that is not a concept. According to [29], this is
a non-definable concept such that $A^{\prime}$ is feasible and A is not. As
mentioned in Section 2.2, the lower and upper approximations of
$(A,A^{\prime})$ as given in [29] are
$\underline{(A,A^{\prime})}:=((\underline{A}_{E_{1}})^{\prime\prime}\cap
A^{\prime\prime},((\underline{A}_{E_{1}})^{\prime\prime}\cap
A^{\prime\prime})^{\prime})=((\underline{A}_{E_{1}})^{\prime\prime},(\underline{A}_{E_{1}})^{\prime})$
and
$\overline{(A,A^{\prime})}=((\overline{A}^{E_{1}})^{\prime\prime},(\overline{A}^{E_{1}})^{\prime})$.
Now observe that $\underline{(A,A^{\prime})}$ and $\overline{(A,A^{\prime})}$
are both concepts. In our case, the lower and upper approximations of a
semiconcept that is not a concept, may not be a concept: recall Example 3. The
lower approximation $(\\{Leech,Bream\\},\\{a,b,g\\})$ of the semiconcept
$(\\{Leech,Bream,Dog\\},\\{a,g\\})$ is not a concept. Furthermore, in our case
(as pointed out in Observation 2), the lower approximation of a semiconcept
lies below the semiconcept, while its upper approximation lies above it –
thereby justifying that these are lower and upper approximations of the
semiconcept. In general, these properties do not hold for the lower and upper
approximations of concepts defined in [29].

##  7 Conclusions

The hyper-sequent calculus PDBL is defined for the class of pdBas and extended
to MPDBL for the class of pdBaos. For any set $\Sigma$ of sequents in MPDBL,
$\textbf{MPDBL}\Sigma$ is defined. As particular cases of
$\textbf{MPDBL}\Sigma$, the logics MPDBL4 (for tpdBas) and MPDBL5 are
obtained. This presents a technique for constructing various logics in
general, that may represent attributes of pdBaos and associated classes of
Kripke contexts.

For conceptual knowledge, it is established that using PDBL and its models,
one can express the basic notion of concept, and the relations ``object
belongs to a concept'', ``property abstracts from a concept'' and ``a concept
is a subconcept of another concept''. Further, the basic notions of objects
and attributes and the relation ``an object has an attribute" are expressible
by using the named PDBL models. When interpretations of PDBL are restricted to
the collection of named models, the logic is sound with respect to the class
of of all contexts. An open question then is, whether a logic may possibly be
derived from PDBL that would be complete with respect to this restricted
collection of models.

This work attempts to highlight the significance of semiconcepts of a context
from different points of view. In particular, the observations of Section 5
indicate that a mathematical model for knowledge related to semiconcepts may
well be defined, akin to the conceptual knowledge system defined in [24, 32].
The significance of such a model may be worth exploring.

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######  Appendix 1.

## Proofs

Proof of Theorem 9: The proofs are straightforward and one makes use of axioms
2a, 3a, 4a, Proposition 7 and the rule $(R4)$ in most cases. $1a$:

$(\alpha\sqcap\beta)\vdash(\beta\sqcap\alpha)~{}(R4)\lx@proof@logical@and
4a~{}(\alpha\sqcap\beta)\vdash(\alpha\sqcap\beta)\sqcap(\alpha\sqcap\beta)(\alpha\sqcap\beta)\sqcap(\alpha\sqcap\beta)\vdash\beta\sqcap\alpha~{}(R6)\lx@proof@logical@and
3a~{}\alpha\sqcap\beta\vdash\beta\alpha\sqcap\beta\vdash\alpha~{}2a$

Interchanging $\alpha$ and $\beta$ in the above, we get
$(\beta\sqcap\alpha)\vdash(\alpha\sqcap\beta)$.

$2a.$

$(\alpha\sqcap\beta)\sqcap\gamma\vdash\beta\sqcap\gamma~{}(R4)~{}\mbox{--~{}(I)}\lx@proof@logical@and
4a~{}(\alpha\sqcap\beta)\sqcap\gamma\vdash((\alpha\sqcap\beta)\sqcap\gamma)\sqcap((\alpha\sqcap\beta)\sqcap\gamma)((\alpha\sqcap\beta)\sqcap\gamma)\sqcap((\alpha\sqcap\beta)\sqcap\gamma)\vdash\beta\sqcap\gamma~{}(R6)\lx@proof@logical@and(R4)~{}(\alpha\sqcap\beta)\sqcap\gamma\vdash\beta\lx@proof@logical@and
2a~{}(\alpha\sqcap\beta)\sqcap\gamma\vdash(\alpha\sqcap\beta)\alpha\sqcap\beta\vdash\beta~{}3a(\alpha\sqcap\beta)\sqcap\gamma\vdash\gamma~{}3a$

Now,
$(\alpha\sqcap\beta)\sqcap\gamma\vdash\alpha\sqcap(\beta\sqcap\gamma)~{}(R4)\lx@proof@logical@and
4a~{}(\alpha\sqcap\beta)\sqcap\gamma\vdash((\alpha\sqcap\beta)\sqcap\gamma)\sqcap((\alpha\sqcap\beta)\sqcap\gamma)((\alpha\sqcap\beta)\sqcap\gamma)\sqcap((\alpha\sqcap\beta)\sqcap\gamma)\vdash\alpha\sqcap(\beta\sqcap\gamma)~{}(R6)\lx@proof@logical@and(\alpha\sqcap\beta)\sqcap\gamma\vdash\alpha~{}(R4)\lx@proof@logical@and
2a~{}(\alpha\sqcap\beta)\sqcap\gamma\vdash\alpha\sqcap\beta\alpha\sqcap\beta\vdash\alpha~{}2a(\alpha\sqcap\beta)\sqcap\gamma\vdash\beta\sqcap\gamma~{}\mbox{(from
(I) above)}$

Similarly we can show that
$\alpha\sqcap(\beta\sqcap\gamma)\vdash(\alpha\sqcap\beta)\sqcap\gamma$.

$3a.$
$(\alpha\sqcap\alpha)\sqcap\beta\vdash\alpha\sqcap\beta~{}(R4)\lx@proof@logical@and
4a~{}(\alpha\sqcap\alpha)\sqcap\beta\vdash((\alpha\sqcap\alpha)\sqcap\beta)\sqcap((\alpha\sqcap\alpha)\sqcap\beta)((\alpha\sqcap\alpha)\sqcap\beta)\sqcap((\alpha\sqcap\alpha)\sqcap\beta)\vdash\alpha\sqcap\beta~{}(R6)\lx@proof@logical@and(R4)~{}(\alpha\sqcap\alpha)\sqcap\beta\vdash\alpha\lx@proof@logical@and
2a~{}(\alpha\sqcap\alpha)\sqcap\beta\vdash\alpha\sqcap\alpha\alpha\sqcap\alpha\vdash\alpha~{}2a(\alpha\sqcap\alpha)\sqcap\beta\vdash\beta~{}3a$

$\alpha\sqcap\beta\vdash(\alpha\sqcap\alpha)\sqcap\beta~{}(R4)\lx@proof@logical@and
4a~{}\alpha\sqcap\beta\vdash(\alpha\sqcap\beta)\sqcap(\alpha\sqcap\beta)(\alpha\sqcap\beta)\sqcap(\alpha\sqcap\beta)\vdash(\alpha\sqcap\alpha)\sqcap\beta~{}(R6)\lx@proof@logical@and(R4)~{}\alpha\sqcap\beta\vdash\alpha\sqcap\alpha\lx@proof@logical@and
4a~{}\alpha\sqcap\beta\vdash(\alpha\sqcap\beta)\sqcap(\alpha\sqcap\beta)(\alpha\sqcap\beta)\sqcap(\alpha\sqcap\beta)\vdash\alpha\sqcap\alpha~{}(R6)\lx@proof@logical@and
2a~{}(\alpha\sqcap\beta)\vdash\alpha(\alpha\sqcap\beta)\vdash\alpha~{}2a\alpha\sqcap\beta\vdash\beta~{}3a$

$(4a)$ follows from axiom 2a and $(R3)$.

$5a.$
$\alpha\sqcap(\alpha\sqcup\beta)\vdash\alpha\sqcap\alpha~{}(R4)\lx@proof@logical@and
4a~{}\alpha\sqcap(\alpha\sqcup\beta)\vdash(\alpha\sqcap(\alpha\sqcup\beta))\sqcap(\alpha\sqcap(\alpha\sqcup\beta))(\alpha\sqcap(\alpha\sqcup\beta))\sqcap(\alpha\sqcap(\alpha\sqcup\beta))\vdash\alpha\sqcap\alpha~{}(R6)\lx@proof@logical@and
2a~{}\alpha\sqcap(\alpha\sqcup\beta)\vdash\alpha\alpha\sqcap(\alpha\sqcup\beta)\vdash\alpha~{}2a$
$6a.$ Proof is identical to that of $5a.$

$(7a),(8a)$ follow from axiom 11a.

Note that the proofs of $(ib),i=1,2,3,4,5,6,7,8,$ are obtained using the
axioms and rules dual to those used to derive $(ia)$.

Proof of Proposition 8 For $1\implies 2$, we make use of $(R1)^{\prime}$,
$(R4)$, axiom 2a and Theorem 9(2a, 3a).

$\alpha\vdash\beta$ $\alpha\sqcap\alpha\vdash\alpha\sqcap\beta$
$\alpha\sqcap\beta\vdash\alpha$
$\alpha\sqcap(\alpha\sqcap\beta)\vdash\alpha\sqcap\alpha~{}~{}~{}\alpha\sqcap\beta\vdash\alpha\sqcap(\alpha\sqcap\beta)$
$\alpha\sqcap\beta\vdash\alpha\sqcap\alpha$

So $\alpha\sqcap\alpha\dashv\vdash\alpha\sqcap\beta$, which implies that
$[\alpha]\sqcap[\alpha]=[\alpha\sqcap\alpha]=[\alpha\sqcap\beta]=[\alpha]\sqcap[\beta]$.
Dually we can show that $[\alpha]\sqcup[\beta]=[\beta]\sqcup[\beta]$.
Therefore $[\alpha]\sqsubseteq[\beta]$.
For $2\implies 1$, suppose $[\alpha]\sqsubseteq[\beta]$. Then
$[\alpha]\sqcap[\beta]=[\alpha]\sqcap[\alpha]$. So
$[\alpha\sqcap\beta]=[\alpha\sqcap\alpha]$. Similarly we can show that
$[\alpha\sqcup\beta]=[\beta\sqcup\beta]$. Therefore
$\alpha\sqcap\beta\dashv\vdash\alpha\sqcap\alpha$ and
$\alpha\sqcup\beta\dashv\vdash\beta\sqcup\beta$. Now using (R5),
$\alpha\vdash\beta$.


# Deep VLBI Observations Challenge Previous Evidence of a Binary Supermassive
Black Hole Residing in the Seyfert Galaxy NGC 7674

Peter Breiding Department of Physics and Astronomy, Johns Hopkins University,
Bloomberg Center, 3400 N. Charles St., Baltimore, MD 21218, USA  Department of
Physics and Astronomy, West Virginia University, P.O. Box 6315, Morgantown, WV
26506, USA  Center for Gravitational Waves and Cosmology, West Virginia
University, Chestnut Ridge Research Building, Morgantown, WV 26505, USA  Sarah
Burke-Spolaor Department of Physics and Astronomy, West Virginia University,
P.O. Box 6315, Morgantown, WV 26506, USA  Center for Gravitational Waves and
Cosmology, West Virginia University, Chestnut Ridge Research Building,
Morgantown, WV 26505, USA  Canadian Institute for Advanced Research, CIFAR
Azrieli Global Scholar, MaRS Centre West Tower, 661 University Ave. Suite 505,
Toronto ON M5G 1M1, Canada  Tao An Shanghai Astronomical Observatory, Key
Laboratory of Radio Astronomy, CAS, 80 Nandan Road, Shanghai 200030, China
Karishma Bansal  SOFIA-USRA, NASA Ames Research Center, MS 232-12, Moffett
Field, CA 94035-0001, USA  Prashanth Mohan Shanghai Astronomical Observatory,
Key Laboratory of Radio Astronomy, CAS, 80 Nandan Road, Shanghai 200030, China
Gregory B. Taylor  Department of Physics and Astronomy, University of New
Mexico, 210 Yale Blvd NE, Albuquerque, NM 87106, USA  Yingkang Zhang  Shanghai
Astronomical Observatory, Key Laboratory of Radio Astronomy, CAS, 80 Nandan
Road, Shanghai 200030, China

(Received October 18, 2021; Revised May 15, 2022; Accepted May 26, 2022)

###### Abstract

Previous Ku-band (15 GHz) imaging with data obtained from the Very Long
Baseline Array (VLBA) had shown two compact, sub-pc components at the location
of a presumed kpc-scale radio core in the Seyfert galaxy NGC 7674. It was then
presumed that these two unresolved and compact components were dual radio
cores corresponding to two supermassive black holes (SMBHs) accreting
surrounding gas and launching radio-bright relativistic jets. However,
utilizing the original VLBA dataset used to claim the detection of a binary
SMBH, in addition to later multi-epoch/multi-frequency datatsets obtained from
both the VLBA and the European VLBI Network, we find no evidence to support
the presence of a binary SMBH. We place stringent upper limits to the flux
densities of any sub-pc-scale radio cores which are at least an order of
magnitude lower than the original VLBI radio-core detections, directly
challenging the original binary SMBH detection claim. With this in mind, we
discuss the possible reasons for the non-detection of any VLBI radio cores in
our imaging, the possibility of a binary SMBH still residing in NGC 7674, and
the prospect of future observations shedding further light on the true nature
of this active galactic nucleus.

Active galactic nuclei (16); Galaxy mergers (608); Gravitational waves (678);
Supermassive black holes (1663)

††facilities: VLBA, EVN, VLA††software: astropy (Astropy Collaboration et al.,
2013, 2018), CASA, AIPS

##  1 Introduction

In the standard $\Lambda$CDM concordance model of cosmology, a major driver of
galaxy growth and evolution is the hierarchical merging of other galaxies,
subsuming each other’s stars, gas, dark matter haloes, and supermassive black
holes (SMBHs, M${}_{\mathrm{BH}}\gtrsim 10^{6}$M⊙) in the process (e.g.,
Kauffmann & Haehnelt, 2000). Since most galaxies are believed to host a SMBH
in their dynamical centers (Kormendy & Ho, 2013), a predicted outcome of these
galaxy mergers is the formation of gravitationally-bound binary SMBHs
(Begelman et al., 1980). First, the SMBHs residing in each galaxy are thought
to sink towards the nuclear environment of the post-merger remnant via
dynamical friction (Chandrasekhar, 1943), ultimately achieving binary
separations on the order of $\sim$ 10 pc within $\sim$100 Myr (Callegari et
al., 2009). The subsequent stage of binary evolution from $\sim$ 10 pc to sub-
pc separations relies on three-body interactions with stars (e.g., Sesana et
al., 2007), gravitational torques from a gaseous circumbinary disk (e.g.,
Escala et al., 2005), and potentially three-body SMBH interactions (e.g.,
Hoffman & Loeb, 2007) to harden, or shrink, the binary until gravitational
radiation can efficiently remove angular momentum from the pair until their
eventual coalescence. Depending on the nuclear environment of the post-merger
host galaxy, this intermediate evolutionary stage between dynamical friction
and gravitational radiation dominating the binary’s orbital dynamics can take
anywhere from 10 Myr (e.g., Khan et al., 2015) to several tens of Gyr (e.g.,
Yu, 2002), where the latter scenario is typically referred to as the “final
parsec problem” (Milosavljević & Merritt, 2003). After surmounting the
evolutionary stage pertinent to the final parsec problem, long-wavelength
gravitational waves emitted by close-orbit ($\ll 0.1$ pc) binary SMBHs in the
local Universe should soon be detectable by current Pulsar Timing Array
experiments (McLaughlin, 2013; Hobbs, 2013; Verbiest et al., 2016; Arzoumanian
et al., 2020) and the upcoming Laser Interferometer Space Antenna (LISA,
Amaro-Seoane et al., 2017), constituting a crucial source class for these
observatories (Burke-Spolaor et al., 2019).

There are several observational approaches other than gravitational-wave
detection used to infer the presence of binary SMBHs at separations
$\lesssim~{}10~{}$pc. The two main methods include the observation of quasi-
periodic light curves of close-orbit ($\ll 0.1$ pc) binaries (e.g., Graham et
al., 2015; Charisi et al., 2016; Liu et al., 2016; Mohan et al., 2016; Liu et
al., 2019, 2020) and velocity-offset broad emission lines for binaries with
separations of $\sim$ 0.1 to tens of pc in nearby quasars (e.g., Bogdanović et
al., 2009; Tsalmantza et al., 2011; Eracleous et al., 2012; Ju et al., 2013;
Shen et al., 2013; Runnoe et al., 2017; Kelley, 2021). The former involves a
periodic modulation of the observed luminosity (at the orbital period of the
binary), while the latter technique relies on the Doppler shifting of spectral
lines emitted by gas gravitationally bound to the SMBH in the so-called broad
line region (BLR, where the Doppler shifts result from the binary’s orbital
dynamics). However, both methods have a variety of more mundane explanations.
In the case of periodic light curves, jet precession not induced by a binary
(e.g., Liska et al., 2018), a warped accretion disk (e.g., Hopkins & Quataert,
2010), helical jet morphologies (e.g., Conway & Murphy, 1993),
magnetohydrodynamic (MHD) instabilities associated with the accretion flow
(e.g., King et al., 2013) or global oscillation of the accretion disk (e.g.,
An et al., 2013a; Wang et al., 2014), and stochastic variability (Vaughan et
al., 2016) are all viable alternative hypotheses to a binary SMBH. In the case
of quasars with velocity-offset broad emission lines, recoiling SMBHs and BLR
outflows are also plausible alternative hypotheses (see e.g., Breiding et al.,
2021).

A more direct observational technique which does not suffer from the above
uncertainties is the spatially-resolved imaging of both SMBHs (e.g., An et
al., 2018), given that they are both actively accreting material and shining
brightly as active galactic nuclei (AGN). This technique has revealed several
dozens of dual AGN (AGN with kpc-scale separations), utilizing telescopes
across the electromagnetic spectrum (see e.g. Table 1 from Rubinur et al. 2018
and De Rosa et al. 2019 for an overview). However, in order to resolve the pc
and sub-pc spatial scales in gravitationally-bound binary SMBHs, telescopes
with milliarcsecond (mas) and sub-mas resolving powers are required111Note
that at a redshift of $\sim$ 0.05, 1 pc corresponds to 1 mas. Even at
distances of $\sim$ a few Mpc (roughly the size of the local group), sub-pc
resolution can only be achieved with $\sim$ 10 mas angular resolution..
Currently, the only approach capable of achieving this type of angular
resolution relies on Very Long Baseline Interferometry (VLBI) in order to
synthesize the large required apertures. There have been several
radio222Although see Gravity Collaboration et al. 2017 for an example of the
promise of high-resolution optical interferometry and synthesis imaging. VLBI
studies searching for pc-scale binary SMBHs systematically in surveys (e.g.,
Burke-Spolaor, 2011; Tremblay et al., 2016), and as follow-up observations of
binary SMBH candidates identified through more indirect methods (e.g., Kharb
et al., 2017; Breiding et al., 2021). However, to date these projects have
been largely unsuccessful in uncovering large numbers of bona fide binary
SMBHs.

In the context of high angular resolution radio imaging, the “smoking gun”
signature of a binary SMBH would be the observation of two compact, and flat
(or inverted) spectrum cores. The physical model used to describe these VLBI
cores is the radio emission from the optical depth, $\tau~{}=~{}1$ surface at
which the base of some relativistic jet become opaque to synchrotron self-
absorption (Sokolovsky et al., 2011), or a standing shock slightly downstream
from this surface (Marscher, 2008). In either case, the VLBI core is located
several-to-tens of pc downstream of the SMBH producing the jet, and is thus a
good marker for its location. At a projected separation of 7 pc, the AGN CSO
0402+379 is a striking example of a nearby ($z~{}=~{}$0.055) binary radio core
confidently confirmed as a binary SMBH system (Rodriguez et al., 2006). Using
the Very Long Baseline Array (VLBA), Bansal et al. (2017) have tracked the
orbital motion of the cores with proper motion measurements, lending further
support to the binary SMBH nature of this system.

NGC 7674 (aka MRK 533) is a nearby ($z$ = 0.03), type II Seyfert galaxy
(Mirabel & Wilson, 1984) home to another purported close-separation binary
SMBH. Kharb et al. (2017), referred to as K17 in the rest of the paper,
claimed the detection of two inverted-spectrum radio cores in the nucleus of
NGC 7674 (at a projected separation of 0.7 pc), presumably the product of a
binary SMBH system in which each black hole is active and hosts a radio jet.
The host galaxy is a luminous infrared galaxy (LIRG, González Delgado et al.,
2001), and the brightest in a group of four interacting galaxies comprising
the Hickson 96 compact galaxy group333The two largest galaxies in this group,
one of which being NGC 7674, are separated by $\sim$ 80 kpc (projected). The
closest galaxy companion to NGC 7674 is at a projected separation of $\sim$ 30
kpc (see Figure 1 from Verdes-Montenegro et al. 1997 for an optical image of
the compact galaxy group). . It is a nearly face-on (inclination angle of
$\sim$ 30∘) spiral galaxy (SBc type, Williams & Rood, 1987), exhibiting tidal
features which are likely the imprint of gravitational interactions with its
neighbors (Verdes-Montenegro et al., 1997).

Deep 2001 S-band ($\sim$ 1.4 GHz) VLBA observations of NGC 7674, in
combination with phased Very Large Array (VLA) and Arecibo observations,
allowed for the first detection of the S (or Z)-shaped morphology of the kpc-
scale radio jet (Momjian et al., 2003). One hypothesized origin for S-shaped
radio jet morphologies is jet precession induced by the orbital dynamics of a
binary SMBH (Begelman et al., 1980). However, S-shaped jets may also be caused
by jet precession induced by a tilted accretion disk (Sarazin et al., 1980;
Lu, 1990) or the gas circulation of the interstellar medium (ISM) (Gopal-
Krishna et al., 2003). The linear extent of the jet is $\sim$ 0.6 kpc
projected on the plane of the sky, thus allowing for the classification of
this source as a compact symmetric object444CSOs are young
($\lesssim~{}10^{5}$ yr) radio-loud AGN classified on the basis of double,
symmetric, radio jets/lobes less than 1 kpc in extent (Wilkinson et al., 1994;
An & Baan, 2012) (CSO). Given their small jet size, CSOs are able to interact
strongly with the narrow line region (NLR) gas within the central kiloparsec
of their host galaxy (O’Dea, 1998). With this in mind, the observations of NLR
gas outflows in NGC 7674 (Unger et al., 1988; Shastri et al., 2006; Smirnova
et al., 2007) are naturally explained by the interaction of the jet with the
surrounding medium (e.g. An et al., 2013b; Jaiswal et al., 2019).

In this paper, we use deep multi-band VLBI imaging from both the VLBA and
European VLBI Network (EVN) to search for evidence of the two putative VLBI
radio cores in NGC 7674 indicative of a binary SMBH. This search includes the
VLBA data used to make the original dual VLBI core detection claim in addition
to VLBI data from approximately a decade and a half later. We also analyze the
VLA data used in K17 to claim the presence of a single, kpc-scale radio core
which corresponds to the unresolved emission from the two VLBI cores. Finally,
we assess the binary SMBH hypothesis and AGN activity of this source by
putting our findings into the larger context of the other multi-wavelength
observations and analyses of this source. Throughout this paper we adopt a
$\Lambda$CDM cosmology, with H${}_{0}=67.74$ km s-1 Mpc -1,
$\Omega_{\lambda}=0.69$, and $\Omega_{m}=0.31$ (Planck Collaboration et al.,
2016).

##  2 VLA Data Reduction & Analysis

We reduced the archival Ku-band (15 GHz) radio data from project 14A-471 taken
with the VLA in its A-array configuration on March 21, 2014. The data was
calibrated using the Common Astronomy Software Applications (CASA, McMullin et
al., 2007) software package automatic pipeline (version 5.4.1). After pipeline
calibration, any radio frequency interference (RFI) was removed via the
automatic rflag routine within CASA. 3C 48 was the flux calibrator for our
source, and several rounds of phase-only self-calibration, followed by a
single round of amplitude and phase self-calibration was employed. Finally,
images were constructed from the visibility data using the CLEAN deconvolution
algorithm employed by the CASA task tclean. The final VLA image of NGC 7674
was made using Briggs weighting, a robust parameter of 0.5, and a multi-term,
multi-frequency synthesis (mtmfs) deconvolution and imaging procedure with two
Taylor terms. This type of analysis relies on modeling the sky intensity
distribution as a Taylor polynomial, expanded about some reference frequency
for the subsequent deconvolution and image reconstruction (Rau & Cornwell,
2011). Including the second Taylor term in the analysis allows for the
measurement of spectral index information in radio maps in addition to the
intensity. We also imaged the source with a uniform weighting scheme, but the
marginal improvement in angular resolution resulted in the unacceptable trade-
off of a much lower signal-to-noise detection.

##  3 VLBI Observations & Data Analysis

Below we describe the VLBI observations employed to create our final high-
sensitivity images, using the VLBA and EVN. All of the VLBI observations used
in this paper are listed in Table 1. Combining these disparate datasets
together from both the VLBA and EVN improved the sensitivity afforded by the
individual epochs. In turn, this allowed us to test the hypothesis that dual,
inverted-spectrum radio cores were present in this source, and furthermore
that they were indicative of a binary SMBH (in which both black holes are
active) residing in this galaxy. All VLBI observations are phase-referenced,
where we show plots of phase calibrator phase before and after calibration is
applied to the high-frequency 15 and 22 GHz data obtained with the VLBA in the
appendix. Importantly, no obvious structure that would impact our phase
calibration was seen in the images of any of our phase calibrators. We show
high-frequency 15 and 22 GHz images of our phase calibrators in the appendix
for both the 2002 K17 detection epoch and follow-up 2018 epochs. These phase
calibrator targets include J2327+0940 and J2329+0834, which are separated from
NGC 7674 by 0.9 degrees and 0.35 degrees, respectively. In the appendix, we
also show the (u,v) plane coverage for each individual 15/22 GHz VLBI epoch,
in addition to the (u,v) coverage corresponding to the combined 15 and 22 GHz
high-frequency datasets.

Table 1:  VLBI Observing Sessions

| Project | Observation | Observing
---|---|---|---
Observatory | Code | Date (UTC) | Band
|  | (YYYY/MM/DD) |
VLBA | BV045 | 2002/08/28 | S
VLBA | BV045 | 2002/08/28 | C
VLBA | BV045 | 2002/08/28 | X
EVN | EA059 | 2018/06/05 | X
VLBA | BV045 | 2002/08/28 | Ku
VLBA | BK212 | 2018/03/31 | Ku
VLBA | BT143 | 2018/11/29 | Ku
VLBA | BK212 | 2018/03/19 | K
VLBA | BK212 | 2018/04/12 | K
VLBA | BT143 | 2018/11/29 | K

Note. — Observing dates mark the start of an observing session if the
observations spill over into subsequent days.

###  3.1 VLBI Observations & Calibration

####  3.1.1 VLBA Project BV045

The original VLBI dataset used to assert the existence of a binary SMBH in NGC
7674 was obtained from phase-referenced VLBA Ku-band (15 GHz) observations
(project ID BV045) taken in 2002. Associated with this project were
observations at S, C, and X bands (2.3, 5, 8 GHz, and 15 GHz). The
experimental setup included only a single polarization recording capability,
4.2 s integration times, and a total bandwidth of 32 MHz, split into four
spectral windows (with 16 channels per spectral window). These observations
included nine VLBA antennas (excluding the Brewster station), and the cycle
time used for the “nodding” mode observations (phase-target-phase scans) was
$\sim$ 11 min ($\sim$ 4 min on the phase calibrator J2329+0834, $\sim$ 7 min
on NGC 7674). The quasar 3C 84 was used as both the fringe finder and bandpass
calibrator for each set of observations (i.e., each observing band) in BV045.

####  3.1.2 VLBA Projects BK212 & BT143

VLBA projects BK212 and BT143 involved follow-up observations of NGC 7674 at
Ku and K bands (15 GHz and 22 GHz, respectively) in 2018. The experimental
setups for both BK212 and BT143 included full polarization capabilities, 2 s
integration times, and a total bandwidth of 256 MHz, split into eight spectral
windows (where BK212 had 16 channels per spectral window and BT143 had 64
channels per spectral window). BK212 included at least nine VLBA antennas, and
BT143 had at least eight VLBA antennas. The cycle time used for the “nodding”
mode observations (phase-target-phase scans) was $\sim$ 7 min for BK212
($\sim$ 4 min on the phase calibrator J2329+0834, $\sim$ 3 min on NGC 7674).
For BT143, the cycle time was $\sim$ 3 min ($\sim$ 2 min on the phase
calibrator J2327+0940, $\sim$ 1 min on NGC 7674). The quasars 3C 454.3 and 3C
345 were used as the fringe finder/bandpass calibrators for BK212 and BT143,
respectively.

####  3.1.3 VLBA calibration

We calibrated all of the VLBA visibility data in AIPS (van Moorsel et al.,
1996) using the standard calibration procedures applied in the pipeline task
VLBARUN for continuum imaging, where appropriate reference antennas were
chosen. Log-based flagging was performed prior to calibration.

####  3.1.4 EVN Observations & Calibration

The target source was also observed by EVN on June 6th, 2018, (project code
EA059A) with the aim of probing the dual cores. The observations were
conducted at 8.4 GHz with a total observing time of 10 hours. The session was
recorded at 1024 Mbps rate (16 MHz $\times$ 8 subbands, 2-bit sampling, dual
polarization). 12 stations participated in session A, which are Ef
(Effelsberg, Germany), Wb (Westerbork, The Netherlands), Mc (Medicina, Italy),
Nt (Noto, Italy), O6 (Onsala, Sweden), Ys (Yebes, Spain), Hh (Hartebeesthoek,
South Africa), Sv (Svetloe, Russia), Zc (Zelenchukskaya, Russia), Bd (Badary,
Russia), Ir (Irbene, Latvia), T6 (Tianma65, China).

During the observations, the phase-reference observing mode was employed due
to the weakness of the target source. J2327+0940 was chosen as the phase-
reference calibrator and several scans on J2329+0834 were observed for finding
fringes. To reduce the impact from the dynamic troposphere, we used a shorter
nodding cycle at shorter wavelengths. We used a cycle time of
’cal(30s)-tar(135s)-cal(45s)’. Due to some operational problems and bad fringe
solutions at a few stations, some antennas did not record well during specific
time periods. These bad data were deleted. After the observations were
completed, the data from each station were transported to JIVE at Dwingeloo,
the Netherlands for correlation. The correlated visibility data were then
downloaded to the China SKA Regional Centre computing clusters (An et al.,
2019) for further analysis and processing. We calibrated the visibility data
using AIPS following a standard procedure used for phase-reference EVN
observations of weak radio sources (e.g., Mohan et al., 2020; Salafia et al.,
2021).

###  3.2 Data Combination & Imaging

Table 2:  VLBI Image Properties Observing | Central | Image |  |  |
---|---|---|---|---|---
Band | Frequency | Noise Level | $\mathrm{B_{maj}}$ | $\mathrm{B_{min}}$ | P.A
| (GHz) | ($\mathrm{\mu}$Jy beam-1) | (mas) | (mas) | (∘)
S | 2.27 | 126 | 11.5 | 4.21 |  $-$14.5
C | 5.00 | 90.0 | 4.40 | 1.74 |  $-$10.0
X | 8.41 | 86.9 | 2.77 | 1.23 |  $-$7.40
Ku | 15.3 | 40.0 | 1.35 | 0.54 |  $-$6.76
K | 22.2 | 45.2 | 1.59 | 0.43 |  $-$14.7

After calibration, we used CASA for further flagging of RFI. Subsequently, we
combined all of the visibility data from the different observatories and
epochs common to a given observing band using the CASA task concat, and then
cleaned/imaged that data with the CASA task tclean. We used a natural
weighting scheme, as this has the highest sensitivity for point sources (which
is the expectation for the two unresolved, compact cores), with cell sizes of
$\sim$ 4 pixels per restoring beam. We used the hogbom deconvolution
algorithm, and an interactive cleaning procedure with CLEAN masks ultimately
created for components C and W during the cleaning process. At no point did
any significant residuals suggest the need for cleaning any image components
near the purported location of the binary SMBH. All cleaning had 3$\sigma$,
rms-based threshold minor cycle stopping points.

We also made wide-field images to search for any VLBI cores on scales up to
$\sim$ 1 arcsec from the phase center. This was accomplished with the wproject
gridding algorithm (Cornwell et al., 2008, which corrects for the effect of
non-coplanar baselines), with the use of wprojplanes=-1 in CASA. This choice
automatically determines the number of planes to use based upon the data and
image size555CASA determines this number based upon the following formula:
$\mathrm{N_{wprojplanes}=0.5\times\frac{W_{max}}{\lambda}\times\frac{imsize}{(radians)}}$.
Here, imsize is the image size, $\lambda$ is the wavelength, and
$\mathrm{W_{max}}$ is the maximum w in the uvw data (i.e., physical extent of
the visibility data that is orthogonal to the image plane)..

##  4 Results

Figure 1:  In the top panel we show the VLA Ku-band (15 GHz) radio image of
NGC 7674 as a contour map, with a base level of 3$\sigma$ and spaced by
factors of two thereafter (RMS noise level is 0.7 $\mathrm{\mu
Jy~{}beam^{-1}}$). Negative contours are shown as dashed lines. Components
labeled C and W correspond to the hot spots, following the naming convention
used in previous studies. We also label the feature claimed to correspond to
the radio core in K17, in addition to lobe plasma east of component C. The
synthesized beam is shown in the bottom left as a filled ellipse. The bottom
panel shows the spectral index map, with white contours corresponding to the
image contours presented in the top panel. The green ellipse shows the region
used to extract the spectral index information for the region corresponding to
the purported radio core. Figure 2:  High-sensitivity VLBI images of NGC 7674
at S (top left panel, 2.3 GHz), C (middle left panel,5 GHz), X (bottom left
panel, 8.4 GHz), Ku (top right panel, 15.4 GHz), and K (middle right panel,
22.2 GHz) bands, using the datasets listed in Table 1. The bottom-right panel
shows the contour-only image of the Ku-band (15 GHz) K17 binary detection
dataset. We show image contours for all images at a base level of 3$\sigma$,
spaced by factors of two thereafter. The synthesized beam shapes are shown in
the bottom left of the images as filled gray ellipses with a cross inside,
except the bottom right contour plot where it is a filled black ellipse. The
left-panel images show the eastern and western hot spots, labeled C and W,
respectively, are detected in our S/C/X-band imaging, as also shown in K17
(although component W was not detected in our C or X-band imaging). The center
position bewteen the dual cores reported by K17 is marked with cross hairs in
each image, where both the dual core separation and errors of the purported
core locations are on the order of the beam sizes shown for our images. As is
evident, these radio cores are not detected in any of these images.

###  4.1 VLA imaging and spectral analysis

In Figure 1 we show the 15 GHz radio image of NGC 7674, synthesized from 2014
A-configuration VLA observations and label the jet features following the
conventions in previous studies of the source. The feature most relevant to
this analysis is between the eastern/western hot spot components (C/W), and is
identified as the radio core in K17. We show in the bottom panel of Figure 1
the in-band spectral index, $\alpha$, map of NGC 7674. For this study, we
assumed a spectral index defined as $F_{\nu}\propto\nu^{-\alpha}$, where
$F_{\nu}$ is the flux density, and $\nu$ is the observing frequency. Using an
elliptical extraction region (with the same shape as the synthesized beam)
across the feature identified as the radio core, we measured the mean spectral
index from this region to be $-0.66~{}\pm~{}0.52$. This index is consistent
with the claimed inverted-spectrum index found in K17.

###  4.2 Non-detections of VLBI radio cores

In Table 2 we give the central observing frequencies, RMS noise levels, major
and minor axes ($\mathrm{B_{maj}}$ and $\mathrm{B_{min}}$, respectively) of
the synthesized beam full width at half maximums (FWHMs), and beam position
angles (P.A.) for the final high-sensitivity images used in our study (refer
to Table 1 for the list of datasets that went into creating each image). In
Figure 2 we show the S, C, and X-band images in which we can report detections
of the hot spot components (C and W). As shown in K17 and found again in our
imaging, these hot spots have a fairly steep spectrum indicative of aged,
optically thin plasma emitting synchrotron radiation. In Figure 2 we also show
the Ku-band (15 GHz) and K-band (22 GHz) synthesized VLBI images for the
combined Ku and K-band data described in section 3.1, in addition to an image
of the original Ku-band K17 detection dataset. We mark the locations of the
dual VLBI core detections from K17 with cross hairs (taking the center of the
cross hair to be the center position between the dual core positions), where
we estimate the error on the core positions from K17 to be no greater than
$\sim~{}0.7$ mas666For this estimate we note that the phase calibrator
position error is negligible (0.15 mas), and the error associated with the
phase-referencing technique is expected to be on the order of $\sim$ 0.1 mas
(Pradel et al., 2006). Thus, these errors would add negligible contributions,
considering the beam size of the tentative detection reported in K17 is 0.7
$\times$ 0.7 mas., a size which is roughly on the order of the beam widths
shown in Figure 2.

In Figure 3, we plot the flux density reported for both VLBI cores from K17
(0.9 $\pm$ 0.4 mJy). We also show a line (with arrows) representing the
4$\sigma$ upper limit reported by K17 on the spectral index for the western
VLBI core, i.e., $\alpha~{}<0.38$ (where this limit is less restrictive than
that of the eastern VLBI core, $\alpha~{}<0.33$). In red arrows, we show our
3$\sigma$ upper limits on the flux density of either VLBI core using the
images displayed in Figure 2, where it is clear that neither core is detected
in our imaging. We also plot as a blue arrow the resulting 3$\sigma$ upper
limit from our reanalysis of the original K17 Ku-band (15 GHz) “binary
detection” dataset. Given the spectral index limit, the K-band non-detections
are especially problematic for the hypothesis of dual, inverted-spectrum VLBI
radio cores with $\sim$ 1 mJy flux densities residing in the heart of NGC
7674.

##  5 Discussion & Concluding Remarks

Figure 3: The radio flux densities of both VLBI radio cores detected at 15 GHz
by K17 are shown in black with associated 1$\sigma$ error bars. The solid
black line with arrows shows the upper limit on the radio spectral index
reported by K17 for the western VLBI core based upon a X-band non-detection.
The red arrows show the 3$\sigma$ upper limits on the flux densities for
either putative VLBI core based upon the non-detections of our high-
sensitivity data combinations at S, C, X, Ku, and K bands. As is apparent from
the radio spectrum, the non-detections at Ku and K bands are especially
problematic for the original detection claim, given the limit on spectral
index and assumed inverted spectrum for the putative VLBI radio cores.

Our upper limits obtained from our VLBI core non-detections at 15 and 22 GHz
are $\sim$ an order of magnitude below the flux densities reported by K17 for
the purported dual, inverted-spectrum VLBI cores. Thus, we consider the
possibility of the K17 detections to be false-alarm detections below (where
the original detection confidence levels were reported to be between 4$\sigma$
and 5$\sigma$). In support of this idea is the fact that the 2002 dataset
analyzed by K17 for the original dual radio core detections was also used in
our analysis (BV045, comprised of S, C, X, and Ku-band VLBA observations).
While we report the high signal-to-noise detection of the hot spot components
(just as in K17, see Figure 2), there is no evidence for the putative binary.
If there was some systematic error in our analysis, we would expect that this
would also affect our detection of the hot spot features (which was not the
case). Furthermore, the Ku-band and K-band observations from 2018 yield non-
detection upper limits well below the expected flux densities for these cores
given their supposed inverted spectrum. One way out of this conclusion is to
presume flux density variability of the cores could allow them to remain
undetected in the recent 2018 K-band and Ku-band imaging. The implied large-
amplitude variations are incompatible with typical intra/inter-day stochastic
variability typically observed in radio light curves of flat spectrum AGN
cores and usually ascribed to interstellar scintillation (e.g., Lovell et al.,
2008; Koay et al., 2018). Furthermore, we note that the luminosity of the core
is relatively weak compared to the hot spot/lobe plasma, implying this source
is not consistent with a highly variable blazar. Rather, this long-term
variability would require both cores to dim by more than an order of magnitude
in luminosity in $\sim$ 16 years. This hypothesis is somewhat compromised
considering the feature detected in the 2014 VLA image (Figure 1) between the
eastern and western hot spot components (C and W, respectively), claimed by
K17 to be the unresolved emission from the dual VLBI cores. Together these
hypotheses imply that the emission from the binary AGN held a relatively
constant 15 GHz luminosity from 2002 to 2014, and then dimmed by more than an
order of magnitude from 2014 to 2018. This scenario can be tested with future,
deep Ku-band (and higher frequency) VLA A-configuration observations, combined
with simultaneous VLBI imaging (possibly in phased VLA and VLBA observations,
as in Momjian et al. 2003). If the feature identified as the kpc-scale core by
K17 is not detected in such an experimental setup, then this lends credence to
the idea that the dual radio jets in NGC 7674 have “turned off”. This type of
scenario could be explained by long-term “geometric” variability induced by
e.g. jet precession (potentially due to a binary), and electromagnetic
radiation which is progressively relativistically beamed (Blandford & Königl,
1979) in a direction further from our line of sight (as the approaching jets
swing away from us), subsequently yielding lower observer-frame radio
luminosities (e.g., Bach et al., 2006). In principle, the jet precession
induced by a binary SMBH system can result from two scenarios. In the first
(disk precession, e.g. Katz, 1997; Liu & Chen, 2007; Nandi et al., 2021), the
accretion disk surrounding the primary SMBH (hosting the jet) precesses due to
torque action owing to a mis-alignment between the orbital plane of the binary
system and the disk plane. In the second (geodetic precession, e.g. Begelman
et al., 1980; Rieger, 2004; Liu & Chen, 2007), the direction of spin of the
primary SMBH may be mis-aligned with the total angular momentum of the binary
system, thus setting up a precession of the spin axis and hence, the jet
hosted by the primary SMBH-accretion disk system. The Keplerian angular
frequency, and those in the cases of disk precession (Katz, 1997) and geodetic
precession (Rieger, 2004) are

$\displaystyle\Omega_{K}$
$\displaystyle=\left(\frac{GM_{\bullet}(1+q)}{d^{3}}\right)^{1/2}$ (1)
$\displaystyle\Omega_{D}$
$\displaystyle=\frac{3q\cos\theta_{0}}{4(1+q)^{1/2}}\left(\frac{r_{D}}{d}\right)^{3/2}\Omega_{K}$
(2) $\displaystyle\Omega_{G}$
$\displaystyle=qM_{\bullet}\left(\frac{4+3q}{1+q}\right)\frac{G\Omega_{K}}{2dc^{2}},$
(3)

where $G$ is the gravitational constant, $M_{\bullet}$ is the mass of the
primary SMBH, $q\leq 1$ is the mass ratio between the secondary companion and
the primary SMBH, $d$ is the binary separation, $\theta_{0}$ is the angle of
inclination between the accretion disk and the binary orbital plane, and
$r_{D}$ is the extent of the accretion disk. We use the choices $M\approx
10^{7}~{}M_{\odot}$ (Woo & Urry, 2002; Kharb et al., 2017), $d=0.35$ pc (Kharb
et al., 2017), $\theta_{0}=20^{\circ}$ (Katz, 1997), and a disk extent (King,
2016; Gaur et al., 2018)

$r_{D}=(5.45\times 10^{16}~{}{\rm
cm})~{}\dot{m}^{-8/27}(M/10^{7}M_{\odot})^{1/27},$ (4)

where $\dot{m}$ is the ratio of the mass accretion rate scaled in terms of the
Eddington rate and is set to 0.1, appropriate for accretion disks in Seyfert
galaxies (e.g. Mohan & Mangalam, 2014). With these assumptions, we obtain
minimum precession periods for the case $q=1$ (equal mass binary), with
$P_{D}=2\pi/\Omega_{D}\geq 2.6\times 10^{5}$ yr. and
$P_{G}=2\pi/\Omega_{G}\geq 1.8\times 10^{9}$ yr. These correspond to angular
precessions of $\sim$ 4.9 arcsec yr-1 and $\sim$ 0.2 mas yr-1, respectively. A
maximal value of $\sim$ 0.02∘ may thus be inferred for the case of disk
precession, considering a luminosity dimming timescale of $\sim$ 16 yr. For a
typical jetted AGN with an inclination angle of a few degrees (e.g. Lister et
al., 2019), this precession is two orders of magnitude smaller; changes due to
relativistic beaming effects may not be discernible within the observation
window. Thus, a binary SMBH enabled jet precession is unlikely to result in a
significant dimming of the luminosity and we place a low credence on this
possibility for our radio core non-detections.

Alternatively, it is possible mass loading mediated by jet-ISM feedback, or
some other jet disruption/diminishment mechanism (potentially due to variable
accretion) is responsible for the radio core non-detections we observed in the
2018 epoch and the short observed duty cycle777Momjian et al. (2003) estimate
the age of the AGN in NGC 7674 to be $\sim$ a few Myr, based upon the time
necessary to inflate its lobes into the ambient medium. See Jurlin et al.
(2020) for a longer discussion of radio-jet duty cycles in AGN. for this AGN
(Croston & Hardcastle, 2014; Shabala et al., 2020).

However, a phased VLA/VLBA setup also allows for a sufficient mixture between
small and large-distance baselines to determine the actual scale (i.e.,
between sub-pc to kpc scales) and morphology of this emission component in the
event that we can detect this feature. The future next-generation Very Large
Array (ngVLA, Murphy, 2018) would naturally allow for this mixture between
small and large-distance baseline lengths to allow for sensitivities to
features of varying size in this radio-loud AGN from sub-pc to kpc scales. Our
analysis of the VLA in-band spectral index of the purported core feature is
consistent with its flat or inverted spectrum, and thus its interpretation as
a synchrotron self-absorbed radio core. However, as suggested by Momjian et
al. (2003), the inverted radio spectrum of this jet region could be a
consequence of free-free absorption (FFA, see models by e.g. Bicknell et al.,
1997; Begelman, 1999). The FFA hypothesis for the inverted spectrum is
strengthened by the work of Gandhi et al. (2017), who find large hydrogen
column densities associated with the Compton-thick X-ray AGN in NGC 7674 from
recent NuStar observations, in addition to ionized Fe-line emission (together
these findings suggest a high density of hydrogen gas and a high intensity of
ionization continuum radiation in the vicinity of the central AGN).
Considering both hot spots and lobes detected from this young radio-loud AGN
(see Figure 1 from Momjian et al. 2003 for a clear detection of lobes on
either side of both hot spots and depiction of the overall S-shape), the naive
expectation would be a relatively misaligned radio jet. This would imply the
core emission could be relativistically beamed out of our line of sight,
leading to the expectation of very weak emission in the direction of the
observer.

If, as we suspect, the dual radio core detections reported by K17 correspond
to spurious noise features, then presumably there is still at least one radio
core of lower luminosity which has yet to be detected. In this vein, even-
deeper high-sensitivity VLBI observations could help detect this feature. One
property of this system which still suggests a binary SMBH may be present is
the overall S-shape of the radio jets. However, we note that the “curve”
portion of the S-shape in this jet is made by the lobe plasma, after it leaves
the working surfaces corresponding to the hot spots where the jet is assumed
to be pushing against the ambient medium. Since the lobe plasma does not have
the momentum or kinetic power carried by the twin radio jets, it is much more
likely the lobes could be pushed into the observed anti-aligned curves by the
circulating ISM gas pressure (see Gopal-Krishna et al. 2003 for a discussion
of this model in generating S-shaped jets).

We thank the anonymous referee for the many useful and constructive comments
which ultimately helped improve the quality of the manuscript. TA thanks the
grant support by the Youth Innovation Promotion Association of CAS. The EVN
data analysis has used the China SKA Regional Center prototype (An et al.,
2019) funded by the National Key R&D Programme of China (under grant number
2018YFA0404603) and Chinese Academy of Sciences (under grant number
114231KYSB20170003). YZ thanks Jun Yang for helping with VLBI data processing.
Below we show the (u,v) plane coverage for the high-frequency Ku (15 GHz) and
K-band (22 GHz) VLBI visibilities used in this study, along with plots
demonstrating the proper phase calibration of these visbilities and high-
frequency images of our phase calibrators. The purpose of these figures is to
demonstrate the improvements in (u,v) plane coverage upon data combination and
the fact that the non-detection of any VLBI radio cores in this work is not a
result of improper calibration.

Figure 4: (U,V) plane coverage for all of the individual epoch, high-frequency
Ku and K-band (15/22 GHz) VLBI vsibility data used in this study. The VLBI
observatory along with project code is given in the top left of each plot, and
the date of the observation is given in the top right as YYYY/MM/DD. The
frequency/band of the observation is also labeled in the bottom center of each
plot. Figure 5: Ku-band (15 GHz) phase of each observation’s phase calibrator
before (left plots) and after (right plots) phase calibration is applied for a
selection of representative VLBA baselines. The data are frequency averaged
and we use a phase-solution interval ranging from 15 s to the scan length for
calibration depending upon which resulted in the best phase calibration.
Observatory and project code are given above each plot, along with date of
observation and the designation “Uncalibrated” or “Calibrated” representing
phase data before or after calibration, respectively. Figure 6: A continuation
of Figure 5 for the K-band, 22 GHz VLBA data. Figure 7: Here we show the high-
frequency VLBA Ku and K-band (15 and 22 GHz) images of our phase calibrators
as contour plots. The contours start at a base level of 5$\sigma$, and are
spaced by factors of two thereafter. The synthesized beams are shown as filled
ellipses, and the image intensity noise level is specified by the RMS quoted
in each contour plot. The frequency and name of phase calibrator are given in
the top of the plots, and project codes are also given along with the date of
the observation in YYYY/MM/DD format. The images are made folllowing the same
CLEAN procedure defined in section 3.2. No relevant structure which would
impact the phase calibrtion is observed for any of our calibrators.

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