#  Realizing convex codes with axis-parallel boxes

Miguel Benitez Siran Chen††footnotemark:  Tianhui Han††footnotemark:
R. Amzi Jeffs††footnotemark:  Kinapal Paguyo11footnotemark: 1 Kevin A.
Zhou11footnotemark: 1 Department of Mathematical Sciences, Carnegie Mellon
University, Pittsburgh, PA 15213, USA.Supported by the National Science
Foundation through Award No. 2103206.

(September 2022)

###### Abstract

Every ordered collection of sets in Euclidean space can be associated to a
combinatorial code, which records the regions cut out by the sets in space.
Given two ordered collections of sets, one can form a third collection in
which the $i$-th set is the Cartesian product of the corresponding sets from
the original collections. We prove a general “product theorem” which
characterizes the code associated to the collection resulting from this
operation, in terms of the codes associated to the original collections. We
use this theorem to characterize the codes realizable by axis-parallel boxes,
and exhibit differences between this class of codes and those realizable by
convex open or closed sets. We also use our theorem to prove that a
“monotonicity of open convexity” result of Cruz, Giusti, Itskov, and Kronholm
holds for closed sets when some assumptions are slightly weakened.

##  1 Introduction

Given a collection $\mathcal{U}=\\{U_{1},\ldots,U_{n}\\}$ of sets in
$\mathbb{R}^{d}$, one can record how they intersect and cover each other using
a combinatorial code on the index set $[n]$, defined as follows:

$\displaystyle\operatorname{code}(\mathcal{U})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{\sigma\subseteq[n]\,|\,\text{there exists $p\in\mathbb{R}^{d}$
with $p\in U_{i}$ if and only if $i\in\sigma$}\\}.$

We say that $\mathcal{U}$ is a _realization_ of
$\operatorname{code}(\mathcal{U})$, and elements of a code are called
_codewords_. A codeword is _maximal_ if it is not properly contained in any
other codeword.

For notational convenience, we define the _intersection pattern_ of
$\mathcal{U}$ at a point $p$ to be the set
$\operatorname{patt}_{\mathcal{U}}(p)\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{i\in[n]\mid p\in U_{i}\\}$. For $\sigma\subseteq[n]$ the _atom_
of $\sigma$ in $\mathcal{U}$ is
$\mathcal{U}^{\sigma}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{p\in\mathbb{R}^{d}\mid\operatorname{patt}_{\mathcal{U}}(p)=\sigma\\}$.
With this notation, $\operatorname{code}(\mathcal{U})$ consists of the sets
$\sigma\subseteq[n]$ with nonempty atoms. Figure 1 shows a realization of a
code by four axis-parallel rectangles in $\mathbb{R}^{2}$. Throughout, we
write maximal codewords in bold.

Figure 1: Four axis-parallel rectangles in $\mathbb{R}^{2}$ which realize the
code $\\{\mathbf{123},\mathbf{24},12,13,23,1,2,3,4,\emptyset\\}$.

#### Convex codes.

In 2013, Curto, Itskov, Veliz-Cuba, and Youngs [5] initiated the study of
_open convex codes_ , which are codes with realizations consisting of convex
open sets. One may also consider _closed convex codes_ , which have
realizations by closed convex sets. Curto, Itskov, Veliz-Cuba, and Youngs were
motivated to study open convex codes by neuroscientific phenomena, and the
study of open and closed convex codes additionally generalizes the more
classical study of $d$-representable complexes (see Tancer’s survey [15]).
Since its inception the study of open and closed convex codes has thus led to
rich results of purely mathematical interest, such as new discrete geometry
theorems, families of extremal examples, and more.

######  Remark 1.

Consistent with the convex codes literature, we only consider bounded sets in
$\mathbb{R}^{d}$. This is equivalent to considering codes that contain
$\emptyset$ as a codeword (if any sets are unbounded, we may intersect them
with a large open or closed ball to obtain a bounded realization of the same
code).

Beyond determining whether a code has an open or closed convex realization, it
is natural to seek realizations that are minimal in dimension. To this end, it
is useful to define the _open_ (respectively, _closed_) _embedding dimension_
of a code. These are

$\displaystyle\operatorname{odim}(\mathcal{C})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\min\\{d\mid\text{$\mathcal{C}$ has a realization by open convex
sets in $\mathbb{R}^{d}$}\\},\text{ and}$
$\displaystyle\operatorname{cdim}(\mathcal{C})$
$\displaystyle\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\min\\{d\mid\text{$\mathcal{C}$ has a realization by closed convex
sets in $\mathbb{R}^{d}$}\\}.$

By convention, the minimum over the empty set is $\infty$.

A complete classification of open or closed convex codes and their embedding
dimensions is currently out of reach, and an efficient classification seems to
be out of the question: results of Kunin, Lienkaemper, and Rosen [12] posit
that recognizing open convex codes in $\mathbb{R}^{2}$ is at least NP-hard. A
further difficulty is the difference between open convex and closed convex
realizations— Lienkaemper, Shiu, and Woodstock [13] described a code which has
a closed convex realization, but no open convex realization, and Cruz, Giusti,
Itskov, and Kronholm [2] described a code with the opposite behavior. Families
of codes which exhibit arbitrary disparities between closed and open embedding
dimension were given more recently by Jeffs [10], but general criteria for
understanding and classifying these differences are lacking.

#### Box codes.

Given the difficulties of classifying open and closed convex codes, we tackle
the simpler problem of which codes can be realized by axis-parallel boxes. An
_axis-parallel box_ in $\mathbb{R}^{d}$ is either the empty set, or the
Cartesian product of nonempty intervals
$[a_{1},b_{1}]\times[a_{2},b_{2}]\times\cdots\times[a_{d},b_{d}]$. For
brevity, we use the term _box_ from here on. Note that we are considering
closed boxes, and we are allowing for boxes that are not full-dimensional—in
particular, a point is a box. We say that a code is a _box code_ or _box-
convex_ if it has a realization consisting of boxes in $\mathbb{R}^{d}$ for
some value of of $d$. Lastly, we define the _box embedding dimension_ of a
code $\mathcal{C}$ to be

$\operatorname{bdim}(\mathcal{C})\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\min\\{d\mid\text{$\mathcal{C}$ has a realization by boxes in
$\mathbb{R}^{d}$}\\}.$

#### Our results.

Our main result is Theorem 3, which applies to any collections of sets, but
proves particularly useful for analyzing box-convex codes and their embedding
dimensions. To state the result we first require a definition.

######  Definition 2.

The _intersection product_ of two codes $\mathcal{C}\subseteq 2^{[n]}$ and
$\mathcal{D}\subseteq 2^{[n]}$ is the code

$\mathcal{C}\hskip 1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\mathcal{D}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}\\{c_{1}\cap
c_{2}\mid c_{1}\in\mathcal{C}\text{ and }c_{2}\in\mathcal{D}\\}.$

######  Theorem 3 (Product theorem).

Let $\mathcal{U}=\\{U_{1},\ldots,U_{n}\\}$ and
$\mathcal{V}=\\{V_{1},\ldots,V_{n}\\}$ be collections of (not necessarily
open, closed, or convex) sets in $\mathbb{R}^{d_{1}}$ and $\mathbb{R}^{d_{2}}$
respectively. Define $\mathcal{W}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{W_{1},\ldots,W_{n}\\}$ where
$W_{i}\stackrel{{\scriptstyle\mbox{\tiny def}}}{{=}}U_{i}\times V_{i}$. Then

$\operatorname{code}(\mathcal{W})=\operatorname{code}(\mathcal{U})\hskip
1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\operatorname{code}(\mathcal{V}).$

Since boxes can be decomposed as Cartesian products of intervals, Theorem 3
will let us reduce the study of box-convex codes to the study of _interval
codes_ which have realizations by collections of closed intervals in the real
line.

Recall that products of open sets are open, and similarly for closed and
convex sets. Likewise, a product of boxes is again a box. Hence we have the
following immediate corollaries.

######  Corollary 4.

For any codes $\mathcal{C}\subseteq 2^{[n]}$ and $\mathcal{D}\subseteq
2^{[n]}$, we have $\operatorname{odim}(\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\mathcal{D})\leq\operatorname{odim}(\mathcal{C})+\operatorname{odim}(\mathcal{D})$.
The analogous inequalities hold for $\operatorname{cdim}(\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D})$ and
$\operatorname{bdim}(\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D})$.

######  Corollary 5.

Let $\mathcal{C}\subseteq 2^{[n]}$ be a code. Then $\mathcal{C}$ is box-convex
in $\mathbb{R}^{d}$ if and only if $\mathcal{C}$ is the intersection product
of $d$-many interval codes.

Corollary 5 yields a brute-force algorithm for recognizing whether or not a
code is box-convex in $\mathbb{R}^{d}$. First, we enumerate all interval codes
on $[n]$, which is possible because each such code is determined by the
relative order of the $2n$ endpoints of the intervals in question (we explain
this fact formally in Section 5). We then take all possible $d$-fold
intersection products of these interval codes, and check whether or not
$\mathcal{C}$ arises in this process.

Although the algorithm described above is far from efficient, this situation
already stands in contrast to open and closed convex codes—Bukh and Jeffs [1]
recently proved that open and closed convex codes realizable in
$\mathbb{R}^{2}$ can be recognized algorithmically, but no such decidability
result is known in dimension three or higher. It remains an open question
whether or not there is an _efficient_ algorithm to recognize box-convex codes
in $\mathbb{R}^{d}$.

The remainder of the paper is devoted to proving Theorem 3 in Section 2, and
then exploring a variety of applications. In Section 3 we classify all codes
$\mathcal{C}\subseteq 2^{\\{1,2,3\\}}$ which are convex, but not box-convex—it
turns out that there are only three such codes up to symmetry. We use one of
these codes to show that a “monotonicity of open convexity” result proved by
Cruz, Giusti, Itskov, and Kronholm [2] does not hold for box-convexity. In
Section 4 we describe a family of codes whose box embedding dimension ranges
over all integer values greater than or equal to two, while the open and
closed embedding dimensions remain fixed and equal to two. We prove in Section
5 that codes $\mathcal{C}\subseteq 2^{[n]}$ realizable by boxes in
$\mathbb{R}^{d}$ always have realizations in which the corners of the boxes
have integer coordinates between 1 and $2n$, and additionally the interiors of
the boxes realize the same code. In particular, there is no difference between
open and closed box codes. Finally, Section 6 applies Theorem 3 beyond box
codes by showing that a weak analog of the aforementioned “monotonicity of
open convexity” result holds for closed convex codes. The analog we prove is
necessarily weak, as Gambacini, Jeffs, Macdonald, and Shiu [6] proved that the
strong analog does not hold.

##  2 Proof and examples of the product theorem (Theorem 3)

We start by proving Theorem 3, and then given two examples. Let $\mathcal{U}$,
$\mathcal{V}$, and $\mathcal{W}$ be as in the statement of Theorem 3. Observe
that the nonempty atoms $\mathcal{U}^{\sigma}$ arising in the realization
$\mathcal{U}$ cover all of $\mathbb{R}^{d_{1}}$, and likewise the nonempty
atoms of the form $\mathcal{V}^{\tau}$ cover $\mathbb{R}^{d_{2}}$. Therefore,
all the Cartesian products $\mathcal{U}^{\sigma}\times\mathcal{V}^{\tau}$ as
$\sigma$ and $\tau$ range over $\operatorname{code}(\mathcal{U})$ and
$\operatorname{code}(\mathcal{V})$ respectively cover the entirety of
$\mathbb{R}^{d_{1}+d_{2}}$. Fix codewords
$\sigma\in\operatorname{code}(\mathcal{U})$ and
$\tau\in\operatorname{code}(\mathcal{V})$. Each point in
$\mathcal{U}^{\sigma}\times\mathcal{V}^{\tau}$ can be written as a pair
$(p,q)$ where $p\in\mathcal{U}^{\sigma}$ and $q\in\mathcal{V}^{\tau}$. We
claim that $\operatorname{patt}_{\mathcal{W}}(p,q)=\sigma\cap\tau$.

#### The inclusion
$\operatorname{patt}_{\mathcal{W}}(p,q)\subseteq\sigma\cap\tau$.

Let $k\in\operatorname{patt}_{\mathcal{W}}(p,q)$. Then $(p,q)\in
W_{k}=U_{k}\times V_{k}$, which implies that $p\in U_{k}$ and $q\in V_{k}$.
This in turn implies that $k\in\operatorname{patt}_{\mathcal{U}}(p)=\sigma$,
and $k\in\operatorname{patt}_{\mathcal{V}}(q)=\tau$. Thus
$k\in\sigma\cap\tau$.

#### The inclusion
$\operatorname{patt}_{\mathcal{W}}(p,q)\supseteq\sigma\cap\tau$.

Let $k\in\sigma\cap\tau$. Then $p\in U_{k}$ and $q\in V_{k}$. Thus $(p,q)\in
U_{k}\times V_{k}=W_{k}$, and so $k\in\operatorname{patt}_{\mathcal{W}}(p,q)$,
as desired.

#### Summary.

Since $\operatorname{patt}_{\mathcal{W}}(p,q)=\sigma\cap\tau$ for all
$(p,q)\in\mathcal{U}^{\sigma}\times\mathcal{V}^{\tau}$, we see that the
codewords of $\operatorname{code}(\mathcal{W})$ are exactly those of the form
$\sigma\cap\tau$, as $\sigma$ ranges over $\operatorname{code}(\mathcal{U})$
and $\tau$ ranges over $\operatorname{code}(\mathcal{V})$. This is equivalent
to the statement that
$\operatorname{code}(\mathcal{W})=\operatorname{code}(\mathcal{U})\hskip
1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\operatorname{code}(\mathcal{V})$, proving the result. ∎

Figure 2: (Left) A realization of a box code in the plane, and its
decomposition into interval codes in each coordinate direction. (Right) The
atoms in the interval codes, and their products in the plane.

######  Example 6.

The left side of Figure 2 shows five boxes
$\mathcal{U}=\\{U_{1},U_{2},U_{3},U_{4},U_{5}\\}$ in the plane, and their
corresponding decompositions into intervals
$\mathcal{I}=\\{I_{1},I_{2},I_{3},I_{4},I_{5}\\}$ and
$\mathcal{J}=\\{J_{1},J_{2},J_{3},J_{4},J_{5}\\}$ in the $x$-direction and the
$y$-direction respectively. We have
$\operatorname{code}(\mathcal{U})=\\{\mathbf{12},\mathbf{13},\mathbf{14},\mathbf{15},1,\emptyset\\}$,
which is the intersection product of
$\operatorname{code}(\mathcal{I})=\\{\mathbf{124},\mathbf{135},1,\emptyset\\}$,
and
$\operatorname{code}(\mathcal{J})=\\{\mathbf{123},\mathbf{145},1,\emptyset\\}$,
as guaranteed by Theorem 3. The right side of the figure shows the atoms of
nonempty codewords in the $x$-direction and $y$-direction, and the Cartesian
products of these atoms, which we used in the proof of Theorem 3. For example,
the top left square $\mathcal{I}^{124}\times\mathcal{J}^{123}$ is contained in
the atom $\mathcal{U}^{12}$ since $12=124\cap 123$.

######  Example 7.

Recall the box-convex code
$\mathcal{C}=\\{\mathbf{123},\mathbf{24},12,13,23,1,2,3,4,\emptyset\\}$ from
Figure 1. We can decompose each box in this realization as a product of two
intervals. Letting $\mathcal{I}=\\{I_{1},I_{2},I_{3},I_{4}\\}$ be the
intervals in the $x$-direction and $\mathcal{J}=\\{J_{1},J_{2},J_{3},J_{4}\\}$
be the intervals in the $y$-direction, we then have
$\mathcal{C}=\operatorname{code}(\mathcal{I})\hskip
1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\operatorname{code}(\mathcal{J})$ by Theorem 3. It turns out that
$\operatorname{code}(\mathcal{I})=\\{\mathbf{123},\mathbf{24},12,23,1,2,4,\emptyset\\}$
and
$\operatorname{code}(\mathcal{J})=\\{\mathbf{1234},134,12,1,2,\emptyset\\}$ in
this case, and Figure 3 shows this decomposition.

Figure 3: The boxes from Figure 1, decomposed into intervals in each
coordinate direction.

##  3 Convex codes that are not box-convex

Every box-convex code is also convex, but the converse fails even for codes on
three indices. Understanding the degree to which the converse fails helps us
understand the difference between convex codes and box codes, and so we use
Corollary 5 to characterize all codes on the index set $\\{1,2,3\\}$ that are
convex, but not box-convex. There are three such codes, up to permuting the
indices:

$\displaystyle\mathcal{C}_{1}$
$\displaystyle=\\{\mathbf{123},12,13,23,\emptyset\\},$
$\displaystyle\mathcal{C}_{2}$
$\displaystyle=\\{\mathbf{123},12,13,23,1,\emptyset\\},\text{ and}$
$\displaystyle\mathcal{C}_{3}$
$\displaystyle=\\{\mathbf{12},\mathbf{13},\mathbf{23},1,2,3,\emptyset\\}.$

The codes $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ are closed and open convex
in $\mathbb{R}^{2}$ since they have only one maximal codeword (see [4, Lemma
2.5]) and $\mathcal{C}_{3}$ is closed and open convex in $\mathbb{R}^{2}$
using three sets that pairwise intersect arranged in a triangular loop.
Arguing that all other convex codes on $\\{1,2,3\\}$ are box-convex requires
us to treat a large number of cases by hand, which we do in an appendix
following the bibliography. In this section, we will establish that the codes
$\mathcal{C}_{1},\mathcal{C}_{2}$, and $\mathcal{C}_{3}$ are not box-convex,
and also give an interesting example of such a code on four indices (Theorem
10).

The reason that $\mathcal{C}_{1}$ and $\mathcal{C}_{3}$ are not box-convex is
that they have the following properties:

  1. (i)

For $i=1,3$, the code $\mathcal{C}_{i}$ is not an interval code, and

  2. (ii)

For $i=1,3$, if $\mathcal{C}_{i}=\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D}$ where
$\mathcal{C}$ and $\mathcal{D}$ are convex codes, then
$\mathcal{C}=\mathcal{C}_{i}$ or $\mathcal{D}=\mathcal{C}_{i}$.

Indeed, any code with these two properties is not box-convex—the first
condition prevents it from being realized in $\mathbb{R}^{1}$, and the second
condition means that it cannot be decomposed as an intersection product of
interval codes since one of the factors would be the code itself. The code
$\mathcal{C}_{2}$ requires a very slightly different argument, which we
outline after our proof that $\mathcal{C}_{1}$ is not box-convex.

######  Theorem 8.

The code $\mathcal{C}_{1}=\\{\mathbf{123},12,13,23,\emptyset\\}$ is not box-
convex.

###### Proof.

It suffices to establish that $\mathcal{C}_{1}$ satisfies the conditions (i)
and (ii) outlined above.

$\mathcal{C}_{1}$ is not an interval code. Suppose for contradiction that
$\mathcal{C}_{1}$ has a realization $\mathcal{I}=\\{I_{1},I_{2},I_{3}\\}$ by
intervals in $\mathbb{R}^{1}$. Since $123$ is a maximal codeword,
$\mathcal{I}^{123}$ is an interval. Since $I_{1},I_{2},$ and $I_{3}$ are
connected and all meet $\mathcal{I}^{123}$, the atoms of the nonempty
codewords in $\mathcal{C}_{1}$ must have a connected union. In particular,
$\mathcal{I}^{123}$ is adjacent to the atom of a nonempty codeword, say on its
right. Without loss of generality, let it be the atom $\mathcal{I}^{12}$. We
have the following situation:

$\mathcal{I}^{123}$$\mathcal{I}^{12}$

Now, note that $I_{3}$ cannot include any points to the right of
$\mathcal{I}^{123}$, lest $I_{3}$ cover $\mathcal{I}^{12}$. Thus the atoms of
$13$ and $23$ must be to the left of $\mathcal{I}^{123}$.

Without loss of generality, let the atom directly to the left of
$\mathcal{I}^{123}$ be $\mathcal{I}^{13}$, as pictured below:

$\mathcal{I}^{123}$$\mathcal{I}^{12}$$\mathcal{I}^{13}$

Now, $I_{2}$ cannot include anything to the left of $\mathcal{I}^{123}$. But
this implies that $I_{2}\cap I_{3}=\mathcal{I}^{123}$, so the codeword $23$
never appears, a contradiction.

$\mathcal{C}_{1}$ satisfies condition (ii). Suppose that
$\mathcal{C}_{1}=\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D}$ where
$\mathcal{C}\subseteq 2^{\\{1,2,3\\}}$ and $\mathcal{D}\subseteq
2^{\\{1,2,3\\}}$ are convex codes. The maximal codeword
$123\in\mathcal{C}_{1}$ is the intersection of codewords from $\mathcal{C}$
and $\mathcal{D}$ respectively, and the only way this can happen is if
$123\in\mathcal{C}$ and $123\in\mathcal{D}$. Since every intersection of
codewords between $\mathcal{C}$ and $\mathcal{D}$ lies in $\mathcal{C}_{1}$,
we conclude that $\mathcal{C}\subseteq\mathcal{C}_{1}$ and
$\mathcal{D}\subseteq\mathcal{C}_{1}$. Moreover, each of the codewords
$\\{12,13,23\\}$ must appear in $\mathcal{C}$ or $\mathcal{D}$. By pigeonhole
principle, two of these codewords appear in one of $\mathcal{C}$ or
$\mathcal{D}$, and the third must appear in the same code lest we obtain a
singleton codeword in $\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D}$. Hence,
without loss of generality we have
$\mathcal{C}=\\{\mathbf{123},12,13,23,\emptyset\\}$, which proves condition
(ii). ∎

A very similar proof shows that $\mathcal{C}_{2}$ is not box-convex. One can
use the same proof to argue that $\mathcal{C}_{2}$ is not an interval code,
and if $\mathcal{C}_{2}=\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D}$, a similar
pigeonhole argument shows that one of $\mathcal{C}$ or $\mathcal{D}$ is equal
to $\mathcal{C}_{2}$ or $\mathcal{C}_{1}$—in either case, $\mathcal{C}_{2}$
cannot be decomposed as the intersection product of interval codes, so it is
not box-convex.

######  Theorem 9.

The code
$\mathcal{C}_{3}=\\{\mathbf{12},\mathbf{13},\mathbf{23},1,2,3,\emptyset\\}$ is
not box-convex.

###### Proof.

$\mathcal{C}_{3}$ is not an interval code. Suppose for contradiction that
$\mathcal{C}_{3}$ has a realization $\mathcal{I}=\\{I_{1},I_{2},I_{3}\\}$ by
intervals in $\mathbb{R}^{1}$. The atoms of the maximal codewords $12$, $13$,
and $23$ in this realization are each intervals. By permuting the indices, we
can assume that they appear in the order $12$, $13$, $23$. But then $I_{2}$
contains points to the left and to the right of $\mathcal{I}^{13}$, and so
$I_{2}$ covers $\mathcal{I}^{13}$. Since the codeword $13$ does not contain 2,
this is a contradiction.

$\mathcal{C}_{3}$ satisfies condition (ii). Suppose that
$\mathcal{C}_{3}=\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D}$ where
$\mathcal{C}\subseteq 2^{\\{1,2,3\\}}$ and $\mathcal{D}\subseteq
2^{\\{1,2,3\\}}$ are convex codes. Since $123$ is not a codeword of
$\mathcal{C}_{3}$, at least one of $\mathcal{C}$ or $\mathcal{D}$ does not
have $123$ as a codeword. Say without loss of generality that $\mathcal{C}$
does not contain $123$. Since $12$, $13$, and $23$ are codewords in
$\mathcal{C}_{3}$, and each is the intersection of some codeword in
$\mathcal{C}$ with a codeword in $\mathcal{D}$, each of $12$, $13$, and $23$
must appear as a codeword in $\mathcal{C}$. But since $\mathcal{C}$ is convex
and does not contain $123$, this means that $\mathcal{C}$ must contain the
singleton codeword $1$—indeed, any line segment from the atom of $12$ to the
atom of $13$ in a convex realization of $\mathcal{C}$ must pass through the
atom of $1$. Similar reasoning shows that the singletons $2$ and $3$ are
codewords of $\mathcal{C}$, so $\mathcal{C}=\mathcal{C}_{3}$ and condition
(ii) holds. ∎

We conclude with one last example of a convex code that is not box-convex,
this time on four indices. Its realizations consist of a central region with
three “flaps” attached, and it is not box-convex because one cannot add three
flaps to a box to obtain a larger box. Such “flaps” will also play a role in
Section 4.

######  Theorem 10.

The code $\mathcal{C}_{4}=\\{\mathbf{1234},12,13,14,\emptyset\\}$ is open and
closed convex in $\mathbb{R}^{2}$, but not box-convex in any dimension.

###### Proof.

The code $\mathcal{C}_{4}$ is open and closed convex in $\mathbb{R}^{2}$ since
it has a unique maximal codeword (recall [4, Lemma 2.5]). To see that
$\mathcal{C}_{4}$ is not box-convex, we will show that $\mathcal{C}_{4}$
satisfies the conditions (i) and (ii) defined at the beginning of the section.

$\mathcal{C}_{4}$ is not an interval code. Assume there exists a realization
$\mathcal{I}=\\{I_{1},I_{2},I_{3},I_{4}\\}$ of $\mathcal{C}_{4}$ in
$\mathbb{R}^{1}$. Note that the atom $\mathcal{I}^{1234}$ is an interval since
$1234$ is a maximal codeword. Since every nonempty codeword contains $1$, the
atoms of $1234$, $12$, $13$, and $14$ must together comprise a connected set.
In particular, one of the atoms of $12$, $13$, or $14$ is adjacent to
$\mathcal{I}^{1234}$. Without loss of generality, suppose that
$\mathcal{I}^{12}$ is adjacent to $\mathcal{I}^{1234}$ on the right, as below:

$\mathcal{I}^{1234}$$\mathcal{I}^{12}$

Now consider $\mathcal{I}^{13}$ and $\mathcal{I}^{14}$. These atoms cannot
contain any points to the right of $\mathcal{I}^{12}$ lest $I_{3}$ or $I_{4}$
cover $\mathcal{I}^{12}$. Thus one of $\mathcal{I}^{13}$ and
$\mathcal{I}^{14}$ is adjacent to $\mathcal{I}^{1234}$ on the left, say
$\mathcal{I}^{13}$ without loss of generality.

$\mathcal{I}^{1234}$$\mathcal{I}^{12}$$\mathcal{I}^{13}$

Note that the atom of $\mathcal{I}^{14}$ cannot be placed anywhere without
$I_{4}$ covering the atom of $13$ or the atom of $12$, both of which lead to
contradictions since $4$ is not contained in either codeword. We conclude that
$\mathcal{C}_{4}$ is not an interval code.

$\mathcal{C}_{4}$ satisfies condition (ii). Suppose that
$\mathcal{C}_{4}=\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{D}$ where
$\mathcal{C}\subseteq 2^{[4]}$ and $\mathcal{D}\subseteq 2^{[4]}$ are convex
codes. Since $1234$ is the intersection of codewords from $\mathcal{C}$ and
$\mathcal{D}$, we must have $1234\in\mathcal{C}$ and $1234\in\mathcal{D}$.
This implies that $\mathcal{C}\subseteq\mathcal{C}_{4}$ and
$\mathcal{D}\subseteq\mathcal{C}_{4}$.

Since every pairwise intersection of elements in $\\{12,13,14\\}$ would
produce the singleton codeword $1$, the only possibility is (up to swapping)
$\mathcal{C}=\\{\mathbf{1234},12,13,14,\emptyset\\}$ and
$\mathcal{D}=\\{\mathbf{1234},\emptyset\\}$. Thus $\mathcal{C}_{4}$ satisfies
property (ii). ∎

###  3.1 Non-monotonicity of box-convexity

Cruz, Giusti, Itskov and Kronholm [2] showed that open convexity is a monotone
property of codes, in the following sense:

######  Theorem 11 (Monotonicity of open convexity, [2]).

Let $\mathcal{C}\subseteq 2^{[n]}$ be an open convex code, and let
$\mathcal{D}\subseteq 2^{[n]}$ be a code with
$\mathcal{C}\subseteq\mathcal{D}\subseteq\Delta(\mathcal{C})$. Then
$\mathcal{D}$ is open convex and
$\operatorname{odim}(\mathcal{D})\leq\operatorname{odim}(\mathcal{C})+1$.

This result fails for closed convex codes, as shown by Gambacini, Jeffs,
Macdonald, and Shiu [6]. It is thus natural to ask whether an analogous result
holds for box-convex codes. It turns out that it does not. The method of proof
used by Cruz, Giusti, Itskov, and Kronholm relies heavily upon the curvature
of an open ball cutting through cylinders over a realization of $\mathcal{C}$.
Such a technique cannot be applied for boxes since they exhibit no curvature
on the boundary.

The codes we constructed earlier in this section provide an explicit
counterexample showing that box-convexity is not monotone. Consider the code
$\mathcal{C}=\\{\mathbf{1234},12,13,\emptyset\\}$. This is certainly box-
convex, and in fact this code can be realized by intervals in
$\mathbb{R}^{1}$. However, the code $\mathcal{C}_{4}$ of Theorem 10 has the
property that
$\mathcal{C}\subseteq\mathcal{C}_{4}\subseteq\Delta(\mathcal{C})$, but
$\mathcal{C}_{4}$ is not box-convex.

##  4 Open embedding dimension versus box embedding dimension

In this section, we explore differences in embedding dimensions between
ordinary convex codes and box-convex codes. The open embedding dimension and
box embedding dimension of a code may be vastly different. Identifying
specific characteristics of these differences allows us to better understand
restrictions on box-convexity.

Our examples in this section are based on codes whose realizations are
sunflowers. We say that $\mathcal{U}=\\{U_{1},\ldots,U_{n}\\}$ is a
_sunflower_ if
$\operatorname{code}(\mathcal{U})=\\{[n],1,2,\ldots,n,\emptyset\\}$. The sets
$U_{i}$ are called _petals_ and $\bigcap_{i\in[n]}U_{i}$ is called the
_center_ of the sunflower. Sunflowers have proved to be highly useful in the
study of open embedding dimensions of convex codes [8, 9], and they will
additionally help us prove that box embedding dimension can be arbitrarily
larger than the closed or open embedding dimensions of a code. Specifically,
we will prove the following theorem.

######  Theorem 12.

For even $n\geq 4$, the code
$\mathcal{F}_{n}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{[n],1,2,\ldots,n,\emptyset\\}$ has
$\operatorname{bdim}(\mathcal{F}_{n})=n/2$, while on the other hand
$\operatorname{odim}(\mathcal{F}_{n})=\operatorname{cdim}(\mathcal{F}_{n})=2$.

###### Proof.

Note that $\operatorname{odim}(\mathcal{F}_{n})$ and
$\operatorname{cdim}(\mathcal{F}_{n})$ are both at most two, since
$\mathcal{F}_{n}$ has a unique maximal codeword. To prove that
$\operatorname{bdim}(\mathcal{F}_{n})=n/2$, we work by induction, starting
from the slightly smaller base case $n=2$.

Let $n=2k$, where $k\geq 1$. We will prove that
$\operatorname{bdim}(\mathcal{F}_{n})=n/2$ by induction on $k$. In the base
case $k=1$, observe that $\mathcal{F}_{2}=\\{\mathbf{12},1,2,\emptyset\\}$,
which can be realized by intervals in $\mathbb{R}^{1}$. For the inductive
step, it suffices to prove that $\mathcal{F}_{n}$ is the intersection product
of $k$-many interval codes, and no fewer. We have
$\mathcal{F}_{n}=\mathcal{C}_{1}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\cdots\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{C}_{k}$, where
$\mathcal{C}_{i}=\\{[n],2i-1,2i,\emptyset\\}$, which establishes that
$\operatorname{bdim}(\mathcal{F}_{n})\leq k$.

To establish the matching lower bound, assume for contradiction that
$\mathcal{F}_{n}=\mathcal{C}_{1}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\cdots\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{C}_{\ell}$
where $\ell<k$ and each $\mathcal{C}_{i}$ is an interval code. Since $[n]$ is
a codeword of $\mathcal{F}_{n}$, it must be a codeword of each
$\mathcal{C}_{i}$. Consequently, we have
$\mathcal{C}_{i}\subseteq\mathcal{F}_{n}$ for each $1\leq i\leq\ell$, and
moreover each singleton codeword must appear in some $\mathcal{C}_{i}$.
However, each $\mathcal{C}_{i}$ can contain at most two singleton codewords,
since it is an interval code consisting of the codeword $[n]$ together with
singletons and the empty set. There are $2k>2\ell$ total singletons that must
appear among the $\mathcal{C}_{i}$, a contradiction by pigeonhole principle.
We conclude that $\mathcal{F}_{n}$ is the intersection product of $k$-many
interval codes and no fewer, proving the result. ∎

##  5 Integer coordinates and open boxes

In this section we discuss a strong application of the product theorem to box
codes. As mentioned in Section 1, the product theorem applies naturally to
boxes since each box in $\mathbb{R}^{d}$ is the Cartesian product of $d$-many
intervals. In particular, Theorem 3 allows us to start with a collection of
$n$ boxes in $\mathbb{R}^{d}$, decompose it into $d$-many collections of $n$
intervals in $\mathbb{R}^{1}$, perform some transformations on each collection
of intervals in $\mathbb{R}^{1}$ that do not change their codes, and then
return to a collection of boxes in $\mathbb{R}^{d}$ by taking products of our
intervals. In this way, we can obtain nicer realizations in $\mathbb{R}^{d}$
by performing simple transformations in $\mathbb{R}^{1}$.

Our main result in this section is Theorem 13, which guarantees realizations
that are “nicer” in the sense that all the boxes have corners with bounded
integer coordinates, and interiors that realize the same code. Moreover, this
theorem guarantees that the family of codes realizable by open boxes (which
are products of nonempty open intervals) is identical to the family of box-
convex codes, i.e. there is no difference between codes realizable with open
boxes and those realizable with closed boxes. As mentioned in Section 1, this
contrasts the situation for convex codes, where open convex codes differ
greatly from closed convex codes.

######  Theorem 13.

Given a realization $\mathcal{U}=\\{U_{1},\ldots,U_{n}\\}$ of a code
$\mathcal{C}\subseteq 2^{[n]}$ by boxes in $\mathbb{R}^{d}$, there exists a
realization $\mathcal{V}=\\{V_{1},\ldots,V_{n}\\}$ of $\mathcal{C}$ by boxes,
with the following properties:

  * (i)

each $V_{i}$ has corners with integer coordinates between 1 and $2n$; that is,
each $V_{i}$ is the Cartesian product of closed intervals with integer
endpoints between $1$ and $2n$, and

  * (ii)

the interiors of the $V_{i}$ also comprise a realization of $\mathcal{C}$.

Moreover, if $\mathcal{C}$ has a realization by interiors of boxes in
$\mathbb{R}^{d}$, then it has a realization by boxes in $\mathbb{R}^{d}$.

###### Proof.

First, we work with just one coordinate. We start with a collection
$\mathcal{I}=\\{I_{1},\ldots,I_{n}\\}$ of closed intervals in
$\mathbb{R}^{1}$. We may express each interval according to its endpoints, as
$I_{1}=[a_{1},a_{2}]$, $I_{2}=[a_{3},a_{4}]$, and so on. Observe that the
relative order of these endpoints completely determines
$\operatorname{code}(\mathcal{I})$. Specifically, for every $p\in\mathbb{R}$
we have $i\notin\operatorname{patt}_{\mathcal{I}}(p)$ if and only if
$a_{2i}<p$ or $p<a_{2i-1}$. The relative order of the endpoints also
determines the code of the interiors of the intervals in $\mathcal{I}$,
according to the fact that a point $p$ does not lie in the interior of $I_{i}$
if and only if $a_{2i}\leq p$ or $p\leq a_{2i-1}$.

To prove item (i) in the theorem, consider the function
$f:\mathbb{R}\to\mathbb{N}$ which sends $a_{i}$ to $k$, where $a_{i}$ is the
$k$-th smallest number out of the set of endpoints
$\\{a_{1},a_{2},\ldots,a_{2n}\\}$. Note that if two endpoints $a_{i}$ and
$a_{j}$ are equal, then $f(a_{i})=f(a_{j})$. So, $f$ sorts the endpoints of
the intervals from least to greatest, and equal endpoints are sent to the same
number. Now, consider the collection of intervals
$\mathcal{J}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\\{J_{1},\ldots,J_{n}\\}$ where

$J_{i}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}[f(a_{2i-1}),f(a_{2i})]\text{\, for $i\in[n]$}.$

The collection $\mathcal{J}$ realizes the same code as $\mathcal{I}$ since the
function $f$ preserves the relative order of the endpoints. Moreover,
$\mathcal{J}$ consists of intervals with integer endpoints between $1$ and
$2n$, since $f(a_{i})$ is always an integer between $1$ and $2n$. This proves
condition (i) for $d=1$.

To prove (ii), consider the collection
$\mathcal{J}^{\prime}=\\{J_{1}^{\prime},\ldots,J_{n}^{\prime}\\}$ where
$J_{i}^{\prime}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}[f(a_{2i-1})-1/4,f(a_{2i})+1/4]$. That is, we extend each interval
in $\mathcal{J}$ by $1/4$ to the left and to the right. This operation
preserves the relative order of the endpoints of the intervals, except when a
left endpoint coincides with a right endpoint, in which case these endpoints
become separated with the new left endpoint strictly smaller than the new
right endpoint. The latter change merely widens the atom of the codeword that
appeared where this left and right endpoint coincided, and so does not remove
or introduce any codewords in the realization. In particular,
$\operatorname{code}(\mathcal{J}^{\prime})=\operatorname{code}(\mathcal{J})$.
See Figure 4 for an example of this.

unitsize(1cm); draw((-0.5,0) – (7.5,0), arrow=Arrow(TeXHead)); draw ((0,0) –
(-0.5,0), arrow=Arrow(TeXHead));

// tick marks for(int i = 0; i ¡ 8; ++i) draw((i,0) – (i,-0.15));
label((string) i, (i,-0.15), align=S);

pair A1,A2,B1,B2,C1,C2,D1,D2; A1 = (1,0.5); A2 = (2,0.5); B1 = (2,0.8); B2 =
(4,0.8); C1 = (3,1.1); C2 = (5,1.1); D1 = (4,1.4); D2 = (6,1.4);

draw(A1–A2); draw(B1–B2); draw(C1–C2); draw(D1–D2);

filldraw(circle(A1,0.06),black,black); filldraw(circle(A2,0.06),black,black);
filldraw(circle(B1,0.06),black,black); filldraw(circle(B2,0.06),black,black);
filldraw(circle(C1,0.06),black,black); filldraw(circle(C2,0.06),black,black);
filldraw(circle(D1,0.06),black,black); filldraw(circle(D2,0.06),black,black);

label(”$\mathcal{J}$”, (-1,1));

unitsize(1cm); draw((-0.5,0) – (7.5,0), arrow=Arrow(TeXHead)); draw ((0,0) –
(-0.5,0), arrow=Arrow(TeXHead));

// tick marks for(int i = 0; i ¡ 8; ++i) draw((i,0) – (i,-0.15));
label((string) i, (i,-0.15), align=S);

pair[] endpts = (0.75,0.5), (2.25,0.5), (1.75,0.8), (4.25,0.8), (2.75,1.1),
(5.25,1.1), (3.75,1.4), (6.25,1.4);

//Draw the lines for(int i = 0; i ¡ 4; ++i) draw(endpts[2*i] – endpts[2*i+1]);

//Draw closed circles for(int i = 0; i ¡ 8; ++i)
filldraw(circle(endpts[i],0.06),black,black);

label(”$\mathcal{J}^{\prime}$”, (-1,1));

Figure 4: Extending the collection of intervals $\mathcal{J}$ to a collection
$\mathcal{J}^{\prime}$ whose closures and interiors both realize the code of
$\mathcal{J}$.

We claim moreover that the interiors of the intervals in
$\mathcal{J}^{\prime}$ realize the same code as $\mathcal{J}^{\prime}$. Let
$\mathcal{J}^{\prime\prime}$ denote the collection of interiors of intervals
in $\mathcal{J}^{\prime}$. To see that
$\operatorname{code}(\mathcal{J}^{\prime})=\operatorname{code}(\mathcal{J}^{\prime\prime})$,
first note that
$\operatorname{patt}_{\mathcal{J}^{\prime}}(p)\neq\operatorname{patt}_{\mathcal{J}^{\prime\prime}}(p)$
only when $p$ is a left or right endpoint of an interval in
$\mathcal{J}^{\prime}$. Moreover, there are no points in $\mathbb{R}^{1}$ that
are both a left and right endpoint of some interval in $\mathcal{J}^{\prime}$.
This implies that every intersection pattern that occurs in
$\mathcal{J}^{\prime}$ occurs at a point that is not an endpoint of any
interval—if a pattern occurs at a left endpoint, it also occurs slightly to
the right of that endpoint, and symmetrically for right endpoints. Hence
$\operatorname{code}(\mathcal{J}^{\prime})\subseteq\operatorname{code}(\mathcal{J}^{\prime\prime})$.
A symmetric argument shows that the intersection patterns that occur in
$\mathcal{J}^{\prime\prime}$ also occur away from endpoints of the intervals,
implying that
$\operatorname{code}(\mathcal{J}^{\prime\prime})\subseteq\operatorname{code}(\mathcal{J}^{\prime})$.
Thus
$\operatorname{code}(\mathcal{J}^{\prime})=\operatorname{code}(\mathcal{J}^{\prime\prime})$
as desired, proving condition (ii).

To obtain conditions (i) and (ii) simultaneously, we may simply apply the
sorting argument used earlier to $\mathcal{J}^{\prime}$, obtaining closed
intervals with integer endpoints between $1$ and $2n$ and the same relative
order of endpoints as in $\mathcal{J}^{\prime}$. Since the relative order of
endpoints also determines the code of the interiors, this collection will have
interiors that realize $\mathcal{C}$ as well.

A symmetric argument shows that if $\mathcal{C}$ has a realization by open
intervals then it has a realization by closed intervals. Again, we can
transform our starting realization so that the open intervals have integer
endpoints. We then shrink each interval by $1/4$ from the left and right,
separating left and right endpoints, and noting that this does not affect the
code of the intervals. With left and right endpoints separated, the closures
of these intervals will realize the same code as their interiors. This proves
the entirety of the theorem for $d=1$.

For $d>1$, note that a collection of $n$-many boxes $\mathcal{U}$ in
$\mathbb{R}^{d}$ can be decomposed into collections
$\mathcal{I}_{1},\ldots,\mathcal{I}_{d}$ of intervals, where $\mathcal{I}_{j}$
consists of the $n$-many intervals obtained by projecting the boxes in
$\mathcal{U}$ onto the $j$-th coordinate. Theorem 3 implies that
$\operatorname{code}(\mathcal{U})=\operatorname{code}(\mathcal{I}_{1})\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\cdots\hskip
1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\operatorname{code}(\mathcal{I}_{d})$. Since we have established
Theorem 13 for $d=1$, we can replace each $\mathcal{I}_{j}$ by a realization
$\mathcal{J}_{j}$ consisting of intervals with integer endpoints between $1$
and $2n$ whose interiors realize the same code as $\mathcal{I}_{j}$. Then
define $\mathcal{V}=\\{V_{1},\ldots,V_{n}\\}$ where $V_{i}$ is the Cartesian
product of the $i$-th interval from each $\mathcal{J}_{j}$. Again applying
Theorem 3 we have
$\operatorname{code}(\mathcal{V})=\operatorname{code}(\mathcal{J}_{1})\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\cdots\hskip
1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\operatorname{code}(\mathcal{J}_{d})$, and since
$\operatorname{code}(\mathcal{J}_{j})=\operatorname{code}(\mathcal{I}_{j})$
for all $j\in[d]$ this proves (i). The fact that
$\operatorname{code}(\mathcal{J}_{j})$ is the same as the code of the
interiors of the $\mathcal{J}_{j}$ proves (ii). Finally, if we start with a
collection of open boxes in $\mathbb{R}^{d}$, we may likewise decompose them
into collections of open intervals, then apply Theorem 13 for $d=1$ to obtain
realizations of the same codes by closed intervals, and finally apply Theorem
3 to combine these collections of closed intervals to obtain a realization of
the original code by closed boxes in $\mathbb{R}^{d}$. ∎

##  6 Weak monotonicity of closed convexity

Our goal in this section is to prove Theorem 14, which provides a weak analog
of monotonicity of open convexity (Theorem 11) for closed convex codes. The
result is “weak” in two senses. First, the new non-maximal codewords that we
add to our code must comprise a downward-closed set, instead of being
arbitrary. This is necessary, as Gambacini, Jeffs, Macdonald, and Shiu [6]
gave an example of a closed convex codes with the property that adding a
certain non-maximal codeword yields a code that is not closed convex in any
dimension. Second, the closed embedding dimension may increase by two in our
theorem, whereas the monotonicity of open convexity theorem established by
Cruz, Giusti, Itskov, and Kronholm [2] requires a dimension increase of at
most one. It would be interesting to find examples of codes where our “$+2$”
term is necessary, if they exist.

Below, a _downward-closed set_ is a set of elements in a poset which contains
every element of the poset that is less than some element of the set.

######  Theorem 14.

Let $\mathcal{C}\subseteq 2^{[n]}$ be a closed convex code. Let $\mathcal{D}$
be a downward-closed subset of $\Delta(\mathcal{C})\setminus\mathcal{C}$. Then
$\mathcal{C}\cup\mathcal{D}$ is closed convex, and moreover

$\operatorname{cdim}(\mathcal{C}\cup\mathcal{D})\leq\operatorname{cdim}(\mathcal{C})+2.$

###### Proof.

By Corollary 4, it suffices to show that $\mathcal{C}\cup\mathcal{D}$ is equal
to $\mathcal{C}\hskip 1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\mathcal{E}$, where $\mathcal{E}$ is a code with
$\operatorname{cdim}(\mathcal{E})\leq 2$. In fact, the choice
$\mathcal{E}\stackrel{{\scriptstyle\mbox{\tiny
def}}}{{=}}\mathcal{D}\cup\\{[n],\emptyset\\}$ will suffice. Certainly we have
$\operatorname{cdim}(\mathcal{E})\leq 2$ since $\mathcal{E}$ has a unique
maximal codeword.

To prove that $\mathcal{C}\cup\mathcal{D}=\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{E}$, we show
containment in each direction. We have $\mathcal{C}\subseteq\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{E}$, since
$[n]$ is a codeword of $\mathcal{E}$. Moreover, if $\tau\in\mathcal{D}$ then
there exists a maximal codeword $\sigma$ of $\mathcal{C}$ with
$\tau\subseteq\sigma$. We then have $\tau=\sigma\cap\tau$, which proves that
$\tau\in\mathcal{C}\hskip 1.49994pt\includegraphics{intprod.eps}\hskip
1.49994pt\mathcal{E}$.

For the reverse containment, let $\sigma\in\mathcal{C}$ and
$\tau\in\mathcal{E}$. If $\tau=[n]$, then $\sigma\cap\tau=\sigma$, which lies
in $\mathcal{C}$ and hence in $\mathcal{C}\cup\mathcal{D}$. If
$\tau=\emptyset$, then $\sigma\cap\tau=\emptyset$, which again lies in
$\mathcal{C}$. Otherwise, $\tau\in\mathcal{D}$. Note that
$\mathcal{C}\cup\mathcal{D}$ contains $\Delta(\mathcal{D})$ since
$\mathcal{D}$ is a downward-closed subset of
$\Delta(\mathcal{C})\setminus\mathcal{C}$. Thus $\mathcal{C}\cup\mathcal{D}$
contains $\sigma\cap\tau$, since $\sigma\cap\tau$ is a subset of $\tau$. We
conclude that $\mathcal{C}\cup\mathcal{D}=\mathcal{C}\hskip
1.49994pt\includegraphics{intprod.eps}\hskip 1.49994pt\mathcal{E}$, which
proves the result. ∎

##  7 Conclusion and future questions

We have developed a useful “product theorem” (Theorem 3) for realizations of
codes, and used it to explain several features of box-convex codes, including
points of difference compared to ordinary convex codes. However, a number of
fundamental questions remain open, and we describe several below that may be
of interest for future research.

  1. 1.

For each fixed $d\geq 2$, is there a polynomial-time algorithm to recognize
whether or not a given code $\mathcal{C}\subseteq 2^{[n]}$ has a realization
by boxes in $\mathbb{R}^{d}$? Every interval code $\mathcal{C}\subseteq
2^{[n]}$ has at most $2n+1$ codewords (see the proof of [14, Proposition
1.5]), which together with Corollary 5 implies that every box-convex code in
$\mathbb{R}^{d}$ has at most $(2n+1)^{d}=O(n^{d})$ codewords. Thus it would
make sense to measure the complexity of such an algorithm as a function of
$n$. Note that [14] has proved that there is an efficient algorithm for
recognizing interval codes.

  2. 2.

Among all box-convex codes $\mathcal{C}\subseteq 2^{[n]}$, which ones have
$\operatorname{bdim}(\mathcal{C})$ as large as possible? What is this largest
value in terms of $n$?

  3. 3.

Can we classify “local obstructions” to box-convexity, specializing the notion
of local obstructions for convex codes (see [4])? This would mean classifying
which simplicial complexes are the nerve of a collection of boxes whose union
is a box, analogous to the “convex union representable” complexes of Jeffs and
Novik [11].

  4. 4.

Can we find further applications of Theorem 3 to general open and closed
convex codes?

  5. 5.

Curry, Jeffs, Youngs, and Zhao [3] recently showed that every “inductively
pierced” code has a realization consisting of open or closed balls. Does every
inductively pierced code also admit a realization by boxes?

  6. 6.

Work of Goldrup and Phillipson [7] classifies all open and closed convex codes
on up to five indices. Our work has only enumerated box-convex codes on up to
three indices. This motivates us to ask: which convex codes on up to four or
five indices are not box-convex?

## References

  * [1] Boris Bukh and R. Amzi Jeffs.  Planar convex codes are decidable.  arXiv e-prints:2207.06290, 2022.
  * [2] Joshua Cruz, Chad Giusti, Vladimir Itskov, and Bill Kronholm.  On open and closed convex codes.  Discrete & Computational Geometry, 61:247–270, 2016.
  * [3] Ryan Curry, R. Amzi Jeffs, Nora Youngs, and Ziyu Zhao.  Recognizing and realizing inductively pierced codes.  arXiv e-prints:2207.06266, 2022.
  * [4] Carina Curto, Elizabeth Gross, Jack Jeffries, Katherine Morrison, Mohamed Omar, Zvi Rosen, Anne Shiu, and Nora Youngs.  What makes a neural code convex?  SIAM Journal on Applied Algebra and Geometry, 1(1):222–238, 2017\.
  * [5] Carina Curto, Vladimir Itskov, Alan Veliz-Cuba, and Nora Youngs.  The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes.  Bulletin of Mathematical Biology, 75(9):1571–1611, 2013.
  * [6] Brianna Gambacini, R. Amzi Jeffs, Sam Macdonald, and Anne Shiu.  Non-monotonicity of closed convexity in neural codes.  Vietnam Journal of Mathematics, 50:359–373, 2022.
  * [7] Sarah Ayman Goldrup and Kaitlyn Phillipson.  Classification of open and closed convex codes on five neurons.  Advances in Applied Mathematics, 112:101948, 2020.
  * [8] R. Amzi Jeffs.  Sunflowers of convex open sets.  Advances in Applied Mathematics, 111:101935, 2019.
  * [9] R. Amzi Jeffs.  Embedding dimension phenomena in intersection complete codes.  Selecta Mathematica, 28, 2021.
  * [10] R. Amzi Jeffs.  Open, closed, and non-degenerate embedding dimensions of neural codes.  arXiv e-prints: 2111.01010, 2021.
  * [11] R. Amzi Jeffs and Isabella Novik.  Convex union representability and convex codes.  International Mathematics Research Notices, pages 7132–7158, 2019\.
  * [12] Alexander Kunin, Caitlin Lienkaemper, and Zvi Rosen.  Oriented matroids and combinatorial neural codes.  arXiv e-prints: 2002.03542, 2020.
  * [13] Caitlin Lienkaemper, Anne Shiu, and Zev Woodstock.  Obstructions to convexity in neural codes.  Advances in Applied Mathematics, 85:31–59, 2017.
  * [14] Zvi Rosen and Yan X. Zhang.  Convex neural codes in dimension 1.  arXiv e-prints: 1702.06907, 2017.
  * [15] Martin Tancer.  Intersection patterns of convex sets via simplicial complexes: a survey.  In Thirty essays on geometric graph theory, pages 521–540. Springer, New York, 2013.

## Appendix

This appendix supplements Section 3, in which we showed that the codes
$\mathcal{C}_{1}$, $\mathcal{C}_{2}$, and $\mathcal{C}_{3}$ are open and
closed convex but not box-convex. Here we will argue that, up to permutation
of indices, every other open or closed convex code $\mathcal{C}\subseteq
2^{\\{1,2,3\\}}$ is box-convex. We must first enumerate the open and closed
convex codes $\mathcal{C}\subseteq 2^{\\{1,2,3\\}}$ up to symmetry, which we
do in Figure 5, partitioning them according to their maximal codewords. The
fact that this is the full list of open and closed convex codes on three
indices follows from the fact that such a code must be max-intersection
complete (i.e. contain all possible intersections of maximal codewords) and
every max-intersection complete code $\mathcal{C}\subseteq 2^{\\{1,2,3\\}}$
has an open and closed convex realization (see [4, Supplementary Text S1]). In
fact, we will do slightly more than show that these codes are box-convex; we
will determine their exact box embedding dimensions.

$\\{\mathbf{1},\emptyset\\}$ | $\\{\mathbf{1},\mathbf{2},\emptyset\\}$ |  $\\{\mathbf{1},\mathbf{2},\mathbf{3},\emptyset\\}$
---|---|---
$\\{\mathbf{12},\emptyset\\}$ |  $\\{\mathbf{12},1,\emptyset\\}$ |  $\\{\mathbf{12},1,2,\emptyset\\}$
$\\{\mathbf{12},\mathbf{3},\emptyset\\}$ | $\\{\mathbf{12},\mathbf{13},1,\emptyset\\}$ |  $\star\,\\{\mathbf{12},\mathbf{13},\mathbf{23},1,2,3,\emptyset\\}$
$\\{\mathbf{12},\mathbf{3},1,\emptyset\\}$ | $\\{\mathbf{12},\mathbf{13},1,2,\emptyset\\}$ |
$\\{\mathbf{12},\mathbf{3},1,2,\emptyset\\}$ | $\\{\mathbf{12},\mathbf{13},1,2,3,\emptyset\\}$ |
$\\{\mathbf{123},\emptyset\\}$ |  $\\{\mathbf{123},1,\emptyset\\}$ |  $\\{\mathbf{123},1,2,\emptyset\\}$
$\dagger\,\\{\mathbf{123},1,2,3,\emptyset\\}$ |  $\\{\mathbf{123},12,\emptyset\\}$ |  $\\{\mathbf{123},12,1,\emptyset\\}$
$\\{\mathbf{123},12,3,\emptyset\\}$ |  $\\{\mathbf{123},12,1,2,\emptyset\\}$ |  $\\{\mathbf{123},12,1,3,\emptyset\\}$
$\dagger\,\\{\mathbf{123},12,1,2,3,\emptyset\\}$ |  $\\{\mathbf{123},12,13,\emptyset\\}$ |  $\\{\mathbf{123},12,13,1,\emptyset\\}$
$\\{\mathbf{123},12,13,2,\emptyset\\}$ |  $\\{\mathbf{123},12,13,1,2,\emptyset\\}$ |  $\\{\mathbf{123},12,13,2,3,\emptyset\\}$
$\dagger\,\\{\mathbf{123},12,13,1,2,3,\emptyset\\}$ |  $\star\,\\{\mathbf{123},12,13,23,\emptyset\\}$ |  $\star\,\\{\mathbf{123},12,13,23,1,\emptyset\\}$
$\dagger\,\\{\mathbf{123},12,13,23,1,2,\emptyset\\}$ |  $\dagger\,\\{\mathbf{123},12,13,23,1,2,3,\emptyset\\}$ |
Figure 5: All open and closed convex codes on the index set $\\{1,2,3\\}$, up
to symmetry. Those marked with $\star$ were shown not to be box-convex in
Section 3. Those marked with $\dagger$ have box embedding dimension equal to
two. The remainder can be realized by intervals.

We leave it as an exercise to verify that the unmarked codes in Figure 5 can
be realized by closed intervals, i.e. have box embedding dimension equal to
one. To prove that the $\dagger$-marked codes have box embedding dimension
two, we require the following theorem.

######  Theorem 15.

Any code $\mathcal{C}\subseteq 2^{\\{1,2,3\\}}$ that contains the codewords
$123,1,2,$ and $3$ has $\operatorname{bdim}(\mathcal{C})\geq 2$, i.e. it is
not realizable with closed intervals in $\mathbb{R}^{1}$.

###### Proof.

Suppose for contradiction that $\mathcal{I}=\\{I_{1},I_{2},I_{3}\\}$ is a
realization of $\mathcal{C}$ by intervals. Note that $\mathcal{I}^{123}$ must
itself be a nonempty interval. Each interval contains points outside of
$\mathcal{I}^{123}$ since we have all possible singleton codewords, and so two
of them contain points on the same side of $\mathcal{I}^{123}$. Without loss
of generality, suppose that $\mathcal{I}^{1}$ and $\mathcal{I}^{2}$ have
points to the left of $\mathcal{I}^{123}$. Again without loss of generality,
we may assume that $\mathcal{I}^{2}$ contains a point strictly to the left of
$\mathcal{I}^{1}$. However, $I_{2}$ is an interval, so all points between
$\mathcal{I}^{2}$ and $\mathcal{I}^{123}$ must be in $I_{2}$. This would imply
that $\mathcal{I}^{1}\subseteq I_{2}$, a contradiction. ∎

This shows that all $\dagger$-marked codes from Figure 5 have box embedding
dimension at least two, except the code
$\\{\mathbf{123},12,13,23,1,2,\emptyset\\}$. However, the same reasoning
implies that this code does not have a realization by intervals—if it did, two
of the codewords $12,13$, and $23$ would appear on the same side of the
codeword $123$, leading to a contradictory containment. Thus it remains to
show that all $\dagger$-marked codes have realizations by boxes in
$\mathbb{R}^{2}$, which we do below in Figure 6.

Figure 6: 2-dimensional box realizations of the $\dagger$-marked codes from
Figure 5.


[a]Fangcheng He

# The calculations of Nucleon Electric Dipole Moment using background field on
Lattice QCD

Michael Abramczyk  Tom Blum  Taku Izubuchi  Hiroshi Ohki  Sergey Syritsyn

###### Abstract

Measurements of nucleon and nuclei Electric Dipole Moments (EDMs) play an
important role in probing CP violation and exploring physics beyond the
Standard Model. We extract the neutron EDM by measuring the energy shift of
the nucleon two-point correlation function in the presence of a background
field. The UV divergence of the topological charge density operator is
mitigated using gradient flow, and the diffusion effect induced by the
gradient flow process is included into the fit ansatz. Our calculations were
carried out on two 2+1 DWF fermion, Iwasaki, gauge field ensembles generated
by the RBC/UKQCD collaborations with inverse lattice spacing 1.73 GeV and pion
masses of about 340 and 420 MeV.

##  1 Introduction

The nucleon electric dipole moment (nEDM) is an important indicator of
CP(T)-symmetry violation. The nEDM prediction from the Standard Model’s CKM
matrix for the neutron is $\approx 10^{-31}$ [$e$ cm], which is much smaller
than the upper-limit determined through experiments on neutrons (ILL) [1] and
199Hg [2]. The SM prediction is also too small to explain the matter-
antimatter asymmetry compared to what is required by the Sakharov conditions.
Hence, the measurement of the nEDM is important in the search for physics
beyond Standard Model.

Strong interactions may be a source of CP violation known as the QCD
"$\theta$-term", $S_{\theta}=\theta Q$, where $Q$ is the topological charge.
Its contributions to the nEDM can be studied systematically from first
principles only using lattice QCD. Many efforts have been made to calculate
the nEDM using Lattice QCD [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. In earlier
calculations, an incorrect definition of the electric dipole form factor
$F_{3}$ resulted in significant mixing with the Pauli form factor [9]. After
subtracting the mixing term, those nEDM results became much smaller,
comparable with phenomenology, and universally dominated by statistical noise
even with unphysical heavy quark masses.

Currently, most of nEDM calculations on a lattice use the traditional form
factor method in which nEDMs are extracted as electric dipole form factors
(EDFF) $F_{3}(Q^{2})$ from ${CP}$-odd corrections to matrix elements of the
quark vector current due to topological charge. This form factor has to be
extrapolated to the forward limit ($Q^{2}\to 0$) to obtain the nEDM. The other
method is to calculate the EDM from the nucleon energy shift $\Delta E\propto
d_{N}(2\vec{S}\cdot\vec{E})$ in a uniform background electric field [14, 15,
9]. This method has significant advantages: no forward limit is required,
there is no parity mixing between $F_{2}$ and $F_{3}$, and one only needs to
calculate the ${CP}$-odd part of the two- instead of three-point function.

We present our $\theta$-nEDM results using the background field method.
Details of this method were first described in Ref. [16]. The background field
method is described in Sec. 2, the results are presented in Sec. 3, and
discussed in Sec 4.

##  2 Electric dipole moment from background field method

In this section, we discuss and compare methods to compute the nEDM on a
lattice. The form factor method has been described in multiple publications
(see, e.g., Ref.[9] and references therein). The EDFF is defined as

$\displaystyle\langle
p^{\prime},\sigma^{\prime}|J^{\mu}|p,\sigma\rangle_{{\cancel{\rm CP}}}$
$\displaystyle=$
$\displaystyle\bar{u}_{p^{\prime},\sigma^{\prime}}\big{[}F_{1}(Q^{2})\gamma^{\mu}+\big{(}F_{2}(Q^{2})+iF_{3}(Q^{2})\gamma_{5}\big{)}\frac{i\sigma^{\mu\nu}q_{\nu}}{2M_{N}}\big{]}u_{p,\sigma}\,$
(1)

where $Q^{2}=-(p^{\prime}-p)^{2}$, and $F_{1,2,3}$ are the Dirac, Pauli, and
electric dipole form factors. The forward limit of the latter yields the nEDM,
$F_{3}(0)=2m_{n}d_{n}$. Although the nucleon states in QCD vacuum with
${CP}$-violation are no longer parity eigenstates, it is crucial to ensure
that their spinors satisfy the positive-parity Dirac equation with real-valued
mass, $({\ooalign{\hfil/\hfil\crcr$p$}}-m_{N})u_{p,\sigma}=0$, otherwise
$F_{3}$ receives a spurious contribution from $F_{2}$ [9],

$\tilde{F}_{3}(Q^{2})=F_{3}\cos(2\alpha)-F_{2}\sin(2\alpha),$ (2)

where $\alpha$ is the parity mixing of the lattice nucleon spinor due to
${CP}$-violation that can be extracted from the two-point function.

Another approach to calculate the nucleon EDM is the background field method
introduced in [14, 15, 17, 16]. It has also been extended to analyze CP-even
properties, such as the electric polarizability and magnetic moments of the
nucleon [18, 19]. Since the proton accelerates in an electric field, and
requires a more sophisticated analysis, we study only the neutron EDM in this
work. After including the background field, the Dirac equation for the parity-
positive neutron spinor $u_{N}$ in the rest frame ($p_{N}=(m_{N},\vec{0})$)
becomes in Euclidean space [9]

$\displaystyle\left[i\not{p}+m_{N}-\big{(}\frac{1}{2}F_{\mu\nu}\sigma^{\mu\nu}\big{)}\frac{\kappa+i\zeta\gamma_{5}}{2m_{N}}\right]\,u_{N}=\left(\begin{array}[]{cc}m_{N}-\frac{(\kappa+i\zeta)\vec{\mathcal{E}}\cdot\vec{\sigma}}{2m_{N}}&-E_{N}\\\
-E_{N}&m_{N}+\frac{(\kappa-i\zeta)\vec{\mathcal{E}}\cdot\vec{\sigma}}{2m_{N}}\end{array}\right)\,u_{N}=0\,,$
(5)

where $E_{N}$ is the neutron energy, $\kappa,\zeta=F_{2,3}(0)$ are magnetic
and electric dipole moments, and $\vec{\mathcal{E}}$ is the Euclidean electric
field. To the order linear in $\kappa,\zeta$, the nucleon energy $E_{N}$ is

$E_{N}^{2}=m_{N}^{2}-i\zeta(\vec{\Sigma}\cdot\vec{\mathcal{E}})+O(\kappa^{2},\zeta^{2})\,,\quad\text{
or }\quad
E_{N}=m_{N}-\frac{i\zeta}{2m_{N}}\,(\vec{\Sigma}\cdot\vec{\mathcal{E}})+O(\kappa^{2},\zeta^{2})$
(6)

where $\zeta/(2m_{N})=d_{N}$ is the electric dipole moment and
$\vec{\Sigma}=\text{diag}[\vec{\sigma},\vec{\sigma}]$ is the spin operator.
Note that the linear part of the energy shift $\delta
E=-\frac{i\zeta}{2m_{N}}\,(\vec{\Sigma}\cdot\vec{\mathcal{E}})$ is imaginary
because of the analytic continuation in electric field on a Euclidean lattice,
implying that the correlation function acquires a complex phase. Expanding the
path integral in $\theta\ll 1$, one obtains the ${CP}$-violating correction to
the nucleon correlation function in the background field ${\mathcal{E}}$ on
one hand, and the $t$-linear correction on the other:

$C_{2pt,{\mathcal{E}},\theta}\approx\langle
N(t)\bar{N}(0)\rangle_{\mathcal{E}}-i\theta\langle
Q\,N(t)\bar{N}(0)\rangle_{\mathcal{E}}\propto e^{-Et}\big{[}1-t\delta
E+O(\theta^{2})\big{]}\,.$ (7)

The nEDM $\zeta=F_{3}(0)$ can be extracted from nucleon correlators in an
electric field $\vec{\mathcal{E}}={\mathcal{E}}_{z}\hat{z}$ as

$\frac{\zeta}{\theta}=i\frac{2m_{N}}{{\mathcal{E}}_{z}}\frac{\delta
E}{\theta}=-\frac{2m_{N}}{{\mathcal{E}}_{z}}\frac{d}{dt}\,\frac{\mathrm{Tr}\big{[}T^{+}_{S_{z}}\langle
Q\,N(t)\bar{N}(0)\rangle_{{\mathcal{E}}_{z}}\big{]}}{\mathrm{Tr}\big{[}T^{+}\langle
N(t)\bar{N}(0)\rangle_{{\mathcal{E}}_{z}}\big{]}}\,,$ (8)

where $T^{+}=\frac{1}{2}(1+\gamma_{4})$ is the positive-parity projector, and
$T^{+}_{S_{z}}=T^{+}\cdot(1+\Sigma_{z})$ is the spin-$\hat{z}$ projector.

So far, we have ignored excited states and negative parity state. In practice,
a multi-state model can be fitted to the time dependence of the EDM estimator
(8). On the other hand, the formula (8) resembles the “summation” method of
computing ground-state matrix elements. This relation is made apparent by the
Feynman-Hellman theorem (recently discussed in Ref. [20] in the context of
lattice QCD), so the EDM can alternatively be calculated from the matrix
element of the local density of topological charge,

$\frac{\zeta}{\theta}=-\frac{2m_{N}}{{\mathcal{E}}_{z}}\langle
N_{\uparrow}|q|N_{\uparrow}\rangle_{{\mathcal{E}}_{z}}\approx-\frac{2m_{N}}{{\mathcal{E}}_{z}}\frac{{\mathrm{Tr}}\big{[}T^{+}_{S_{z}}\langle
N(t_{f})\,q(\tau)\,\bar{N}(0)\rangle_{{\mathcal{E}}_{z}}\big{]}}{{\mathrm{Tr}}\big{[}T^{+}\langle
N(t_{f})\,\bar{N}(0)\rangle_{{\mathcal{E}}_{z}}\big{]}}\,.$ (9)

Thus, the problem is reduced from computing a 4-point function between the
nucleon fields $N,\,\bar{N}$, the global topological charge $Q=\int
d^{4}xq(x)$, and the vector current $J_{\mu}=\bar{\psi}\gamma_{\mu}\psi$ to a
correlator of $N,\,\bar{N}$ and the _local topological charge density_ $q(x)$
in uniform field:

$C_{3}(t_{f},\tau)={\mathrm{Tr}}\big{[}T^{+}_{S_{z}}\langle
N(t_{f})\,q(\tau)\,\bar{N}(0)\rangle_{{\mathcal{E}}_{z}}\big{]}=\big{[}T^{+}_{S_{z}}\big{]}_{\beta\alpha}\sum_{\vec{y},\vec{z}}\langle
N_{\alpha}(t_{f},\vec{y})\,q(\tau,\vec{z})\,\bar{N}_{\beta}(0)\rangle\,.$ (10)

This point is crucial to reducing fluctuations in correlators with any
operator that leads to “disconnected” Wick contractions: topological charge,
Weinberg 3-gluon interaction, isoscalar 2-quark and 4-quark interactions, and
so on. In this paper we concentrate on the topological charge but the
methodology can be readily extended to these other ${\cancel{\rm CP}}$
interactions.

Chiral symmetry is important for calculations involving topological charge. We
use gauge configurations generated with $N_{f}=2+1$ flavors of domain wall
fermions (DWF) [21]. (Tab. 1).

Table 1:  Gauge ensembles used in this work [21]. | size | $m_{l}/m_{s}$ | $m_{\pi}\,MeV$ | $N_{cfg}$
---|---|---|---|---
I24_m010 | $24^{3}\times 64$ | 0.01/0.04 | 432 | 1100
I24_m005 | $24^{3}\times 64$ | 0.005/0.04 | 340 | 1400

Periodicity on a lattice in space and time requires quantization of abelian
field flux $Q_{q}E_{k}L_{k}L_{4}=2\pi n$, where $Q_{u}=\frac{2}{3}$ and
$Q_{d}=-\frac{1}{3}$ are quark charges. A uniform electric field on the
lattice is introduced by multiplying the gauge links by the $U(1)$ phase [18],

$\displaystyle A_{x,4}=n_{k4}\Phi_{k4}x_{k},\quad\quad
A_{x,k}|_{x_{k}=L_{k}-1}=-n_{k4}\Phi_{k4}L_{k}x_{4},$ (11)

where $n_{\mu\nu}$ is the number of quanta and
$\Phi_{\mu\nu}=\frac{6\pi}{L_{\mu}L_{\nu}}$ is the unit of flux through
plaquette ($\mu\nu$). Such potentials result in a uniform electric field
$\vec{E}=(n_{14}\Phi_{14},n_{24}\Phi_{24},n_{34}\Phi_{34})$.

##  3 Numerical results

To calculate nucleon correlators, we use all-mode-averaging (AMA) [22] by
combining high-precision and truncated-CG samples computed with low-mode
deflation of the preconditioned Dirac operator. We also employ low-mode
averaging whereby we approximate the full-volume average of nucleon
correlators from low modes and combine it with AMA samples to correct bias,
which have resulted in a $50\%$ reduction of statistical errors:

$\langle{\mathcal{O}}\rangle\approx\langle{\mathcal{O}}^{\text{LMA}}\rangle_{V_{4}}+\langle{\mathcal{O}}^{\text{approx}}-{\mathcal{O}}^{\text{LMA}}\rangle_{N_{approx}}+\langle{\mathcal{O}}^{\text{exact}}-{\mathcal{O}}^{\text{approx}}\rangle_{N_{exact}}\,.$
(12)

In addition, we average over spin orientations $2S_{z}=\pm 1$ and electric
fields ${\mathcal{E}}_{z}=\pm 1,\pm 2$.

Figure 1: Plateaus of neutron matrix elements of topological charge density
converted to EDM (9), depending on the gradient flow time. The vertical dashed
lines of respective colors show the locations of the nucleon fields, which
lead to contact terms with the gluon operator on the same time slices. As the
gradient flow time increases, the contact terms “diffuse” into the EDM
plateaus.

To use the Feynman-Hellman method, we need to determine the local topological
charge density $q(x)$. We calculate $q(x)$ using 5-loop improved
discretization [23] and gradient flow [24]. Gradient flow helps suppress
$O(a)$ size dislocations that contribute to fluctuation of the global
topological charge. At large enough gradient flow time, the _global
topological charge_ $Q$ becomes constant. However, the _local density_ (summed
over the spatial volume) entering the correlator in Eq. (9), becomes
delocalized (“diffused”) in the Euclidean time, which complicates analysis of
its ground-state matrix elements. This effect is visible in the plateaus for
$d_{n}/\theta$ (9) shown in Fig. 1 for source-sink separations $t=(6\ldots
10)a$: as the gradient flow time $t^{\text{gf}}$ is increased, the
$\tau$-dependent features in the ratios (9) become more diffused. In
particular, it becomes hard to isolate the ground-state plateau from the
contact term contributions where the density $q(x)$ overlaps with the nucleon
operators.

Figure 2:  (Left) correlation of the topological charge density $\langle
q(r)q(0)\rangle$ computed with $t^{\text{gf}}$ and (right) diffusion kernel
$K(t^{\text{gf}}_{2}-t^{\text{gf}}_{1};\tau)$ (14).

We model this Euclidean-time diffusion effect of the gradient flow by

$q(t^{\text{gf}}_{2};\tau)=\int
dt^{\prime}K(t^{\text{gf}}_{2}-t^{\text{gf}}_{1};|\tau-\tau^{\prime}|)\,q(t^{\text{gf}}_{1};\tau^{\prime})$
(13)

where $K(t^{\text{gf}}_{2}-t^{\text{gf}}_{1};|\tau-\tau^{\prime}|)$ is the
diffusion kernel that can be extracted from Fourier-transformed correlations
of the topological charge

$K(t^{\text{gf}}_{2}-t^{\text{gf}}_{1};\tau)=\frac{1}{L_{t}}\sum_{k}e^{i\omega_{k}\tau}\sqrt{\frac{\tilde{\chi}(t^{\text{gf}}_{2};\omega_{k})}{\tilde{\chi}(t^{\text{gf}}_{1};\omega_{k})}}\,,\quad\tilde{\chi}(t^{\text{gf}};\omega_{k})=\sum_{\tau}e^{-i\omega_{k}\tau}\langle
q(t^{\text{gf}};\tau)q(t^{\text{gf}};0)\rangle\,,$ (14)

where the frequency $\omega_{k}=\frac{2\pi}{L_{t}}k$. Numerical results for
kernel $K(t^{\text{gf}}_{2}-t^{\text{gf}}_{1};|\tau|)$ are shown in Fig. 2,
illustrating how the diffusion profile widens with increasing gradient flow
time $t^{\text{gf}}$. Similarly, the effect of “diffusion” on the three-point
correlation function (10) can be written as

$\tilde{C_{3}}(t^{\text{gf}};t_{f},\tau)=\sum_{\tau^{\prime}}K(t^{\text{gf}};|\tau-\tau^{\prime}|)C_{3}(t_{f},{\tau}^{\prime})\,,$
(15)

where $C_{3}(t_{f},\tau)$ is the correlation function that would be computed
with a strictly local definition of the topological charge $q(\tau)$.
Unfortunately, this diffusion mixes the neutron EDM signal with contributions
from unwanted regions $\tau\leq 0$ and $\tau\geq t_{f}$, which we model as

$C_{3pt}(\tau,t_{f})=\left\\{\begin{array}[]{ll}a_{0}e^{-E_{0}t_{f}}\left(d_{n}+c_{1}e^{-\Delta
E_{1}\,\tau}+c_{1}e^{-\Delta E_{1}\,(t_{f}-\tau)}+c_{2}e^{-\Delta
E_{1}\,t_{f}}\right)\,,&0<\tau<t_{f}\,,\\\
C_{ext}e^{-E_{0}t_{f}}e^{E_{ext}\tau}\,,&\tau\leq 0\,,\\\
C_{ext}e^{-E_{0}t_{f}}e^{-E_{ext}(\tau-t_{f})}\,,&t_{f}\leq\tau\,.\\\
\end{array}\right.$ (16)

The first line corresponds to the matrix element where the gluon operator is
located between the source and sink. The second and the third lines represent
the contributions from contact terms ($\tau=0$ or $\tau=t_{f}$) and “crossed”
regions with $q(\tau)$ located outside of the source-sink time interval.

Figure 3:  Fits of combined Euclidean-time (16) and gradient flow-time
dependence (15) of lattice nEDM correlation functions. The band is the fit
result using Eq. (16).

We perform a fit of three-point (10) data to ansatz (16) joint with a two-
state fit of two-point neutron data to constrain ground- and excited-state
energies $E_{0}$, $E_{0}+\Delta E_{1}$. For the former, we select data with
with longer range $8a\leq t_{f}\leq 10a$ and $5a^{2}\leq t^{\text{gf}}\leq
8a^{2}$. The fit result for the ground-state EDM $d_{n}/\theta$ is shown in
Fig. (3). We obtain the following values at the two pion masses

$\displaystyle[d_{n}/\theta]_{340\,\text{MeV}}$
$\displaystyle=-0.0072(20)^{\text{stat}}\,,$
$\displaystyle[d_{n}/\theta]_{432\,\text{MeV}}$
$\displaystyle=-0.0060(20)^{\text{stat}}\,.$

Using chiral prediction $d_{n}(m_{\pi}^{2})\sim
c_{1}m_{\pi}^{2}+c_{2}m_{\pi}^{2}\log(m_{\pi}^{2})$, we can obtain only a
tentative estimate of the value at the physical point
$(d_{n}/\theta)^{\text{phys}}_{\text{lin.+log.}}=-0.0036(21)^{\text{stat}}$.
The uncertainty of the extrapolation is substantially constrained by chiral
symmetry in our EDM calculation.

##  4 Summary

In this work, we have calculated the neutron EDM employing the background
field method on two gauge ensembles with chirally symmetric quarks and pion
masses $\approx 340$ and 420 MeV. Unlike in the traditional form-factor
approach, we calculate the forward matrix element of the topological charge
density in a simultaneously spin- and electrically-polarized nucleon state.
This method is instrumental to limit the growth of the stochastic noise from
the global topological charge as the physical volume of a lattice increases.
Without the need to extrapolate in the momentum transfer $Q^{2}\to 0$, the
systematic uncertainty can be further dramatically reduced.

The main obstacle in using this method is the difficulty of determining
topological charge density. In this work, we have used the field-theoretical
definition from the gluon fields combined with gradient flow. However,
gradient flow leads to “diffusion” in Euclidean time and mixing of the nEDM
signal with contact terms and contributions from $n\bar{n}$-pair states. We
have implemented the numerical procedure to extract the diffusion kernel from
lattice data and incorporate it in the fits of correlators of the nucleon and
the topological charge density. With only two pion mass points available, we
could perform only a tentative estimate of the physical-point value. Another
point at lighter pion mass $m_{\pi}\approx 250$ MeV is currently being
investigated.

We plan to explore the fermionic definition [11, 25], which may be especially
advantageous when combined with the low-mode approximation for neutron
correlation function. Further, this method can be easily applied to other
${\cancel{\rm CP}}$ operators, and might be especially beneficial for the nEDM
induced by Weinberg’s three-gluon operator and 4-quark ${\cancel{\rm CP}}$
operators.

## Acknowledgments

The research reported in this work made use of computing and long-term storage
facilities provided by the USQCD Collaboration, which are funded by the Office
of Science of the U.S. Department of Energy, and the Hokusai supercomputer of
the RIKEN ACCC facility. We are grateful for the gauge configurations provided
by the RBC/UKQCD collaboration. SS and FH are supported by the National
Science Foundation under CAREER Award PHY-1847893. Any opinions, findings, and
conclusions or recommendations expressed in this material are those of the
author(s) and do not necessarily reflect the views of the National Science
Foundation. HO is supported in part by JSPS KAKENHI Grants No. 21K03554 and
No. 22H00138. TB and MA were partially supported by the US DOE under the award
DE-SC001033. TI is supported by U.S. Department of Energy (DOE) under award
DE-SC0012704, SciDAC-5 LAB 22-2580, and also Laboratory Directed Research and
Development (LDRD No. 23-051) of BNL and RIKEN BNL Research Center.

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