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\hypersetup{pdftitle={Theta bases for rank-two quantum cluster algebras of geometric type},pdfauthor={Qiyue Tang},
 pdfsubject={Quantum theta and greedy bases with geometric and principal coefficients},
 pdfkeywords={quantum cluster algebra, theta basis, greedy basis, scattering diagram}}
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\title[Quantum theta bases with coefficients]{Theta bases for rank-two quantum cluster algebras of geometric type}
\author{Qiyue Tang}
\address{Zhili College, Tsinghua University, Beijing 100084, China}
\email{tangqy24@mails.tsinghua.edu.cn}
\date{4 October 2026}
\subjclass[2020]{Primary 13F60; Secondary 16S35, 16S85}
\keywords{Quantum cluster algebra, theta basis, greedy basis, scattering diagram}

\begin{document}
\begin{abstract}
We prove that the coefficient-free quantum theta and greedy bases
coincide for all positive exchange parameters. Their homogeneous lifts
form an integral theta basis for arbitrary geometric coefficients and
compatible integral quantizations, with frozen variables inverted.
For nonnegative initial coefficient blocks, they also form a basis
with polynomial frozen coefficients. A signed coefficient example
separates the ordinary and upper algebras.
\end{abstract}
\maketitle

\section{Introduction}

Throughout, \(\mathbb N=\mathbb Z_{\geq0}\), \(v\) is an indeterminate,
and \([r]_+=\max(r,0)\) for an integer \(r\).
Let \(b,c\) be positive integers. The coefficient-free quantum cluster algebra
\(\A_v(b,c)\) is generated by variables \(X_j\), \(j\in\Z\), with
\begin{equation}\label{eq:exchange}
 X_{j+1}X_{j-1}=
 \begin{cases}
 v^bX_j^b+1,&j\text{ odd},\\
 v^cX_j^c+1,&j\text{ even},
 \end{cases}
 \qquad X_{j+1}X_j=v^2X_jX_{j+1}.
\end{equation}
We use the normalization of Lee--Li--Rupel--Zelevinsky \cite{LLRZ}.
Their quantum greedy basis is indexed by \(a=(a_1,a_2)\in\Z^2\); its
element \(G[a]\) has a finite expansion
\[
 G[a]=\sum_{p,q\in\mathbb N}e_a(p,q)
 X^{(-a_1+bp,-a_2+cq)},\qquad e_a(0,0)=1,
 \quad X^{(r,s)}=v^{rs}X_1^rX_2^s.
\]
This is the meaning of pointedness used below. The coefficients are
specified by \eqref{eq:greedyrec}.

The classical greedy and theta bases agree in rank two
\cite{CGMMRSW}. Burcroff--Lee give quantum-weighted bijections
in type \(A_2\) and the Kronecker case \cite{BL}.
Davison--Mandel construct quantum theta bases in the
skew-symmetric setting \cite{DM}; their Remark~4.4 also discusses the
skew-symmetrizable construction and the expected comparison with quantum
greedy elements. Gao's thesis \cite{Gao}, Theorems~3.12 and~3.15, states the
comparison in the symmetric and finite rank-two cases. Its
Conjecture~5.2 asks for the comparison when one exchange parameter
divides the other.

We use the following scattering convention. In the lattice
\(M=\Z^2\), the initial walls of the \(g\)-diagram carry
\begin{equation}\label{eq:ginputintro}
 \Psi_{v^b}(X^{(-b,0)}),\qquad
 \Psi_{v^c}(X^{(0,c)})
\end{equation}
on the horizontal and vertical axes, respectively. Here
\(\Psi_w(z)=\exp\bigl(\sum_{j\geq1}(-1)^{j+1}z^j/
(j(w^j-w^{-j}))\bigr)\) is a formal series over \(\mathbb Q(w)\).
A broken line carries a based monomial on each straight segment;
its velocity is the negative exponent, and at a wall it selects a
term of the wall-crossing action. The theta function is the sum of
its terminal monomials for a fixed initial exponent and endpoint.
Section~\ref{sec:scattering} defines the completion, crossing signs,
and genericity used in this construction.
For \(g=(g_1,g_2)\), set
\begin{equation}\label{eq:rhointro}
 T(g)=(g_1+b\min(g_2,0),g_2),\qquad
 \rho(g)=-T(g).
\end{equation}

\begin{theorem}\label{thm:main}
For all \(b,c\in\Z_{>0}\) and \(g\in\Z^2\), the broken-line theta function
\(\vartheta_{g,Q}\) of the diagram \eqref{eq:ginputintro}, with \(Q\)
generic in the first quadrant, is a Laurent polynomial and
\[
 \vartheta_{g,Q}=G[\rho(g)].
\]
Equivalently, put \(g(a)=\bigl(b\pos{a_2}-a_1,-a_2\bigr)\);
then \(\vartheta_{g(a),Q}=G[a_1,a_2]\).
The family of these theta functions is a
\(\Z[v^{\pm1}]\)-basis of \(\A_v(b,c)\).
\end{theorem}

We next allow arbitrary geometric coefficients in the sense of
Berenstein--Zelevinsky and Davison--Mandel: a finite frozen lattice,
arbitrary integral frozen exchange rows, and a compatible quantum
form. Write
\[
 B_0=\begin{pmatrix}0&-b\\c&0\end{pmatrix},\qquad
 \widetilde B=\begin{pmatrix}B_0\\ C\end{pmatrix},\qquad
 \widetilde B^{\mathsf T}\widetilde\Lambda=(D\ \ 0),\qquad
 D=\operatorname{diag}(d_1,d_2)>0,
\]
where \(C\in\operatorname{Mat}_{f\times2}(\Z)\),
\(\widetilde\Lambda\) is integral and skew, and
\(R=\Z[v^{\pm1}]\). Full based monomials multiply by
\(X^mX^{m'}=v^{\widetilde\Lambda(m,m')}X^{m+m'}\).
Compatibility forces
\(\ell=d_1/c=d_2/b\in\Q_{>0}\); this scale need not be integral.
Here \(f\in\mathbb N\) is the number of frozen variables, and the first
two coordinates of \(m\in\mathbb Z^{2+f}\) are mutable. The frozen
quantum Laurent torus is
\(\mathcal R_F=\bigoplus_{z\in\mathbb Z^f}R X^{(0,z)}\), with
multiplication inherited from the full torus. A balanced Laurent
coefficient means the coefficient of a based monomial \(X^m\).
The bar operation is the anti-involution fixing every \(X^m\) and
sending \(v\) to \(v^{-1}\). The ordinary algebra is generated by the
mutable cluster variables and the chosen frozen coefficient ring;
the upper algebra is the intersection of the corresponding seed
Laurent charts, all inside the same skew field.
The initial ice walls are \(\{x_i=0\}\), \(i=1,2\), with factors
\(\Psi_{v^{d_i}}(X^{\widetilde B e_i})\), where \(e_i\) is the
\(i\)-th coordinate vector. Their consistent completion defines
\(\Theta_m\), with initial exponent \(m\), as in
Section~\ref{sec:coefficients}. We evaluate it at a generic endpoint
whose mutable projection lies in the initial first quadrant.

\begin{theorem}\label{thm:coeff-main}
For every such ice seed, the following statements hold.
\begin{enumerate}
\item All broken-line theta functions \(\Theta_m\),
\(m\in\Z^{2+f}\), are finite integral Laurent polynomials. They
form a seed-independent \(R\)-basis of the ordinary quantum
cluster algebra, which equals its upper algebra, when all frozen
variables are inverted. The functions \(\Theta_{(g,0)}\),
\(g\in\Z^2\), form both a left and a right
\(\mathcal R_F\)-basis.
\item The theta functions equal the homogeneous lifts of the
quantum greedy elements given in
\eqref{eq:coefficient-greedy-formula}. Their balanced Laurent
coefficients lie in \(\Z[v^{\pm\gcd(d_1,d_2)}]\), and every
theta function is bar invariant.
\item If \(C\geq0\) entrywise, including \(C=I_2\), the ordinary
and upper algebras also agree without inverting any frozen
variable. Their \(R\)-basis is
\(\{\Theta_{(g,z)}:g\in\Z^2,\ z\in\mathbb N^f\}\), and
\(\{\Theta_{(g,0)}\}\) is a left and right basis over the frozen
quantum polynomial algebra.
\end{enumerate}
\end{theorem}

The first two assertions are proved in
Section~\ref{sec:coefficients}, and the third in
Theorem~\ref{thm:polynomial-coefficients}. Proposition~\ref{prop:disconnected}
treats the disconnected principal part. For signed coefficient rows,
Proposition~\ref{prop:frozen-counterexample} gives an example whose
ordinary and upper algebras differ with polynomial frozen coefficients.

The coefficient-free proof uses the denominator-coordinate geometry of
\cite[Sections~4--5]{CGMMRSW}. We verify the quantum coordinate change
and bound broken lines by their angular momentum. Grouping lines at
their last axis crossing gives Gaussian divisibility and strict bounds
on the quotient degrees. These bounds imply the greedy recurrences.

For geometric coefficients, projection identifies the broken lines
with their rank-two counterparts. An orthogonal finite lattice
extension separates the mutable and frozen factors; taking the
original lattice degree and descending the parameter ring give the
integral basis. A frozen-valuation argument with acyclic standard
monomials treats nonnegative coefficient blocks.

\section{Quantum tori and support rigidity}\label{sec:algebra}

\subsection{Normalization}

Put \(R=\Z[v^{\pm1}]\), \(K=\Q(v)\), and
\[
 \T_R=R\langle X_1^{\pm1},X_2^{\pm1}
          \mid X_2X_1=v^2X_1X_2\rangle.
\]
For \(u=(u_1,u_2)\in M\), define
\[
 X^u=v^{u_1u_2}X_1^{u_1}X_2^{u_2},\qquad
 \Lambda(u,w)=u_2w_1-u_1w_2.
\]
Then
\begin{equation}\label{eq:torus}
 X^uX^w=v^{\Lambda(u,w)}X^{u+w}.
\end{equation}
The anti-involution determined by
\(\overline v=v^{-1}\), \(\overline{X_i}=X_i\) fixes every \(X^u\).
For example, the exchange matrix
\[
 B_0=\begin{pmatrix}0&-b\\c&0\end{pmatrix}
\]
is compatible with the matrix of \(\Lambda\), since
\(B_0^{\mathsf T}\Lambda=\operatorname{diag}(c,b)\).
Changing the orientation changes the accompanying sign convention.

For \(n\geq0\), write
\begin{equation}\label{eq:P}
 P_{n,w}(z)=\prod_{j=1}^n(1+w^{n+1-2j}z)
           =\sum_{k=0}^n\qb{n}{k}{w}z^k .
\end{equation}
The coefficient of \(z^k\) in this product defines the symmetric
quantum binomial coefficient \(\qb{n}{k}{w}\) for \(0\leq k\leq n\);
we set it to zero for \(k<0\) or \(k>n\), with \(n\geq0\).
In particular, \(P_{0,w}=1\). We shall use
\begin{equation}\label{eq:inverseP}
 P_{n,w}(z)^{-1}
 =\sum_{k\geq0}(-1)^k\qb{n+k-1}{k}{w}z^k\quad(n>0).
\end{equation}
When \(n=0\), the inverse is \(1\); coefficients of positive powers
on the right are interpreted as zero.

\subsection{The greedy input}

A finite element of \(\T_K\) is pointed at \(a\in\Z^2\) if it has the
form
\begin{equation}\label{eq:pointed}
 F=\sum_{p,q\geq0}e(p,q)X^{(-a_1+bp,-a_2+cq)},\qquad e(0,0)=1.
\end{equation}
The existence and basis theorems in \cite[Theorems~1.7 and~1.9]{LLRZ}
give integral elements \(G[a]\) characterized by the following
recurrences, imposed only for \(p+q>0\):
\begin{equation}\label{eq:greedyrec}
e(p,q)=
\begin{cases}
\displaystyle\sum_{k=1}^{p}(-1)^{k-1}
 \qb{\pos{a_2-cq}+k-1}{k}{v^b}e(p-k,q),
 &ca_1q\leq ba_2p,\\[6pt]
\displaystyle\sum_{\ell=1}^{q}(-1)^{\ell-1}
 \qb{\pos{a_1-bp}+\ell-1}{\ell}{v^c}e(p,q-\ell),
 &ca_1q\geq ba_2p.
\end{cases}
\end{equation}
On the equality line both formulas apply. These elements form an
\(R\)-basis of \(\A_v(b,c)\), are bar invariant, contain the normalized
cluster monomials, and specialize to the classical greedy basis.
The basis is independent of the initial cluster.

For a pointed element define row and column polynomials
\[
 E_q(z)=\sum_{p\geq0}e(p,q)z^p,\qquad
 F_p(z)=\sum_{q\geq0}e(p,q)z^q.
\]
The divisibility conditions are
\begin{align}
 P_{a_2-cq,v^b}(z)&\mid E_q(z)
       &&(q\geq0,\ cq<a_2),\label{eq:rowdiv}\\
 P_{a_1-bp,v^c}(z)&\mid F_p(z)
       &&(p\geq0,\ bp<a_1).\label{eq:coldiv}
\end{align}
Divisibility is in \(K[z]\), where \(z\) is an auxiliary commuting
indeterminate. The degree of a nonzero polynomial is its largest
exponent with nonzero coefficient; we set \(\deg0=-\infty\). These are also the conditions for
Laurentness in the adjacent tori; see \cite[Lemma~2.2]{LLRZ}.
For our application we prove the conditions directly from broken
lines, before asserting membership in a cluster algebra.

\subsection{A support bound sufficient for uniqueness}

Assume \(a_1,a_2>0\), and let
\[
 O=(0,0),\quad A=(a_2,0),\quad
 B=(a_1/b,a_2/c),\quad C=(0,a_1).
\]
Define \(\Omega_a\) to be the union of the closed triangles \(OAB\)
and \(OBC\), with the segments \(AB\) and \(BC\) removed and the
points \(A,C\) restored. Thus \(B\) is excluded and the two
coordinate-axis edges are included. This definition also covers
degenerate triangles.

\begin{proposition}[Support rigidity]\label{prop:rigidity}
Let \(a_1,a_2>0\). A finite pointed element \eqref{eq:pointed}
satisfying \eqref{eq:rowdiv}--\eqref{eq:coldiv} and
\[
 \{(p,q):e(p,q)\ne0\}\subseteq\Omega_a
\]
equals \(G[a]\).
\end{proposition}

\begin{proof}
Set \(\gamma=ca_1/(ba_2)\). For \(0<q<a_2/c\), every point of
\(\Omega_a\) on row \(q\) satisfies
\begin{equation}\label{eq:rowbound}
 p<a_2-cq+\gamma q.
\end{equation}
Indeed, this is the strict bound from \(AB\) on the first triangle;
on the second triangle \(p\leq\gamma q\), which is smaller than
the right side. If \(q\geq a_2/c\), every point on that row
satisfies \(p<\gamma q\). At equality the first triangle could
contribute only the excluded point \(B\), and the second triangle
lies on the other side of \(OB\).

For \(0<q<a_2/c\), put \(n=a_2-cq\). By divisibility,
\(E_q=P_{n,v^b}H_q\), where \(H_q\in K[z]\). Since \(P_{n,v^b}\)
is monic of degree \(n\), \eqref{eq:rowbound} implies
\[
 \deg H_q\leq\lceil\gamma q\rceil-1.
\]
The coefficient of \(z^p\) in \(H_q=E_q/P_{n,v^b}\) is consequently
zero for every \(p\geq\gamma q\). Formula \eqref{eq:inverseP}
gives the first recurrence in \eqref{eq:greedyrec}.
For \(q\geq a_2/c\) and \(p\geq\gamma q\), the coefficient \(e(p,q)\)
is zero by the support bound, and the same recurrence is zero because
\(\pos{a_2-cq}=0\).

The row \(q=0\) has degree at most \(a_2\). Its divisibility and
constant coefficient imply \(E_0=P_{a_2,v^b}\), which gives the
first recurrence on that axis for \(p>0\).

The corresponding column bounds are
\[
 q<a_1-bp+\gamma^{-1}p\quad(0<p<a_1/b),\qquad
 q<\gamma^{-1}p\quad(p\geq a_1/b).
\]
The column \(p=0\) is the segment \(0\leq q\leq a_1\).
The same division argument, now using \eqref{eq:coldiv}, gives
the second recurrence, including the column axis.

In particular, a nonzero lattice point on \(p=\gamma q\) satisfies
both recurrences. Below \(B\), both quotient-degree bounds are
strict. At or beyond \(B\), the point is outside \(\Omega_a\)
and both positive parts in the recurrences vanish.
Thus the equality line contributes no free coefficient.
Each recurrence expresses a coefficient in terms of smaller
\(p+q\). Induction, with \(e(0,0)=1\), proves equality with
the greedy family.
\end{proof}

\begin{remark}\label{rem:rigidityprior}
The sharper support characterization in
\cite[Proposition~3.7]{LLRZ} already implies Gao's
Proposition~3.11. The proposition above also permits a larger
support region in the two trapezoid cases. The comparison in
Section~\ref{sec:lastaxis} uses degree bounds on the last-axis
quotients directly.
\end{remark}

\section{Rank-two quantum scattering}\label{sec:scattering}

\subsection{Completions and wall crossings}

For a strongly convex monoid \(P\subset M\), write \(P^+=P\setminus\{0\}\).
We use either
\[
 P_d=\Z_{\geq0}(b,0)+\Z_{\geq0}(0,c),\qquad
 P_g=\Z_{\geq0}(-b,0)+\Z_{\geq0}(0,c).
\]
Both have the degree \(p+q\) on their displayed generators.
The Lie algebra
\[
 \g_P=\bigoplus_{n\in P^+}KX^n,\qquad
 [X^n,X^m]=(v^{\Lambda(n,m)}-v^{-\Lambda(n,m)})X^{n+m}
\]
has a degree completion \(\widehat{\g}_P\).
The Lie algebra quotients discard homogeneous terms of degree greater
than \(N\); they are finite dimensional and nilpotent. For the action
on series supported in \(m+P\), truncation instead means retaining
\(X^{m+n}\) with the increment \(n\in P\) of degree at most \(N\).
Products use the induced increment filtration on \((m+m')+P\).
This defines the inverse limits on translated supports; it does not
require an ideal filtration of the full Laurent torus.
The exponential group \(G_P=\exp(\widehat{\g}_P)\) acts by
conjugation on the completed quantum Laurent algebra consisting
of formal sums whose support lies in a finite union of translates
of \(P\). Strong convexity makes multiplication of these sums
well defined.

Define
\begin{equation}\label{eq:dilog}
 \Psi_w(z)=
 \exp\left(\sum_{j\geq1}
       \frac{(-1)^{j+1}}{j(w^j-w^{-j})}z^j\right).
\end{equation}
This is a formal series over a rational function field.
It satisfies
\begin{equation}\label{eq:dilogfunctional}
 \Psi_w(w^2z)=(1+wz)\Psi_w(z).
\end{equation}
The infinite-product notation
\(\prod_{j\geq0}(1+w^{2j+1}z)^{-1}\) denotes this same formal
series.

A wall is a line through the origin or a ray from the origin,
parallel to a nonzero \(n\in P\), together with
\[
 f\in\exp\left(\prod_{j\geq1}KX^{jn_0}\right),
\]
where \(n_0\) is a primitive lattice vector along \(n\); only
exponents belonging to \(P\) are used. A diagram has finitely many
nontrivial walls at every degree. Coincident walls are combined
by multiplying their commuting factors. A crossing with velocity
\(\xi\) acts by
\begin{equation}\label{eq:crosssign}
 \Ad_{f^\epsilon},\qquad
 \epsilon=-\sgn\Lambda(n,\xi).
\end{equation}
Path-ordered products compose these automorphisms in the order
of crossing. A generic path avoids the origin and crosses each relevant wall
transversely. All such statements are interpreted in each finite-degree
truncation. A diagram is consistent if every closed generic path has
trivial product.

\begin{lemma}\label{lem:axis}
Let \(n=(b,0)\), \(u=(r,s)\), and \(N=|s|\).
When a broken line with exponent \(u\) crosses the horizontal
initial wall \(\Psi_{v^b}(X^n)\), its possible next exponents and
coefficients are
\[
 u+kn,\qquad \qb{N}{k}{v^b},\qquad 0\leq k\leq N.
\]
For the vertical wall \(\Psi_{v^c}(X^{(0,c)})\), replace \(N\)
by \(|r|\), \(n\) by \((0,c)\), and \(v^b\) by \(v^c\).
\end{lemma}

\begin{proof}
Write \(\lambda=\Lambda(n,u)\). Commutation gives
\[
 \Ad_{\Psi_{v^b}(X^n)}X^u
 =X^u\frac{\Psi_{v^b}(v^{2\lambda}X^n)}
                  {\Psi_{v^b}(X^n)}.
\]
Since the broken-line velocity is \(-u\), the crossing exponent
in \eqref{eq:crosssign} is \(\sgn\lambda\).
If \(\lambda=bN>0\), the quotient is
\(\prod_{j=1}^N(1+v^{b(2j-1)}X^n)\).
Its \(k\)-th coefficient is
\(v^{bNk}\qb{N}{k}{v^b}\), and multiplication by \(X^u\)
contributes \(v^{-bNk}\). The factors cancel.
If \(\lambda=-bN<0\), the inverse adjoint produces the analogous
product with negative odd powers, with the same cancellation.
The vertical calculation uses \(|\Lambda((0,c),u)|=c|r|\).
\end{proof}

\subsection{The denominator diagram}

\begin{proposition}\label{prop:dexist}
There is a unique consistent central diagram \(\D^d(b,c)\),
up to equivalence, with initial walls
\[
 (\R(1,0),H),\quad(\R(0,1),V),\qquad
 H=\Psi_{v^b}(X^{(b,0)}),\quad
 V=\Psi_{v^c}(X^{(0,c)}),
\]
and with all other walls on rays
\(\R_{\leq0}(bp,cq)\), \(p,q>0\).
\end{proposition}

\begin{proof}
In the degree completion for \(P_d\), form
\[
 C=H^{-1}VHV^{-1}.
\]
Its logarithm has no pure horizontal or vertical term: projection
to either axis makes this commutator equal to \(1\).
Factor \(C\) as an ordered product of elements supported on
individual mixed rays, in increasing order of their geometric slopes
\(cq/(bp)\).

This factorization exists and is unique by induction on degree.
After degrees below \(N\) have been fixed, the remaining
degree-\(N\) error is a finite sum of monomials. At this degree
these corrections commute with all positive-degree terms.
Assign each correction to its ray and continue.
This also proves finiteness modulo every degree.

Put the resulting factor on the negative ray parallel to its
exponents. A counterclockwise circuit, with the crossing convention
\eqref{eq:crosssign}, has trivial product precisely when
\[
 VH=HCV,
\]
which holds by construction. At each finite degree, all walls are
central and the only joint is the origin. Cancellation of opposite
crossings and the circuit identity prove consistency for every
closed path. The ordered factorization also proves uniqueness.
\end{proof}

\subsection{Broken lines}

For a generic endpoint \(Q\) away from the walls, a broken line
with initial exponent \(m\in M\) is a continuous piecewise affine map
\(\gamma:(-\infty,0]\to M_\R\setminus\{0\}\) ending at \(Q\).
It has finitely many bends; inserting or removing a subdivision
with unchanged attached monomial does not give a different line.
Each segment carries \(aX^u\), with velocity \(-u\).
The initial segment carries \(X^m\).
At a crossing, the next monomial is a term of the crossing
automorphism applied to the incoming monomial. The unchanged term
is allowed. A bend means a change of exponent.
Set \(\vartheta^d_{0,Q}=1\), and for other \(m\) set
\[
 \vartheta^d_{m,Q}=\sum_\gamma \operatorname{Mono}(\gamma),
\]
where the summand is the last attached monomial.

For the quantum-torus Lie algebra above, formal broken-line sums
are well defined degree by degree and satisfy endpoint transport
\cite[Example~2.1(iii), Proposition~2.13, and Theorem~2.14]{Mandel}.
We check the structural hypotheses of that result. Commutators are
derivations of the quantum torus, preserve the exponent grading,
and satisfy
\[
 [X^n,X^u]=0\quad\text{if }\Lambda(n,u)=0.
\]
A wall with direction \(n\) has normal functional
\(u\mapsto\Lambda(n,u)\). For two directions \(n,n'\), the normal
in the bracket condition of that reference is
\[
 \Lambda(n,n')\Lambda(n+n',\,\cdot\,).
\]
It is parallel to the normal of the bracket degree \(n+n'\);
if \(\Lambda(n,n')=0\), the bracket itself is zero. Thus its
wall-compatible Lie-action condition holds. For the unsaturated monoids \(P_d,P_g\), extend the grading to the
saturated lattice cone by assigning zero Lie components to missing degrees.
The action on each translated support is completed in increment
degree as above.

For fixed initial exponent and total increment, there are finitely
many relevant walls and bend sequences: each bend increases degree.
For a generic endpoint, reconstructing a prescribed sequence backward
intersects a straight segment with a specified wall at each step,
so gives at most one path. Endpoint transport identifies the sums by
the path-ordered product. In particular they are constant in a chamber.
We apply the theorem to the consistent completion of the initial walls.

We may choose \(Q\) away from the countably many exceptional
directions that occur in the construction. In particular,
\(Q_1/Q_2\) may be chosen irrational.

\section{Geometric support and polynomiality}\label{sec:geometry}

Every bend in \(\D^d\) adds a nonzero exponent in \(P_d\).
This follows from the ray subgroup containing its wall factor,
and holds for a factor or its inverse. Consequently both exponent
coordinates are nondecreasing along a broken line.
The unbent line is the unique contribution with final exponent \(m\).
Thus
\begin{equation}\label{eq:thetapointed}
 \vartheta^d_{m,Q}
 =\sum_{p,q\geq0}e(p,q)X^{m+(bp,cq)},\qquad e(0,0)=1
\end{equation}
as a formal series.

\begin{lemma}\label{lem:easy}
If \(Q\) lies in the first quadrant, then
\[
 \vartheta^d_{m,Q}=
 \begin{cases}
 X^m,&m_1,m_2\geq0,\\[2pt]
 \displaystyle\sum_{k=0}^{-m_2}\qb{-m_2}{k}{v^b}
     X^{m+(bk,0)},&m_1\geq0>m_2,\\[6pt]
 \displaystyle\sum_{k=0}^{-m_1}\qb{-m_1}{k}{v^c}
     X^{m+(0,ck)},&m_2\geq0>m_1.
 \end{cases}
\]
\end{lemma}

\begin{proof}
If \(m_1\geq0\), the first exponent stays nonnegative and the
spatial first coordinate is nonincreasing. A line ending at
\(Q_1>0\) therefore stays to the right of the vertical axis.
It meets neither that axis nor an additional wall in the third
quadrant. If \(m_2\geq0\), it remains in the first quadrant.
If \(m_2<0\), it crosses only the horizontal wall, and
Lemma~\ref{lem:axis} gives the second formula. Interchanging the
coordinates gives the third formula. This argument includes
zero coordinates of \(m\).
\end{proof}

It remains to consider \(m_1,m_2<0\).
For a segment with position \(z\) and exponent \(u\), put
\[
 L=z_2u_1-z_1u_2.
\]

\begin{lemma}\label{lem:momentum}
The number \(L\) is constant along a broken line.
A line starting in the third quadrant and ending in the first
quadrant enters the first quadrant through the positive horizontal
axis if \(L>0\), and through the positive vertical axis if \(L<0\).
For generic \(Q\), these are the only possibilities.
\end{lemma}

\begin{proof}
On a segment, \(z'=-u\), so the derivative of \(L\) vanishes.
At a bend, \(u\) changes to \(u+kn\) and \(z\) is parallel to \(n\).
The change in \(L\) is \(k(z_2n_1-z_1n_2)=0\).

If a spatial coordinate is nonpositive, its exponent must still
be negative for the line eventually to reach the corresponding
positive coordinate of \(Q\). Otherwise the exponent can never
decrease, and that spatial coordinate can never increase again.
The line therefore passes through quadrant II or quadrant IV
before entering quadrant I. At a positive horizontal crossing,
\(L=-z_1u_2>0\); at a positive vertical crossing,
\(L=z_2u_1<0\).
After entering quadrant I there are no further walls, and the
line cannot leave and return.
Finally \(L=Q_2u_1-Q_1u_2\) for the final exponent \(u\).
Genericity excludes zero unless \(u=0\). A zero final exponent
would give a stationary final segment, forcing \(Q\) to be its
last bend point on a wall, which is impossible.
\end{proof}

\begin{lemma}\label{lem:bounds}
Let \(m_1,m_2<0\), and let \(e=(e_1,e_2)\) be the final exponent
of a broken line ending in the first quadrant.
If \(L>0\), then
\begin{equation}\label{eq:boundpositive}
 m_2\leq e_2<0,\qquad
 m_1\leq e_1\leq (m_1/m_2-b)e_2.
\end{equation}
Equality in the last bound is possible only for
\(e=(m_1-bm_2,m_2)\).
If \(L<0\), then
\begin{equation}\label{eq:boundnegative}
 m_1\leq e_1<0,\qquad
 m_2\leq e_2\leq (m_2/m_1-c)e_1,
\end{equation}
with equality in the last bound only for
\(e=(m_1,m_2-cm_1)\).
\end{lemma}

\begin{proof}
Suppose \(L>0\). Before the final horizontal crossing, both
exponent coordinates are negative. They must be negative in
quadrant III; the crossing into quadrant IV changes only the
second exponent, and this exponent must remain negative in order
to reach the upper half-plane.

Every earlier bend takes place at \(z=-tn\), \(t>0\), and
changes \(u\) to \(u'=u+kn\), \(k>0\).
Then
\[
 u_1u'_2-u_2u'_1
 =k(u_1n_2-u_2n_1)=-kL/t<0.
\]
As \(u_1,u'_1<0\), this implies
\(u'_2/u'_1<u_2/u_1\).
If \(u\) is the exponent immediately before the last horizontal
crossing, it follows that
\[
 u_2/u_1\leq m_2/m_1,
\]
and equality is possible only if there has been no earlier bend.
The last crossing has
\[
 e=(u_1+bk,u_2),\qquad 0\leq k\leq-u_2.
\]
Hence
\[
 e_1\leq u_1-bu_2
       =(u_1/u_2-b)e_2
       \leq(m_1/m_2-b)e_2.
\]
The monotonicity of exponent coordinates gives the lower bounds.
Equality requires \(u=m\) and \(k=-m_2\).
The argument for \(L<0\) interchanges the coordinates.
\end{proof}

\begin{proposition}\label{prop:finite}
For every \(m\in M\), \(\vartheta^d_{m,Q}\) is a Laurent polynomial
over \(K\).
\end{proposition}

\begin{proof}
The cases in Lemma~\ref{lem:easy} are finite.
In the remaining case, \eqref{eq:boundpositive} and
\eqref{eq:boundnegative} are alternatives. Their union is bounded:
in the first alternative \(e_2\) lies in a bounded interval and
both bounds on \(e_1\) are finite; the second alternative has
the transposed property. There are therefore only finitely many possible final lattice exponents.
Each such exponent has a fixed finite degree increment from \(m\),
so its coefficient is a finite broken-line sum.
Thus the theta series is a Laurent polynomial.
\end{proof}

\section{Divisibility from the final axis crossing}\label{sec:lastaxis}

\begin{proposition}\label{prop:packet}
The coefficients of \(\vartheta^d_{m,Q}\) in
\eqref{eq:thetapointed} satisfy
\[
 P_{-m_2-cq,v^b}(z)\mid E_q(z)\quad(m_2+cq<0),
 \qquad
 P_{-m_1-bp,v^c}(z)\mid F_p(z)\quad(m_1+bp<0).
\]
\end{proposition}

\begin{proof}
Fix a row with \(s=m_2+cq<0\).
By chamber independence we may choose a generic \(Q\) satisfying
\begin{equation}\label{eq:endpointchoice}
 Q_1>\max(0,-m_1)Q_2.
\end{equation}
Every broken line in this row has negative second exponent on
every segment, since exponents are nondecreasing and its final
second exponent is \(s<0\). Its vertical coordinate increases
strictly. It crosses the horizontal axis exactly once.

Its final first exponent \(r\) is at least \(m_1\), and \(-s\geq1\).
Its angular momentum therefore satisfies
\[
 L=Q_2r-Q_1s\geq Q_2m_1+Q_1>0.
\]
Entry into the first quadrant through the positive vertical axis
would give \(L<0\). Thus the horizontal crossing is on the positive
axis and is the last wall crossing. This also excludes a horizontal
crossing on the negative axis, which would have to be followed
by a positive vertical crossing.

Cut such a line immediately before the horizontal crossing.
Write its terminal prefix monomial as \(C X^{(r_0,s)}\), with
\(r_0=m_1+bp_0\).
Positive dilation of space, with the corresponding time change,
preserves the condition \(\gamma'=-u\), all central walls,
and all attached coefficients. Normalize the prefix endpoint
to \((1,0)\).

Conversely, fix such a normalized prefix and let \(N=-s\).
For every \(0\leq k\leq N\), set
\[
 x_k=Q_1-\frac{Q_2(r_0+bk)}s.
\]
It is positive by \eqref{eq:endpointchoice}, since
\(r_0+bk\geq m_1\) and \(-s\geq1\).
Dilate the prefix to end at \((x_k,0)\), choose the \(k\)-th
term in the horizontal expansion of Lemma~\ref{lem:axis}, and
attach the straight segment to \(Q\). That segment lies entirely
in the first quadrant after the crossing. It meets no further wall.
Cutting and this extension are inverse constructions.

Each normalized prefix therefore contributes the whole polynomial
\[
 C z^{p_0}P_{N,v^b}(z)
\]
to the row. The sum of prefix monomials is a polynomial: the
choice \(k=0\) extends every prefix to a valid full broken line,
so the individual bounds in Lemma~\ref{lem:bounds}, or the
easy cases in Lemma~\ref{lem:easy}, bound \(p_0\); degreewise
finiteness bounds the number of contributions for each \(p_0\).
Consequently
\[
 E_q(z)=P_{N,v^b}(z)
        \sum_{\text{normalized prefixes}}Cz^{p_0}
\]
in \(K[z]\).

For the column assertion choose
\(Q_2>\max(0,-m_2)Q_1\), cut before the positive vertical crossing,
and normalize to \((0,1)\). The possible increments are
\((0,c\ell)\), \(0\leq\ell\leq-r\), for
\(r=m_1+bp<0\), with coefficients
\(\qb{-r}{\ell}{v^c}\). This gives the stated column divisor.
Chamber independence identifies the polynomials obtained from the
separate row and column endpoint choices.
\end{proof}

\begin{remark}
The rows \(q=0\) and columns \(p=0\) are included.
If the relevant exponent is zero, its putative divisor is
\(P_0=1\). 
\end{remark}

\begin{proposition}[Quotient-degree bounds]\label{prop:quotient}
Suppose \(m=-a\), \(a_1,a_2>0\), and put
\(\gamma=ca_1/(ba_2)\).
The quotients in Proposition~\ref{prop:packet} satisfy
\[
 E_0=P_{a_2,v^b},\qquad
 E_q=P_{a_2-cq,v^b}H_q,\quad
 \deg H_q<\gamma q\quad(0<q<a_2/c),
\]
and
\[
 F_0=P_{a_1,v^c},\qquad
 F_p=P_{a_1-bp,v^c}J_p,\quad
 \deg J_p<\gamma^{-1}p\quad(0<p<a_1/b).
\]
Also \(e(p,q)=0\) whenever \(bp\geq a_1\) and \(cq\geq a_2\).
\end{proposition}

\begin{proof}
Consider a prefix occurring in the row packet construction.
Immediately before its positive horizontal crossing its exponent
is \((r_0,s)\), with both entries negative. The slope calculation
in the proof of Lemma~\ref{lem:bounds}, applied before this last
crossing, gives
\[
 \frac{r_0}{s}\geq\frac{m_1}{m_2}=\frac{a_1}{a_2}.
\]
Equality is possible only if the prefix has no bend.
For \(q>0\) there has been a positive change in the second
exponent, so at least one bend has occurred. Multiplication by
\(s<0\) therefore gives
\[
 -a_1+bp_0=r_0<
 \frac{a_1}{a_2}(-a_2+cq),
 \qquad p_0<\gamma q.
\]
This bounds every individual monomial of the prefix quotient
\(H_q\), hence its degree. 

When \(q=0\), a prefix cannot bend: an additional wall or a
vertical-axis bend would increase its second exponent, while
its unique horizontal crossing has not yet occurred.
The normalized unbent prefix is unique, has coefficient \(1\),
and has \(p_0=0\). Thus \(H_0=1\).
For the column construction a normalized vertical prefix has
exponent \((r,s_0)\), \(r<0\), and negative angular momentum.
The coordinate-interchanged slope calculation gives
\[
 \frac{s_0}{r}\geq\frac{a_2}{a_1},
\]
strictly if \(p>0\). Therefore
\[
 -a_2+cq_0=s_0<
 \frac{a_2}{a_1}(-a_1+bp),\qquad
 q_0<\gamma^{-1}p.
\]
For \(p=0\) the same no-bend argument gives the unique prefix
of coefficient \(1\), hence \(J_0=1\).

Finally a line starting in quadrant III enters quadrant I
through one of its positive axes. If it enters horizontally,
its final second exponent is negative; if it enters vertically,
its final first exponent is negative. It cannot have both final
coordinates nonnegative. This proves the last assertion.
\end{proof}

\begin{theorem}\label{thm:dequality}
For every \(b,c>0\), \(m\in\Z^2\), and generic first-quadrant
endpoint \(Q\),
\[
 \vartheta^d_{m,Q}=G[-m_1,-m_2].
\]
\end{theorem}

\begin{proof}
Suppose both coordinates of \(m\) are negative and set \(a=-m\).
For \(0<q<a_2/c\), Proposition~\ref{prop:quotient} makes the
coefficient of \(z^p\) in \(E_q/P_{a_2-cq,v^b}\) zero whenever
\(p\geq\gamma q\). Expanding the inverse by
\eqref{eq:inverseP} gives the first greedy recurrence in
\eqref{eq:greedyrec}. The row \(q=0\) follows from
\(E_0=P_{a_2,v^b}\).
For \(q\geq a_2/c\) and \(p\geq\gamma q\), one also has
\(p\geq a_1/b\). The last assertion of
Proposition~\ref{prop:quotient} makes the coefficient zero,
as is the recurrence with \(\pos{a_2-cq}=0\).
The column bounds give the second recurrence for
\(p\leq\gamma q\). On equality both recurrences hold.
Induction on \(p+q\), starting at \(e(0,0)=1\), identifies
the coefficients with those of \(G[a]\).

If one coordinate of \(m\) is nonnegative,
Lemma~\ref{lem:easy} gives the elementary greedy formulas.
For example, when \(m_1\geq0>m_2\), the only coefficients are
\(\qb{-m_2}{p}{v^b}\) on \(q=0\); these satisfy
\eqref{eq:greedyrec} with the prescribed constant coefficient.
The other mixed quadrant is the column version, and when
both coordinates are nonnegative the element is \(X^m\).
\end{proof}

\section{The change from g- to denominator coordinates}\label{sec:shear}

We now identify the preceding construction with the quantum cluster
diagram specified in \eqref{eq:ginputintro}.
Put
\[
 A=\Psi_{v^b}(X^{(-b,0)}),\qquad
 B=\Psi_{v^c}(X^{(0,c)}).
\]
Use \(P_g\), and complete the two initial full axes by outgoing
rays in quadrant IV. The same ordered-factorization construction
as in Proposition~\ref{prop:dexist} defines a unique consistent
diagram \(\D^g\). With its factors ordered by increasing \(p/q\),
the circuit identity reads
\begin{equation}\label{eq:gfactor}
 ABA^{-1}=B\prod_i C_i,
\end{equation}
where \(C_i\) is supported in positive multiples of
\((-bp_i,cq_i)\), \(p_i,q_i>0\).
Its theta functions use the same broken-line definition as before,
with \(\vartheta_{0,Q}=1\).

\begin{lemma}\label{lem:wedge}
Every nontrivial outgoing factor of \(\D^g\) has
\(0<p_i\leq cq_i\).
The factor on the extremal ray is
\[
 \Psi_{v^c}(X^{(-bc,c)}).
\]
\end{lemma}

\begin{proof}
The adjoint calculation gives
\begin{equation}\label{eq:conj}
 \Ad_A X^{(0,c)}
 =\sum_{k=0}^c\qb{c}{k}{v^b}X^{(-bk,c)}.
\end{equation}
Conjugation commutes with products and formal logarithms.
Thus \(\Ad_A\log B=\log(ABA^{-1})\) is supported in the cone
generated by \((0,c)\) and \((-bc,c)\).
The degree-induction factorization stays in this cone, which
proves \(p_i\leq cq_i\).
On its extremal ray, only the \(k=c\) term of
\eqref{eq:conj} can contribute. Its coefficient is \(1\),
and repeated products of this monomial have no additional
commutation factor. The restriction of the logarithm to that
ray is therefore \(\log\Psi_{v^c}(X^{(-bc,c)})\).
\end{proof}

Let \(H_+=\{y\geq0\}\), \(H_-=\{y\leq0\}\), and let \(T\) be
the piecewise linear map in \eqref{eq:rhointro}.
Its restrictions are \(T_+=\id\) and
\[
 T_-=S,\qquad S(x,y)=(x+by,y).
\]
Both preserve \(\Lambda\). Their monomial maps are therefore
quantum torus automorphisms.

\begin{lemma}\label{lem:cut}
Let \(H=\Psi_{v^b}(X^{(b,0)})\), and let \(S\) also denote
the monomial automorphism \(X^u\mapsto X^{Su}\). Then, as
automorphisms of the quantum fraction field,
\begin{equation}\label{eq:cut}
 \Ad_H\circ S=\Ad_{A^{-1}}.
\end{equation}
\end{lemma}

\begin{proof}
Both sides fix \(X_1\). On \(X_2\), they equal
\[
 X_2(1+v^bX_1^{-b})^{-1}
 =X^{(b,1)}(1+v^{-b}X_1^b)^{-1}.
\]
The equality uses \(X^{(b,1)}=v^bX_1^bX_2
=v^{-b}X_2X_1^b\). Since the fraction field is generated by
\(X_1,X_2\), this proves the identity.
\end{proof}

\begin{proposition}\label{prop:shear}
The map \(T\) gives a coefficient-preserving correspondence
between broken lines of \(\D^g\) with initial exponent \(g\)
and endpoint \(Q\), and broken lines of \(\D^d\) with initial
exponent \(T(g)\) and endpoint \(T(Q)\).
In particular, if \(Q\) is in the first quadrant,
\[
 \vartheta_{g,Q}=\vartheta^d_{T(g),Q}.
\]
\end{proposition}

\begin{proof}
Transform wall supports and exponent labels by \(T_+\) in the
upper half-plane and by \(S\) in the lower half-plane.
Lemma~\ref{lem:wedge} sends every outgoing exponent to
\[
 (-bp+bcq,cq)=\bigl(b(cq-p),cq\bigr),
\]
which has nonnegative coordinates. The extremal outgoing ray
becomes the negative vertical half-line with label
\(\Psi_{v^c}(X^{(0,c)})\). It joins the unchanged positive
vertical half-line to form the full vertical initial wall.
The old negative vertical half-line becomes an outgoing ray.
Replace the horizontal label \(A\) by \(H\).

Write the ordered product in \eqref{eq:gfactor} as
\(\prod_i C_i=C_{<}C_*\), where
\(C_*=\Psi_{v^c}(X^{(-bc,c)})\) is the extremal factor and
\(C_{<}\) is the product of the remaining factors. Put
\(W=S(B)=\Psi_{v^c}(X^{(bc,c)})\).
Applying \(S\) to \eqref{eq:gfactor} gives
\[
 AWA^{-1}=W S(C_{<})B,
\]
since \(S(A)=A\) and \(S(C_*)=B\). In particular, \(S\) commutes
with \(\Ad_A\). The cut identity implies
\(\Ad_A\circ S=\Ad_{H^{-1}}\), hence
\(AWA^{-1}=H^{-1}BH\). Therefore
\[
 BH=H\bigl(W S(C_{<})\bigr)B.
\]
This is the denominator-diagram circuit identity. Its outgoing
factors are ordered by increasing slope: \(W\) has slope \(1/b\),
and the transformed mixed ray has slope
\(cq/[b(cq-p)]\), strictly increasing in \(p/q<c\).
Interpret these infinite products in the quantum skew power-series
algebra in \(X_2^c\) over \(K(X_1)\). All nonhorizontal factors have positive
\(X_2\)-degree. The wedge bound gives finitely many horizontal
powers at each such degree. The maps \(S,\Ad_A,\Ad_H\) act
continuously on this algebra, and \eqref{eq:cut} holds there.
The transformed labels lie in the \(P_d\)-completion.
Moreover a bound on the new degree \(p'+q\), where
\(p'=cq-p\geq0\), bounds both \(q\) and \(p\leq cq\);
degreewise local finiteness is preserved.
The transformed diagram is therefore the consistent completion
of the two positive initial axes. Uniqueness in
Proposition~\ref{prop:dexist} identifies it with \(\D^d\).

Within either open half-plane the coordinate map is linear
and preserves the quantum torus product, so it preserves
broken-line velocities and attached coefficients. Only a
horizontal crossing needs a separate check.
For a crossing from \(H_-\) to \(H_+\), write the incoming
exponent as \(u=(u_1,-N)\), \(N>0\).
In \(\D^g\), the possible exponents after the crossing are
\[
 u-k(b,0),\qquad 0\leq k\leq N,
\]
with coefficients \(\qb{N}{k}{v^b}\).
The transformed incoming exponent is
\(Su=u-N(b,0)\). In \(\D^d\) the possible exponents are
\[
 Su+j(b,0)=u-(N-j)(b,0),\qquad0\leq j\leq N.
\]
Taking \(j=N-k\) gives the same outgoing exponent and the
same coefficient, by quantum-binomial symmetry.
For a crossing from \(H_+\) to \(H_-\), write \(u=(u_1,N)\),
\(N>0\). The \(g\)-crossing has outgoing exponent
\(u-k(b,0)\) and coefficient \(\qb{N}{k}{v^b}\).
Its transformed outgoing exponent is
\[
 S(u-k(b,0))=u+(N-k)(b,0).
\]
The \(d\)-crossing from incoming exponent \(u\) has outputs
\(u+j(b,0)\) and coefficients \(\qb{N}{j}{v^b}\).
Again \(j=N-k\) matches both exponents and coefficients.
This is the downward cut identity
\(\Ad_{H^{-1}}=S\circ\Ad_A\).
Crossings without a bend are included:
their transformed images can bend at the cut, and vice versa.
If \(g_2=0\), the initial exponent is fixed by both branches.
The constructions are inverse on all segments and crossings.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Proposition~\ref{prop:shear} and Theorem~\ref{thm:dequality} give
\[
 \vartheta_{g,Q}=\vartheta^d_{T(g),Q}=G[-T(g)].
\]
The map \(T\) is an integral piecewise linear bijection, with
inverse
\[
 T^{-1}(m_1,m_2)=(m_1-b\min(m_2,0),m_2).
\]
Thus \(-T\) indexes all greedy elements exactly once.
Their integral basis theorem proves the stated basis assertion.
\end{proof}

\begin{corollary}\label{cor:properties}
The theta basis in Theorem~\ref{thm:main} is bar invariant and
independent, as a set of elements of the cluster algebra, of
the initial cluster. Every theta element is universally
Laurent and specializes at \(v=1\) to the corresponding
classical theta element.
\end{corollary}

\begin{proof}
Bar invariance, seed independence, universal Laurentness,
and specialization to the classical greedy element follow
from the corresponding properties of the quantum greedy
basis. The classical identification with theta functions,
in precisely the coordinates \(T\), is
\cite[Theorems~4.3 and~5.1]{CGMMRSW}.
\end{proof}

\section{Arbitrary geometric coefficients}\label{sec:coefficients}

\subsection{Compatible ice seeds}

Let \(f\geq0\), let \(\widetilde M=\Z^2\oplus F\), where
\(F=\Z^f\), and write \(\pi:\widetilde M\to\Z^2\) for projection.
Fix an extended exchange matrix and an integral skew form
\begin{equation}\label{eq:ice-data}
 \widetilde B=\begin{pmatrix}B_0\\ C\end{pmatrix},\qquad
 B_0=\begin{pmatrix}0&-b\\c&0\end{pmatrix},\qquad
 \widetilde B^{\mathsf T}\widetilde\Lambda=(D\ \ 0),\quad
 D=\operatorname{diag}(d_1,d_2),\quad d_i>0.
\end{equation}
Here \(b,c>0\) and \(C\) is an arbitrary integral \(f\times2\) matrix.
Write \(\beta_i=\widetilde B e_i\) and \(C_i=C e_i\).
The based quantum torus has multiplication
\[
 X^mX^{m'}=v^{\widetilde\Lambda(m,m')}X^{m+m'}.
\]
Its frozen subalgebra is the quantum Laurent torus
\(\mathcal R_F=\bigoplus_{z\in F}R X^{(0,z)}\).
It need not be commutative or central.

We use the ordinary and upper quantum cluster algebras
\(\widetilde\A\) and \(\widetilde{\mathcal U}\) of
\cite[Definition~4.12 and Section~5]{BZ}. In particular, every frozen
generator is inverted. This is also the convention in
\cite[Section~4.2]{DM}. For a seed toric frame \(M_t\), mutation at
\(i\in\{1,2\}\) replaces its \(i\)-th variable by
\begin{equation}\label{eq:ice-mutation}
 M_t\bigl(-e_i+[\widetilde b_{t,i}]_+\bigr)
 +M_t\bigl(-e_i+[-\widetilde b_{t,i}]_+\bigr),
\end{equation}
where positive parts are taken coordinatewise. The algebra
\(\widetilde\A\) is generated by all mutable cluster variables and
\(\mathcal R_F\); the upper algebra is the intersection of all seed
quantum Laurent tori in their common skew field.

Compatibility gives
\begin{equation}\label{eq:ice-pairing}
 \widetilde\Lambda(\beta_i,m)=d_i\pi_i(m).
\end{equation}
Since \(DB_0\) is skew, \(bd_1=cd_2\). Put
\begin{equation}\label{eq:ice-scale}
 \ell=\frac{d_1}{c}=\frac{d_2}{b}
      =\frac{\delta}{\kappa},\qquad
 \kappa=\gcd(b,c),\quad\delta=\gcd(d_1,d_2).
\end{equation}
The number \(\ell\) can be rational. We temporarily adjoin a suitable
root of \(v\) and use \(w=v^\ell\) for the coefficient-free comparison.
Let \(\Lambda_0(u,u')=u_2u'_1-u_1u'_2\). Then
\begin{align}
 \widetilde\Lambda(\widetilde Bn,m)
   &=\ell\Lambda_0(B_0n,\pi m),\label{eq:ice-action}\\
 \widetilde\Lambda(\widetilde Bn,\widetilde Bn')
   &=\ell\Lambda_0(B_0n,B_0n').\label{eq:ice-wall-pairing}
\end{align}
These identities hold for all integral arguments and extend rationally.

The initial quantum scattering walls in \(\widetilde M_\R\) are
\begin{equation}\label{eq:ice-walls}
 \bigl(\{x_1=0\},\Psi_{v^{d_1}}(X^{\beta_1})\bigr),\qquad
 \bigl(\{x_2=0\},\Psi_{v^{d_2}}(X^{\beta_2})\bigr).
\end{equation}
The positive monoid is \(\widetilde B\mathbb N^2\), graded by the sum
of the two coordinates in \(\mathbb N^2\). Completions use the degree \(n_1+n_2\) of the wall exponent
\(\widetilde Bn\). In this higher-dimensional setting a wall support
is a rational polyhedral cone of codimension one, possibly with
lineality, contained in
\(\{x:\widetilde\Lambda(\widetilde Bn,x)=0\}\).
At velocity \(\xi\), its factor \(f\) acts by
\(\Ad_{f^\epsilon}\), with
\(\epsilon=-\operatorname{sgn}\widetilde\Lambda(\widetilde Bn,\xi)\).
Coincident factors are multiplied. Joints are boundaries and
intersections of distinct wall supports of codimension at least two;
generic paths avoid these and meet walls transversely. Broken lines
have velocity equal to the negative full exponent. For the cylindrical
diagram below the joint is \(\pi_\mathbb R^{-1}(0)\).
Equation~\eqref{eq:ice-pairing} makes the wall actions the quantum
mutations with compatibility diagonal \(D\).

\subsection{The diagram and its broken lines}

\begin{lemma}\label{lem:cylindrical}
Let \(\D_0\) be the coefficient-free \(g\)-diagram at parameter
\(w=v^\ell\). Its consistent completion lifts to the completion of
\eqref{eq:ice-walls} by replacing every wall support \(\sigma\) by
\(\pi_\R^{-1}(\sigma)\) and every wall monomial
\(X_0^{B_0n}\) by \(X^{\widetilde Bn}\).
For every \(m\in\widetilde M\), the linear map
\begin{equation}\label{eq:coset-lift}
 \iota_m(X_0^{\pi m+B_0n})=X^{m+\widetilde Bn}
\end{equation}
intertwines all wall actions on the corresponding completed cosets.
\end{lemma}

\begin{proof}
The map \(B_0n\mapsto\widetilde Bn\) is an isomorphism of the wall
monoids because \(B_0\) is invertible. Equation~\eqref{eq:ice-wall-pairing}
identifies their quantum products, Lie brackets, exponentials, and
degree completions. It takes the initial factors to
\eqref{eq:ice-walls}. Equation~\eqref{eq:ice-action} proves the claimed
intertwining first for homogeneous brackets and then by exponentiation.

If a wall has exponent \(B_0n\), its inverse image lies in the
\(\widetilde\Lambda\)-orthogonal hyperplane of \(\widetilde Bn\).
The inverse image of an outgoing ray is outgoing, since both the wall
direction and its support project to the corresponding base data.
Lineality in \(F_\R\) is allowed in a wall support. Crossing signs and
path-ordered products are preserved by projection. Hence the lifted
diagram is consistent and has only outgoing added walls. Uniqueness
of consistent completion identifies it with the required diagram.
\end{proof}

Fix \(Q\in\widetilde M_\R\) such that \(Q_0=\pi Q\) is generic in
the first quadrant. Write
\[
 \vartheta^0_{g,Q_0}
   =\sum_{n\in\mathbb N^2} A_{g,n}(w)X_0^{g+B_0n}
\]
for the coefficient-free theta function.

\begin{proposition}[Theta lifting formula]\label{prop:theta-lift}
The broken-line theta function of \eqref{eq:ice-walls} is
\begin{equation}\label{eq:theta-lift}
 \Theta_{m,Q}
  =\sum_{n\in\mathbb N^2}A_{\pi m,n}(v^\ell)X^{m+\widetilde Bn}.
\end{equation}
It is a finite Laurent polynomial. Every balanced Laurent coefficient
belongs to \(\Z[v^{\pm\delta}]\), and \(\Theta_{m,Q}\) is bar invariant.
\end{proposition}

\begin{proof}
Suppose first that \(\pi m\ne0\). A base broken line has segment
exponents \(a(t)=\pi m+B_0n(t)\), with \(n(t)\in\mathbb N^2\).
Its unique lift with endpoint \(Q\) is
\begin{equation}\label{eq:lifted-path}
 \widetilde\gamma(t)
   =Q+\int_t^0\bigl(m+\widetilde Bn(s)\bigr)\,ds.
\end{equation}
Its derivative is the negative of the full segment exponent, and
its projection is the original broken line. Since every wall and
joint is a cylinder, the crossing times, transversality conditions,
and avoidance of joints agree. Lemma~\ref{lem:cylindrical} preserves
the coefficient of each chosen monomial. Conversely, projection
recovers \(n(t)\) uniquely, and \eqref{eq:lifted-path} recovers the
entire lifted line. A projected exponent cannot first become zero:
a bend \(a\mapsto a+ku=0\) would have \(\Lambda_0(u,a)=0\), so
the wall action would fix \(X_0^a\) and that nonconstant term would vanish.
This proves a coefficient-preserving bijection.

If \(\pi m=0\), the initial monomial commutes with all wall factors.
Only the straight frozen-direction line contributes, giving
\(\Theta_m=X^m\); for \(m=0\) this is the convention \(\Theta_0=1\).
Thus \eqref{eq:theta-lift} holds in this case as well.

Theorem~\ref{thm:main} makes the sum finite. Its greedy coefficients
use only Gaussian parameters \(w^b\) and \(w^c\). The recurrence
therefore puts them in \(\Z[w^{\pm\kappa}]\), which becomes
\(\Z[v^{\pm\delta}]\) by \eqref{eq:ice-scale}.
Bar invariance follows from the same property of the greedy
coefficients and of based monomials.
\end{proof}

In denominator coordinates, retain \(g(a)=(b[a_2]_+-a_1,-a_2)\), and let
\(e_a(p,q;w)\) be the coefficient in \eqref{eq:pointed} for the
quantum greedy element at parameter \(w\). Then
\begin{equation}\label{eq:coefficient-greedy-formula}
 \boxed{\displaystyle
 \Theta_{(g(a),z),Q}
  =\sum_{p,q\geq0} e_a(p,q;v^\ell)
    X^{(-a_1+bp,\,-a_2+cq,\,
            z+qC_1+([a_2]_+-p)C_2)} .}
\end{equation}
Indeed, the wall-monoid coordinates are
\(n=(q,[a_2]_+-p)\). Whenever a coefficient is nonzero,
\(p\leq[a_2]_+\), as follows from the coefficient-free
\(g\)-pointed expansion. The sum in
\eqref{eq:coefficient-greedy-formula} thus uses nonnegative wall
coordinates.

The coefficients satisfy \eqref{eq:greedyrec} with Gaussian
parameters \(v^{d_2}\) and \(v^{d_1}\), respectively. These
recurrences characterize the homogeneous greedy lift on its projected
support.

\subsection{Membership and the integral basis}

We prove algebra membership by an orthogonal lattice extension.
This is a rank-two instance of coefficient variation; see
\cite[Section~4, Lemmas~4.7, 4.9, and~4.10]{QinCoefficients}.

Write \(m_F\) for the frozen component of \(m\). Define rational
linear maps \(J:\mathbb Q^2\to F_\mathbb Q\),
\(s:\mathbb Q^2\to\widetilde M_\mathbb Q\), and
\(h:\widetilde M_\mathbb Q\to F_\mathbb Q\) by
\begin{equation}\label{eq:orthogonal-splitting}
 J=CB_0^{-1},\qquad s(u)=(u,Ju),\qquad
 h(m)=m_F-J\pi m.
\end{equation}
Thus \(m=s(\pi m)+(0,h(m))\) and \(h(\widetilde Bn)=0\).
Let \(\Omega\) be the restriction of \(\widetilde\Lambda\) to
\(F_\Q\). Compatibility implies
\begin{align}
 \widetilde\Lambda(s(u),(0,z))&=0,\label{eq:orthogonal-zero}\\
 \widetilde\Lambda(m,m')
  &=\ell\Lambda_0(\pi m,\pi m')+
       \Omega(h(m),h(m')).\label{eq:orthogonal-form}
\end{align}
For the first identity write \(s(u)=\widetilde B B_0^{-1}u\);
the second follows from \eqref{eq:ice-wall-pairing} and the first.

Set
\[
 F'=F+J\Z^2\subset F_\Q,\qquad
 L'=s(\Z^2)\oplus(0,F'),\qquad
 \Gamma=L'/\widetilde M.
\]
The lattice \(F'\) contains \(F\) with finite index, so
\(\Gamma\) is finite. The \(\Gamma\)-degree of \(X^m\), \(m\in L'\),
is the class of \(m\) modulo \(\widetilde M\). Choose \(N>0\) clearing \(\ell\) and all
pairings on \(L'\), and put
\(R'=\Z[v^{\pm1/N}]\). This is a finite free, faithfully flat
extension of \(R\). Write \(\mathcal T_{F'}\) for the frozen
quantum torus over \(R'\), with monomials \(Z^z=X^{(0,z)}\).
Equations~\eqref{eq:orthogonal-zero}--\eqref{eq:orthogonal-form} give
\begin{equation}\label{eq:torus-split}
 R'[L']_{\widetilde\Lambda}
   \simeq \mathcal T^0_{v^\ell,R'}\otimes_{R'}\mathcal T_{F'},
 \qquad X^m=U^{\pi m}Z^{h(m)},\quad U^u=X^{s(u)}.
\end{equation}
The two tensor factors commute with one another. The degree-zero part
for the \(\Gamma\)-grading
is precisely the original lattice torus over \(R'\).

\begin{lemma}\label{lem:split-mutation}
Let \(\A^0\) be the coefficient-free rank-two quantum cluster algebra
at parameter \(v^\ell\), with scalars extended to \(R'\).
The algebra generated by \(R'\otimes_R\widetilde\A\) and
\(\mathcal T_{F'}\) inside \eqref{eq:torus-split} is
\begin{equation}\label{eq:extended-algebra}
 \A^{\rm ext}=\A^0\otimes_{R'}\mathcal T_{F'}.
\end{equation}
Moreover,
\begin{equation}\label{eq:degree-zero-algebra}
 (\A^{\rm ext})_0=R'\otimes_R\widetilde\A.
\end{equation}
\end{lemma}

\begin{proof}
At every seed \(t\), its toric frame has a factorization
\begin{equation}\label{eq:seed-factorization}
 M_t(a)=U_t(a_{\rm uf})Z^{h_t(a)},
 \qquad h_t:\Z^{2+f}\longrightarrow F',\qquad
 h_t\widetilde B_t=0,
\end{equation}
where \(U_t\) is the coefficient-free toric frame for the same
mutation sequence. Initially this is \eqref{eq:torus-split}, with
\(h_0(a)=a_F-Ja_{\rm uf}\).
Assume \eqref{eq:seed-factorization} at \(t\). The two exponent
vectors in \eqref{eq:ice-mutation} have equal \(h_t\)-image,
since their difference is the mutated column \(\widetilde b_{t,i}\).
Their mutable parts are the two coefficient-free exchange
vectors. Thus the new variable is the new coefficient-free
variable times their common frozen monomial.

The seed form is the sum of the form of \(U_t\) and the form
\((a,a')\mapsto\Omega(h_t(a),h_t(a'))\). Hence the normalized
ordered product factors into the two normalized ordered products,
giving \eqref{eq:seed-factorization}.
If \(E\) has columns \(e_j\) for \(j\ne i\) and
\(-e_i+[\widetilde b_{t,i}]_+\) for \(j=i\), then
\(h_{t'}=h_tE\). Matrix mutation has the form
\(\widetilde B_{t'}=E\widetilde B_t H\), where \(H\) is the identity
except in row \(i\), with \(H_{ii}=-1\) and
\(H_{ij}=[-b_{t,ij}]_+\) for \(j\ne i\). Since \(E^2=1\),
\(h_{t'}\widetilde B_{t'}=0\). This completes the induction.

Each mutable variable in either algebra is therefore a variable
in the other times an invertible element of \(\mathcal T_{F'}\),
proving \eqref{eq:extended-algebra}. Every \(Z^z\), \(z\in F'\),
normalizes every cluster variable of \(\widetilde\A\) by a scalar unit, by
\eqref{eq:seed-factorization}. Consequently any word in the
extended generators can be written as a scalar times a word in the
generators of \(\widetilde\A\) times one \(Z^z\). The word in
\(\widetilde\A\) has degree zero in \(\Gamma\), so the product has
degree zero precisely when \((0,z)\in\widetilde M\), or equivalently
\(z\in F\). Taking homogeneous components of a finite sum proves
\eqref{eq:degree-zero-algebra}.
\end{proof}

\begin{theorem}[Integral coefficient theorem]\label{thm:integral-coefficients}
For the ice seed \eqref{eq:ice-data}, all \(\Theta_{m,Q}\),
\(m\in\widetilde M\), belong to \(\widetilde\A\) and form an
\(R\)-basis. Moreover,
\[
 \widetilde\A=\widetilde{\mathcal U}.
\]
The family \(\{\Theta_{(g,0),Q}:g\in\Z^2\}\) is both a left and
a right \(\mathcal R_F\)-basis. The full \(R\)-basis is independent
of the initial seed and contains every normalized extended cluster
monomial, with arbitrary integral frozen exponents.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:main} gives an \(R'\)-basis
\(\{\vartheta^0_g(U):g\in\Z^2\}\) of \(\A^0\).
Tensoring with \(\mathcal T_{F'}\) gives the basis
\[
 \{\vartheta^0_g(U)Z^z:g\in\Z^2,\ z\in F'\}
\]
of \(\A^{\rm ext}\). Each element is homogeneous for \(\Gamma\),
because its exponent differences are \(\widetilde Bn\in\widetilde M\).
It has degree zero exactly when \(s(g)+(0,z)\in\widetilde M\).
Proposition~\ref{prop:theta-lift} identifies these degree-zero
elements with the \(\Theta_m\). By
\eqref{eq:degree-zero-algebra}, they form an \(R'\)-basis of
\(R'\otimes_R\widetilde\A\).

Proposition~\ref{prop:theta-lift} already puts every \(\Theta_m\)
in the original torus over \(R\). Faithful flatness descends its
membership from \(R'\otimes\widetilde\A\) to
\(\widetilde\A\): apply it to the quotient of the original torus
by the \(R\)-submodule \(\widetilde\A\). Applying it next to the
kernel and cokernel of the map
\(\bigoplus_m R\to\widetilde\A\), \(e_m\mapsto\Theta_m\),
descends the basis assertion.

To prove the upper-algebra equality, let \(x\in\widetilde{\mathcal U}\).
Equation~\eqref{eq:seed-factorization} puts its image in
\(\mathcal T^0_t\otimes\mathcal T_{F'}\) for every seed \(t\).
Expand it in the finitely many frozen monomials occurring in its
initial torus expression. These monomials are linearly independent
over the coefficient-free skew field: multiplying a proposed
finite relation by a common Ore denominator reduces it to monomial
independence in the split torus. Their coefficients therefore lie
in every \(\mathcal T^0_t\). Coefficient-free rank two is acyclic, so
\(\bigcap_t\mathcal T^0_t=\A^0\) by
\cite[Theorem~7.5]{BZ}, applied over \(R'\) to the integral form
\(N\ell\Lambda_0\) in parameter \(v^{1/N}\).
Hence \(x\in\A^{\rm ext}\), and
\eqref{eq:degree-zero-algebra} followed by the same scalar descent
gives \(x\in\widetilde\A\). The reverse inclusion is the quantum
Laurent phenomenon.

For \(z\in F\), compatibility gives the normalized shift identity
\begin{equation}\label{eq:frozen-shift}
 \Theta_{m+(0,z)}
    =v^{-\widetilde\Lambda((0,z),m)}X^{(0,z)}\Theta_m
    =v^{-\widetilde\Lambda(m,(0,z))}\Theta_m X^{(0,z)}.
\end{equation}
The \(R\)-basis assertion therefore gives the left and right
\(\mathcal R_F\)-basis assertions.

Finally, \eqref{eq:seed-factorization} identifies a theta function
computed from any seed with a coefficient-free theta function
times one frozen monomial in the extension. The coefficient-free basis is seed independent. If a seed has the
opposite ordered orientation, interchange its mutable indices before
applying the rank-two statement; the skew form and parameters are
relabelled at the same time. The resulting coefficient-free factor
is one original \(\vartheta^0_g\). Membership, already proved for
the seed in question, and the Laurent phenomenon put the product
in the original lattice torus. Since \(B_0\) is injective and the
coefficient-free expansion has leading coefficient one, its leading
full exponent \(s(g)+(0,z)\) cannot cancel and belongs to
\(\widetilde M\). The lifting formula identifies the product with
\(\Theta_{s(g)+(0,z)}\). Applying the inverse mutation gives equality of
the basis sets. A normalized cluster monomial has the analogous
factorization with a coefficient-free cluster monomial; this is
a coefficient-free theta function and gives the last assertion.
\end{proof}

The full \(R\)-basis in the theorem is seed independent. A chosen
set of zero-frozen-index representatives is a basis over
\(\mathcal R_F\); a change of seed can change those representatives
by the normalized frozen units in \eqref{eq:frozen-shift}.

\subsection{Multiplication and changes of coefficients}

The arbitrary frozen form affects multiplication through an
explicit scalar. Write
\[
 \vartheta^0_g\vartheta^0_{g'}
       =\sum_k\alpha_{g,g'}^k(w)\vartheta^0_k.
\]
The coefficient-free algebra is graded by \(\Z^2/B_0\Z^2\),
and its theta basis is homogeneous. Thus
\(\alpha_{g,g'}^k\ne0\) implies \(k-g-g'\in B_0\Z^2\).

\begin{proposition}[Multiplication formula]\label{prop:coeff-products}
Let \(g=\pi m\), \(g'=\pi m'\), and, for every nonzero term, set
\[
 m_k=m+m'+\widetilde B B_0^{-1}(k-g-g')\in\widetilde M.
\]
Then
\begin{equation}\label{eq:coeff-products}
 \Theta_m\Theta_{m'}
  =\sum_k v^{\Omega(h(m),h(m'))}
          \alpha_{g,g'}^k(v^\ell)\Theta_{m_k}.
\end{equation}
The displayed coefficients belong to \(R\). Their signs, and the
signs of Laurent coefficients in corresponding seed charts, are
the same as in the coefficient-free case. In particular, strong
positivity is independent of the geometric coefficient matrix
and of the additional compatible frozen form.
\end{proposition}

\begin{proof}
In \eqref{eq:torus-split} one has
\(\Theta_m=\vartheta^0_g(U)Z^{h(m)}\). Multiply these factorizations
and use the quantum product in \(\mathcal T_{F'}\).
The resulting term has frozen degree \(h(m)+h(m')\), which is
exactly the degree of \(m_k\). This proves
\eqref{eq:coeff-products}; different \(k\) give different \(m_k\).
The integral basis theorem puts each coefficient in \(R\), although
the two factors in its displayed expression may separately involve
fractional powers. An injective positive rational rescaling of
exponents and multiplication by a monomial preserve every scalar
coefficient sign. Formula~\eqref{eq:theta-lift}, applied in each
seed using \eqref{eq:seed-factorization}, proves the assertion for
Laurent expansions. Taking all \(m,m'\) proves both directions of
the strong positivity statement.
\end{proof}

Here positivity means nonnegative integral coefficients in the
quantum parameter, and strong positivity refers to multiplication
in the full \(R\)-basis. The smaller ring
\(\Z[v^{\pm\delta}]\) in Proposition~\ref{prop:theta-lift} need not
contain the multiplication coefficients: even two spectator frozen
monomials can multiply with an extra factor \(v\).

There is a version of \eqref{eq:coeff-products} with manifestly
integral exponents. Since the theta coefficients belong to
\(\Z[w^{\pm\kappa}]\), every monomial coefficient of
\(\vartheta^0_g\vartheta^0_{g'}\) lies in
\(w^{\Lambda_0(g,g')}\Z[w^{\pm\kappa}]\): shifting either exponent
by \(B_0\Z^2\) changes its pairing by a multiple of \(\kappa\).
Separating quantum powers modulo \(\kappa\) in the unique theta
basis expansion gives
\[
 \alpha_{g,g'}^k(w)
   =w^{\Lambda_0(g,g')}P_{g,g'}^k(w^\kappa),\qquad
 P_{g,g'}^k(t)\in\Z[t^{\pm1}].
\]
Thus the corresponding coefficient in \eqref{eq:coeff-products} is
\begin{equation}\label{eq:integral-product-factor}
 v^{\widetilde\Lambda(m,m')}P_{g,g'}^k(v^\delta).
\end{equation}
This also proves its integrality directly from
\eqref{eq:orthogonal-form}.

Suppose two seeds
with the same \(B_0,D\) admit a lattice map
\(L:\widetilde M\to\widetilde M'\) satisfying
\[
 \pi'L=\pi,\qquad L\widetilde B=\widetilde B',\qquad
 \widetilde\Lambda'(Lm,Lm')=\widetilde\Lambda(m,m').
\]
Then the based quantum-torus homomorphism \(X^m\mapsto(X')^{Lm}\)
sends \(\Theta_m\) to \(\Theta'_{Lm}\), directly by
\eqref{eq:theta-lift}. Dropping frozen coordinates is such a
homomorphism only when it preserves the skew forms. In particular,
setting a noncentral frozen variable equal to \(1\) is generally
not a permissible quantum coefficient specialization.

\begin{proposition}[Adapted principal specialization]\label{prop:adapted-principal}
Every seed \eqref{eq:ice-data} is obtained from a compatible seed
with principal coefficients and \(f\) additional frozen variables
by a central coefficient quotient preserving theta functions.
\end{proposition}

\begin{proof}
On \(\widetilde M^\sharp=\Z^2\oplus\Z^2\oplus F\), set
\[
 \widetilde B^\sharp=\begin{pmatrix}B_0\\I_2\\0\end{pmatrix},
 \qquad L(u,n,z)=(u,Cn+z),\qquad
 \widetilde\Lambda^\sharp(x,y)=\widetilde\Lambda(Lx,Ly)
 \quad(x,y\in\widetilde M^\sharp).
\]
Then \(L\widetilde B^\sharp=\widetilde B\), and the pullback
form is integral and compatible with the same \(D\).
The map \(L\) is onto and satisfies the preceding conditions,
so it sends \(\Theta^\sharp_m\) to \(\Theta_{Lm}\).
The integral basis theorem makes the induced ordinary-algebra
homomorphism surjective.

Every \(k\in\ker L\) is frozen and lies in the radical of the
pullback form. Thus \(X^k\) is central and
\(\Theta^\sharp_{m+k}=X^k\Theta^\sharp_m\).
Grouping a finite theta expansion by fibers of \(L\), independence
of the target basis shows that the kernel is generated by
\(X^k-1\), \(k\in\ker L\). This proves the assertion.
\end{proof}

With signed coefficient rows, a cluster generator can map to a target generator times a frozen
unit, while the displayed theta mapping remains exact.

\subsection{Principal coefficients}

For \(C=I_2\) and \(z=0\), formula
\eqref{eq:coefficient-greedy-formula} reads
\begin{equation}\label{eq:principal-formula}
 \Theta^{\rm prin}_{g(a)}
 =\sum_{p,q\geq0}e_a(p,q;v^\ell)
       X^{(-a_1+bp,-a_2+cq,q,[a_2]_+-p)}.
\end{equation}
Both frozen exponents are nonnegative. These functions are
polynomials in the frozen variables in the initial chart, and
their full frozen translates form the basis of the cluster algebra
with inverted frozen variables. For example, the first mutations are
\[
 \Theta^{\rm prin}_{(-1,0)}
     =X^{(-1,0,0,0)}+X^{(-1,c,1,0)},\qquad
 \Theta^{\rm prin}_{(b,-1)}
     =X^{(b,-1,0,0)}+X^{(0,-1,0,1)}.
\]
A useful compatible principal form is
\begin{equation}\label{eq:principal-lambda}
 \widetilde\Lambda_{\rm prin}
   =\begin{pmatrix}0&-D\\D&-DB_0\end{pmatrix}.
\end{equation}
Its mutable variables commute; the frozen directions supply the
required compatibility pairings. When \(b=c=2\) and \(D=I_2\),
this integral form has \(\ell=1/2\). All coefficients in
\eqref{eq:principal-formula} nevertheless lie in \(R\), since the
Gaussian parameters are both \(v\).

For this example, write \(Y_1=X^{(0,0,1,0)}\) for the first
frozen generator. The relation
\(X_1Y_1=v^{-2}Y_1X_1\) prevents a homomorphism setting
\(Y_1=1\) while retaining the indeterminate \(v\) and an invertible
\(X_1\). An integral monomial rescaling from the coefficient-free
torus would require an integral frozen correction matrix satisfying
\(SB_0=I_2\). Here \(B_0^{-1}\) has half-integral entries, so such
a rescaling exists over the extended lattice \(L'\).

\section{Finite type and positivity}\label{sec:consequences}

\subsection{Finite rank two}

If \(bc\leq3\), all greedy elements are normalized cluster
monomials. For \(b=1\), their denominator fans are as follows.
The other orientation interchanges
the coordinates.
\[
\begin{array}{c|l}
 A_2&
 (-1,0),(0,-1),(1,0),(1,1),(0,1)\\
 B_2&
 (-1,0),(0,-1),(1,0),(1,1),(1,2),(0,1)\\
 G_2&
 (-1,0),(0,-1),(1,0),(1,1),(2,3),(1,2),(1,3),(0,1).
\end{array}
\]
Consecutive rays \(a_j,a_{j+1}\) are unimodular and their
closed cones cover the plane. Index each displayed list starting at \(a_1=(-1,0)\), so that
\(a_j\) is the denominator vector of \(X_j\), and extend cyclically.
For \(a=r a_j+s a_{j+1}\), \(r,s\in\mathbb N\), the corresponding element is
\[
 G[a]=v^{rs}X_j^rX_{j+1}^s .
\]
On a common ray one exponent is zero, so neighboring cone
descriptions agree. The origin gives \(1\).

The finite quantum scattering identities can also be checked
directly. For example, with our convention and
\(\psi_h(u)=\Psi_{v^h}(X^u)\), the \(B_2\) identity is
\[
 \psi_2(0,2)\psi_1(1,0)
 =\psi_1(1,0)\psi_2(2,2)\psi_1(1,2)\psi_2(0,2).
\]
For \(G_2\) it is
\begin{align*}
 \psi_3(0,3)\psi_1(1,0)
 ={}&\psi_1(1,0)\psi_3(3,3)\psi_1(2,3)\\
   &{}\cdot\psi_3(3,6)\psi_1(1,3)\psi_3(0,3).
\end{align*}
Here \(\psi_h(r,s)\) abbreviates \(\psi_h((r,s))\).
These are the identities of \cite[Section~3.2]{Nakanishi}
in the present normalization; they follow from the pentagon
relation and
\[
 \Psi_w(z)=\prod_{j=1}^r
       \Psi_{w^r}(w^{r+1-2j}z).
\]

\subsection{Positivity and negative coefficients}

\begin{example}\label{ex:negative}
In \(\A_v(2,3)\), the coefficient of \(X^{(1,-1)}\)
in \(G[3,4]\) is
\[
 e(2,1)=v^2-1+v^{-2}.
\]
This follows directly from \eqref{eq:greedyrec}:
\[
 e(2,1)=\qb{4}{1}{v^2}-\qb{3}{1}{v^3}
       =v^2-1+v^{-2}.
\]
Consequently the same coefficient occurs in
\(\vartheta_{(5,-4),Q}\).


There is also a negative multiplication coefficient. The product
\(X^{(3,4)}G[3,4]\) has only nonnegative exponent vectors
\((2p,3q)\), so its initial-cluster monomial expansion is already
an expansion in the theta basis. Its coefficient at
\(X^{(4,3)}\) is
\[
 v^7(v^2-1+v^{-2})=v^9-v^7+v^5.
\]
Thus this basis is not strongly positive.
\end{example}


\begin{example}[A principal-coefficient product]\label{ex:principal-negative}
Take \(b=2,c=3\), \(C=I_2\), \(D=\operatorname{diag}(3,2)\),
and the compatible form \eqref{eq:principal-lambda}. Then
\(\Theta_{(3,4,0,0)}=X^{(3,4,0,0)}\). In its product with
\(\Theta_{(5,-4,0,0)}\), the coefficient of
\(\Theta_{(4,3,1,2)}\) is
\[
 v^{-25}(v^2-1+v^{-2})=v^{-23}-v^{-25}+v^{-27}.
\]
Indeed, the contributing term of the second factor has exponent
\((1,-1,1,2)\), and
\[
 \widetilde\Lambda\bigl((3,4,0,0),(1,-1,1,2)\bigr)=-25.
\]
Every product exponent has the form
\((2p,3q,q,4-p)\geq0\), so each term is already a theta
monomial. The coefficient is thus a genuine negative structure
constant. The two factors and all these monomials belong to the
polynomial-frozen theta basis of
Theorem~\ref{thm:polynomial-coefficients} as well.
\end{example}

For \(b=c\), take the common dilogarithm parameter \(t=v^b\)
and the ambient skew form \(\Lambda/b\). Then
\(t^{(\Lambda/b)(u,u')}=v^{\Lambda(u,u')}\), so the
skew-symmetric strong positivity theorem of Davison--Mandel
\cite{DM} applies to the present torus and diagram.
Under Theorem~\ref{thm:main}, both Laurent positivity and
positivity of multiplication coefficients transfer to the
quantum greedy basis in this range. Proposition~\ref{prop:coeff-products}
then transfers strong positivity to every compatible geometric
coefficient system with that principal part.
For nondividing \(b,c\), nonpositive theta coefficients are
also established in \cite[Theorem~5.10 and
Proposition~5.11]{Nakanishi}.

\begin{remark}
When coefficients of a quantum theta element are nonnegative,
specialization at \(v=1\) preserves its exact support.
Without nonnegativity, specialization alone gives only an
inclusion: a nonzero Laurent polynomial may vanish at \(1\).
\end{remark}

\section{Disconnected rank two and frozen regularity}\label{sec:frozen-boundary}

\subsection{The disconnected principal part}

An exchange matrix with two mutable indices has either the
connected form considered above, up to reversing orientation,
or zero principal part.

\begin{proposition}\label{prop:disconnected}
Let \((\widetilde B,\widetilde\Lambda)\) be an integral compatible
ice seed with two mutable indices and \(\widetilde B=(0;C)\). Let
\(\widetilde B^{\mathsf T}\widetilde\Lambda=(D\ \ 0)\), and let
\(d_1,d_2>0\). Then \(C\) has rank two. Put
\(\beta_i=(0,C_i)\), \(r_i=[-C_i]_+\), and
\(N_i=[-a_i]_+\). For \(m=(a_1,a_2,z)\), the theta function is
\begin{equation}\label{eq:disconnected-theta}
 \Theta_m=
 \sum_{k_1=0}^{N_1}\sum_{k_2=0}^{N_2}
 \qb{N_1}{k_1}{v^{d_1}}\qb{N_2}{k_2}{v^{d_2}}
 X^{m+k_1\beta_1+k_2\beta_2}.
\end{equation}
These functions form a bar-invariant, universally Laurent positive,
strongly positive \(R\)-basis of the ordinary quantum cluster
algebra with inverted frozen variables. The ordinary and upper
algebras agree. If frozen variables are instead not inverted, the
theta functions with
\begin{equation}\label{eq:disconnected-regularity}
 z\geq N_1r_1+N_2r_2
\end{equation}
coordinatewise form an \(R\)-basis of that ordinary algebra.
In this noninverted case the ordinary and upper algebras also
agree, and equal the intersection of the frozen-localized
algebra with the initial mixed Laurent chart.
\end{proposition}

\begin{proof}
Compatibility gives
\(\widetilde\Lambda(\beta_i,m)=d_i a_i\).
Pairing a proposed linear dependence between \(\beta_1,\beta_2\)
with the two mutable basis vectors proves their independence.
Moreover, \(\widetilde\Lambda(\beta_1,\beta_2)=0\), and each
\(\beta_i\) is orthogonal to the frozen lattice. The initial
dilogarithms therefore commute, so their two hyperplanes already
form a consistent diagram. Every bend changes only frozen
exponents. A broken line crosses the \(i\)-th hyperplane once
if \(a_i<0\), and never otherwise. The two independent Gaussian
choices give \eqref{eq:disconnected-theta}.

The only other variable in direction \(i\) is
\[
 X_i'=X^{-e_i+(0,r_i)}+X^{-e_i+(0,r_i)+\beta_i}.
\]
The two directions mutate independently, giving four clusters.
For fixed \(a\), let \(Z_i(a_i)=X_i^{a_i}\) when \(a_i\geq0\),
and \(Z_i(a_i)=(X_i')^{-a_i}\) otherwise. Put
\(\mu_i=a_i e_i+(0,N_i r_i)\). Quantum binomial expansion and
\(\widetilde\Lambda(\beta_i,\mu_j)=0\) for \(i\ne j\) show that
\[
 S_a:=v^{-\widetilde\Lambda(\mu_1,\mu_2)}
       Z_1(a_1)Z_2(a_2)
       =\Theta_{(a,N_1r_1+N_2r_2)}.
\]
Frozen monomials and factors from different mutable directions
quasi-commute. In a fixed direction, if
\(\eta_i=\widetilde\Lambda(e_i,(0,r_i))\), then
\[
 X_iX_i'=v^{\eta_i}X^{(0,r_i)}
             +v^{\eta_i-d_i}X^{(0,r_i)+\beta_i},
\]
and the reversed product has the reversed scalar exponents.
Repeated use of these relations reduces the number of mutable
factors and expresses every word as a linear combination of
\(X^{(0,z)}S_a\). These elements are independent: different
\(a\) have disjoint mutable-exponent supports, and at fixed
\(a\) a relation is \(P S_a=0\) in a domain, with \(P\) in
the frozen quantum torus. The normalized frozen shifts identify
this basis with \eqref{eq:disconnected-theta}.

The displayed Gaussian coefficients are positive, and the same
straightening uses only positive two-term relations and quantum
monomial factors. It proves strong positivity. The four seed
descriptions prove universal Laurent positivity. Bar invariance
is immediate in based monomials; ordinary equals upper also
follows from \cite[Theorem~7.5]{BZ}.

If the frozen variables are not inverted, the straightening
relations still use only nonnegative frozen exponents. Hence
\(X^{(0,z)}S_a\), \(z\in\mathbb N^f\), give an independent
spanning set. Its normalized labels are exactly
\eqref{eq:disconnected-regularity}. Equivalently, the minimum
\(j\)-th frozen exponent in \eqref{eq:disconnected-theta} is
\(z_j-N_1(r_1)_j-N_2(r_2)_j\); endpoint Gaussian coefficients
show that this minimum is attained.

To prove the last equality, expand an element of the localized
algebra uniquely as \(\sum_a P_aS_a\), with frozen Laurent
coefficients \(P_a\). The least exponent in each frozen
coordinate of \(S_a\) is zero by the endpoint calculation.
Least coordinate degrees add under multiplication: the two
least-degree components have nonzero product in the quantum
Laurent domain. Moreover, different \(a\) have disjoint mutable
supports. Hence \(\sum_aP_aS_a\) lies in the initial mixed
chart precisely when every \(P_a\) is a frozen polynomial.
This proves that the intersection is the ordinary polynomial
coefficient algebra. The mixed quantum Laurent phenomenon
\cite[Definition~2.14 and Theorem~2.15]{GY}, together with the
localized equality already proved, then gives equality with
the noninverted upper algebra.
\end{proof}

\subsection{Polynomial frozen coefficients with nonnegative initial rows}

We now allow frozen variables to be noninvertible and assume that
the initial coefficient block is entrywise nonnegative.

Let \(\mathcal R_F^+=\bigoplus_{z\in\mathbb N^f}R X^{(0,z)}\)
be the frozen quantum polynomial algebra, and let
\(\mathcal T_0^+\) be the initial chart with mutable variables
inverted and frozen variables not inverted. Write
\(\A_{\rm pol}\) for the algebra generated by all mutable cluster
variables and \(\mathcal R_F^+\), and \(\mathcal U_{\rm pol}\)
for the intersection of the analogous mixed charts in all seeds.
For a finite Laurent polynomial \(x\), let \(\nu_j(x)\) be its
least exponent in the \(j\)-th frozen coordinate, with
\(\nu_j(0)=+\infty\).

\begin{lemma}\label{lem:frozen-valuation}
Let \(H_i\in\mathcal T_0^+\) be a family whose components of
frozen degree zero, denoted \(H_i^0\), are \(R\)-linearly
independent. For a finite sum with \(P_i\in\mathcal R_F\),
\begin{equation}\label{eq:frozen-valuation}
 \nu_j\left(\sum_i P_iH_i\right)=\min_i\nu_j(P_i).
\end{equation}
Zero coefficients are assigned valuation \(+\infty\); for the
all-zero family the minimum is \(+\infty\).
The same formula holds with the coefficients on the right.
\end{lemma}

\begin{proof}
If every \(P_i=0\), the assertion is immediate. Otherwise expand
every \(P_i\) in based frozen monomials. Order their
finitely many exponent vectors lexicographically, placing
coordinate \(j\) first, and choose the least exponent \(\alpha\)
that occurs with nonzero coefficient. A nonzero nonnegative
frozen exponent from \(H_i\) can contribute to the \(\alpha\)-layer
only from a strictly smaller coefficient exponent, which is
absent. That layer is consequently
\[
 X^{(0,\alpha)}\sum_i p_{i,\alpha}H_i^0,
 \qquad p_{i,\alpha}\in R.
\]
It is nonzero by the asserted independence. No exponent in the
sum has \(j\)-th coordinate smaller than \(\alpha_j\), proving
\eqref{eq:frozen-valuation}. For right coefficients the common
frozen monomial is factored on the right. 
\end{proof}

\begin{theorem}[Nonnegative coefficient rows]\label{thm:polynomial-coefficients}
Assume \eqref{eq:ice-data} and \(C\geq0\) entrywise. Then
\begin{equation}\label{eq:polynomial-equality}
 \A_{\rm pol}=\mathcal U_{\rm pol}
             =\widetilde\A\cap\mathcal T_0^+.
\end{equation}
The family \(\{\Theta_{(g,0)}:g\in\Z^2\}\) is both a left and
a right \(\mathcal R_F^+\)-basis. Equivalently,
\begin{equation}\label{eq:polynomial-theta-basis}
 \{\Theta_{(g,z)}:g\in\Z^2,\ z\in\mathbb N^f\}
\end{equation}
is an \(R\)-basis.
\end{theorem}

\begin{proof}
Let \(X_1',X_2'\) denote the first mutations of the initial
variables. For \(a\in\Z^2\), put
\(\xi(a)=(a_1+b[-a_2]_+,a_2)\). Let \(E_a\) be the multiple of
\[
 X_1^{[a_1]_+}X_2^{[a_2]_+}
 (X_1')^{[-a_1]_+}(X_2')^{[-a_2]_+}
\]
by a power of \(v\) for which the coefficient of
\(X^{(\xi(a),0)}\) is one. This normalization exists: since \(C\geq0\),
\[
 X_1'=X^{(-1,0,0)}+X^{(-1,c,C_1)},\qquad
 X_2'=X^{(b,-1,0)}+X^{(0,-1,C_2)}.
\]
Multiplying these expansions gives
\begin{equation}\label{eq:standard-pointed}
 E_a=X^{(\xi(a),0)}+
    \sum_{0\ne n\in\mathbb N^2}e_{a,n}
         X^{(\xi(a)+B_0n,Cn)},\qquad e_{a,n}\in R.
\end{equation}
The sum is finite and \(a\mapsto\xi(a)\) is a bijection.
The normalized standard monomials form a left
\(\mathcal R_F\)-basis of \(\widetilde\A\), by
\cite[Theorems~7.3 and~7.5]{BZ}. Their exponent differences
are exchange-lattice vectors, which are orthogonal to the
frozen lattice. They therefore quasi-commute with every frozen
monomial by a fixed scalar unit, giving a right basis as well.

We check the hypothesis of Lemma~\ref{lem:frozen-valuation}
even when \(C\) has zero columns. On taking frozen degree zero
in \eqref{eq:standard-pointed}, only \(n\) with \(Cn=0\)
survive. The linear functional
\[
 \omega(u)=-u_1/b+u_2/c
\]
satisfies \(\omega(B_0n)=n_1+n_2>0\) for \(n\ne0\).
In a proposed finite relation among the \(E_a^0\), choose the
smallest \(\omega(\xi(a))\) among its nonzero summands.
At that weight only their distinct leading monomials occur,
so they cannot cancel. Thus the \(E_a^0\) are independent.
Exactly the same argument applies to \(\Theta_{(g,0)}^0\),
using \eqref{eq:theta-lift} and its leading coefficient one.

Apply Lemma~\ref{lem:frozen-valuation} to the unique finite
standard-monomial expansion of an element of \(\widetilde\A\).
It belongs to \(\mathcal T_0^+\) exactly when all its frozen
coefficients belong to \(\mathcal R_F^+\). Hence
\[
 \widetilde\A\cap\mathcal T_0^+
   =\bigoplus_a\mathcal R_F^+E_a\subseteq\A_{\rm pol}.
\]
The quantum Laurent phenomenon allowing noninverted frozen
variables, \cite[Definition~2.14 and Theorem~2.15]{GY}, gives
\[
 \A_{\rm pol}\subseteq\mathcal U_{\rm pol}
       \subseteq\widetilde{\mathcal U}\cap\mathcal T_0^+.
\]
In this application its field is \(\Q(v)\), its ground subring
is \(R\), its quantum commutation units lie in \(R^\times\),
and its chosen set of inverted frozen variables is empty.
Theorem~\ref{thm:integral-coefficients} identifies
\(\widetilde{\mathcal U}=\widetilde\A\); the two inclusions
prove \eqref{eq:polynomial-equality}.

Apply the same valuation lemma to the left and right frozen
theta expansions from Theorem~\ref{thm:integral-coefficients}.
This proves both polynomial module basis assertions. The
normalized frozen shift formula \eqref{eq:frozen-shift}
then gives \eqref{eq:polynomial-theta-basis}.
\end{proof}

This \(R\)-basis is the intersection of the full theta basis
with \(\A_{\rm pol}\), so it is intrinsic to that algebra.
The hypothesis on \(C\) may be imposed at any seed; thus it
suffices that a seed with nonnegative coefficient block exists.


If only a chosen subset of the frozen variables is inverted,
the same proof imposes the valuation conditions only on the
remaining coordinates. For \(C\geq0\), it gives the ordinary
and upper algebra equality and the corresponding theta basis
for every such partial choice of frozen inverses.

\subsection{Signed coefficient rows: a counterexample}

For signed coefficient rows, the ordinary algebra with polynomial
frozen coefficients can be strictly smaller than its upper algebra.

\begin{proposition}\label{prop:frozen-counterexample}
There is a compatible acyclic rank-two quantum seed with one
central frozen variable \(Y\) such that
\[
 \A_{\rm pol}\subsetneq\mathcal U_{\rm pol}.
\]
Here \(\A_{\rm pol}\) is generated by all mutable cluster variables
and \(Y\), and \(\mathcal U_{\rm pol}\) is the intersection of
seed charts with mutable variables inverted and \(Y\) not inverted.
A theta function of the frozen-localized algebra lies in
\(\mathcal U_{\rm pol}\setminus\A_{\rm pol}\), while its product
with \(Y\) belongs to \(\A_{\rm pol}\).
\end{proposition}

\begin{proof}
Take
\[
 \widetilde B=\begin{pmatrix}0&-2\\2&0\\-1&1\end{pmatrix},
 \qquad
 \widetilde\Lambda=
 \begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}.
\]
Then \(\widetilde B^{\mathsf T}\widetilde\Lambda=(2I_2\ \ 0)\).
Mutation at the first mutable index, followed by interchange of
the two mutable indices, returns these matrices. Label the
consecutive clusters by \((X_m,X_{m+1})\), \(m\in\Z\). Their
relations are
\begin{equation}\label{eq:central-coefficient-exchange}
 X_{m+1}X_m=v^2X_mX_{m+1},\qquad
 X_{m+1}X_{m-1}=v^2X_m^2+Y.
\end{equation}
Put \(M_m(r,s)=v^{rs}X_m^rX_{m+1}^s\), and define
\[
 Z_m=M_m(-1,1)+M_m(1,-1)+Y M_m(-1,-1).
\]
The neighboring mutations are
\[
 X_{m+2}=M_m(-1,2)+Y M_m(-1,0),\qquad
 X_{m-1}=M_m(2,-1)+Y M_m(0,-1).
\]
Using based multiplication gives
\[
 Z_mX_{m+1}=Z_{m+1}X_{m+1}
             =vX_{m+2}+v^{-1}X_m.
\]
Cancellation proves that all \(Z_m\) are the same element \(Z\).
Its displayed formula in every seed has frozen exponents
\(0,0,1\), so \(Z\in\mathcal U_{\rm pol}\). The identities
\[
 ZX_m=vX_{m+1}+v^{-1}X_{m-1}
\]
also express all cluster variables recursively in any one mixed
chart, in both directions. Thus
\(\A_{\rm pol}\subseteq\mathcal U_{\rm pol}\) here follows
directly.

Give the initial mixed Laurent chart the direct-sum grading
\(\deg X_1=\deg X_2=1\), \(\deg Y=2\), and \(\deg v=0\).
The three terms of \(Z\) have degree zero, and the preceding
linear recurrence proves that every \(X_m\) has degree one.
Therefore the zero-degree component of the algebra generated
by all \(X_m\) and \(Y\) is exactly \(R\). Since \(Z\) is a
nonconstant sum of three distinct based monomials,
\(Z\notin\A_{\rm pol}\).

Finally, direct multiplication in the initial torus gives
\[
 X_0X_3=v^{-2}X_1X_2+vYZ,
 \qquad
 YZ=v^{-1}X_0X_3-v^{-3}X_1X_2\in\A_{\rm pol}.
\]
Formula~\eqref{eq:coefficient-greedy-formula}, with
\(a=(1,1)\), \(C=(-1,1)\), and \(z=0\), identifies
\(Z=\Theta_{(1,-1,0)}\). This proves all assertions.
\end{proof}

In this example the least nonnegative power of \(Y\) that makes
the theta function regular in all mixed seed charts is zero,
whereas the least power that puts it in the ordinary polynomial
coefficient algebra is one.

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\end{document}