% arXiv v1; presentation derivative of general v5, dated 6 October 2026.
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\hypersetup{pdftitle={An analytic lower bound for the cubic-lattice dimer constant},
  pdfauthor={Yair Lavi}}
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\title{An analytic lower bound for the cubic-lattice dimer constant}
\author{Yair Lavi}
\date{6 October 2026}

\begin{document}
\maketitle

\begin{abstract}
We prove that the close-packed dimer entropy per site $\ell_3$ of the
simple cubic lattice satisfies
\[
\ell_3\ge \frac12\log\frac{3125}{1296}+\frac{47}{10000}
=0.444775842629\ldots.
\]
This exceeds the previous best published lower bound,
$0.441075842629\ldots$, obtained by Csikv\'ari. The proof uses finite matching
inequalities and elementary integration, without numerical quadrature or a
computational certificate.
\end{abstract}

\section{Introduction and result}

For an even positive integer $L$, let $G_L$ be the graph with vertex set
$\{0,\ldots,L-1\}^3$, where two vertices are adjacent precisely when they
differ by one in one coordinate and agree in the other two. A matching is
a set of edges with no shared vertices; it is perfect if every vertex belongs
to one of those edges. A perfect matching describes a close-packed dimer
covering: each edge represents a dimer covering its two endpoint sites.
Write $\PM(H)$ for the number of perfect matchings of a finite graph $H$.
The close-packed dimer entropy per site of the simple cubic lattice, also
called the cubic-lattice dimer constant, is
\begin{equation}\label{eq:constant}
\ell_3=\lim_{\substack{L\to\infty\\L\ \mathrm{even}}}
             L^{-3}\log\PM(G_L).
\end{equation}
For even $L\ge4$, let $F_L$ be the periodic counterpart of $G_L$: it has
the same vertices and edges, together with the edges between vertices that
agree in two coordinates and have $0$ and $L-1$ in the third.
All logarithms are natural, and entropy is normalized per vertex.
Existence of the limit \eqref{eq:constant} and its equality with the periodic
limit $\lim_{\substack{L\to\infty\\L\ \mathrm{even}}}L^{-3}\log\PM(F_L)$ follow
from Friedland and Peled \cite[Theorem~4.1 and (4.7)]{FP},
using Hammersley's density-limit theorem \cite{Hammersley}.

\subsection*{Known bounds and approximations}

Every vertex of $F_L$ has six neighbors, and the parity of the sum of its
coordinates divides the vertices into two classes with every edge joining
different classes. Schrijver's theorem \cite{Schrijver} states that every
$k$-regular bipartite graph on $2n$ vertices, for $k\ge2$, has at least
$\bigl((k-1)^{k-1}/k^{k-2}\bigr)^n$ perfect matchings. Applied to $F_L$
with $k=6$ and $2n=L^3$, and combined with the Friedland--Peled equality
of free-box and periodic limits stated above, it gives the lower bound
\[
B_0:=\frac12\log\frac{3125}{1296}=0.440075842629\ldots.
\]
Csikv\'ari \cite[Remark~3.4]{Csikvari} improves this to
$\ell_3\ge B_0+1/1000=0.441075842629\ldots$.
For comparison, a tensor-network calculation of Vanderstraeten, Vanhecke
and Verstraete \cite{VVV} estimates $\ell_3$ as $0.44988452$, while He's
computer-assisted theorem \cite{He} proves $\ell_3\le0.452130$.
The tensor-network value is a numerical estimate without rigorous error
control; Csikv\'ari's bound, He's bound and our Theorem~\ref{thm:main} below
are proved inequalities.

\begin{theorem}\label{thm:main}
For the limit in \eqref{eq:constant},
\begin{equation}\label{eq:main}
\ell_3\ge B_0+\frac{47}{10000}=0.444775842629\ldots.
\end{equation}
\end{theorem}

The gain over Csikv\'ari's displayed bound is exactly $37/10000$.
The proof follows Csikv\'ari's entropy-gap framework
\cite[Theorem~1.4]{Csikvari}: a finite Heilmann--Lieb
identity \cite{HL,Csikvari} represents the relevant defect as a sum of
nonnegative path terms, which controls the derivative of the entropy gap.
Our new ingredient is a reflection bound, uniform in the dimer activity,
for the probability that two opposite edges of a coordinate unit square
are simultaneously occupied. It gives a sharper lower bound for the
four-cycle terms in the path sum. Reflection methods for dimer systems have established
precedents; see, for example, Starr and Williams \cite{SW}. We give the
needed argument directly. Retaining a quadratic term in the elementary
logarithm estimate gives an integrable gain over $B_0$; the final comparison
with $47/10000$ is a rational identity.

\section{Finite matching inequalities}

Fix an even integer $L\ge4$, and put $F=F_L$ and $N=L^3$, with coordinates
interpreted modulo $L$. As observed in the introduction, $F$ is six-regular
and bipartite. Pairing consecutive vertices in the first coordinate gives
a perfect matching.

For any finite graph $H$ and $t>0$, define
\[
Z_H(t)=\sum_{M\text{ a matching of }H}t^{|M|}.
\]
Here $|M|$ is the number of edges in $M$, and uncovered vertices are called
monomers. This is the monomer--dimer partition function of $H$
\cite{HL,Csikvari}. The parameter $t$ is known as the dimer activity.
It specifies the probability
$\Prob_t(M)=t^{|M|}/Z_F(t)$ on matchings of $F$.
Write $\E_t$ for expectation with respect to this probability.
For a vertex set $A$, $H-A$ means the induced graph after deleting $A$;
we also write $H-u-v$ for $H-\{u,v\}$.
Set
\begin{equation}\label{eq:density}
p=p(t)=\frac{2\E_t|M|}{N}
       =\frac{2tZ_F'(t)}{NZ_F(t)},\qquad q=\frac p6.
\end{equation}
Translations, coordinate permutations and coordinate sign changes make
all vertices equivalent and all edges equivalent. Conditioning on $|M|=k$
preserves these symmetries. A matching of size $k$ covers exactly $2k$ of
the $N$ vertices and occupies exactly $k$ of the $6N/2=3N$ edges. Hence,
for any vertex $v$, any edge $e$, and $k=0,\ldots,N/2$,
\[
\Prob_t(v\text{ is covered}\mid |M|=k)=\frac{2k}{N},\qquad
\Prob_t(e\in M\mid |M|=k)=\frac{k}{3N}.
\]
The law of total probability therefore gives
\[
\begin{aligned}
\Prob_t(v\text{ is covered})
 &=\sum_{k=0}^{N/2}\frac{2k}{N}\Prob_t(|M|=k)
   =\frac{2\E_t|M|}{N}=p,\\
\Prob_t(e\in M)
 &=\sum_{k=0}^{N/2}\frac{k}{3N}\Prob_t(|M|=k)
   =\frac{\E_t|M|}{3N}=\frac p6=q.
\end{aligned}
\]

Fix an edge $uv$. Removing $uv$ from a matching containing it gives a
matching of $F-u-v$; conversely, every matching of $F-u-v$ extends uniquely
by adding $uv$. Adding this edge multiplies the weight by $t$, so the
total weight of matchings containing $uv$ is $tZ_{F-u-v}$. A matching
leaves $u$ uncovered precisely when it uses no edge incident to $u$.
Such matchings are exactly those of $F-u$, with their weights unchanged,
so their total weight is $Z_{F-u}$. Applying the same argument to $v$
and dividing by $Z_F$ gives
\begin{equation}\label{eq:deletions}
q=t\frac{Z_{F-u-v}}{Z_F},\qquad
1-p=\frac{Z_{F-u}}{Z_F}=\frac{Z_{F-v}}{Z_F}
\end{equation}
for every edge $uv$. The common argument $t$ is suppressed in partition
functions when no confusion is possible. At finite positive $t$,
$0<p<1$ and $0<q<1/6$.

Let $e_i,e_j$ be two distinct coordinate unit vectors and
$x\in(\mathbb Z/L\mathbb Z)^3$. A coordinate plaquette $C$ is the
four-vertex cycle with vertices, in cyclic order,
\[
v_1=x,\qquad v_2=x+e_i,\qquad
v_3=x+e_i+e_j,\qquad v_4=x+e_j,
\]
where coordinates are interpreted modulo $L$. Its two pairs of opposite
edges are
\[
D_1=\{v_1v_2,v_3v_4\},\qquad D_2=\{v_2v_3,v_4v_1\}.
\]
Define $a:=\Prob_t(D_1\subseteq M)$, the probability that both edges of
the first pair are occupied.

For either $k=1$ or $k=2$, removing the two edges of $D_k$ from a matching
containing them leaves a matching of $F-C$, where $F-C$ denotes the graph
obtained by deleting all four vertices of $C$. Conversely, adding $D_k$
to any matching $M'$ of $F-C$ yields a unique matching containing that
pair, with weight $t^{|M'|+2}$. Therefore
\[
\Prob_t(D_k\subseteq M)
 =\frac{\sum_{M'\text{ a matching of }F-C}t^{|M'|+2}}{Z_F}
 =t^2\frac{Z_{F-C}}{Z_F}.
\]
In particular, both pairs have the same probability $a$, and
\begin{equation}\label{eq:a}
a=t^2\frac{Z_{F-C}}{Z_F}.
\end{equation}
The two pair events are mutually exclusive, so the probability that either
pair is occupied is $2a$. Translations, coordinate permutations and
coordinate sign changes carry any coordinate plaquette to any other while
preserving matching weights. Hence $a$ is independent of the plaquette chosen.

The preceding definitions and identities are standard in the monomer--dimer
literature. Our first new ingredient is the following quantitative inequality
for the joint occupancy of opposite plaquette edges.

\begin{lemma}\label{lem:reflection}
For every $t>0$,
\begin{equation}\label{eq:reflection}
a(1-2q+a)\ge q^2.
\end{equation}
\end{lemma}

\begin{proof}
We split the torus along two cuts into halves exchanged by a reflection.
Once the occupied crossing edges are fixed, the remaining matchings in the
two halves are independent and have identical distributions after reflection.
This lets us compare the occupancy probabilities of an edge and its reflected
copy using the Cauchy--Schwarz inequality.

By the symmetry noted after \eqref{eq:a}, we may take $C$ to be the
plaquette with $x=(0,0,0)$, $e_i=(1,0,0)$ and $e_j=(0,1,0)$, so that
$v_1=(0,0,0)$, $v_2=(1,0,0)$, $v_3=(1,1,0)$ and $v_4=(0,1,0)$. Put
$e=v_1v_2$ and $e'=v_4v_3$; these are the two edges of $D_1$, and they lie
in the layers $y=0$ and $y=1$ respectively. The two edges $b_1=v_1v_4$ and
$b_2=v_2v_3$ of $D_2$ join these layers.

Reflect vertices by $\theta(x,y,z)=(x,1-y,z)$ modulo $L$; this is an
automorphism of $F$ with $\theta e=e'$. Let $H$ be the set of vertices
whose $y$ coordinate lies in $\{0,L/2+1,L/2+2,\ldots,L-1\}$; then
$\theta H$ consists of the vertices with $y\in\{1,2,\ldots,L/2\}$, and
$H$, $\theta H$ partition the vertex set. The edge $e$ lies inside $H$ and
$e'$ inside $\theta H$. Let $B$ be the set of edges with one endpoint in
$H$ and the other in $\theta H$. These are exactly the $y$-direction edges
across the two cuts of the layer cycle, between layers $0,1$ and between
layers $L/2,L/2+1$. Each edge of $B$ joins a vertex to its reflection,
so $\theta$ fixes every edge of $B$; the edges of $B$ are pairwise
vertex-disjoint because $L\ge4$; and $b_1,b_2$ are the only edges of $B$
at the endpoints of $e$. Every other edge of $F$ lies inside $H$ or inside
$\theta H$. Write $F[H]$ for the subgraph induced by $H$; then $\theta$
maps $F[H]$ isomorphically onto $F[\theta H]$.

Condition on $S=M\cap B$, the set of occupied crossing edges. Let
$V_H(S)$ and $V_{\theta H}(S)$ denote the endpoints of $S$ in the two
halves. Given $S$, the rest of $M$ consists of a matching of
$F[H]-V_H(S)$ and a matching of $F[\theta H]-V_{\theta H}(S)$; conversely,
every such pair of
matchings together with $S$ is a matching of $F$. Since $\theta$ fixes $S$
and maps $F[H]-V_H(S)$ isomorphically onto $F[\theta H]-V_{\theta H}(S)$,
both residual graphs have the same partition function, which we call
$A_S$. Therefore
\[
Z_F=\sum_S t^{|S|}A_S^2,
\qquad w_S:=\Prob_t(M\cap B=S)=\frac{t^{|S|}A_S^2}{Z_F},
\]
and, given $S$, the two residual matchings are independent, each
distributed with weights $t^{|\cdot|}$ on its residual graph.
Let $f_S$ be the conditional probability, given $M\cap B=S$, that $e$ is
occupied; by the same isomorphism, $f_S$ is also the conditional
probability that $e'$ is occupied. If $S$ covers an endpoint of $e$, then
$f_S=0$. Conditional independence gives
\[
q=\sum_S w_Sf_S,\qquad a=\sum_S w_Sf_S^2.
\]
Let $U$ be the set of $S$ containing neither $b_1$ nor $b_2$; these are
exactly the $S$ covering no endpoint of $e$. Each of $b_1,b_2$ is occupied
with probability $q$, and both are occupied with probability $a$, so
inclusion--exclusion gives $\sum_{S\in U}w_S=1-2q+a$.
Since $f_S=0$ for $S\notin U$, the Cauchy--Schwarz inequality gives
\[
q=\sum_{S\in U}w_Sf_S
 \le\Bigl(\sum_{S\in U}w_S\Bigr)^{1/2}\Bigl(\sum_{S\in U}w_Sf_S^2\Bigr)^{1/2}
 \le\bigl((1-2q+a)\,a\bigr)^{1/2},
\]
and squaring yields \eqref{eq:reflection}.
\end{proof}

Put $Q=q(1-q)$ and $r=Q-t(1-p)^2$. Here $t$ is the dimer activity and
$p=p(t)$ is the mean fraction of covered vertices.
The quantity $r$ measures departure from the relation
$t(1-p)^2=q(1-q)$, which holds on the infinite six-regular tree---the
connected graph with no cycles and six neighbors at every vertex
\cite[Proposition~6.5]{Csikvari}.
The path identity \eqref{eq:path-sum} below, the edge-transitive
specialization of \cite[Proposition~2.5(c)]{Csikvari}, expresses this
departure through nonnegative contributions from cycles.
The following lemma retains four plaquette terms of that sum and bounds
them through the plaquette probability $a$. In Section~3 the resulting
lower bound $r\ge4a^2$ improves the entropy lower bound.

\begin{lemma}\label{lem:paths}
For every $t>0$,
\begin{equation}\label{eq:defect}
4a^2\le r<Q.
\end{equation}
\end{lemma}

\begin{proof}
For adjacent $u,v$, let $P:u\leadsto v$ denote a simple path, that is,
a path with no repeated vertex. Its vertex set is $V(P)$, and $|V(P)|$
counts vertices, not edges. The finite Heilmann--Lieb identity, in the
bipartite form of \cite[Lemma~2.4(c)]{Csikvari}, states that
\[
Z_F Z_{F-u-v}-Z_{F-u}Z_{F-v}
 =\sum_{P:u\leadsto v}t^{|V(P)|-1}Z_{F-V(P)}^2.
\]
All terms have the displayed positive sign because a path between the two
parity classes has an odd number of edges. Multiply by $t/Z_F^2$ and
subtract the term for the single-edge path $uv$. By \eqref{eq:deletions},
\begin{equation}\label{eq:path-sum}
r=\sum_{\substack{P:u\leadsto v\\P\ne(uv)}}
       t^{|V(P)|}\left(\frac{Z_{F-V(P)}}{Z_F}\right)^2.
\end{equation}
The edge $uv$ belongs to four coordinate plaquettes: two choices of the
other coordinate and two choices of its sign; since $L\ge4$, these four
plaquettes are distinct. Going around each plaquette gives a distinct
three-edge path, whose vertex set is the plaquette and whose contribution
is $t^4(Z_{F-C}/Z_F)^2=a^2$ by \eqref{eq:a}. Retaining these four terms and
dropping all other paths, which contribute nonnegative terms (for $L=4$
these include the straight three-edge path around the torus), proves the
lower inequality.
The upper inequality follows from $t>0$ and $p<1$.
\end{proof}

Solving \eqref{eq:reflection} as a quadratic inequality in $a$ gives
$a\ge a_*(q)$, where
\[
a_*(q)=\frac{\sqrt{(1-2q)^2+4q^2}-(1-2q)}2.
\]
We use the polynomial lower bound below to make the final integration
elementary. The coefficient $29/10$ is a convenient rational choice:
the bound holds throughout the required density interval, and its
integrated gain exceeds $47/10000$.

\begin{lemma}\label{lem:polynomial}
For every $t>0$, let $h(q)=1+2q+\frac{29}{10}q^2$. Then
\begin{equation}\label{eq:polynomial}
a\ge q^2h(q).
\end{equation}
\end{lemma}

\begin{proof}
For $0<q\le1/6$, the function
$f_q(z)=z^2+(1-2q)z-q^2$ is strictly increasing on $z\ge0$.
Writing $s=q^2h(q)$, direct expansion gives
\[
f_q(s)=q^4T(q),\qquad
T(q)=-\frac1{10}-\frac95q+\frac{49}5q^2
                  +\frac{58}5q^3+\frac{841}{100}q^4.
\]
Since
\[
T''(q)=\frac{98}5+\frac{348}5q+\frac{2523}{25}q^2>0,
\quad T(0)=-\frac1{10},\quad
T(1/6)=-\frac{8759}{129600},
\]
convexity puts $T$ below its negative endpoint chord.
Consequently $f_q(s)<0\le f_q(a)$ for $q>0$, by
\eqref{eq:reflection}, so $a>s$. In particular, since $h(q)\ge1$, we have
$a\ge q^2$.
\end{proof}

\section{Entropy integration and proof of the theorem}

The finite-volume entropy identity turns the estimate $r\ge4a^2$ from
Lemma~\ref{lem:paths} into an integrable gain over $B_0$.
For a random variable $X$, write
$\operatorname{Var}_t(X)=\E_t(X^2)-(\E_tX)^2$.
Differentiating $\E_t|M|=tZ_F'(t)/Z_F(t)$ gives
$(tZ_F'/Z_F)'=\E_t(|M|^2)/t-(\E_t|M|)^2/t$, so the density in
\eqref{eq:density} obeys
\[
p'(t)=\frac{2}{Nt}\operatorname{Var}_t(|M|)>0.
\]
The variance is strictly positive because the matching with no edges and
any one-edge matching both have positive probability, so $|M|$ is not
constant.

The constant term of $Z_F(t)$ is $1$, and its highest-degree term is
$\PM(F)t^{N/2}$. Consequently, the density identity \eqref{eq:density}
gives $p(t)\to0$ as $t\downarrow0$ and $p(t)\to1$ as $t\to\infty$.

Thus $p$ is a strictly increasing bijection from $(0,\infty)$ onto $(0,1)$;
let $t_F(p)$ be its inverse, which is continuously differentiable on $(0,1)$.
Following Csikv\'ari \cite[Sections~2--3]{Csikvari}, we define
\begin{align}
\lambda_F(p)&=N^{-1}\log Z_F(t_F(p))-\frac p2\log t_F(p),\label{eq:lambda}\\
G_6(p)&=\frac12\left[p\log\frac6p+(6-p)\log(1-p/6)
                        -2(1-p)\log(1-p)\right],\label{eq:G}\\
g_F(p)&=\lambda_F(p)-G_6(p).\nonumber
\end{align}
Here $G_6(p)$ is the matching entropy per vertex of that infinite tree at
covered-vertex density $p$ \cite[Section~3]{Csikvari}. Thus $g_F(p)$ compares
the entropy on the periodic cubic graph with this tree benchmark.
The bound $r\ge4a^2$ turns the plaquette occupancy estimate into a lower
bound on the derivative of this entropy gap.
Use continuous endpoint values for products $x\log x$.
For $t=t_F(p)$, the matching probabilities satisfy
$\log\Prob_t(M)=|M|\log t-\log Z_F(t)$. Averaging this identity and using
$\E_t|M|=Np/2$ gives
\[
-\frac1N\sum_M\Prob_t(M)\log\Prob_t(M)
 =\frac1N\log Z_F(t)-\frac p2\log t=\lambda_F(p).
\]
Thus $\lambda_F(p)$ is the entropy of the matching distribution, normalized
per vertex.

As $p\to0$, the distribution concentrates on the single empty matching,
whose entropy is zero. As $p\to1$, it becomes uniform on the $\PM(F)$
perfect matchings, since all have the same weight $t^{N/2}$; this
distribution has entropy $\log\PM(F)$. Together with $G_6(0)=0$ and
$G_6(1)=B_0$, these continuous endpoint values give
\begin{equation}\label{eq:endpoints}
g_F(0)=0,\qquad
g_F(1)=N^{-1}\log\PM(F)-B_0.
\end{equation}

For $0<p<1$, differentiating \eqref{eq:lambda} by the chain rule yields
\[
\lambda_F'(p)=\left[\frac{Z_F'(t)}{NZ_F(t)}-\frac p{2t}\right]t_F'(p)
                  -\frac12\log t,\qquad t=t_F(p).
\]
The bracket vanishes by the density identity \eqref{eq:density}, leaving
$\lambda_F'(p)=-\tfrac12\log t_F(p)$. Direct differentiation of
\eqref{eq:G} gives
\[
G_6'(p)=\frac12\log\frac{(1-p)^2}{q(1-q)},\qquad q=p/6.
\]
Subtracting these derivatives therefore gives
\begin{equation}\label{eq:gap-derivative}
g_F'(p)=\frac12\log\frac{q(1-q)}{t_F(p)(1-p)^2}
        =-\frac12\log\left(1-\frac rQ\right).
\end{equation}
The last equality uses $Q=q(1-q)$ and $t_F(p)(1-p)^2=Q-r$, with $r,Q$
evaluated at $t=t_F(p)$.
By Lemma~\ref{lem:paths}, $x:=r/Q$ lies in $[0,1)$. The inequality
$-\log(1-x)\ge x+x^2/2$ for $0\le x<1$ follows by integrating
$(1-u)^{-1}\ge1+u$ over $0\le u\le x$. Applying it in
\eqref{eq:gap-derivative} and then using Lemmas~\ref{lem:paths}
and~\ref{lem:polynomial} gives
\begin{align}
g_F'(p)&\ge\frac{r}{2Q}+\frac{r^2}{4Q^2}\nonumber\\
&\ge\frac{2q^3h(q)^2}{1-q}+\frac{4q^6}{(1-q)^2}
 \ge\frac{2q^3h(q)^2}{1-q}+4q^6.\label{eq:derivative-lower}
\end{align}
Csikv\'ari uses the linear estimate $g_F'(p)\ge r/(2Q)$; here we retain
the additional quadratic term. The first contribution in
\eqref{eq:derivative-lower} uses $r\ge4a^2$ and $a\ge q^2h(q)$; the second
uses $r\ge4a^2$ and $a\ge q^2$. The function $g_F$ is continuous on
$[0,1]$ with the endpoint values \eqref{eq:endpoints} and continuously
differentiable on $(0,1)$. Integrating \eqref{eq:derivative-lower} over
$[\varepsilon,1-\varepsilon]$, letting $\varepsilon\downarrow0$, and
substituting $p=6q$ gives
\begin{equation}\label{eq:J-integral}
N^{-1}\log\PM(F)\ge B_0+J,\qquad
J:=12\int_0^{1/6}\frac{q^3h(q)^2}{1-q}\,dq
                              +24\int_0^{1/6}q^6\,dq.
\end{equation}

To evaluate $J$, expand
$h(q)^2=1+4q+\frac{49}{5}q^2+\frac{58}{5}q^3+\frac{841}{100}q^4$ and use
\[
\int_0^{1/6}\frac{q^m}{1-q}\,dq
 =\log\frac65-\sum_{j=1}^{m}\frac{1}{j6^j}\qquad(m\ge1).
\]
Substitution in \eqref{eq:J-integral} gives
\begin{equation}\label{eq:J-value}
J=\frac{10443}{25}\log\frac65
                     -\frac{3108938251}{40824000}+\frac{1}{81648}.
\end{equation}
Integrating the geometric series for $2/(1-u^2)$ from $0$ to $1/z$
gives, for real $z>1$,
\[
\log\frac{z+1}{z-1}
 =2\sum_{k=0}^{\infty}\frac{1}{(2k+1)z^{2k+1}}.
\]
At $z=11$, the first four positive terms give the strict bound
\[
\log\frac65>\frac{373057792}{2046152955}.
\]
Substituting this bound into \eqref{eq:J-value} gives
\begin{equation}\label{eq:margin}
J-\frac{47}{10000}
 >\frac{10443}{25}\frac{373057792}{2046152955}
    -\frac{3108938251}{40824000}+\frac{1}{81648}-\frac{47}{10000}
 =\frac{11298741211}{3977721344520000}>0.
\end{equation}
For orientation, $J\approx0.00470288014$, exceeding
$47/10000=0.0047$ by approximately $2.88\times10^{-6}$.
These decimal values illustrate the margin; the strict comparison is
proved by the rational bound in \eqref{eq:margin}.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Equations~\eqref{eq:J-integral}--\eqref{eq:margin} give the same lower
bound on every even periodic cubic graph $F_L$, $L\ge4$.
Friedland--Peled \cite[Theorem~4.1 and (4.7)]{FP} identify
the periodic perfect-matching entropy with the free-box limit whenever
the volume is even and every side tends to infinity. Our boxes have three
even sides $L\to\infty$, and their free and periodic matchings are exactly
the dimer tilings in that theorem. Hence
\[
\ell_3=\lim_{\substack{L\to\infty\\L\ \mathrm{even}}}
                 L^{-3}\log\PM(F_L)\ge B_0+J>B_0+\frac{47}{10000}.
\]
This proves \eqref{eq:main}.
\end{proof}

\section*{Acknowledgments}

The proof of this lower bound was carried out by GPT-5.6-sol, GPT-6 Astra,
Claude Fable 5, and Claude Fable 5.1, under the guidance of the author.
The author has reviewed the resulting proof arguments. Responsibility for
the final text rests with the author.

\begin{thebibliography}{8}
\small
\raggedright

\bibitem{Schrijver}
A. Schrijver, ``Counting 1-factors in regular bipartite graphs,''
\emph{Journal of Combinatorial Theory, Series B} \textbf{72} (1998), 122--135.
\href{https://doi.org/10.1006/jctb.1997.1798}{doi:10.1006/jctb.1997.1798}.

\bibitem{Csikvari}
P. Csikv\'ari, ``Matchings in vertex-transitive bipartite graphs,''
\emph{Israel Journal of Mathematics} \textbf{215} (2016), 99--134.
\href{https://doi.org/10.1007/s11856-016-1375-9}{doi:10.1007/s11856-016-1375-9}.
The cited numbering refers to the
\href{https://csikvarip.web.elte.hu/matchings_vertex-transitive_V5}{author's V5
version}, dated 31 March 2015.

\bibitem{HL}
O. J. Heilmann and E. H. Lieb, ``Theory of monomer-dimer systems,''
\emph{Communications in Mathematical Physics} \textbf{25} (1972), 190--232.
\href{https://doi.org/10.1007/BF01877590}{doi:10.1007/BF01877590}.

\bibitem{FP}
S. Friedland and U. N. Peled, ``Theory of computation of multidimensional
entropy with an application to the monomer-dimer problem,''
\emph{Advances in Applied Mathematics} \textbf{34} (2005), 486--522.
\href{https://doi.org/10.1016/j.aam.2004.08.005}{doi:10.1016/j.aam.2004.08.005}.
The cited numbering refers to
\href{https://arxiv.org/abs/math/0402009v1}{arXiv:math/0402009v1} (2004).

\bibitem{Hammersley}
J. M. Hammersley, ``Existence theorems and Monte Carlo methods for the
monomer-dimer problem,'' in \emph{Research Papers in Statistics:
Festschrift for J. Neyman}, Wiley, 1966, 125--146.

\bibitem{VVV}
L. Vanderstraeten, B. Vanhecke and F. Verstraete, ``Residual entropies for
three-dimensional frustrated spin systems with tensor networks,''
\emph{Physical Review E} \textbf{98} (2018), 042145.
\href{https://doi.org/10.1103/PhysRevE.98.042145}{doi:10.1103/PhysRevE.98.042145}.

\bibitem{He}
Q. He, ``A new upper bound on the dimer constant of $\mathbb Z^3$,''
\href{https://arxiv.org/abs/2607.28810v2}{arXiv:2607.28810v2} (2026).

\bibitem{SW}
S. Starr and S. Williams, ``Rough bounds for emptiness formation probability
in the 2d dimer model using reflection positivity,''
\href{https://arxiv.org/abs/1810.08846}{arXiv:1810.08846} (2018).

\end{thebibliography}
\end{document}