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\title{Finite quantum theta functions for skew-symmetrizable seeds}
\author{Qiyue Tang}
\date{}
\begin{document}
\maketitle
\begingroup\renewcommand{\thefootnote}{}\footnotetext{\textit{2020 Mathematics Subject Classification.} Primary 13F60; Secondary 16W60, 52B20.\newline\textit{Keywords.} Quantum cluster algebra, theta function, scattering diagram, Newton polytope, skew-symmetrizable matrix.}\endgroup
\begin{abstract}
For integral compatible skew-symmetrizable seeds, finite quantum theta functions form a bar-invariant, mutation-compatible basis of an algebra between the cluster and upper cluster algebras. We prove parity, positivity of highest coefficients, valuative independence, and unit Newton vertex coefficients, with independence preserved by base change. We obtain upper theta bases for large cluster complexes, reddening classes, and Markov matrices and their one-way acyclic extensions. In mutation-acyclic classes, these are also bases of the cluster algebra. Further results include freezing and specialization criteria and a Chebyshev theta subalgebra in a mixed-sign extension.
\end{abstract}

\begingroup
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\section*{AI use and scholarly credit}
The results of this paper were developed through AI-assisted exploration, drawing on our earlier work and the research materials collected for this project. The arguments underwent multiple rounds of dialogue and review, and the manuscript was organized using the built-in mathematical writing skill.

By posting this manuscript on this platform, I do not seek to assert ownership of the results or claim priority. I welcome anyone interested in preparing a finished account or checking the arguments; the eventual scholarly credit should go to those who undertake that work.
\par
\endgroup

\section{Introduction}
Broken lines associate formal Laurent series to tropical lattice points. To obtain a basis of the upper cluster algebra, one must establish Laurent finiteness and finite theta expansions of regular functions. Quantum coefficients may contain negative terms, so classical specialization can shrink their support.

Let $v$ be an indeterminate. Gross--Hacking--Keel--Kontsevich \cite{GHKK} construct classical theta functions. Mandel's refined scattering framework \cite{Mandel} and the skew-symmetrizable initial diagrams discussed in \cite[Remark 4.4]{DM} precede the construction used here. Davison--Mandel prove coefficientwise quantum positivity for skew-symmetric cluster data. For general skew-symmetrizable seeds, we use the ordering of $\Q(v)$ determined by the highest term. It permits coefficients such as $v^2-1+v^{-2}$, whose Laurent coefficients are not all nonnegative. Nakanishi \cite[Theorem 5.10 and Corollary 5.12]{Nakanishi} proves nonpositivity for certain nonskew-symmetric quantum cluster scattering diagrams.

Set $R=\Z[v^{\pm1}]$ and $K=\Q(v)$. A seed with $n\ge1$ mutable and $f\ge0$ frozen variables has exponent lattice $M=\Z^{n+f}$; write $M_\R=M\otimes_\Z\R$. For $m\in M$, let $X^m$ be its based quantum monomial and $\Theta_m$ its broken-line theta series. The multiplication and broken-line conventions are given in Section~\ref{sec:data}. Let $\Theta_{\rm fin}\subseteq M$ consist of the labels whose theta series has finitely many nonzero monomial coefficients in a seed chart. Bar fixes each $X^m$, sends $v$ to $v^{-1}$, and reverses products. Define
\[
 \Mt=\bigoplus_{m\in\Theta_{\rm fin}}R\Theta_m.
\]
A normalized cluster monomial is a based monomial in any seed with nonnegative mutable exponents and arbitrary frozen exponents. Here $\A$ is the $R$-algebra generated by all mutable cluster variables and the frozen variables and their inverses in the common skew field; $\U$ is the intersection of all seed quantum tori in that skew field.

\begin{theorem}\label{thm:main}
For every integral compatible seed with positive diagonal skew-symmetrizer, the theta series have coefficients in $R$, are invariant under bar, and are transported by quantum seed mutation. Laurent finiteness is independent of the seed. The module $\Mt$ is an algebra and
\[
 \A\subseteq\Mt\subseteq\U.
\]
Its structure constants have positive highest coefficient whenever nonzero. The set $\Theta_{\rm fin}$ is closed under addition and is saturated: $km\in\Theta_{\rm fin}$ for a positive integer $k$ if and only if $m\in\Theta_{\rm fin}$.
\end{theorem}

For a nonzero finite Laurent polynomial $f=\sum_{b\in M}f_bX^b$ with $f_b\in K$, set $\supp f=\{b:f_b\ne0\}$ and $\Newt(f)=\conv(\supp f)$, where $\conv$ denotes the real convex hull in $M_\R$. For a real linear functional $\ell:M_\R\to\R$, set $\val_\ell(f)=\min_{b\in\supp f}\ell(b)$.

\begin{theorem}\label{thm:newton}
Fix a seed. If $J\subset\Theta_{\rm fin}$ is finite and nonempty and $c_m\in K\setminus\{0\}$, then
\[
 \Newt\left(\sum_{m\in J}c_m\Theta_m\right)
 =\conv\bigcup_{m\in J}\Newt(\Theta_m).
\]
Every vertex of the polytope on the right belongs to the support of exactly one participating theta function. Its coefficient in that theta function is one.
For every nonzero commutative unital ring $S$ and homomorphism $R\to S$, the specialized finite theta functions remain linearly independent in the specialized based quantum torus. For each finite set of theta functions, the matrix of their monomial coefficients has a triangular maximal minor with diagonal entries one.
\end{theorem}

Cheung--Magee--Mandel--Muller prove valuative independence and Newton vertex monicity in \cite[Theorems 3.8, 3.10 and 3.15]{CMMM}. Their Remark 1.3 extends these results to the quantum setting covered by Davison--Mandel positivity. Theorem~\ref{thm:newton} treats finite theta functions for general compatible skew-symmetrizable seeds and gives coefficientwise base-change independence on the fixed exponent lattice. Its proof follows the rational broken-line and reverse-reconstruction approach, using ordered wall-action supports and finite local certificates.

For an exchange matrix $B=(b_{ij})$, its directed graph has an arrow $i\to j$ when $b_{ij}>0$. It is acyclic when this graph has no directed cycle, and mutation-acyclic when some matrix in its mutation class is acyclic. To define reddening, start with the auxiliary extended matrix $\binom B{I_n}$ and apply the same matrix mutations. The bottom block is its $c$-matrix, and its columns are the $c$-vectors. A sequence is reddening when all final $c$-vectors are entrywise nonpositive; it is green when every selected $c$-vector is entrywise nonnegative before its mutation. For index sets $I,F$, $B_{F,I}$ denotes the submatrix with rows in $F$ and columns in $I$. Freezing $F$ means retaining its variables but allowing mutation only in the complementary mutable set $I$.

\begin{theorem}\label{thm:applications}
\begin{enumerate}
\item A reddening sequence gives finite support boxes for every theta label and an $R$-basis of $\U$. In a mutation-acyclic class this is also a basis of $\A$.
\item The principal enlargement in Section~\ref{sec:coeff} has a central frozen kernel, and quotienting the enlarged algebras by the ideal generated by the central kernel monomials minus one gives $\U$ and $\Mt$, respectively.
\item Suppose the classical upper algebra has a basis of Laurent-polynomial theta functions satisfying valuative independence and each basis label has a finite quantum theta function. Then these quantum theta functions form an $R$-basis of $\U$, and each nonzero summand in the expansion of a nonzero $f\in\U$ has support in $\Newt(f)$.
\item These hypotheses hold for the Markov matrix with any compatible integral coefficient realization. Thus $\Mt=\U$ in that class, although it admits no reddening sequence.
\item Partition the mutable indices as $I\sqcup F$, with $B_{F,I}\ge0$ entrywise and $B_{F,F}$ acyclic. All-label finiteness and the equality $\Mt=\U$ each pass from the frozen core to the full seed. In particular the connected Markov extensions of Section~\ref{sec:markov-extension} have upper theta bases.
\item If the cluster complex is large in the sense of Section~\ref{sec:large-complex}, all theta labels are finite and $\Mt=\U$.
\end{enumerate}
\end{theorem}

Berenstein--Zelevinsky \cite[Theorem 7.5]{BZ} prove equality of the quantum cluster and upper cluster algebras in mutation-acyclic classes, including skew-symmetrizable data. Equality with the intermediate theta algebra then follows from Theorem~\ref{thm:main}. The reddening application uses Qin's criterion \cite[Theorem 6.2.1(1)]{Qin}, and the Markov application uses Zhou's classical principal-coefficient basis \cite[Theorem 5.1]{Zhou}.

Gao \cite[Chapter 3]{Gao} develops skew-symmetrizable quantum theta functions and their relation to quantum greedy bases. In Section 3.3, the term ``finite type'' is defined by requiring the basis multipliers to take values from a pair of positive integers with product at most three; Theorems 3.17 and 3.19 are stated in this multiplier-based setting.

Qin's freezing operators identify theta functions on charge-coordinate faces and transfer upper bases to frozen seeds under the hypotheses of \cite[Theorems 5.2, 6.13 and 6.18]{QinFreeze}. In the classical quiver-with-potential setting, Fei--Weyman \cite[Corollary 2.13 and Theorem 2.14]{FW} establish source--sink and optimized-vertex extensions of cluster models under full-rank and Noetherian hypotheses. Our unfreezing theorem transfers finite quantum theta upper bases to one-way extensions over $R$, including the nonskew-symmetric Markov extensions of Section~\ref{sec:markov-extension}.

The formal theta coefficients and finite structure constants satisfy parity formulas in the original quantum parameter, including unequal symmetrizer entries (Theorem~\ref{thm:parity} and Corollary~\ref{cor:structure-parity}). Integral specialization embeds the quantum upper algebra modulo $v-1$ into its classical counterpart; classical unfreezing gives an isomorphism for the upper-basis classes above (Theorem~\ref{thm:integer-injection} and Corollary~\ref{cor:integer-classes}).

\section{Compatible lattices and scattering conventions}\label{sec:data}
Let $n\ge1$ and $f\ge0$ be integers. Put $N=\Z^n$ and $M=\Z^{n+f}$, with standard bases denoted by $e_i$ in the indicated lattice. Write $m=(m_{\rm mut},m_{\rm fr})\in M$, where $m_{\rm mut}\in\Z^n$ and $m_{\rm fr}\in\Z^f$. We write $M_\R=M\otimes_\Z\R$ and $M_\Q=M\otimes_\Z\Q$. A compatible datum consists of integer matrices
\[
 p=\binom BC:N\longrightarrow M,\qquad
 \Lambda=-\Lambda^T,\qquad
 p^T\Lambda=(D\ \ 0),\quad D=\diag(d_1,\ldots,d_n),\ d_i>0.
\]
Here $B$ is $n\times n$, $C$ is $f\times n$, and $\Lambda$ is $(n+f)\times(n+f)$. If $pa=0$, then $a^Tp^T\Lambda=(a^TD,0)=0$, so $a=0$. Thus $p$ is injective. The identity $p^T\Lambda p=DB$ makes $DB$ skew-symmetric. The sublattice $pN$ may be nonsaturated. For an extended matrix $p=(p_{ij})$, mutation $\mu_k$ at $1\le k\le n$ is defined by
\[
 (\mu_k(p))_{ij}=
 \begin{cases}
 -p_{ij},&i=k\text{ or }j=k,\\
 p_{ij}+[p_{ik}]_+[p_{kj}]_+-[-p_{ik}]_+[-p_{kj}]_+,&i\ne k,\ j\ne k,
 \end{cases}
\]
where $[a]_+=\max(a,0)$; on vectors and matrices this notation applies entrywise.

The based quantum torus is the free $R$-module
\[
 \T_R=\bigoplus_{m\in M}RX^m,\qquad
 X^mX^{m'}=v^{\Lambda(m,m')}X^{m+m'}.
\]
Here $\Lambda(m,m')=m^T\Lambda m'$ and $X_i=X^{e_i}$ for $1\le i\le n+f$. For a commutative unital $R$-algebra $S$, put $\T_S=S\otimes_R\T_R$, with the same based monomials and the image of $v$ in its multiplication law. For a seed $s$, subscripts $s$ indicate its exponent lattice $M_s$, real span $M_{s,\R}$, charge map $p_s$, compatible form $\Lambda_s$, and torus $\T_{s,R}$; write $\Theta_{\rm fin,s}$ for its finite theta labels. For a scalar localization $L$ of $R$, write $\U_L$ for the intersection of the seed tori $\T_{s,L}$ in the common skew field; thus $\U_R=\U$. Bar is the $\Z$-linear anti-involution $\overline{v}=v^{-1}$, $\overline{X^m}=X^m$. For $F=\sum_{b\in M}c_bX^b$, define $[X^m]F=c_m$, with value zero when $m$ is absent. Products are first expressed in the based monomials using the displayed multiplication rule.

Let $P=\N^n$, including zero, and $N^+=P\setminus\{0\}$. For $a\in P$ set $\htg(a)=\sum_i a_i$. For a series, its support is again the set of exponents with nonzero coefficients. Formal series below have support in a finite union of translates of $pP$. On a fixed translate $b+pP$, truncation at height $H$ retains the terms $X^{b+pa}$ with $\htg(a)\le H$. For a finite union, take these truncations in each translate; their retained exponent sets are finite. Convergence on a fixed support set means eventual agreement on every such truncation. Products are defined coefficientwise: injectivity of $p$ makes each coefficient a finite sum, since a fixed nonnegative charge has finitely many decompositions in $P$.

Set $G=M/pN$, and give $X^m$ degree $[m]\in G$ and every scalar degree zero. Thus $\T_R=\bigoplus_{\gamma\in G}\T_\gamma$, where $\T_\gamma$ is the span of the monomials of degree $\gamma$. A finite union of translated positive cones meets only finitely many $G$-classes.

Define, for $a,b\in N$ and $m\in M$,
\[
 \nu_a(m)=\Lambda(pa,m)=\sum_i d_i a_i m_i,\qquad
 \Omega(a,b)=\Lambda(pa,pb)=a^TDBb.
\]
The map $a\mapsto\nu_a$ is injective. In particular, at a general point of a wall, all wall degrees with the same supporting hyperplane lie on the same ray in $N$.

Extend each $\nu_a$ linearly to $M_\R$, and write $\nu_a^\perp=\{Q\in M_\R:\nu_a(Q)=0\}$. For a primitive charge $a\in N^+$, a wall of charge ray $\R_{>0}a$ has rational polyhedral support of codimension one in $M_\R$, contained in $\nu_a^\perp$ and a group element in $1+X^{pa}K[[X^{pa}]]$. For a wall element $g$, write $\Ad_g(F)=gFg^{-1}$. A transverse path uses $\Ad_g$ when $\nu_a$ decreases along the crossing, and $\Ad_g^{-1}$ when it increases. Both preserve the $G$-degree, since their exponent shifts lie in $pP$. On a monomial $X^m$, the favorable action is $\Ad_g$ when $\nu_a(m)>0$ and $\Ad_g^{-1}$ when $\nu_a(m)<0$. A wall is central when its support is a cone with vertex at the origin. Such a wall is incoming when its support contains $pa$, and outgoing otherwise. For an affine support, incoming means that translation by $tpa$ preserves the support for every $t\ge0$; equivalently, $pa$ lies in its recession cone. A path-ordered product is the composition, in crossing order, of the wall actions along a path transverse to the walls and disjoint from wall boundaries and intersections of distinct supporting hyperplanes, computed at each height. A diagram is consistent when this product is the identity around every such loop. Diagrams are equivalent when their path-ordered products agree. They have finitely many nontrivial walls at every bounded positive height. Boundaries of walls and intersections of distinct supporting hyperplanes are called seams. Their codimension-two cells are joints. A joint is parallel to a charge $a$ if $pa$ belongs to its tangent space, and perpendicular to that charge otherwise. We use the rational existence and uniqueness theorem for scattering diagrams \cite[Theorem 2.13]{DM}, or \cite[Theorem 2.6]{Mandel}, and endpoint transport \cite[Theorem 2.14]{Mandel}.

For independent indeterminates $w,z$, define $\Psi_w(z)\in\Q(w)[[z]]$ by
\[
 \Psi_w(z)=\exp\left(\sum_{j\ge1}
 \frac{(-1)^{j+1}z^j}{j(w^j-w^{-j})}\right).
\]
The initial wall of direction $e_i$ is the full hyperplane $\nu_{e_i}^\perp$ with label $\Psi_{v^{d_i}}(X^{pe_i})$. The cluster diagram is the consistent completion obtained by adding outgoing walls. An endpoint is general if it lies outside all walls. It is lattice general if it lies outside every proper rational linear hyperplane in $M_\R$, as in \cite[Section 3.1]{DM}. A general wall point lies in the relative interior of a wall and outside the seams. A broken line ending at a point $Q$ is a continuous piecewise affine path on $(-\infty,0]$ with finitely many genuine bends, whose segment decorated by $cX^m$ has velocity $-m$. It begins with $X^m$, avoids seams, and at a transverse wall crossing selects one nonzero monomial from the wall action on its current decoration. Taking the constant action term is allowed. For lattice-general $Q$, the sum of final monomials over these paths defines $\Theta_{m,Q}$; set $\Theta_{0,Q}=1$. At fixed height these sums are finite. At other general endpoints, $\Theta_{m,Q}$ denotes the coefficientwise continuation constructed below. The positive seed chamber is $Q_i>0$ for $1\le i\le n$. We omit $Q$ when it is fixed, and write $\Theta_m^s$ for the theta function in the positive chart of seed $s$. Each $\Theta_{m,Q}$ is homogeneous of degree $[m]\in G$, because every bend adds a charge in $pP$.

\begin{lemma}\label{lem:endpoint-continuation}
The theta sums at lattice-general endpoints extend uniquely to all general endpoints by local constancy at each finite charge height. The extension has support in $m+pP$, coefficient one at $X^m$, and satisfies endpoint transport.
\end{lemma}
\begin{proof}
First let $Q$ be lattice general. Every seam of the central cluster diagram has rational linear span $V$ of codimension at least two. If the straight path $Q-tm$, $t\le0$, met that seam, then $Q\in V+\R m$, a proper rational subspace. This contradicts lattice generality. If $\nu_a(m)=0$, the path stays off $\nu_a^\perp$ because $\nu_a(Q)\ne0$; all other wall crossings are transverse. Selecting the constant action terms gives the straight contribution $X^m$. Any genuine bend adds a nonzero charge in $P$, so injectivity of $p$ excludes another contribution to this exponent. For $m=0$, use the constant path.

Now let $Q$ be any general endpoint and fix an integer height $H\ge0$. Only finitely many closed wall supports are nontrivial through height $H$. Choose an open ball $B_H$ centered at $Q$ disjoint from these supports, and a lattice-general point $Q_H\in B_H$. Such points are dense, since the proper rational hyperplanes form a countable family. Define the coefficients through height $H$ at $Q$ to be those of $\Theta_{m,Q_H}$. Endpoint transport for generic endpoints, \cite[Theorem 2.14]{Mandel}, makes them independent of $Q_H$: a path in $B_H$ has identity wall product through height $H$. The same argument in the intersection of two such balls proves independence of the ball. Choose the balls nested as $H$ increases. The truncations then agree at every lower height and define a series supported in $m+pP$. Each coefficient, including the coefficient one at $X^m$, is inherited from a lattice-general endpoint.

For a path between general endpoints, fix a height and perturb its endpoints within their corresponding balls to lattice-general points. The added short paths have identity products at that height. Generic endpoint transport therefore gives the asserted transport identity through that height. Passing through all heights proves the identity for the extended series. The same local constancy determines every coefficient, proving uniqueness.
\end{proof}

For an integer $r\ge0$, the centered Gaussian polynomials are defined by
\[
 \prod_{j=0}^{r-1}(1+w^{r-1-2j}z)
  =\sum_{s=0}^r\gb r s w z^s.
\]
For integers $s$, set $\gb r s w=0$ when $s<0$ or $s>r$. For $r\in\Z$ and $s\ge0$, define
\[
 C_r(s;w)=
 \begin{cases}
 \gb r s w,&r\ge0,\\
 (-1)^s\gb{-r+s-1}s w,&r<0.
 \end{cases}
\]
Since $\Psi_w(w^2z)=(1+wz)\Psi_w(z)$, the initial wall action is
\begin{equation}\label{eq:gaussian-action}
 \Ad_{\Psi_{v^{d_i}}(X^{pe_i})}(X^m)
 =\sum_{s\ge0}C_{m_i}(s;v^{d_i})X^{m+spe_i},
\end{equation}
These coefficients belong to $R$ and are bar invariant. The inverse action uses $C_{-m_i}$.

\section{Integral actions and mutation}\label{sec:integral}
\subsection{Integral ray factorization}
Let $G_K$ be the group obtained by exponentiating the completed positive Lie algebra $\prod_{a\in N^+}KX^{pa}$, with commutator bracket $[X^{pa},X^{pb}]=(v^{\Omega(a,b)}-v^{-\Omega(a,b)})X^{p(a+b)}$, acting by conjugation on the completed torus. The exponential and group products are taken degreewise in positive height. Let $G_R$ consist of those elements whose action coefficients on every $X^m$ lie in $R$. The identity term is required to be $X^m$; every other term has positive charge. Composition preserves $G_R$. Solving the identity $gg^{-1}=1$ one height at a time proves that inversion also preserves it.

\begin{lemma}\label{lem:ray}
Choose any total order on the positive rational rays of $P$. Every $g\in G_K$ has a unique degreewise ordered factorization $g=\prod_\rho g_\rho$ with $g_\rho\in\exp(\prod_{a\in\rho\cap N^+}KX^{pa})$. If $g\in G_R$, all $g_\rho$ belong to $G_R$. If $g$ commutes with bar, so do its ray factors.
\end{lemma}
\begin{proof}
Modulo height greater than one, each homogeneous component belongs to a unique ray. Having determined factors below height $h$, compare their ordered product with $g$ in height $h$. The quotient is central in that truncation and splits uniquely by ray. This determines the next coefficient and proves existence and uniqueness by induction. At a fixed shift $pa$ of height $h$, the coefficient of $g(X^m)$ is the coefficient of the factor on the ray of $a$, plus products of coefficients of smaller heights from the other factors, with powers of $v$. No other factor can contribute a single term of shift $pa$. Subtracting these integral terms gives the height-$h$ action coefficient of the ray factor in $R$. Applying bar and uniqueness proves the last assertion.
\end{proof}

\begin{proposition}\label{prop:integral}
The cluster diagram has a representative with integral, bar-compatible wall actions. Its theta series have integral bar-invariant coefficients, leading term $X^m$, and classical specialization equal to the classical theta series.
\end{proposition}
\begin{proof}
Use full ray factors in the usual joint induction. At a joint $\mathfrak j$ where the diagram is consistent below height $h$, the full local monodromy is in $G_R$. Factor it by Lemma~\ref{lem:ray}, and attach the required full ray factors, with the orientation prescribed by the loop, on the outgoing supports $\mathfrak j-\R_{\ge0}pa$. Only factors whose leading height is $h$ are needed at this stage.

Walls are cylindrical in the frozen subspace. At a codimension-two joint the charges annihilating its tangent space form a saturated rank-two sublattice. The restriction of $\Omega$ is either zero, in which case all local actions commute, or nondegenerate, in which case $pa$ is transverse to the joint for every nonzero incident charge. Brackets have the sum charge and vanish when $\Omega$ does. These are precisely the properties used in \cite[Lemmas C.6--C.7]{GHKK}.

A newly introduced wall passing through a new joint away from its boundary contributes twice with inverse signs in height $h$. A side boundary of $\mathfrak j-\R_{\ge0}pa$ is tangent to $pa$ and hence cannot be a perpendicular joint. Thus all perpendicular local products are trivial. Comparing with the rational consistent diagram leaves central walls of height $h$. Their labels glue across perpendicular facets; every remaining facet is parallel to their direction. Such a polyhedral support contains its direction and is incoming. There is no new incoming datum, so the residual diagram is empty. Bar is preserved by Lemma~\ref{lem:ray}.

At a lattice-general endpoint, each genuine bend consumes positive height. At bounded height there are finitely many walls, shifts, and bend sequences; a transverse sequence determines at most one path ending at $Q$. A tangent segment selects only the constant term. Thus the theta coefficients are finite sums of products of integral bar-invariant action coefficients. Lemma~\ref{lem:endpoint-continuation} gives the unique zero-charge contribution $X^m$ and extends these coefficient assertions to every general endpoint.

The integral actions specialize at $v=1$. On a ray they fix the kernel of its normal and multiply a normal coordinate by a one-variable unit; hence their specialization lies in the classical tropical-vertex group. The initial actions specialize to the classical initial actions. Consistency and rational uniqueness identify the specialized diagram. Degreewise finite broken-line sums commute with specialization at lattice-general endpoints. Applying the same finite-height continuation to both diagrams gives the assertion at all general endpoints.
\end{proof}

\subsection{Transverse completion and the slab}
Fix $k$ and write $z=X^{pe_k}$. The transverse degree of a charge $a$ is $\sum_{i\ne k}a_i$. Invert all nonzero polynomials in $z$. Monomials normalize this Ore set by $z\mapsto v^{2\Lambda(m,pe_k)}z$. A choice of representatives for $M/\Z pe_k$ identifies the localized torus, as a vector space, with a direct sum of copies of $K(z)$.

Complete in positive transverse degree, allowing only finitely many integral $e_k$-coordinates at each fixed transverse degree in wall labels. Let $\bar G$ be the associated group; let $\widetilde G$ be its subgroup with all original charge coordinates nonnegative. The group $\widetilde G$ embeds in both the original completion and the transverse completion. Denote the slab action by $S=\Ad_{\Psi_{v^{d_k}}(z)}$.

\begin{lemma}\label{lem:slab}
The cluster diagram admits a representative in the transverse completion with the pure $e_k$ wall retained as a separate slab. Its other labels belong to $\widetilde G$, and there are finitely many nontrivial walls at bounded transverse degree. This representative is consistent on the Ore-localized torus.
\end{lemma}
\begin{proof}
Let $a_i\ge0$ and let $a$ lie off the $e_k$ axis. Formula~\eqref{eq:gaussian-action} gives a finite polynomial for $S(X^{pa})$ when $(Ba)_k\ge0$, and for $S^{-1}(X^{pa})$ when $(Ba)_k\le0$. All resulting charges remain nonnegative. These are the favorable conjugations used at a slab joint.

Induct on transverse degree. At a general slab joint, the two possible oriented local products can be written $g_2Sg_1S^{-1}$ when $(Ba)_k>0$, and $S^{-1}g_2Sg_1$ when $(Ba)_k<0$. Insert the ray correction on the opposite half-wall on the unconjugated side. The favorable conjugation just established puts this correction in $\widetilde G$ and makes its $e_k$ support finite. When $(Ba)_k=0$ it commutes with the slab. Away from the slab, use the ordinary central joint correction.

At a perpendicular slab joint, all incident non-slab positive charges project to one positive ray modulo $\Z e_k$. Writing them as $\alpha e_k+\beta n$ with $\beta>0$ shows that $(Ba)_k=\beta(Bn)_k$ has one common sign. Hence the chosen favorable conjugation works for the whole local product. The new support lies on the negative side when $(Ba)_k>0$ and on the positive side when $(Ba)_k<0$, precisely the unconjugated side in the displayed loop words. Its intersection with the slab is only its source joint, so it creates no further transverse slab joint. Away from the slab the three-case perpendicular-joint argument of Proposition~\ref{prop:integral} applies.

Parallel joints require a separate comparison. Fix transverse height $J$ and use the original positive-charge completion, further truncated at original height $H$. Induct on $H$ against the rational consistent diagram. A first discrepancy is central at that original height; perpendicular products are trivial, so the facet-gluing argument of Proposition~\ref{prop:integral} makes any residual support incoming, a contradiction. Letting $H$ tend to infinity proves agreement in this positive-charge completion. The transverse-positive subgroup embeds in it: at each transverse height its $e_k$ support is finite. Thus the non-slab parallel products are trivial there as well. At parallel slab joints, cancel the commuting slab and its inverse first.

For uniqueness in the transverse group, where negative $e_k$ charges are allowed, take the first transverse height $J$ of a discrepancy and minimize its finite polyhedral support, separately for each charge $a$. If $(Ba)_k\ge0$, a discrepancy in the positive half-space can be followed along $+pa$ until its last exit from the finite union of outgoing cones. The exit remains on that side and its central local loop would force the discrepancy to vanish, a contradiction. Hence its support is in the negative half-space. For $(Ba)_k\le0$ it is in the positive half-space. When $(Ba)_k=0$ these statements place it in the slab, impossible for a non-slab normal with positive transverse charge. At a perpendicular slab joint, the remaining discrepancy pieces consequently all lie on the unconjugated side of the favorable loop. Consistency then makes their local discrepancy product trivial. The same last-exit argument now removes all remaining pieces, proving uniqueness through $J$.

Taking the transverse limit gives the asserted representative. Products involving the slab act on $K(z)$; equality of their expansions is equality of these rational coefficients, since the Laurent expansion of a rational function is injective. Thus consistency also holds on the stated localized torus. This is the geometric construction of \cite[Appendix C.2]{GHKK} with finite Gaussian conjugations.
\end{proof}

\subsection{Mutation and Laurent finiteness}
For a matrix $B$, write $B_{k,*}$ for row $k$ and $B_{\bullet k}$ for column $k$. For mutation in direction $k$, let $E$ be the identity on $M$ except for column $k$, which is $-e_k+[-pe_k]_+$, and let $F$ be the identity on $N$ except for row $k$, which is $-e_k^T+[B_{k,*}]_+$. Then
\[
 p'=EpF,\qquad\Lambda'=E^T\Lambda E,\qquad
 T_k(m)=m+[m_k]_+pe_k.
\]
A linear lattice map $L$ acts on based series by $L(\sum_m c_mX^m)=\sum_m c_mX^{Lm}$, fixing the coefficients; for a piecewise linear map this convention applies to each linear piece. Both $E$ and $F$ are involutions; the quantum mutation pullback is $S^{-1}E$. The piecewise linear map on labels and endpoints is $ET_k$. On a half-space containing $Q$ its linear part is denoted $ET_{k,Q}$. Since $F$ is integral and invertible, $EpN=p'N$, so $E$ induces an isomorphism $G\to M/p'N$. Moreover,
\[
 ET_k(m)-Em=[m_k]_+Epe_k=[m_k]_+p'Fe_k\in p'N.
\]
Thus tropical label transport carries degree $[m]$ to degree $[Em]$.

\begin{proposition}\label{prop:mutation}
Theta series satisfy
\[
 \Theta'_{ET_k(m),\,ET_k(Q)}=ET_{k,Q}(\Theta_{m,Q}).
\]
For fixed transverse charge $(a_i)_{i\ne k}$, only finitely many values of $a_k$ occur in $\Theta_{m,Q}$. If a theta function is finite in one positive seed chart, it is finite in every seed chart and belongs to $\U$.
\end{proposition}
\begin{proof}
First take $Q$ lattice general. The two linear pieces of $ET_k$ are invertible integral maps, so $ET_k(Q)$ is lattice general as well. The linear shear $\tau(m)=m+m_kpe_k$ is symplectic. Writing $S_{\pm pe_k}=\Ad_{\Psi_{v^{d_k}}(X^{\pm pe_k})}$, the dilogarithm functional equation gives $S_{-pe_k}=\tau S_{pe_k}^{-1}$. Apply this identity to the slab representative of Lemma~\ref{lem:slab}. On the two half-spaces, $T_k$ changes wall supports and labels by the corresponding linear maps; the identity identifies their slab products. The resulting diagram has the mutated incoming walls, and rational uniqueness identifies it with the mutated cluster diagram.

A broken line is transformed segment by segment. At the slab, a favorable action with pairing $r\ge0$ has indices $0\le j\le r$. The transformed bend index is $r-j$, and $\gb r j {v^{d_k}}=\gb r {r-j}{v^{d_k}}$. Straight crossings are included as $j=0$ or $r$. Away from the slab the linear maps preserve the quantum pairing and the selected action coefficients. This is a bijection, also after every finite truncation, and proves the displayed identity.

The coefficient identity extends to every general endpoint by Lemma~\ref{lem:endpoint-continuation}. On a neighborhood with fixed sign of $Q_k$, the endpoint map is linear. The preimages of finitely many target charge positions form a finite set, so the identity at any fixed target height follows from a sufficiently high source truncation.

For slice finiteness, bound the transverse height by $J$. There are at most $J$ non-slab bends and at most $J+1$ slab crossings. The $k$th mutable exponent on every segment is bounded in absolute value by
\[
 |m_k|+J\max\bigl(\{0\}\cup\{|b_{ki}|:i\ne k\}\bigr),
\]
since slab bends do not change that coordinate. Each slab action is a finite Gaussian with that bound; at the other bends, the transverse construction supplies finite $e_k$ support. Finitely many sequences and choices remain, with an $e_k$ bound independent of the endpoint. The same bound therefore holds for the continued coefficients at general endpoints.

Suppose now that the original positive-chart theta is finite. Move the endpoint to the adjacent seed chamber using the rational slab action. This preserves the finite set of transverse charge positions, although it can a priori introduce infinitely many $e_k$ terms. The series after transport is a theta series, so slice finiteness makes each of these finitely many slices finite. The mutation identity then gives a finite Laurent polynomial in the adjacent torus. Iterate along any finite mutation sequence. The intersection of these tori is $\U$.
\end{proof}
\subsection{Parity of theta coefficients}\label{sec:parity}
Define the quadratic function $q_{B,D}:N\to\Z/2\Z$ by
\[
 q_{B,D}(a)=\sum_i d_i a_i^2+\sum_{i<j}d_i b_{ij}a_i a_j\pmod2.
\]
Its polarization is $q_{B,D}(a+b)-q_{B,D}(a)-q_{B,D}(b)=\Omega(a,b)$ modulo two: the diagonal cross terms are even, and the remaining terms give $a^TDBb$ since $DB$ is skew-symmetric. We use $(-1)^e$ for the sign determined by $e\in\Z/2\Z$. The following is the weighted version of the parity property in \cite[Proposition 3.11]{DM}.

\begin{theorem}\label{thm:parity}
For $m\in M$, $a\in P$ and a general endpoint $Q$, put $c_{m,a,Q}(v)=[X^{m+pa}]\Theta_{m,Q}$. Then
\[
 c_{m,a,Q}(-v)=(-1)^{\nu_a(m)+q_{B,D}(a)}c_{m,a,Q}(v).
\]
\end{theorem}
\begin{proof}
On the positive charge completion over $K$, define, for $f\in K$ and $a\in P$,
\[
 \mathscr J\bigl(f(v)X^{pa}\bigr)=f(-v)(-1)^{q_{B,D}(a)}X^{pa}.
\]
The polarization identity compensates for the sign acquired by $v^{\Omega(a,b)}$ under $v\mapsto-v$. Thus $\mathscr J$ is a semilinear algebra involution preserving every charge and height truncation. In the degree-$je_i$ term of the initial wall logarithm, parameter substitution gives the sign $(-1)^{d_i j}$, while $q_{B,D}(je_i)=d_i j^2\equiv d_i j$. Hence each initial wall element is fixed.

Apply $\mathscr J$ to every wall label, leaving its support unchanged. Path products are transformed by this algebra involution, so the new diagram is consistent and has the same incoming data and outgoing supports. Rational scattering uniqueness makes it equivalent to the original diagram.

On series supported in $m+pN$ with the prescribed positive-cone bounds, define
\[
 \mathscr J_m\bigl(f(v)X^{m+pa}\bigr)
 =f(-v)(-1)^{\nu_a(m)+q_{B,D}(a)}X^{m+pa},\qquad a\in N.
\]
Injectivity of $p$ makes the charge $a$ unique. For charge series $g,h$ and a series $F$ in this exponent coset, direct multiplication gives
\[
 \mathscr J_m(gFh)=\mathscr J(g)\mathscr J_m(F)\mathscr J(h).
\]
For example, left multiplication by $X^{pb}$ has phase $\nu_b(m)+\Omega(b,a)$, and
\[
 \nu_b(m)+\Omega(b,a)+\nu_{a+b}(m)+q_{B,D}(a+b)
 \equiv q_{B,D}(b)+\nu_a(m)+q_{B,D}(a)\pmod2.
\]
Right multiplication has phase $-\nu_b(m)+\Omega(a,b)$ and gives the same identity. Consequently $\mathscr J_m$ intertwines each wall action, including its inverse, with the action in the transformed diagram.

For lattice-general $Q$, since $\mathscr J_m(X^m)=X^m$, applying it to the decorations of a broken line gives the corresponding decorations in the transformed diagram. The geometry and nonzero term choices are unchanged. At total charge $a$, genuine bends consume at most $\htg(a)$ steps and use only finitely many walls and shifts in that truncation. A transverse decorated bend sequence is uniquely reconstructed backwards from $Q$; a tangent crossing contributes only the constant term. Thus the coefficient sum is finite, as in Proposition~\ref{prop:integral}, and $\mathscr J_m$ carries the theta series to the theta series of the transformed diagram. Equivalence of diagrams identifies the latter with $\Theta_{m,Q}$. Comparing coefficients proves the formula. Each continued coefficient at a general endpoint is a coefficient at a lattice-general endpoint at a sufficiently high truncation, so the same parity identity holds there. Integrality follows from Proposition~\ref{prop:integral}.
\end{proof}

\begin{proposition}\label{prop:parity-mutation}
For mutation at $k$, use the matrices $E,F$ of Section~\ref{sec:integral} and fix a general endpoint $Q$. Let $\sigma=1$ when $Q_k>0$ and $\sigma=0$ when $Q_k<0$, and fix $m\in M$ and $a,b\in N$. Write $B'$ for the mutable block of $p'$ and $\nu'$ for the pairing defined by the mutated compatible datum. Set
\[
 m'=ET_k(m),\qquad
 a'=Fa+\bigl([m_k]_+-\sigma m_k-\sigma(Ba)_k\bigr)e_k.
\]
Then, modulo two,
\[
 q_{B',D}(b)=q_{B,D}(Fb),\qquad
 \nu'_{a'}(m')+q_{B',D}(a')=\nu_a(m)+q_{B,D}(a),
\]
Hence the coefficient parity is preserved by the term transport of Proposition~\ref{prop:mutation}.
\end{proposition}
\begin{proof}
The two quadratic functions in the first equality have polarization $F^TDBF=DB'$. At $e_k$ both have value $d_k$. For $i\ne k$, $Fe_i=e_i+[b_{ki}]_+e_k$, so
\[
 q_{B,D}(Fe_i)=d_i+d_k\bigl([b_{ki}]_+^2-b_{ki}[b_{ki}]_+\bigr)=d_i\pmod2.
\]
Their difference is linear and vanishes on the standard basis, proving the first equality.

Put $t=\sigma((Ba)_k+m_k)-[m_k]_+$, the change of the $k$th coordinate in $Fa'=a+te_k$. Compatibility and the first equality give
\begin{align*}
 &\nu'_{a'}(m')+q_{B',D}(a')-\nu_a(m)-q_{B,D}(a)\\
 &\hspace{2em}\equiv d_k\bigl(tm_k-[m_k]_+(Ba)_k+t^2-t(Ba)_k\bigr)\pmod2.
\end{align*}
For each $\sigma\in\{0,1\}$, substitution reduces the parenthesis modulo two to $[m_k]_+([m_k]_+-m_k)=0$. Finally, subtract $m'$ from $ET_{k,Q}(m+pa)$ and use $p'=EpF$ to obtain the displayed charge $a'$.
\end{proof}

For a frozen exponent $f$, $\nu_a(f)=0$, so the parity formula is unchanged by $m\mapsto m+f$. Changing the compatible form with $p,D$ fixed also leaves $\nu_a(m)$ and $q_{B,D}$ unchanged.

\section{Two-wall factorization and the ordering at infinity}\label{sec:twowall}
For a nonzero rational function $c(v)\in\Q(v)$, let $\deg_\infty c$ be its numerator degree minus its denominator degree, and let $\lc_\infty c$ be the quotient of the leading coefficients. Set $\deg_\infty0=-\infty$. Write $c>_\infty0$ when $\lc_\infty c>0$. Together with zero these elements form the positive cone of an ordered field. Finite sums of positive elements cannot vanish. Multiplication by any integral power of $v$ preserves this order.

\subsection{Framed Hall series}
Let $q$ be a prime power, $k=\mathbb F_q$, $t=\sqrt q$, and let $\mathcal C$ be an essentially small hereditary $k$-linear length category with finite-dimensional Hom and Ext spaces. Suppose it has a positive additive integer degree $j(E)$ and finitely many isomorphism classes of bounded degree. Write
\[
 \chi(E,F)=\dim_k\Hom(E,F)-\dim_k\Ext^1(E,F).
\]
Assume for the moment that $\chi$ is symmetric. Write $[E]$ for the isomorphism class of an object $E$. Let $F^E_{M,N}$ count the subobjects of $E$ isomorphic to $N$ with quotient $M$, and use the Hall product $[M]*[N]=\sum_E F^E_{M,N}[E]$. For a formal variable $z$, define
\[
 I([E])=\frac{t^{\chi(E,E)}}{|\Aut E|}z^{j(E)},\qquad
 G(z)=\sum_{[E]}I([E]).
\]
The extension-counting identity
\[
 \sum_E\frac{F^E_{M,N}}{|\Aut E|}
 =\frac{q^{-\chi(M,N)}}{|\Aut M|\,|\Aut N|}
\]
shows that $I$ is an algebra homomorphism. Without symmetry the same identity gives the quantum phase $t^{\chi(N,M)-\chi(M,N)}$.

\begin{lemma}\label{lem:framing}
For each simple $S$, write $\End S=\mathbb F_{q^{\eps_S}}$. Suppose $\lambda\in\Z_{>0}$ and $\eps_S\mid\lambda j(S)$ for every simple. There is an exact $k$-linear functor $F_\lambda$ to finite-dimensional vector spaces with $\dim F_\lambda(E)=\lambda j(E)$. If $\operatorname{Gen}_\lambda(E)$ denotes vectors lying in no $F_\lambda(U)$ for a proper subobject $U$, then
\begin{equation}\label{eq:framing}
 \frac{G(t^\lambda z)}{G(t^{-\lambda}z)}
 =\sum_{[E]}t^{\chi(E,E)-\lambda j(E)}
  \frac{|\operatorname{Gen}_\lambda(E)|}{|\Aut E|}z^{j(E)},
\end{equation}
and each orbit count on the right is a nonnegative integer.
\end{lemma}
\begin{proof}
The category $\operatorname{Ind}(\mathcal C)$ of filtered colimits of objects of $\mathcal C$ is a Grothendieck category and has injective envelopes; see \cite[Section 2, following Theorem 2.3]{Krause}. Let $J_S$ be the injective envelope of $S$. Essentiality gives $\Hom(T,J_S)=0$ for a different simple $T$, and $\Hom(S,J_S)=\End S$. Writing $[E:S]$ for the multiplicity of $S$ in a composition series of $E$, exactness gives
\[
 \dim_k\Hom(E,J_S)=\eps_S[E:S].
\]
Thus the direct sum of $\lambda j(S)/\eps_S$ copies of $\Hom(-,J_S)^*$, where $*$ denotes the $k$-linear dual, over simple isomorphism classes, is the required functor. At any fixed object only finitely many summands are nonzero.

Exactness preserves finite intersections of subobjects. Each object has finitely many subobjects, by the bounded-degree and finite-Hom assumptions. A vector $x\in F_\lambda(E)$ therefore has a unique minimal generating subobject. Consequently
\[
 q^{\lambda j(E)}=\sum_{U\subset E}|\operatorname{Gen}_\lambda(U)|.
\]
If $g\in\Aut E$ fixes a generating vector, that vector belongs to $F_\lambda(\ker(g-1))$, so $g=1$. The action on generating vectors is free. The corresponding identity in the completed Hall algebra is
\[
 \left(\sum_{[E]}[E]\right)*
 \left(\sum_{[E]}|\operatorname{Gen}_\lambda(E)|[E]\right)
 =\sum_{[E]}q^{\lambda j(E)}[E].
\]
Applying $I$ gives
\[
 G(z)\sum_{[E]}|\operatorname{Gen}_\lambda(E)|I([E])
 =G(t^{2\lambda}z).
\] Substitute $t^{-\lambda}z$ to obtain \eqref{eq:framing}.
\end{proof}

If a rational wall series has these Hall realizations at unbounded prime powers, every nonzero coefficient of its favorable action is $>_\infty0$: a nonzero rational function has only finitely many zeros and poles, and its eventual sign on positive real arguments is its leading sign. Converting $X^m z^j$ to based monomials supplies $v^{-\lambda j}$, giving the centered ratio $G(v^\lambda z)/G(v^{-\lambda}z)$.

\subsection{Two-wall ray factors}
Fix positive integers $d_1,d_2,e$ with $d_1\mid e$ and $d_2\mid e$, and use $\Omega(a,b)=e(a_1b_2-a_2b_1)$. Factor
\begin{equation}\label{eq:hall-rays}
 \Psi_{v^{d_1}}(X^{(1,0)})\Psi_{v^{d_2}}(X^{(0,1)})
 =\prod_a^{\mu(a)\text{ increasing}}G_a(X^a),
 \qquad\mu(a)=\frac{a_1}{a_1+a_2},
\end{equation}
Here $X^aX^b=v^{\Omega(a,b)}X^{a+b}$, and $a$ runs over the primitive nonzero vectors of $\N^2$. Thus $(0,1)$ comes first. Products and factorization are interpreted at each total height. Put
\[
 q(a)=d_1a_1^2+d_2a_2^2-ea_1a_2,\qquad
 \delta(a)=\gcd(d_1a_1,d_2a_2),
\]
with the usual convention $\gcd(r,0)=r$ for $r>0$.

\begin{proposition}\label{prop:hallray}
For $\lambda\in\delta(a)\Z_{>0}$, every nonzero coefficient of $G_a(v^\lambda z)/G_a(v^{-\lambda}z)$ is positive at infinity. If $G_a=1+\sum_{j\ge1}g_jz^j$ and $g_j\ne0$, then
\[
 \deg_\infty g_j=-j^2q(a),\qquad\lc_\infty g_j=1.
\]
If $q(a)>0$ and $G_a\ne1$, then $q(a)=\delta(a)$ and $G_a=\Psi_{v^{\delta(a)}}$. If $q(a)\le0$, every nonzero coefficient of $\log G_a$ is positive at infinity.
\end{proposition}
\begin{proof}
Over $\mathbb F_q$, take $K_i=\mathbb F_{q^{d_i}}$ and the $(K_2,K_1)$-bimodule $A=\mathbb F_{q^l}^{\oplus e/l}$, where $l=\operatorname{lcm}(d_1,d_2)$. Representations are maps $A\otimes_{K_1}V_1\to V_2$. The vertex-arrow exact sequence for Hom and Ext gives heredity and Euler form
\[
 \chi(\alpha,\beta)=d_1\alpha_1\beta_1+d_2\alpha_2\beta_2-e\alpha_1\beta_2.
\]
The difference $\chi(\beta,\alpha)-\chi(\alpha,\beta)$ is $\Omega(\alpha,\beta)$. Vertex vector-space Hall series are exactly the two dilogarithms, because their degree-$j$ coefficients are
\[
 \frac{t^{d_i j^2}}{|\operatorname{GL}_j(\mathbb F_{q^{d_i}})|}
 =\frac{t^{-d_i j^2}}{\prod_{h=1}^j(1-t^{-2d_i h})}.
\]
Every representation has subobject $(0,V_2)$ and quotient $(V_1,0)$, fixing the order on the left of \eqref{eq:hall-rays}.

Use slope $\mu$, with semistability meaning that every nonzero subobject has slope at most that of the object. Stability requires strict inequality for every nonzero proper subobject. The unique Harder--Narasimhan filtration puts the highest slope subobject on the right in our Hall convention. Its integrated factorization is precisely \eqref{eq:hall-rays}. The category $\mathcal C_a$ of semistable objects of slope $\mu(a)$ is abelian and extension closed. It is hereditary: an ambient lift killing a Yoneda two-extension has middle term an extension of same-slope objects, hence lies in $\mathcal C_a$. It is a length category with positive degree $j$ for dimension vector $ja$, and its Euler form is symmetric. A stable object $S$ of dimension $ja$ has division endomorphism ring of degree $\eps_S$ over $\mathbb F_q$. Its action on every nonzero vertex space is faithful, so $\eps_S\mid jd_i a_i$ for both vertices, and hence $\eps_S\mid j\delta(a)$. Lemma~\ref{lem:framing} proves the action assertion. Rational factorization uses only additions and multiplications of initial coefficients, so it has no poles at $v=\sqrt q>1$.

The representation space of dimension $ja$ is affine of dimension $ej^2a_1a_2$. Its geometric semistable locus is open: each destabilizing dimension vector gives a closed incidence image under a projective Grassmannian projection. To compare geometric and rational semistability, split the species over an algebraic closure and give each split vertex the original slope and rank weights divided by $d_i$. The geometric Harder--Narasimhan filtration is unique and Frobenius invariant, hence descends. Thus this open set counts exactly $\mathcal C_a$.

For a nonzero $g_j$, choose a prime power $q_0$ at which it is nonzero. Along extensions $q=q_0^h$ with $h\equiv1\pmod l$, the same species and nonempty semistable open set base change correctly. This open set has $q^{ej^2a_1a_2}(1+O(q^{-1}))$ points, by the elementary polynomial zero bound. Dividing by the two general linear groups and multiplying by $q^{j^2q(a)/2}$ gives $g_j(\sqrt q)=q^{-j^2q(a)/2}(1+O(q^{-1}))$. This proves its asserted rational degree and leading coefficient. The set of nonzero degrees is additive, since direct sums preserve semistability and two nonzero rational coefficients are simultaneously nonzero for all sufficiently large prime powers.

If $q(a)>0$, then $\delta(a)\mid q(a)$. For a stable constituent of dimension $ja$, heredity gives
\[
 j^2q(a)=\eps_S-\dim\Ext^1(S,S)\le\eps_S\le j\delta(a).
\]
Thus $j=1$, $q(a)=\eps_S=\delta(a)$, and $S$ is rigid. Two different stable objects of this dimension would have zero Hom and positive Euler pairing, which is impossible. The category is therefore the direct-sum category of this unique simple, giving the dilogarithm. This holds at all sufficiently large prime powers where the ray is nonzero and consequently as a rational identity.

If $q(a)<0$, all terms of a nontrivial partition $j=j_1+\cdots+j_s$ in the logarithm have strictly smaller degree, since $\sum j_i^2<j^2$. The leading term of $g_j$ survives; when $g_j=0$, additivity of its support excludes such partitions altogether.

Finally suppose $q(a)=0$ and both coordinates are positive. Set $b=e/d_1$, $c=e/d_2$. Rationality of a zero direction requires $bc(bc-4)$ to be a square. Writing $r\ge0$ for the integer square root gives $(bc-2-r)(bc-2+r)=4$, which forces $bc=4$. The cases $(b,c)=(2,2),(1,4),(4,1)$ have primitive zero directions $(1,1),(1,2),(2,1)$. Stable objects exist: use a nonzero arrow in the first case, a surjection from the larger field to a two-dimensional space over the smaller field in the second, and the dual injection in the third. The subobject slope criterion verifies stability directly. Direct sums give every $j\ge1$. All $g_j$ have degree zero and leading coefficient one, so the degree-zero leading series is $(1-z)^{-1}$. Its logarithm has coefficient $1/j>0$ at $z^j$.
\end{proof}

\subsection{Dilogarithm and exponential factors}
For a charge $u\in N^+$ define $\delta_D(u)=\gcd_i(d_i u_i)$. We shall use factors of the two forms
\[
 \Psi_{v^d}(v^rX^{pu}),\quad d\in\Z_{>0},\ d\mid\delta_D(u),\ r\in\Z,
 \qquad\exp(c(v)X^{pu}),\quad c>_\infty0.
\]
The charge $u$ need not be primitive. An atomic factorization means a product of these factors on a fixed ray, finite at every bounded height.

\begin{lemma}\label{lem:twoatoms}
Two such initial factors have a two-wall completion whose ray labels admit atomic factorizations. Every new dilogarithm parameter divides the pairing of its charge with every exponent of the original lattice.
\end{lemma}
\begin{proof}
When the pairing vanishes the factors commute. Otherwise order the two directions so that their pairing is $e>0$. For two dilogarithms with parameters $d_1,d_2$, Proposition~\ref{prop:hallray} applies. Multiplying the two initial variables by $v^{r_1}$ and $v^{r_2}$ multiplies the coefficient of degree $j(a_1,a_2)$ by $v^{j(r_1a_1+r_2a_2)}$, preserving all conclusions.

Write the two original charges as $u,w$. For each original coordinate $i$, apply to a species representation the exact vector-space functor consisting of $d_i u_i/d_1$ copies of its first vertex space and $d_i w_i/d_2$ copies of its second. These are nonnegative integers. The division endomorphism ring of the rigid simple acts on every nonzero resulting space. Its degree $\delta(a)$ therefore divides $d_i(a_1u_i+a_2w_i)$ for each $i$.

For the remaining cases consider series with all positive-degree coefficients nonzero, leading degree $sj^2+\ell j$, positive leading coefficient, and $s\ge0$. Exponentials belong to this class with $s=0$. In the product of two such series the coefficient of $(a,b)$ has leading degree
\[
 s_1a^2+s_2b^2+eab+\ell_1a+\ell_2b.
\]
For increments $(a_i,b_i)$ on distinct increasing-slope rays, its excess over their reconstructed product is
\[
 2\sum_{i<j}(s_1a_ia_j+s_2b_ib_j+e b_i a_j)>0.
\]
Thus height induction in the ray factorization leaves the positive leading coefficient unchanged. Every mixed ray has quadratic coefficient $s_1a^2+s_2b^2+eab>0$, so its logarithm is positive at infinity by the strict square inequality used above. This treats two exponentials.

For a dilogarithm of parameter $d$ followed by such a series, move the dilogarithm to the right by conjugation. Write the second series as $\sum_{b\ge0}h_bX^{bpw}$. Here $e/d\in\Z_{>0}$, and the coefficient of $(a,b)$ is
\[
 h_bv^{ra}\gb{eb/d}a{v^d},\qquad b\ge1,\quad0\le a\le eb/d,
\]
of leading degree $sb^2+eab-da^2+ra+\ell b$. For distinct increasing-slope increments the excess degree is
\[
 2\sum_{i<j}\bigl(a_j(e b_i-d a_i)+s b_i b_j\bigr)>0,
\]
since $a_i/b_i<a_j/b_j\le e/d$. The finite band is closed under addition, so factorization stays in it. With an exponential input, the interior rays have positive quadratic coefficient; the upper boundary is exactly $\exp(c(v)v^{re/d}X^{p((e/d)u+w)})$, since its extreme Gaussian coefficient is one. The lower boundary is the original exponential. If the dilogarithm is second, move it to the left using the favorable inverse action. The band is $0\le b\le ea/d$ and the excess is $2\sum_{i<j}(b_i(ea_j-db_j)+s a_i a_j)>0$.
\end{proof}

\section{Positivity in arbitrary rank}\label{sec:propagation}
The monoid lift and perturbation construction of \cite[Sections 5.1--5.2 and Lemma 2.11]{DM} reduces higher-valency joints to the two-wall factorizations of Lemma~\ref{lem:twoatoms} by induction on degree.

Denote the original charge lattice, symmetrizer and skew form by $N_0$, $D_0$ and $\Omega_0$. An auxiliary free positive lattice $N$ carries a map $\rho:N\to N_0$ sending every positive generator to a nonzero positive charge, with form $(\rho^*\Omega_0)(a,b)=\Omega_0(\rho a,\rho b)$ for $a,b\in N$. Divisibility for a dilogarithm of working charge $u$ always means $d\mid\delta_{D_0}(\rho u)$. Working height is the sum of coordinates in $N$. If distinct working charge rays have the same geometric normal, their group elements commute. We separate their logarithms by working degree before applying the argument and multiply them only after projection. The local two-dimensional model lies in the effective normal quotient.

\begin{lemma}\label{lem:lift}
A finite set of positive incoming factors may be replaced by independent positive generators, preserving the original quantum coefficients, wall actions after projection, and incoming and outgoing conditions. If their original heights are $h_j$, a lifted charge $a$ has original height $\sum_jh_ja_j$.
\end{lemma}
\begin{proof}
If the old Hamiltonian lattice is $L$ with form $\omega$, and the shifts are $U_j\in L$, set
\[
 \widetilde L=L\oplus\bigoplus_j\Z\epsilon_j,\quad
 \pi(x,\textstyle\sum b_j\epsilon_j)=x+\sum b_jU_j,
 \quad\widetilde\omega=\pi^*\omega.
\]
Here $\widetilde\omega(x,y)=\omega(\pi x,\pi y)$ for $x,y\in\widetilde L$. The positive working generators $\epsilon_j$ act by their inclusions in $\widetilde L$. Take inverse images of the initial wall hyperplanes. Since $\ker\pi$ is in the radical, the rational scattering construction can be made cylindrical along it. Projection then preserves codimension one, path products, and recession membership: a support saturated along $\ker\pi$ contains a working direction in its recession cone exactly when its projected support contains the projected direction. This verifies the saturation and surjectivity hypotheses of \cite[Lemma 2.11]{DM}.

The rule $X^x\mapsto X^{\pi x}$ is a homomorphism because the form is pulled back. Positive external charges ensure that only finitely many lifted charges can contribute below a fixed old height. Projection therefore respects completed products. Rational uniqueness identifies the projected completion with the old one. Coincident rays are multiplied after projection. \end{proof}

\begin{theorem}\label{thm:ordered}
At every bounded height the cluster diagram has a representative whose merged ray labels admit atomic factorizations. Every nonzero coefficient of a favorable wall action and of a theta series is positive at infinity.
\end{theorem}
\begin{proof}
We prove the factorization assertion simultaneously for all auxiliary charge maps and for central and generically translated initial diagrams. First apply Lemma~\ref{lem:lift} to make the initial factors independent. At any finite truncation, generic translations of the initial hyperplanes make each codimension-two joint lie on at most two initial walls. The associated affine consistent diagram and its central asymptotic diagram have the same initial factors. This is the coefficient-independent perturbation construction in \cite[Section 5.2]{DM}.

Induct on working height $H$. Height one is the initial diagram. Suppose the assertion holds through $H-1$ and take a finite subdivided representative through $H$. At a joint, positive decompositions of incoming labels have at most two factors of height one: they are the original independent initial factors. A ray whose leading height is at least two cannot contain a height-one factor in a positive decomposition, because the first coefficients of all factors have positive leading sign.

Consider a local completion with these positive incoming factors. Lift them to independent generators of weights $h_j$. A lifted charge of old height at most $H$ using any $h_j\ge2$ has lifted height at most $H-1$. For each fixed lifted ray using such a generator, every charge on that ray of old height at most $H$ has working height at most $H-1$. The induction hypothesis controls this truncated ray. The remaining coordinate face has at most two generators and is controlled at all heights by Lemma~\ref{lem:twoatoms}. The span of charges using another coordinate is a Lie ideal, so setting those coordinates to zero gives the group quotient for this face. Pushing back proves that this local positive completion is positive through $H$.

Fix now a working charge $w$ of height $H$. Pure height-$H$ changes to incoming labels are central modulo greater height. A change on any other direction travels straight and cannot alter the coefficient of $X^w$. If an outgoing piece on the ray of $w$ is not positively factorizable through $H$, its incoming piece on that same direction must also fail: otherwise keep its positive factorization through $H$, replace all other incoming labels by their positive factorizations through $H-1$, and use the preceding local argument. The central changes on other rays do not affect the chosen ray. For zero local pairing the factors commute.

Suppose a wall piece fails to admit a positive factorization through height $H$. From a general point of that piece, move in the positive working shift direction of $w$. This ray avoids the codimension-two skeleton within the wall and crosses finitely many transverse facets. The preceding paragraph propagates the failure of factorization across each facet. Every noninitial outgoing support is a translated closed rational polyhedral cone whose recession cone does not contain this direction. Its intersection with the ray is bounded. After leaving their finitely many bounded intersections, only initial full walls and the identity remain, all positively factorizable. This is a contradiction. If the direction is radical in a local quotient, its labels are unchanged across the joint; if it is entirely cylindrical, no new outgoing support in that direction is possible. Taking the asymptotic diagram proves the central statement and completes induction.

For a dilogarithm factor with positive pairing $\lambda$, divisibility gives
\[
 \frac{\Psi_{v^d}(v^{r+\lambda}z)}{\Psi_{v^d}(v^{r-\lambda}z)}
 =\sum_{j=0}^{\lambda/d}v^{rj}\gb{\lambda/d}j{v^d}z^j.
\]
For an exponential factor the ratio is $\exp(c(v)(v^\lambda-v^{-\lambda})z)$. Both have positive leading signs. Reverse favorable crossings give the same formula with the absolute pairing. Products and finite coefficient sums preserve those signs. A fixed theta coefficient only uses a bounded-height diagram, so this proves its positivity. Invariance under equivalent representatives identifies the coefficients obtained at different truncation heights.
\end{proof}

\section{The finite theta algebra}\label{sec:finite}
The leading-term order is $m\preceq m+pa$ for $a\in P$. A series is $m$-pointed if its support lies in $m+pP$ and its coefficient at $X^m$ is one. Pointed series are compatibly pointed across seeds when their leading labels are related by the tropical mutation maps. Finite collections of theta series are independent: choose a minimal label in this order, read its coefficient one, and remove it. For infinite expansions, the support condition gives the following recursion.

\begin{lemma}\label{lem:formal-expansion}
Fix a general endpoint $Q$ and a finite union $S=\bigcup_{j=1}^r(m_j+pP)$, with $r\ge1$ and $m_j\in M$. Every formal series $f$ with coefficients in $R$ and support in $S$ has a unique coefficientwise expansion
\[
 f=\sum_{m\in S}c_m\Theta_{m,Q},\qquad c_m\in R.
\]
On this fixed support set, the coefficient transformation and its inverse are continuous for the height truncations.
\end{lemma}
\begin{proof}
Lemma~\ref{lem:endpoint-continuation} gives support in $m+pP$ and coefficient one at $X^m$ for every theta series used here, including the continued values. Partition the translating vectors $m_j$ by their classes in $G$. In one class choose a representative $r_0\in M$ and write $m_j=r_0+pb_j$, with $b_j\in\Z^n$. Put $b=r_0+p(\min_j(b_j)_1,\ldots,\min_j(b_j)_n)^T$. Then the part of $S$ in this class is contained in $b+pP$. Injectivity of $p$ identifies its exponent positions with $b+pa$, $a\in P$.

Define the coefficients in increasing order of $\htg(a)$ by
\[
 c_{b+pa}=[X^{b+pa}]f-
 \sum_{\substack{d\in P,\ d\le a\\d\ne a}}
 c_{b+pd}[X^{b+pa}]\Theta_{b+pd,Q},
\]
where $d\le a$ is coordinatewise. The sum is finite, and every $d$ in it has smaller height. The diagonal theta coefficient is one, so this recursion gives existence and uniqueness. If $b+pa\notin S$, then $[X^{b+pa}]f=0$ and no predecessor $b+pd\in S$ can contribute, since $S+pP=S$. Induction therefore gives $c_{b+pa}=0$ outside $S$. Apply the recursion in each of the finitely many classes.

Both coefficient transformations through height $H$ use only the finitely many input positions through that height. The common lower translate used above gives the same truncation topology on $S$: each original translating vector differs from $b$ by a fixed nonnegative charge, and there are finitely many such vectors. This proves continuity.
\end{proof}

\begin{lemma}\label{lem:detect}
Let $J\subset M$ be finite and fix $m\in M$. Define
\[
 H=\max\bigl(\{0\}\cup\{\htg(a):a\in P,\ m-pa\in J\}\bigr).
\]
Choose a point $Q$ outside all proper rational linear hyperplanes and satisfying $\nu_a(Q)\nu_a(m)>0$ whenever $0<\htg(a)\le H$ and $\nu_a(m)\ne0$. Then
\[
 [X^m]\sum_{j\in J}c_j\Theta_{j,Q}=c_m,
\]
where $c_j\in K$ for $j\in J$ and $c_m=0$ if $m\notin J$. Such points exist in an open neighborhood of $m$.
\end{lemma}
\begin{proof}
Injectivity of $p$ and finiteness of $J$ make the set defining $H$ finite. The strict conditions hold at $m$ and persist nearby. If a path from $j$ ends with exponent $m$, its total bend charge $a$ satisfies $m-pa=j$, so every bend has height at most $H$. Its last genuine bend would lie on the reverse ray $Q+\R_{\ge0}m$. For a possible wall charge $u$, the value $\nu_u(Q)+s\nu_u(m)$ is never zero: its two terms have the same strict sign when $\nu_u(m)\ne0$, and the first is nonzero otherwise. Thus no last bend exists. Only the straight path from $m$ remains, with coefficient one. When $m=0$ use the constant path.
\end{proof}

This scattering-atlas coefficient detection follows \cite[Proposition 6.4(3)]{GHKK} and \cite[Theorem 1]{Atomic}. Its chart depends on the finite label set and target and may lie outside the cluster atlas.

\begin{proposition}\label{prop:finmult}
If $\Theta_m$ and $\Theta_n$ are finite, their product has a finite theta expansion with coefficients in $R$ and all output theta functions finite. Each nonzero coefficient is positive at infinity. Moreover, every output support is contained in the support of the product in a fixed positive seed chart.
\end{proposition}
\begin{proof}
Lemma~\ref{lem:formal-expansion}, applied to the support set $m+n+pP$, gives a formal expansion with labels $m+n+pa$, coefficients in $R$, and leading coefficient $v^{\Lambda(m,n)}$. For a fixed output label, only finitely many preceding labels can affect it. Apply Lemma~\ref{lem:detect} to this finite truncation and use endpoint transport. Its expansion coefficient is a monomial coefficient of the product in the detecting chart. By Theorem~\ref{thm:ordered}, it is zero or positive at infinity; quantum multiplication adds only powers of $v$.

In the fixed seed chart, each nonzero output contributes its label monomial with its nonzero expansion coefficient. All other contributions to that monomial also have positive leading sign. Therefore the label lies in the finite support of the product. Only finitely many output labels occur. The same noncancellation argument at every exponent shows that each entire output support is contained in that finite support. Proposition~\ref{prop:mutation} gives membership in $\U$.
\end{proof}

\begin{corollary}\label{cor:structure-parity}
For finite theta functions, write the finite expansion from Proposition~\ref{prop:finmult} as
\[
 \Theta_m\Theta_n=\sum_{a\in P}\alpha_a(v)\Theta_{m+n+pa}.
\]
Then
\[
 \alpha_a(-v)=(-1)^{\Lambda(m,n)+\nu_a(m+n)+q_{B,D}(a)}\alpha_a(v).
\]
\end{corollary}
\begin{proof}
Use the involutions $\mathscr J_m$ defined in the proof of Theorem~\ref{thm:parity}. For input charges $b,c\in P$, polarization gives
\begin{align*}
 &\nu_b(m)+q_{B,D}(b)+\nu_c(n)+q_{B,D}(c)+\Lambda(m+pb,n+pc)\\
 &\hspace{2em}\equiv\Lambda(m,n)+\nu_{b+c}(m+n)+q_{B,D}(b+c)\pmod2.
\end{align*}
Thus $\mathscr J_{m+n}(\Theta_m\Theta_n)=(-1)^{\Lambda(m,n)}\Theta_m\Theta_n$. Theorem~\ref{thm:parity} and the same polarization identity give
\[
 \mathscr J_{m+n}(\Theta_{m+n+pa})
 =(-1)^{\nu_a(m+n)+q_{B,D}(a)}\Theta_{m+n+pa}.
\]
Apply $\mathscr J_{m+n}$ to the finite expansion and compare its independent theta coordinates.
\end{proof}

\begin{proposition}\label{prop:scaling}
For subsets $S,T\subseteq M_\R$, write $kS=\{ks:s\in S\}$ and $S+T=\{s+t:s\in S,\ t\in T\}$. At a fixed endpoint $Q$ outside every proper rational linear hyperplane, and for every positive integer $k$,
\[
 k\supp\Theta_m\subseteq\supp\Theta_{km}
 \subseteq\underbrace{\supp\Theta_m+\cdots+\supp\Theta_m}_{k\text{ times}}.
\]
In particular $\Newt(\Theta_{km})=k\Newt(\Theta_m)$ whenever $\Theta_m$ is finite, and $\Theta_{\rm fin}$ is an additive saturated submonoid of $M$.
\end{proposition}
\begin{proof}
At a general wall point with primitive charge $u\in N$, let $H(z)$ be the merged ray label, $z=X^{pu}$, and $\delta=\gcd_i(d_i u_i)$. For $\lambda,\mu\in\delta\Z_{>0}$ put $A_\lambda(z)=H(v^\lambda z)/H(v^{-\lambda}z)$. Cancellation of the middle factors gives
\begin{equation}\label{eq:cocycle}
 A_{\lambda+\mu}(z)=A_\lambda(v^\mu z)A_\mu(v^{-\lambda}z).
\end{equation}
For a series in $z$, let $\supp_z$ be the set of powers with nonzero coefficients. All nonzero coefficients have positive leading sign, hence $\supp_zA_{k\delta}=\underbrace{\supp_zA_\delta+\cdots+\supp_zA_\delta}_{k\text{ times}}$. Stretching the slopes of a broken line by $k$, keeping its geometric image, multiplies each selected bend by $k$ and is permitted by this support identity. The resulting contribution cannot cancel. This proves the first inclusion.

The formal expansion of $\Theta_m^k$ has coefficient one at $\Theta_{km}$: the total leading charge is zero and $\Lambda(m,m)=0$. For each output charge $a$, endpoint transport preserves the charge filtration, and only the finitely many labels $km+pb$ with $0\le b\le a$ can affect its coefficient. Apply Lemma~\ref{lem:detect} to these labels at its detecting endpoint. The coefficient is then a monomial coefficient of the transported product, hence is zero or positive at infinity by Theorem~\ref{thm:ordered}. Transport back to $Q$. The nonnegative leading signs prevent cancellation of any support term of $\Theta_{km}$ in the product, proving the second inclusion. The two inclusions imply scaling of convex hulls and equivalence of finiteness for $m$ and $km$. Additivity follows from Proposition~\ref{prop:finmult}, since its leading output is $\Theta_{m+n}$.
\end{proof}

\begin{corollary}\label{cor:finite-cone}
For a seed $s$, let $\mathscr C_s$ consist of all finite nonnegative real linear combinations of labels in $\Theta_{\rm fin,s}$. Then
\[
 \mathscr C_s\cap M_s=\Theta_{\rm fin,s}.
\]
For seeds $s,t$, let $\phi_{ts}:M_{s,\R}\to M_{t,\R}$ be the tropical map along a mutation sequence from $s$ to $t$. Then $\phi_{ts}$ satisfies $\phi_{ts}(\mathscr C_s)=\mathscr C_t$.
\end{corollary}
\begin{proof}
If an integral $m$ lies in the cone generated by finitely many finite labels, the rational linear system expressing $m$ as their nonnegative combination has a rational solution. Clearing denominators and using additivity gives $km\in\Theta_{\rm fin,s}$ for some positive integer $k$. Saturation gives $m\in\Theta_{\rm fin,s}$.

For $x\in\mathscr C_s$, intersect a finitely generated rational cone containing $x$ and generated by finite labels with a closed rational linearity cone of $\phi_{ts}$ containing $x$. This intersection is a rational polyhedral cone. Scale its finitely many rational generators to lattice points; the first assertion makes them finite labels. Their images are finite by mutation compatibility. Linearity on this cone gives $\phi_{ts}(x)\in\mathscr C_t$. Apply the same argument to the inverse map.
\end{proof}

\begin{corollary}\label{cor:frozen-translation}
Let $M_{\rm fr}\subset M$ consist of the exponents with zero mutable coordinates. For $f\in M_{\rm fr}$,
\[
 \Theta_{m+f}=v^{-\Lambda(f,m)}X^f\Theta_m
 =v^{-\Lambda(m,f)}\Theta_mX^f.
\]
In particular $\Theta_{\rm fin}+M_{\rm fr}=\Theta_{\rm fin}$.
\end{corollary}
\begin{proof}
Compatibility gives $\Lambda(pa,f)=0$ for every charge $a$. If $\gamma(t)$ is a broken line, replace it by $\gamma(t)-tf$ and add $f$ to every segment exponent. Wall supports are cylindrical in frozen directions, so this preserves crossing times, endpoint, and action coefficients. The charge coefficients of the two theta series agree. Since $\Lambda(f,pa)=0$, based-monomial multiplication supplies exactly the displayed phases. Multiplication by a monomial preserves finiteness.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Integrality, bar invariance, and mutation transport were established in Section~\ref{sec:integral}. Propositions~\ref{prop:finmult} and \ref{prop:scaling} give the algebra and label assertions. To include $\A$, take $m_i\ge0$ for all mutable indices. Suppose a path has successive genuine bend charges $a_1,\ldots,a_q$. After the first bend it lies in $\nu_{a_1}\le0$. If its previous segment lies in $\nu_{a_1+\cdots+a_{j-1}}\le0$, the next bend point also has $\nu_{a_j}=0$. On the following segment the cumulative normal has derivative
\[
 \nu_{a_1+\cdots+a_j}(-m-p(a_1+\cdots+a_j))
 =-\nu_{a_1+\cdots+a_j}(m)\le0.
\]
Thus the last segment cannot reach the positive chamber. A tangent first segment cannot select a nonconstant action term. Only the straight path remains, so $\Theta_{m,Q}=X^m$. The same statement in every seed, using Proposition~\ref{prop:mutation}, identifies normalized cluster monomials and frozen units with finite theta functions. These generate $\A$.
\end{proof}

\section{Valuative independence}\label{sec:vit}
Fix a seed and a lattice-general endpoint $Q$. In this section and Section~\ref{sec:vertices}, supports are evaluated at this same $Q$; finiteness here means finiteness at $Q$. At a general wall point $y$, let $u$ be the primitive charge in $N$, and let $\delta=\gcd_i(d_i u_i)$. In particular $\nu_u(M)=\delta\Z$ and $\nu_u(pu)=0$. If $H_y(z)$ is the merged ray label at $y$, put $A_{y,\lambda}(z)=H_y(v^\lambda z)/H_y(v^{-\lambda}z)$ for $\lambda\in\delta\Z_{>0}$, and set
\[
 S_y=\supp_zA_{y,\delta}\subseteq\N,\qquad
 C_y=\conv_\Q(S_y/\delta).
\]
Here $\conv_\Q$ denotes finite rational convex combinations. The set $C_y$ is $[0,L_y]\cap\Q$ with $L_y=\max S_y/\delta$ if bounded, and $\Q_{\ge0}$ otherwise. It includes zero.

A rational broken line has segment exponents in $M_\Q$ and velocity equal to minus the segment exponent. At a transverse crossing at $y$, its incoming and outgoing exponents $x,x^{\prime}$ satisfy
\begin{equation}\label{eq:rbend}
 x'=x+|\nu_u(x)|c\,pu,\qquad c\in C_y.
\end{equation}
It has finitely many genuine bends and avoids seams. A crossing with $c=0$ may be inserted as a record. Its shear is
\[
 T=I+\operatorname{sgn}(\nu_u(x))c\,pu\otimes\nu_u,\qquad
 T^{-1}=I-\operatorname{sgn}(\nu_u(x))c\,pu\otimes\nu_u.
\]
Here $I$ is the identity on $M_\Q$, $\operatorname{sgn}$ is the sign of a nonzero real number, and $(pu\otimes\nu_u)(b)=\nu_u(b)pu$ for $b\in M_\Q$. Since $\nu_u(pu)=0$, the shear fixes the supporting hyperplane pointwise and has determinant one.

\begin{lemma}\label{lem:rational}
For $m\in M$, the final exponents of rational broken lines with initial exponent $m$ and endpoint $Q$ are exactly
\[
 \bigcup_{k\ge1}k^{-1}\supp\Theta_{km,Q}.
\]
If $\Theta_m$ is finite, their convex hull is $\Newt(\Theta_m)$, and every real linear functional attains its minimum on this set.
\end{lemma}
\begin{proof}
At bend $i$, let $y_i$ be the crossing point, $c_i\in C_{y_i}$ its bend parameter, $x_i$ the incoming exponent, and $u_i$ the primitive charge. Put $\nu_i=\nu_{u_i}$ and $\delta_i=\gcd_j(d_j(u_i)_j)$. Choose a finite expression $\delta_i c_i=\sum_j a_{ij}s_{ij}$, with $a_{ij}\in\Q_{\ge0}$, $s_{ij}\in S_{y_i}$ and $\sum_j a_{ij}=1$. Choose a common $k>0$ making all $kx_i$ integral and all
\[
 N_i=\frac{k|\nu_i(x_i)|}{\delta_i},\qquad N_i a_{ij}
\]
integers. The identity $k|\nu_i(x_i)|c_i=\sum_j(N_i a_{ij})s_{ij}$ expresses this increment as a sum of $N_i$ members of $S_{y_i}$. Equation~\eqref{eq:cocycle} permits this actual bend. Reparameterizing the path by $t\mapsto kt$ gives an actual broken line from $km$. Ordered positivity prevents cancellation of its final term. Conversely, divide all exponents of an actual path from $km$ by $k$. Each selected integer in $\supp A_{N\delta}$ is a sum of $N$ members of $S_y$, so the divided rule is \eqref{eq:rbend}. This proves the equality. Proposition~\ref{prop:scaling} gives the convex-hull assertion. The finite support for $k=1$ is included and attains every minimum.
\end{proof}

\begin{lemma}\label{lem:localcertificate}
For a general wall point $y$ and a finite subset $F\subset S_y$, there is a relative neighborhood $W$ in its hyperplane such that $F\subset S_{y'}$ for every general $y'\in W$. Finite rational broken-line records with fixed initial exponent and endpoint vary continuously with their rational bend parameters while transversality, time order, and membership in the chosen relative neighborhoods persist.
\end{lemma}
\begin{proof}
Choose a height at least $s\htg(u)$ for every $s\in F$. In that finite diagram the merged truncated action is constant on the relative open cell containing $y$. Its nonzero coefficients therefore persist in a smaller relative neighborhood. Higher walls cannot change them.

For the continuity assertion, let $\nu_i=\nu_{u_i}$ be the normal of the $i$th recorded wall, where $u_i$ is its primitive charge. Let $T_1,\ldots,T_r$ be the recorded rational shears, $P_i=T_i\cdots T_1$ and $P_0=I$. Unfolding gives the line $P_r^{-1}Q-tm$. The crossing times and points are
\begin{equation}\label{eq:unfold}
 t_i=\frac{\nu_i(P_{i-1}P_r^{-1}Q)}{\nu_i(P_{i-1}m)},\qquad
 q_i=P_{i-1}(P_r^{-1}Q-t_i m).
\end{equation}
The nonzero denominators, strict inequalities $t_1<\cdots<t_r<0$, and the required locations are open conditions. Formula~\eqref{eq:unfold} is continuous wherever they hold.

These rational records avoid the seams of the full diagram for lattice-general $Q$. At a transverse hyperplane, the crossing-point map in \eqref{eq:unfold} is a surjective rational map of rank $\rank M-1$ onto that hyperplane. The preimage of any rational codimension-two seam span is a proper rational subspace. The same projection argument applies to an unrecorded crossing on any segment. A segment contained in a wall would itself require a proper rational linear condition on $Q$. There are countably many such conditions. The chosen $Q$ avoids all of them. Additional wall crossings select the constant term.
\end{proof}

Fix a real functional $\ell$. Transport it backwards along a recorded path: just after bend $i$ it is $\ell_i=\ell\circ T_r\cdots T_{i+1}$. Call a crossing taut if its parameter minimizes $\ell_i(x+|\nu_u(x)|c\,pu)$ over $c\in C_y$.

\begin{lemma}\label{lem:taut}
A rational path attaining the minimum of $\ell$ among all rational paths with its initial exponent and endpoint is taut at every transverse crossing, including those where it does not bend.
\end{lemma}
\begin{proof}
If a crossing fails, insert it as a zero-bend record if necessary, and choose a rational $c'\in C_y$ giving a smaller value. Choose finite support certificates for all existing parameters and for $c'$. Lemma~\ref{lem:localcertificate} provides relative neighborhoods preserving them. Replace the selected parameter by $(1-\tau)c+\tau c'$ for a sufficiently small positive rational $\tau$, leaving all other parameters and crossing signs fixed. Reconstruct the path from \eqref{eq:unfold}, keeping $m$ and $Q$ fixed. All open conditions persist, and the combined finite certificates make the new parameters legal. The full-diagram seam condition also persists by lattice generality and rationality of the new record.

The earlier exponent is unchanged, as are all later shears. Hence the change of final value is exactly
\[
 \tau|\nu_i(x)| (c'-c)\,
 \ell(T_r\cdots T_{i+1}pu_i)<0,
\]
contradicting minimality. The full cocycle identity and preserved finite certificates justify the bends after clearing denominators.
\end{proof}

\begin{lemma}\label{lem:reverse}
Suppose $\ell(w)\ne0$ for every $0\ne w\in M_\Q$. Two minimal rational paths with endpoint $Q$ and the same final exponent have the same initial exponent and the same genuine bend record.
\end{lemma}
\begin{proof}
At a reverse crossing, the outgoing exponent $b$ determines $|\nu_u(x)|=|\nu_u(b)|$. The transported functional remains nonzero on nonzero rational vectors because the shears are invertible and rational. Thus minimization has a unique parameter: zero if $\ell_i(pu)>0$, and $L_y$ if $\ell_i(pu)<0$ and $C_y$ is bounded. If it is unbounded in the latter case, no minimum exists, so that state cannot occur on a minimal path. The incoming exponent is uniquely $b-|\nu_u(b)|c\,pu$.

The two paths have the same final straight segment. Compare the finite union of their genuine bends, starting from $Q$. At the first possible difference they meet the same wall with the same outgoing exponent and future functional. The other path, even if straight there, must be taut. The unique parameter forces equality. Repeating over the finite union proves the claim. A zero final exponent forces all exponents to be zero by invertibility of the shears.
\end{proof}

\begin{proof}[Proof of the Newton assertion in Theorem~\ref{thm:newton}]
First take $\ell$ as in Lemma~\ref{lem:reverse}. Each finite theta support has a unique minimal exponent. A path contributing it is minimal among rational paths by Lemma~\ref{lem:rational}. Two distinct labels cannot have equal minimum values: genericity would make their minimal exponents equal, and Lemma~\ref{lem:reverse} would make the labels equal. A finite combination therefore has a unique theta summand supplying its smallest term, irrespective of the signs of its nonzero coefficients. This proves
\[
 \val_\ell\left(\sum c_m\Theta_m\right)=\min_m\val_\ell(\Theta_m).
\]
Both sides are minima of finitely many linear functions and hence continuous in $\ell$. Such generic functionals are dense, so the equality holds for every real $\ell$. Convex separation gives the asserted Newton equality. Choosing a generic $\ell$ in the interior of a vertex normal cone also proves unique ownership of every joint vertex.
\end{proof}

\begin{corollary}\label{cor:window}
For every convex set $P_0\subset M_\R$,
\[
 (K\otimes_R\Mt)\cap\operatorname{span}_K\{X^a:a\in P_0\cap M\}
 =\bigoplus_{\substack{m\in\Theta_{\rm fin}\\\supp\Theta_m\subset P_0}}K\Theta_m.
\]
The spans consist of finite sums. If $P_0$ is bounded, the right side has finitely many labels. Any collection of linear valuation lower bounds holds for a finite theta combination exactly when it holds for every nonzero summand.
\end{corollary}
\begin{proof}
The Newton equality puts every participating support inside the convex hull of the support of the sum. A bounded set has finitely many lattice points, and each theta contains its label. The valuation assertion follows from the minimum identity.
\end{proof}

\section{Vertex coefficients and base change}\label{sec:vertices}
\begin{lemma}\label{lem:2L}
At a general wall point $y$, write $u$ for its primitive charge, $\delta=\gcd_i(d_i u_i)$, and $A_\lambda=A_{y,\lambda}$. Suppose the basic-pairing action $A_\delta(z)$ is a polynomial of degree $L$. It is a finite product of Gaussian dilogarithm actions, has coefficients in $\N[v^{\pm1}]$, and has leading $z$ coefficient one. The same leading-coefficient assertion holds for $A_{h\delta}$, $h\ge1$.
\end{lemma}
\begin{proof}
For $L=0$ the action is one. For $L>0$, choose the atomic representative of Theorem~\ref{thm:ordered} through charge height $2L\htg(u)$. On the ray, its factors are $\Psi_{v^d}(v^t z^k)$, with $d\mid k\delta$, or $\exp(cz^k)$ with $c>_\infty0$. Their actions are respectively
\[
 \sum_{j=0}^{k\delta/d}v^{tj}\gb{k\delta/d}j{v^d}z^{kj},
 \qquad \exp\bigl(c(v)(v^{k\delta}-v^{-k\delta})z^k\bigr).
\]
Discard factors with $k>2L$. A remaining $k>L$ would give a nonzero coefficient above $L$ and at most $2L$, impossible by noncancellation. Thus $k\le L$. An exponential factor would give such a coefficient at the smallest multiple of $k$ above $L$, which is at most $L+k\le2L$. It too is impossible. Any remaining Gaussian factor of degree greater than $L$ gives the same contradiction. Finally, if their degrees sum to more than $L$, the first partial sum exceeding $L$ is at most $2L$, and taking these top terms gives another forbidden nonzero coefficient. Hence their entire product has degree at most $L$. It agrees with $A_\delta$ through $2L$, so the polynomial equality is exact.

Its leading coefficient is a monomial $v^e$ with integer coefficient one. The wall action commutes with bar by Proposition~\ref{prop:integral}. Since $\delta\in\nu_u(M)$, this applies to $A_\delta$ itself, and forces $v^e=v^{-e}$, hence $e=0$. Iterating \eqref{eq:cocycle} gives
\[
 A_{h\delta}(z)=\prod_{j=0}^{h-1}A_\delta(v^{(h-1-2j)\delta}z).
\]
Its leading phase is $v^{L\delta\sum_{j=0}^{h-1}(h-1-2j)}=1$.
\end{proof}

\begin{proposition}\label{prop:monic}
Every vertex coefficient of a finite theta function is one. If $b$ is a vertex of $\conv\bigcup_{c_m\ne0}\Newt(\Theta_m)$ for a finite combination $\sum c_m\Theta_m$, then $[X^b]\sum c_m\Theta_m=c_{m_b}$ for its unique owner $m_b$.
\end{proposition}
\begin{proof}
Choose a generic functional in the interior of the vertex's minimum normal cone; this cone has nonempty interior even for a lower-dimensional polytope. Every path to the vertex is rationally minimal. Lemmas~\ref{lem:taut} and \ref{lem:reverse} show that there is exactly one merged-wall bend record. At each crossing its parameter is zero or the finite right endpoint of $C_y$. The former selects coefficient one. The latter means $S_y$ is bounded, so Lemma~\ref{lem:2L} applies and its maximal action coefficient is also one. The product of these based action coefficients is one. Unique vertex ownership, proved in Section~\ref{sec:vit}, gives the final assertion.
\end{proof}

\begin{proposition}\label{prop:basechange}
For a finite nonempty set $J\subset\Theta_{\rm fin}$ let $E=\bigcup_{m\in J}\supp\Theta_m$. The matrix $([X^b]\Theta_m)_{b\in E,m\in J}$ has a unit triangular square minor of size $|J|$. Consequently the map
\[
 S\otimes_R\Mt\longrightarrow\T_S
\]
is injective for every nonzero commutative unital $R$-algebra $S$, where the based monomial lattice is unchanged. A finite nonzero combination over $S$ has Newton polytope equal to the convex hull of the Newton polytopes of its nonzero summands after specialization.
\end{proposition}
\begin{proof}
Choose a vertex of the joint polytope. Its row has entry one in its unique owner column and zero in every other column. Remove that column and choose a vertex of the remaining joint polytope. Repeat. Chosen rows are distinct, since an earlier row vanishes in every remaining column. In this order the selected square matrix is triangular with diagonal ones. It remains invertible after every base change, proving finite and hence arbitrary direct-sum independence. Vertex coefficients of each theta remain one, so its polytope does not shrink. A joint vertex still has only its unique owner's nonzero scalar as coefficient, which remains nonzero even if $S$ has zero divisors. Thus the combination's joint vertices survive. All other terms lie in their convex hull.
\end{proof}

Propositions~\ref{prop:monic} and \ref{prop:basechange} complete Theorem~\ref{thm:newton}. In particular, for a finite theta function the quantum and classical Newton polytopes agree.

\subsection{The ordered scattering cone}
Let $R_{\ge0}^\infty=\{0\}\cup\{a\in R:\lc_\infty a>0\}$. In $\Mt$, require every scattering-chart coefficient of an element to lie in this set. Lemma~\ref{lem:detect} and Theorem~\ref{thm:ordered} give exactly
\[
 \bigoplus_{m\in\Theta_{\rm fin}}R_{\ge0}^\infty\Theta_m.
\]
This cone has no nonzero additively indecomposable element: every nonzero $f$ splits as $v^{-1}f+(1-v^{-1})f$, with both summands nonzero in the cone.

\begin{proposition}\label{prop:barcone}
In the bar-invariant part of the ordered scattering cone, the additively indecomposable nonzero elements are exactly the finite theta functions. They do not in general generate that cone additively.
\end{proposition}
\begin{proof}
Bar invariance of a finite combination is equivalent to bar invariance of every coefficient, by theta independence. The scalar semiring is therefore $\{0\}\cup\{a\in\Z[v+v^{-1}]:\lc_\infty a>0\}$. Its only nonzero additive indecomposable is $1$. A nonconstant member decomposes as $1+(a-1)$, since subtraction of the constant does not change the positive highest term. A constant is a positive integer and is indecomposable only when it is one. Conversely a sum of nonconstant positive-leading bar-invariant polynomials cannot be the constant one: their highest degrees cannot cancel. Independence now gives the asserted indecomposables. For example $(v+v^{-1})\Theta_m$ is in the cone but cannot be a finite sum of theta functions with nonnegative integral multiplicities.
\end{proof}


\subsection{Formal series with a finite valuation bound}\label{sec:formal-vit}
For a nonzero formal series $f$ and a real linear functional $\ell$ on $M_\R$, define
\[
 \val_\ell(f)=\inf\{\ell(b):[X^b]f\ne0\}\in\R\cup\{-\infty\},
 \qquad \val_\ell(0)=+\infty.
\]
Every nonzero coefficient has valuation zero. For an $R$-algebra $A$, the notation $\Theta_{j,Q}\otimes_R A$ below means applying $R\to A$ to each monomial coefficient. Lemma~\ref{lem:formal-expansion} holds over $A$: its recursion divides only by the diagonal coefficient one.

\begin{lemma}\label{lem:formal-minimum}
Let $Q$ be lattice general and let $\lambda:M_\R\to\R$ be a real linear functional nonzero on every nonzero vector of $M_\Q$. If $\lambda$ attains a minimum on $\supp\Theta_{m,Q}$, the coefficient at its unique minimum exponent is one. If two theta functions attain the same minimum exponent, and that exponent minimizes $\lambda$ on both supports, their labels are equal.
\end{lemma}
\begin{proof}
Proposition~\ref{prop:scaling} and the clearing-denominators proof of Lemma~\ref{lem:rational} give
\[
 \conv\bigcup_{k\ge1}k^{-1}\supp\Theta_{km,Q}
 =\conv\supp\Theta_{m,Q}
\]
without boundedness of the support. An actual path to an attained minimum therefore minimizes $\lambda$ among rational paths. Lemmas~\ref{lem:taut} and \ref{lem:reverse} give its unique merged-wall bend record and the assertion about labels. At each bend, the minimizing parameter is zero or a finite right endpoint of $C_y$. Its coefficient is one in the first case, and Lemma~\ref{lem:2L} gives one in the second. Multiplication of these based action coefficients proves the claim.
\end{proof}

\begin{theorem}\label{thm:formal-vit}
Fix a lattice-general endpoint $Q$, a finite union
$S=\bigcup_{i=1}^r(b_i+pP)$ with $r\ge1$ and $b_i\in M$, a nonempty set $J\subseteq S$, and a real linear functional $\ell:M_\R\to\R$.
Let $A$ be a nonzero commutative unital $R$-algebra and $c_j\in A\setminus\{0\}$ for $j\in J$. If
\[
 \alpha=\inf_{j\in J}\val_\ell(\Theta_{j,Q})\in\R,
\]
where the valuations on the right are taken before base change, then the coefficientwise sum is defined and
\[
 \val_\ell\left(\sum_{j\in J}c_j(\Theta_{j,Q}\otimes_R A)\right)=\alpha.
\]
In particular a finite quantum valuation of a single theta function is preserved by every such base change.
\end{theorem}
\begin{proof}
Every exponent in $S$ has finitely many predecessors in $S$, by the coset decomposition in Lemma~\ref{lem:formal-expansion}. This proves local finiteness of the sum and its nonvanishing by triangular independence.
Choose a real functional $h$ with $h(pe_i)>0$ for all $i$ and $h(w)\ne0$ for every $0\ne w\in M_\Q$. The first conditions define a nonempty open cone, since $p$ is injective, and the latter exclude countably many proper hyperplanes. Put
\[
 \eta=\min_i h(pe_i)>0,\qquad H_0=\min_i h(b_i),
 \qquad E=\bigcup_{j\in J}\supp\Theta_{j,Q}.
\]
For $x=b_i+pa\in E$ and $\epsilon>0$,
\[
 (\ell+\epsilon h)(x)\ge\alpha+\epsilon H_0+\epsilon\eta\htg(a).
\]
Thus each sublevel set in $E$ is finite. Outside a countable set of positive $\epsilon$, the functional $\ell+\epsilon h$ is nonzero on all nonzero rational vectors. Its minimum on $E$ has one theta owner by Lemma~\ref{lem:formal-minimum}, and that theta coefficient is one. The corresponding coefficient in the sum is its nonzero $c_j$, even if $A$ has zero divisors. Hence the minimum of $\ell+\epsilon h$ on the support of the sum equals its minimum on $E$.

For any nonempty subset $E'\subseteq E$ with $q=\inf_{E'}\ell\in\R$, one has
\[
 q+\epsilon H_0\le\inf_{E'}(\ell+\epsilon h)
 \le\ell(x_\delta)+\epsilon h(x_\delta),
 \qquad \ell(x_\delta)<q+\delta,
\]
where $x_\delta\in E'$ is fixed before letting $\epsilon$ tend to zero. Taking this limit and then $\delta\downarrow0$ proves convergence to $q$. Apply this to $E$ and to the nonempty support of the sum. The latter has $\ell$ bounded below by $\alpha$ and bounded above at any chosen support point. Their equal perturbed minima therefore have equal limits.
\end{proof}

\section{Support certificates and reddening sequences}\label{sec:reddening}
For a mutation at $k$, let $\sigma=1$ if the current endpoint coordinate $Q_k>0$, and $\sigma=0$ if it is negative. Direct subtraction of the transformed label from the transformed final exponent in Proposition~\ref{prop:mutation} gives
\begin{equation}\label{eq:charge-rec}
 a'=J_{k,\sigma}(B)a+([m_k]_+-\sigma m_k)e_k,
\end{equation}
Here $J_{k,\sigma}(B)$ is the $n\times n$ matrix equal to the identity off row $k$, with entries $-1$ at $(k,k)$ and $[b_{kj}]_+-\sigma b_{kj}$ at $(k,j)$ for $j\ne k$. The frozen rows enter the exponent identity, while the charge recurrence depends only on the mutable matrix.

\begin{proposition}\label{prop:cert}
Choose finitely many mutation words along which all selected endpoint coordinates are nonzero. For a word $w$, let $a_w$ be the transported charge. Iterating \eqref{eq:charge-rec} defines an integer matrix $A_w$ and an integer offset $b_w(m)$ by $a_w=A_wa+b_w(m)$. If nonnegative rational weights satisfy
\[
 -\sum_{w,i}\lambda_{w,i}e_i^TA_w=\eta^T,\qquad\eta_i>0,
\]
then every support charge obeys
\[
 0\le a_j\le\left\lfloor\frac{\sum_{w,i}\lambda_{w,i}(b_w(m))_i}{\eta_j}\right\rfloor.
\]
For a fixed finite family $W$ of words, such weights exist if and only if
\[
 \{a\in\R_{\ge0}^n:A_wa\ge0\text{ for every }w\in W\}=\{0\}.
\]
There exists some finite family $W$ with such weights if and only if the same cone, with inequalities for all admissible finite words, is zero. All words use the same chosen endpoint and its successive transforms.
\end{proposition}
\begin{proof}
Actual transport preserves each nonzero coefficient and the target positive charge condition, so $a_w\ge0$. Weighted addition gives the bound. For fixed $W$, a certificate forces its cone to vanish. Conversely, finite-dimensional separation, equivalently the alternative theorem for linear inequalities, gives a nonnegative combination of the rows of the $A_w$ and the coordinate rows that is strictly negative. Dropping the nonnegative coordinate rows preserves strict negativity. Rational approximation of the positive weights, with zero weights fixed, gives the required rational certificate. For the all-word assertion, intersect all homogeneous conditions with the compact simplex $a\ge0$, $\sum a_i=1$. If the intersection is empty, compactness gives a finite subfamily with empty intersection. Apply the fixed-family result to it.
\end{proof}

For one mutation from the positive chamber, \eqref{eq:charge-rec} yields
\begin{equation}\label{eq:onehop}
 a_k\le[-m_k]_++\sum_{i\ne k}[-b_{ki}]_+a_i.
\end{equation}
If $B$ is acyclic, $H=[-B]_+$ is nilpotent after ordering its vertices. Iterating $a\le[-m_{\rm mut}]_++Ha$ gives the finite bound
\[
 0\le a\le(I+H+\cdots+H^{n-1})[-m_{\rm mut}]_+.
\]

For the sharper reddening bound, suppose a mutation word takes $r$ to $s$ with auxiliary principal $c$-matrix $C^r(s)=-P$, where $P$ is the permutation matrix defined by $Pe_j=e_{\sigma(j)}$ for a permutation $\sigma$ of $\{1,\ldots,n\}$. Let $\phi:M_s\to M_r$ be the tropical label map along the inverse word, and $L$ its integral linear piece on the positive chamber of $s$.

\begin{lemma}\label{lem:opposite}
For each $m\in M_s$ there is a vector $b(m)\in\Z^n$, computable by \eqref{eq:charge-rec}, such that
\[
 Lp_s=-p_rP,\qquad Lm-\phi(m)=p_rb(m).
\]
The mutable block of $L$ is $-P$ and $DP=PD$.
\end{lemma}
\begin{proof}
On nonnegative mutable labels the theta functions are cluster monomials. Their degrees add under normalized multiplication, so $\phi$ is linear on the positive chamber with mutable columns the full $g$-vectors and frozen columns the identity columns. In this paragraph let $G$ denote the mutable block of $L$. Tropical duality gives $G^TDC^r(s)=D$; see \cite[Theorem 2.2.6 and Proposition 2.3.3]{Qin}. With $C^r(s)=-P$ this gives $G=-D^{-1}PD$. Its nonzero entries have absolute values $d_j/d_{\sigma(j)}$. They are positive integers whose product along each permutation cycle is one. Hence all are one, proving $DP=PD$ and $G=-P$.

The exponent recurrence gives $Lp_s=p_rA$ for an integral product of $J$ matrices, and $L$ is symplectic. Compatibility then yields
\[
 (D\ \ 0)=p_s^T\Lambda_s
 =A^T(D\ \ 0)L=(-A^TDP\ \ 0),
\]
so $A=-P$. Summing the integral affine offsets in \eqref{eq:charge-rec} gives the second equality.
\end{proof}

\begin{theorem}\label{thm:box}
Put $d_s(m)=P^{-1}b(m)$. Then $d_s(m)\in\N^n$ and
\[
 \Theta^s_m=\sum_{0\le a\le d_s(m)}t_{m,a}(v)X^{m+p_sa},
 \qquad t_{m,a}(v)\in R,\quad t_{m,0}=t_{m,d_s(m)}=1.
\]
All labels are finite, and these theta functions are an $R$-basis of $\U$. If the mutation class is acyclic at some seed, $\A=\U$.
\end{theorem}
\begin{proof}
A term of charge $a$ transports to a term with charge $b(m)-Pa$ at endpoint $\phi(Q)$ in the $r$ diagram. At this endpoint the theta series has nonnegative charges. Hence $0\le a\le P^{-1}b(m)$. The initial term gives $b(m)\ge0$. The target theta's leading term pulls back to the corner $a=P^{-1}b(m)$, with coefficient one. This proves finiteness and the two corner assertions.

Proposition~\ref{prop:mutation} now gives upper-algebra membership and compatible pointedness at every seed. In Qin's notation take $q^{1/2}=v$, $p^*=p$, and the dominance order $g'=g+pa$. Full rank follows from compatibility, and the reddening word gives injective reachability. Thus \cite[Theorem 6.2.1(1), referring to Theorem 1.2.1(1)]{Qin} applies. In the mutation-acyclic case, \cite[Theorem 7.5]{BZ} gives $\A=\U$.
\end{proof}

The corner $m+p_sd_s(m)=L^{-1}\phi(m)$ is a bijective function of $m$. The support box and its occupied far corner show that $d_s$ is independent of the chosen opposing word. It is integer piecewise linear, homogeneous, and unchanged by frozen translation.

\begin{proposition}\label{prop:prod-box}
In the situation of Theorem~\ref{thm:box}, every nonzero output of $\Theta_m\Theta_n$ has label $g=m+n+p_sa$ satisfying
\[
 a\ge0,\qquad a+d_s(g)\le d_s(m)+d_s(n).
\]
The function $d_s$ is componentwise subadditive, and its continuous piecewise linear extension to $M_{s,\R}$ is convex.
\end{proposition}
\begin{proof}
The product is supported in the rectangle of charges from zero to $d_s(m)+d_s(n)$. Proposition~\ref{prop:finmult} puts the entire support of each output, including both its corners, inside that rectangle. Its leading label gives $a\ge0$, and its far corner gives the second inequality. The output $g=m+n$ occurs, so $d_s(m+n)\le d_s(m)+d_s(n)$. Positive homogeneity and continuous piecewise linearity extend this inequality to real arguments and imply convexity.
\end{proof}


\subsection{Finite expansions after freezing}\label{sec:finite-extension}
Let $I\subseteq\{1,\ldots,n\}$ and $F=\{1,\ldots,n\}\setminus I$.
Write $\Theta^I_{m,Q}$ for the theta series obtained by retaining only the initial walls indexed by $I$, on the same exponent lattice $M$ and at the same endpoint $Q$. Reordering the mutable coordinates as $I,F$ makes the restricted columns $p_I$ compatible with $\Lambda$, with diagonal $D_I=\diag(d_i)_{i\in I}$. Put $P_I=\{a\in P:a_k=0\text{ for }k\in F\}$. On series supported in $m+pP$, let $\pi_{m,I}$ retain just the coefficients at exponents in $m+pP_I$.

\begin{lemma}\label{lem:general-face}
For every label $m$ and lattice-general endpoint $Q$,
\[
 \pi_{m,I}\Theta_{m,Q}=\Theta^I_{m,Q}.
\]
\end{lemma}
\begin{proof}
The charge spaces indexed by $P\setminus P_I$ form a closed Lie ideal: adding a nonnegative charge cannot remove a positive coordinate outside $I$. The quotient of the completed positive-charge Lie algebra retains exactly the initial factors indexed by $I$. At each height its path products remain consistent, so scattering uniqueness identifies the quotient with the diagram for $I$. A broken line of total charge in $P_I$ has every bend charge in $P_I$. Consequently taking this quotient commutes with its finite coefficient sum at each height, proving the identity. This is the charge-face mechanism of \cite[Lemma 5.1 and Theorem 5.2]{QinFreeze}, here applied with the individual parameters $v^{d_i}$.
\end{proof}

\begin{theorem}\label{thm:finite-extension}
Suppose $Q$ is lattice general and
\[
 \Theta^I_{m,Q}=\sum_{j\in J}c_j\Theta_{j,Q}
\]
is a finite coefficientwise identity with $J\subset M$ finite, distinct labels, nonzero $c_j\in K$, and every participating $\Theta_{j,Q}$ Laurent finite. Then $\Theta^I_{m,Q}=\Theta_{m,Q}$, this series is Laurent finite, and the expansion consists of the single term with label $m$ and coefficient one.
\end{theorem}
\begin{proof}
For any minimal label $u$ of $J$ in the positive-charge order, the coefficient at $X^u$ on the right is $c_u\ne0$. Thus $u\in m+pP$. Every label of $J$ lies above a minimal label, so $J\subseteq m+pP$. Comparing the coefficient at $X^m$ now forces $m\in J$ and $c_m=1$. In particular $\Theta_{m,Q}$ is one of the finite participants.

Define
\[
 \ell_I(x)=-\sum_{k\in F}d_k^{-1}\Lambda(x,e_k)\qquad(x\in M_\R).
\]
Compatibility gives $\ell_I(pe_i)=0$ for $i\in I$ and $-1$ for $i\in F$. Hence
$\val_{\ell_I}(\Theta^I_{m,Q})=\ell_I(m)$.
If $\Theta_{m,Q}$ differs from $\Theta^I_{m,Q}$, Lemma~\ref{lem:general-face} supplies an actual nonzero coefficient at $m+pa$ with $\sum_{k\in F}a_k>0$. Then
\[
 \val_{\ell_I}(\Theta_{m,Q})<\ell_I(m)
 =\val_{\ell_I}(\Theta^I_{m,Q})
 =\min_{j\in J}\val_{\ell_I}(\Theta_{j,Q}),
\]
contradicting $m\in J$. The last equality is finite valuative independence at $Q$. Finally pointed independence removes all other coefficients.
\end{proof}

Theorem~\ref{thm:finite-extension} is the finite-participant counterpart of the classical extension statements in \cite[Theorems 3.14 and 4.12]{CMMM}. The quantum comparison in \cite[Lemma 6.12]{MQ} assumes nonnegative specialized expansion coefficients.

\begin{corollary}\label{cor:controlled-extension}
The same-label conclusion holds for an infinite coefficientwise expansion with labels in a finite union of translates of $pP$, provided the participating quantum theta functions have a uniform finite lower bound for $\val_{\ell_I}$. It also holds if such an identity is given only after base change to a nonzero commutative unital $R$-algebra, under the same bound on the original quantum series. If the original participating quantum series are all Laurent finite and the expansion is finite, the common series is Laurent finite.
\end{corollary}
\begin{proof}
Apply the unique triangular recursion of Lemma~\ref{lem:formal-expansion} in a finite union of cones containing the labels and $m+pP$. The pointed left side forces all coordinates outside $m+pP$ to be zero and its $m$ coordinate to be one. Theorem~\ref{thm:formal-vit}, with the bound taken before base change, replaces the last minimum identity in the preceding proof by an infimum identity. An actual nonzero off-face coefficient of the original $\Theta_{m,Q}$ would give the same strict contradiction. The face identity proves equality of the original series. Triangular independence over the coefficient ring proves the assertion about the expansion.
\end{proof}

\begin{corollary}\label{cor:eventual-shift}
In the positive seed chart, suppose $\Theta_m^I\in\A_I$, where $\A_I$ is the cluster algebra with mutation restricted to $I$. There exists $f_0\in\N^F$ such that every $f\in\N^F$ with $f\ge f_0$ coordinatewise satisfies
\[
 \Theta_{m+f}=\Theta^I_{m+f}
 =v^{-\Lambda(f,m)}X^f\Theta_m^I,
 \qquad m+f\in\Theta_{\rm fin}.
\]
Here vectors in $\N^F$ have zero coordinates outside $F$.
\end{corollary}
\begin{proof}
Choose a finite cluster-word expression for $\Theta_m^I$. Every cluster variable obtained by mutation in $I$ is the same variable in the full seed. For $k\in F$, compatibility gives $\Lambda(e_k,p_Ia)=0$, so $X_k$ quantum commutes with each old cluster variable. Move its inverses to the left in each word, absorbing the scalar powers of $v$ into the coefficient. This gives a finite expression $\sum_w r_wX^{c_w}U_w$, where $r_w\in R$, $c_w\in\Z^F$, and $U_w$ is a word in full cluster generators and original frozen inverses. Choose $(f_0)_k=\max(0,\max_w(-(c_w)_k))$. Then $X^f\Theta_m^I\in\A$ for every $f\ge f_0$.
Frozen translation in the seed with mutable set $I$ gives the displayed normalization. Theorem~\ref{thm:main} supplies a finite expansion in full finite theta functions, and Theorem~\ref{thm:finite-extension} identifies its label.
\end{proof}

\subsection{Finiteness under unfreezing}\label{sec:unfreezing}
Partition the mutable indices as $I\sqcup F$, where $F$ is nonempty, and let $s^{\rm fr}$ denote the seed obtained by freezing $F$. Its exponent lattice remains $M$; its charge lattice is $\Z^I$, with charge map $p_I$ given by the columns indexed by $I$. Write $p_F$ for the remaining columns. Subscripts $I$ and $F$ on vectors denote their corresponding coordinate subvectors. Assume
\begin{equation}\label{eq:oneway}
 B_{F,I}\ge0\quad\text{entrywise},\qquad B_{F,F}\text{ is acyclic}.
\end{equation}
Put $H=[-B_{F,F}]_+$ and define, for each label $m$,
\[
 t(m)=\sum_{j=0}^{|F|-1}H^j[-m_F]_+.
\]
The matrix $H$ becomes strictly triangular after ordering $F$, so $H^{|F|}=0$ and $(\operatorname{id}-H)t(m)=[-m_F]_+$.

\begin{lemma}\label{lem:frozen-face}
Every support charge of $\Theta_m^s$ satisfies $0\le a_F\le t(m)$. If $m_F\ge0$, then
\[
 \Theta_m^s=\Theta_m^{s^{\rm fr}}
\]
as formal series in the same based monomials.
\end{lemma}
\begin{proof}
For $k\in F$, equation~\eqref{eq:onehop} and $B_{F,I}\ge0$ give
\[
 a_k\le[-m_k]_++\sum_{j\in F}[-b_{kj}]_+a_j.
\]
Iteration of $a_F\le[-m_F]_++Ha_F$ through $|F|$ steps gives the bound.

When $m_F\ge0$, the bound gives $a_F=0$ for every support charge. Lemma~\ref{lem:general-face} identifies this zero-$F$-charge part with $\Theta_m^{s^{\rm fr}}$, proving the equality.
\end{proof}

\begin{theorem}\label{thm:unfreezing}
Assume \eqref{eq:oneway}. For a label $m$, define
\[
 g^{(k)}=-e_k+[-pe_k]_+\quad(k\in F),\qquad
 b=m-\sum_{k\in F}t_k(m)g^{(k)}.
\]
If $\Theta_b^{s^{\rm fr}}$ is finite, then $\Theta_m^s$ is finite. Writing
\[
 A_b=\{a_I\in\N^I:[X^{b+p_Ia_I}]\Theta_b^{s^{\rm fr}}\ne0\},
\]
one has
\begin{equation}\label{eq:unfreezing-bound}
 \supp\Theta_m^s\subseteq
 \{m+p_Ia_I+p_Fa_F:a_I\in A_b,\ 0\le a_F\le t(m)\}.
\end{equation}
Consequently, if every frozen theta label is finite, every theta label of $s$ is finite.
\end{theorem}
\begin{proof}
The $F$-coordinate columns of $g^{(k)}$ form $H-\operatorname{id}$. Thus
\[
 b_F=m_F+(\operatorname{id}-H)t(m)=[m_F]_+.
\]
Lemma~\ref{lem:frozen-face} identifies $\Theta_b^s$ with the finite frozen function. For each $k\in F$, the one-step cluster variable is
\[
 \Theta_{g^{(k)}}^s=X^{g^{(k)}}+X^{g^{(k)}+pe_k}.
\]
Fix an order on $F$ and multiply $\Theta_b^s$ by $(\Theta_{g^{(k)}}^s)^{t_k(m)}$ in that order. Its leading exponent is $m$, with coefficient a power of $v$. Iterating Proposition~\ref{prop:finmult} gives a finite theta expansion whose leading output is $\Theta_m^s$. Ordered noncancellation puts its entire support in the support of this product. The first factor contributes a charge in $A_b$, and the $k$th power contributes between zero and $t_k(m)$ copies of $e_k$. This proves \eqref{eq:unfreezing-bound} and finiteness.
\end{proof}

\section{Central coefficient quotients and pairing twists}\label{sec:coeff}
Define the principal enlargement
\[
 M^+=\Z_g^n\oplus\Z_h^n\oplus\Z_z^f,\quad
 p^+a=(Ba,a,0),\quad
 \varphi(g,h,z)=(g,Ch+z),\quad\Lambda^+(a,b)=\Lambda(\varphi a,\varphi b).
\]
The subscripts name the coordinate blocks. Then $\varphi p^+=p$ and $(p^+)^T\Lambda^+=(D,0,0)$. The kernel
\[
 K_0=\{(0,h,-Ch):h\in\Z^n\}
\]
consists of central frozen exponents. Superscript $+$ on a torus, algebra, or theta function refers to this enlarged seed. The based-monomial map $\Phi(X^{m^+})=X^{\varphi m^+}$ is a torus algebra surjection. Along a fixed translated positive cone it is injective on exponent positions, because $p$ is injective. Pullback of wall supports and their actions therefore identifies every fixed-label theta coefficient and its finiteness on the two sides. This follows either directly by lifting broken lines modulo the cylindrical kernel or by the saturated projection in Lemma~\ref{lem:lift}.

Berenstein--Zelevinsky \cite[Theorem 5.1 and Lemmas 5.4--5.5]{BZ} give the following adjacent-torus description. For mutation at $k$, let $\T_k$ be the adjacent torus and set $X'_k=X^{-e_k+[-pe_k]_+}+X^{-e_k+[pe_k]_+}$. Then
\begin{equation}\label{eq:adjacent}
 S_k=\T_R\cap\T_{k,R}
 =R\langle X_i^{\pm1}\ (i\ne k),X_k,X'_k\rangle,
 \qquad\U_R=\bigcap_{k=1}^nS_k.
\end{equation}
Here $R\langle\,\rangle$ denotes the generated subalgebra of the common skew field. Every element of $S_k$ has a unique finite Poincar\'e--Birkhoff--Witt (PBW) expression
\begin{equation}\label{eq:pbw}
 c_0+\sum_{r>0}\bigl(c_rX_k^r+c'_r(X'_k)^r\bigr),
 \qquad c_0,c_r,c'_r\in\T_{\widehat k,R},
\end{equation}
where $\T_{\widehat k,R}$ is the torus generated by all variables other than $X_k$. Coefficients are placed on the left. These statements remain valid after scalar localization. Each adjacent algebra $S_k$ is $G$-graded: the generators $X_i^{\pm1}$ are homogeneous, and the two exponents of $X'_k$ differ by $pe_k$. Taking homogeneous components in the finite intersection therefore makes $\U_R$ a $G$-graded subalgebra. The finite theta algebra is graded by $\deg\Theta_m=[m]$.

\begin{theorem}\label{thm:central}
The map $\Phi$ induces
\[
 \U^+/(X^\kappa-1:\kappa\in K_0)\simeq\U,
 \qquad
 \Mt^+/(X^\kappa-1:\kappa\in K_0)\simeq\Mt.
\]
Moreover $\Mt^+=\U^+$ if and only if $\Mt=\U$.
\end{theorem}
\begin{proof}
Put $G^+=M^+/p^+N$. Injectivity of $p$ gives the exact sequence
\[
 0\longrightarrow K_0\longrightarrow G^+\longrightarrow G\longrightarrow0.
\]
Define $\T^+_\alpha$ for $\alpha\in G^+$ by the same degree convention as $\T_\gamma$. For any $\alpha\in G^+$ mapping to $\gamma\in G$, the map $\Phi:\T^+_\alpha\to\T_\gamma$ is an $R$-module isomorphism: choosing a representative $a^+$ identifies the exponents on the two sides as $a^++p^+b$ and $\varphi a^++pb$, respectively, with $b\in\Z^n$.

The exchange variable requires a frozen correction. Let $a_k=-e_k+\sum_{i\le n}[-b_{ik}]_+e_i$, with zero frozen component, and $t_k=(0,[-C_{\bullet k}]_+)$. In the target it is $X^{a_k+t_k}+X^{a_k+t_k+pe_k}$. In the source the exchange variable has exponents $a_k^+=(a_{k,\rm mut},0,0)$ and $a_k^++p^+e_k$. If $\widetilde t_k=(0,0,[-C_{\bullet k}]_+)$, its corrected lift is
\[
 v^{-\Lambda^+(\widetilde t_k,a_k^+)}X^{\widetilde t_k}X_k^{\prime+}.
\]
Its two terms acquire the same phase because $\Lambda(t_k,pe_k)=0$. Its image is exactly $X'_k$. All other generators also lift, so $\Phi(S_k^+)=S_k$.

For $f\in S_k\cap\T_\gamma$, express it as a finite sum of homogeneous generator words and lift these words. Multiply each lift by its unique central kernel monomial to bring its degree to $\alpha$. The result lies in $S_k^+\cap\T^+_\alpha$ and maps to $f$. By the torus isomorphism in that degree this lift is unique, independently of the expression and of $k$. It therefore belongs to every $S_k^+$ when $f\in\U$. Equation~\eqref{eq:adjacent} proves surjectivity on upper algebras.

Choose one degree lift $\alpha(\gamma)$ for each $\gamma$. A finite kernel element can be written $F=\sum_{\gamma,\kappa}X^\kappa F_{\gamma,\kappa}$, with $F_{\gamma,\kappa}\in\U^+\cap\T^+_{\alpha(\gamma)}$. Its zero image implies $\sum_\kappa F_{\gamma,\kappa}=0$ by degreewise injectivity. Thus $F=\sum_{\gamma,\kappa}(X^\kappa-1)F_{\gamma,\kappa}$, proving the stated kernel.

Fixed-label coefficient correspondence gives $\Phi(\Theta^+_{m^+})=\Theta_{\varphi m^+}$ and equivalence of finiteness. Central frozen translation gives $\Theta^+_{m^++\kappa}=X^\kappa\Theta^+_{m^+}$. The same degreewise argument proves the quotient for $\Mt$. If $\Mt=\U$, lift each finite homogeneous theta expansion uniquely to a chosen degree $\alpha$ to obtain $\Mt^+=\U^+$. The opposite implication follows by quotienting.
\end{proof}

\begin{proposition}\label{prop:twist}
For fixed $p,D$, changing the compatible form preserves all theta coefficients and upper-algebra generation, under the based-monomial identification. More precisely, the change is a $G$-graded cocycle twist. Consequently
\[
 \Mt(p,\Lambda)=\U(p,\Lambda)
 \quad\Longleftrightarrow\quad
 \Mt^{\rm prin}(B,D)=\U^{\rm prin}(B,D),
\]
where the right side uses $p^{\rm prin}=\binom BI$ and $\Lambda^{\rm prin}=\left(\begin{smallmatrix}0&-D\\D&-DB\end{smallmatrix}\right)$. All-label finiteness is also equivalent on the two sides.
\end{proposition}
\begin{proof}
Let $I$ denote the coefficientwise identification of their based monomials. For two forms write $\Delta=\Lambda'-\Lambda$. Then $p^T\Delta=\Delta p=0$, so it descends to an integral skew form on $G$, automatically zero on torsion. If $F,H$ have degrees $\gamma,\eta$, their based-monomial images satisfy
\[
 I(F)I(H)=v^{\Delta(\gamma,\eta)}I(FH).
\]
The exchange generators are homogeneous, so \eqref{eq:adjacent} identifies the upper algebras as modules with this twisted product. Wall brackets use only $\Lambda(pa,pb)$, and wall actions only $\Lambda(pa,m)$; neither changes. Thus all theta coefficients agree.

On the principal enlargement compare $\Lambda^+$ with
\[
 \Lambda^{\rm st}=\begin{pmatrix}0&-D&0\\D&-DB&0\\0&0&0\end{pmatrix}.
\]
Their difference is such a twist. For $\Lambda^{\rm st}$ the $z$ variables are central and absent from the walls. Taking the common direct-sum decomposition by $z$ exponent in every seed shows that both $\U$ and $\Mt$ are obtained from their standard principal versions by adjoining these Laurent variables. Combine this with Theorem~\ref{thm:central}.
\end{proof}

\begin{corollary}\label{cor:coefficient-labels}
For fixed $B,D$, the coefficient of charge $a$ in $\Theta_{(g,h)}$ depends only on $g\in\Z^n$ and $a\in\N^n$, independently of the frozen label $h\in\Z^f$ and compatible realization $(C,\Lambda)$. In particular there is a subset $F(B,D)\subseteq\Z^n$ such that
\[
 \Theta_{\rm fin}(B,C,\Lambda)=F(B,D)\times\Z^f.
\]
Replacing $D$ by a positive integer multiple preserves this set.
\end{corollary}
\begin{proof}
Corollary~\ref{cor:frozen-translation} preserves charge coefficients under frozen translation. The coefficient correspondence in Theorem~\ref{thm:central} is injective on the charge positions of each fixed label. Proposition~\ref{prop:twist} identifies those coefficients with the standard principal ones. In the standard principal model, replacing $D$ by $rD$ substitutes $v^r$ for $v$ in every coefficient. This substitution is injective on $R$, so it preserves the nonzero charge positions.
\end{proof}

\section{Theta bases of upper cluster algebras}\label{sec:deformation}
\begin{lemma}\label{lem:integral-descent}
If a finite combination $\sum_{m\in J}c_m\Theta_m$, with $c_m\in K$, has all monomial coefficients in $R$, then all $c_m\in R$.
\end{lemma}
\begin{proof}
Choose a minimal label of the finite set $J$ in the positive-charge order. Its coefficient is precisely its scalar $c_m$, since its leading coefficient is one. Subtract this integral multiple and repeat. The process has $|J|$ steps.
\end{proof}

\subsection{Integral specialization}\label{sec:integral-specialization}
Write $\T_{s,\mathrm{cl},\Z}$ for the classical Laurent ring over $\Z$ in seed $s$, with variables $x_i$ obtained from the quantum seed variables at $v=1$ and monomials $x^m=\prod_{i=1}^{n+f}x_i^{m_i}$ for $m\in M$. Set $\U_{\mathrm{cl},\Z}=\bigcap_s\T_{s,\mathrm{cl},\Z}$ in the common classical fraction field. We write $\theta_m$ for the classical theta function obtained by coefficientwise specialization $v=1$. Geiss--Leclerc--Schr\"oer prove the upper-algebra specialization injection over a characteristic-zero field in \cite[Proposition 3.5 and Corollary 3.6]{GLS}.

\begin{theorem}\label{thm:integer-injection}
For every seed $s$, coefficientwise specialization gives
\[
 \U_R\cap(v-1)\T_{s,R}=(v-1)\U_R,\qquad
 \U_R/(v-1)\U_R\lhook\joinrel\longrightarrow\U_{\mathrm{cl},\Z}.
\]
\end{theorem}
\begin{proof}
For two adjacent seed tori, use the finite expression \eqref{eq:pbw}. Its existence can be read by the degree of $X_k$ in the skew Laurent ring over the fraction field of $\T_{\widehat k,R}$: nonnegative degrees come from the original torus and negative degrees from powers of $X'_k$. The latter have degree $-1$, and both exchange variables belong to both tori. Different degrees give uniqueness.

Modulo $v-1$, the ring $\T_{\widehat k,R}$ becomes an integer Laurent domain. The term $(X'_k)^r$ has $x_k$-degree $-r$, with a nonzero classical exchange-polynomial factor. Hence a PBW expression whose initial monomial coefficients vanish modulo $v-1$ has every PBW coefficient divisible by $v-1$. Dividing those coefficients gives
\[
 (\T_{s,R}\cap\T_{s',R})\cap(v-1)\T_{s,R}
 =(v-1)(\T_{s,R}\cap\T_{s',R})
\]
for adjacent $s,s'$.

If $F=(v-1)H\in\U_R$ with $H\in\T_{s,R}$, this equality puts $H$ in each successive torus along any finite mutation path. Thus $H\in\U_R$. The same PBW expressions identify the reductions of $F$ in adjacent charts as one classical rational function, since the exchange polynomials remain nonzero. Following paths shows that this reduction belongs to every $\T_{s,\mathrm{cl},\Z}$. The kernel equality proves injectivity.
\end{proof}

\begin{lemma}\label{lem:matching-integral}
Let $J\subseteq M$. Suppose $\{\theta_j:j\in J\}$ is a basis of the classical upper algebra over $\Q$, and every $\theta_j$ is an integer Laurent polynomial in every seed with unit leading term $x^j$ and other exponents in $j+p(P\setminus\{0\})$. Then it is a $\Z$-basis of $\U_{\mathrm{cl},\Z}$. If $\{\Theta_j:j\in J\}$ is also an $R$-basis of $\U_R$ and specializes to this basis, the map in Theorem~\ref{thm:integer-injection} is an isomorphism.
\end{lemma}
\begin{proof}
Expand an integer upper-algebra element in the given rational basis. In the finite set of participating labels, a minimal label in the positive-charge order receives no contribution from another label. Its monomial coefficient is its basis coordinate and is integral. Subtract that integer multiple and repeat. This gives a finite integer expansion, and independence is inherited from the rational basis. For the final assertion, lift every integer basis expansion using the same coefficients; injectivity is Theorem~\ref{thm:integer-injection}.
\end{proof}


\subsection{Frozen denominators and the specialization image}\label{sec:frozen-denominators}
Let $T_{\rm cl}\subseteq M$ be the labels of classical theta functions that are Laurent polynomials in every seed, and put
\[
 \mathcal M_{\rm cl}=\bigoplus_{m\in T_{\rm cl}}\Z\theta_m,
 \qquad \mathcal B=\bigoplus_{m\in\Theta_{\rm fin}}\Z\theta_m.
\]
Let $\A_{\rm cl,\Z}$ be the integer algebra generated by all classical cluster variables and the original frozen variables and their inverses. With $M_{\rm fr}$ as in Corollary~\ref{cor:frozen-translation}, put $C_{\rm fr}=\Z[x^u:u\in M_{\rm fr}]$, the original frozen Laurent ring. Classical finite theta multiplication makes $\mathcal M_{\rm cl}$ an algebra; $\Theta_{\rm fin}\subseteq T_{\rm cl}$ by specialization.

\begin{proposition}\label{prop:middle-image}
Coefficientwise specialization induces an algebra isomorphism
\[
 \Mt/(v-1)\Mt\simeq\mathcal B,
 \qquad \A_{\rm cl,\Z}\subseteq\mathcal B.
\]
After choosing one representative of each frozen translation orbit, there is a $C_{\rm fr}$-module isomorphism
\[
 \mathcal M_{\rm cl}/\mathcal B
 \simeq\bigoplus_{[m]\in(T_{\rm cl}\setminus\Theta_{\rm fin})/M_{\rm fr}} C_{\rm fr}.
\]
Thus $\mathcal B$ is a $C_{\rm fr}$-module direct summand of $\mathcal M_{\rm cl}$. For $f\in\mathcal M_{\rm cl}$ and $0\ne h\in C_{\rm fr}$, the condition $hf\in\mathcal B$ implies $f\in\mathcal B$.
\end{proposition}
\begin{proof}
A finite specialized theta relation is zero only when all its coordinates are zero, by the minimal-label argument. Hence $\sum r_m(v)\Theta_m$ specializes to zero exactly when every $r_m(1)=0$, equivalently when every $r_m$ is divisible by $v-1$ in $R$. Proposition~\ref{prop:finmult} gives the algebra assertion. Every classical cluster generator lifts to its quantum counterpart; lifting the finitely many words of a classical cluster expression, in any fixed order, proves the inclusion of $\A_{\rm cl,\Z}$.

Corollary~\ref{cor:frozen-translation} and its specialization give $\theta_{m+u}=x^u\theta_m$. Both label sets are unions of free $M_{\rm fr}$-orbits. Choose orbit representatives for $T_{\rm cl}$, using the same representatives for the subset $\Theta_{\rm fin}$. Pointed independence gives the corresponding direct sums of copies of $C_{\rm fr}$. Their quotient is the displayed free module. Since $C_{\rm fr}$ is a domain, multiplication by any nonzero frozen coefficient is injective on this quotient.
\end{proof}

\begin{corollary}\label{cor:frozen-denominator}
If $m\in M$ and a nonzero $h\in C_{\rm fr}$ satisfy the coefficientwise identity $h\theta_m\in\A_{\rm cl,\Z}$, then $m\in\Theta_{\rm fin}$. In particular this holds if $\theta_m$ is classical Laurent finite and belongs to $\operatorname{Frac}(C_{\rm fr})\otimes_{C_{\rm fr}}\A_{\rm cl,\Z}$.
\end{corollary}
\begin{proof}
Write $h=\sum_{u\in E}b_ux^u$ with $E\subset M_{\rm fr}$ finite and $b_u\ne0$. Proposition~\ref{prop:middle-image} writes $h\theta_m$ as a finite combination of $\theta_j$ with $j\in\Theta_{\rm fin}$. Its other expression is $\sum_{u\in E}b_u\theta_{m+u}$. Finite pointed independence, valid also for formal theta series, forces every $m+u$ to occur among the finite quantum labels. Frozen translation gives $m\in\Theta_{\rm fin}$. The last assertion follows by clearing the finitely many frozen denominators.
\end{proof}

\subsection{Finite support and specialization}

Let $\mathscr O=\Q[v^{\pm1}]_{(v-1)}$, a discrete valuation ring with uniformizer $\pi=v-1$, fraction field $K$, and residue field $\Q$. For finite $E\subset M$ write $W_E(L)=\bigoplus_{a\in E}LX^a$ when $L=K,\mathscr O,\Q$. When $L=\Q$, the map $R\to\Q$ sends $v$ to one. Let $\U_{\rm cl}$ be the classical upper algebra over $\Q$ for the same extended matrix. Write $S_k(L)=\T_L\cap\T_{k,L}$ for scalar localizations $L$ of $R$.

\begin{theorem}\label{thm:dim}
For every finite $E\subset M$,
\[
 \dim_K(\U_K\cap W_E(K))
 \le\dim_\Q(\U_{\rm cl}\cap W_E(\Q)).
\]
\end{theorem}
\begin{proof}
Reduce the PBW expression \eqref{eq:pbw} modulo $\pi$. Terms $X_k^r$ have positive $X_k$ degree $r$; terms $(X'_k)^r$ have degree $-r$, with a nonzero exchange-polynomial factor in the other variables. The classical Laurent ring is a domain. Hence this expression is still linearly independent over $\T_{\widehat k,\Q}$ after reduction. If an element of $S_k(\mathscr O)$ has every initial monomial coefficient divisible by $\pi$, all its PBW coefficients are divisible by $\pi$. Thus
\[
 S_k(\mathscr O)\cap\pi\T_{\mathscr O}=\pi S_k(\mathscr O),
 \qquad\U_{\mathscr O}\cap\pi\T_{\mathscr O}=\pi\U_{\mathscr O}.
\]
The second equality uses the finite intersection \eqref{eq:adjacent}. Scalar localization commutes with that finite intersection by clearing finitely many scalar denominators.

Put $V_E=\U_K\cap W_E(K)$ and $L_E=V_E\cap W_E(\mathscr O)$. If $f\in L_E$, finitely many adjacent expressions let us multiply by $\pi^a$ to obtain an element of $\U_{\mathscr O}$. Repeated saturation then gives $f\in\U_{\mathscr O}$. The module $L_E$ is a full finite free lattice in $V_E$ and $L_E\cap\pi W_E(\mathscr O)=\pi L_E$. Its reduction therefore injects into $W_E(\Q)$ with dimension $\dim_K V_E$. Since its elements belong to every quantum torus over $\mathscr O$, reduction gives the same classical rational function in every classical torus; exchange polynomials remain nonzero. Its image lies in $\U_{\rm cl}$, proving the inequality.
\end{proof}

\begin{theorem}\label{thm:deform}
Suppose $\U_{\rm cl}$ has a basis $\{\theta_j:j\in J\}$ of Laurent-polynomial theta functions satisfying valuative independence for finite combinations. Suppose the quantum theta function $\Theta_j$ is finite for each $j\in J$. Then these quantum theta functions form an $R$-basis of $\U_R$, and $\Mt=\U_R$. Every nonzero summand in the theta expansion of $f\in\U_R$ has support in $\Newt(f)$.
\end{theorem}
\begin{proof}
Proposition~\ref{prop:integral} identifies the specialization of $\Theta_j$ with $\theta_j$. Every vertex coefficient of $\Theta_j$ is one by Proposition~\ref{prop:monic}, so all its vertices survive specialization. Consequently
\[
 \Newt(\Theta_j)=\Newt(\theta_j).
\]
For a bounded rational polytope $P_0$, set $E=P_0\cap M$ and
\[
 J(P_0)=\{j\in J:\supp\theta_j\subseteq P_0\}.
\]
This set is finite because every theta contains its label. The classical basis and valuative independence give
\[
 \U_{\rm cl}\cap W_E(\Q)=\bigoplus_{j\in J(P_0)}\Q\theta_j.
\]
The Newton equality places the corresponding quantum functions in $W_E(K)$. They belong to $\U_K$ by Proposition~\ref{prop:mutation} and are independent by their leading terms. Theorem~\ref{thm:dim} gives
\[
 |J(P_0)|\le\dim_K(\U_K\cap W_E(K))
 \le\dim_\Q(\U_{\rm cl}\cap W_E(\Q))=|J(P_0)|.
\]
Thus they form a basis of the quantum window. For nonzero $f\in\U_R$, take $P_0=\Newt(f)$ and apply Lemma~\ref{lem:integral-descent} to its finite expansion. This gives coefficients in $R$ and the required support inclusion. The zero element has the empty expansion. Since $\Mt\subseteq\U_R$ and the displayed basis lies in $\Mt$, the two algebras coincide.
\end{proof}

For skew-symmetric cluster data, the positive quantum coefficients and classical specialization of \cite[Theorems 1.1--1.2 and Appendix A]{DM} give finite quantum lifts of the finite classical theta functions. Classical valuative independence is supplied by \cite[Theorems 3.8 and 3.10]{CMMM}; for principal coefficients the finite-support situation uses its unspecialized version, and finite frozen translations retain the full lattice. Theorem~\ref{thm:deform} therefore transfers a classical theta basis to the quantum upper algebra in this setting.

Specialization can remove an infinite tail from a formal wall action. For example, the wall series
\[
 H(z)=\frac{1-v^{-1}z}{1-vz},\qquad
 \frac{H(vz)}{H(v^{-1}z)}
 =\frac{(1-z)^2}{(1-v^2z)(1-v^{-2}z)}
\]
gives an action with infinitely many nonconstant coefficients, each bar invariant and with positive leading coefficient, whose specialization is one. Parity imposes an additional divisibility condition on coefficients that vanish at $v=1$.

\begin{lemma}\label{lem:vanishing-parity}
If $c\in R$ satisfies $c(v^{-1})=c(v)$, $c(-v)=\pm c(v)$ and $c(1)=0$, then $(v-v^{-1})^2$ divides $c$ in $R$.
\end{lemma}
\begin{proof}
An even bar-invariant Laurent polynomial has the form $P(v^2+v^{-2})$, with $P\in\Z[T]$. An odd one has the form $(v+v^{-1})P(v^2+v^{-2})$: this follows by induction from
\[
 (v^2+v^{-2})(v^r+v^{-r})=(v^{r+2}+v^{-r-2})+(v^{r-2}+v^{2-r})
\]
for odd $r\ge1$. In both cases $c(1)=0$ gives $P(2)=0$. The factor theorem and $v^2+v^{-2}-2=(v-v^{-1})^2$ prove the claim.
\end{proof}

\subsection{Computing theta expansions}
Under the hypotheses of Theorem~\ref{thm:deform}, take $0\ne f\in\U_R$, put $P_0=\Newt(f)$ and choose a rational linear functional $h$ on $M_\R$ with $h(pe_i)=1$ for every $i$; this exists because $p$ is injective. Set
\[
 N_0=\left\lceil\max_{x\in P_0}h(x)-\min_{x\in P_0}h(x)\right\rceil.
\]
For every candidate label $m\in P_0\cap M$, compute the theta series to charge height $N_0$, retain its terms in $P_0$, and test the resulting finite polynomial in all adjacent algebras using \eqref{eq:pbw}. Precisely the labels whose entire theta is supported in $P_0$ pass this test. Indeed, the truncation contains every possible support term in that window. If it is in $\U$, the window basis from Theorem~\ref{thm:deform} expresses it using full theta functions supported there. Triangular independence of the truncated coefficient columns, each with its distinct unit leading term in $P_0$, then identifies it with the theta of the candidate label. Finally solve the finite unit triangular system.

\subsection{Upper bases under unfreezing}\label{sec:unfreezing-bases}
Work first in the standard principal realization
\[
 M=\Z^{2n},\qquad p=\binom B{I_n},\qquad
 \Lambda=\begin{pmatrix}0&-D\\D&-DB\end{pmatrix}.
\]
Write $x_i=X^{e_i}$ and $y_i=X^{e_{n+i}}$. For $J\subseteq\{1,\ldots,n\}$, let $\U_R(J)$ and $\Mt(J)$ be the upper and finite theta algebras obtained by allowing mutation only in $J$. Write $\Theta_b^J$ for the theta function with label $b$ in this restricted seed. All these algebras lie in the original skew field.

\begin{theorem}\label{thm:pole-descent}
Let $k\notin J$ and suppose $B_{k,J}\ge0$ entrywise. If $\Mt(J)=\U_R(J)$, then
\[
 \Mt(J\cup\{k\})=\U_R(J\cup\{k\}).
\]
For a nonzero Laurent polynomial $f$, let $\nu_k(f)$ be its minimum initial $x_k$ exponent, and set $\nu_k(0)=+\infty$. Each nonzero element $f$ of the latter upper algebra has a finite integral theta expansion obtained in at most $\max(0,-\nu_k(f))$ steps of strict pole-order descent.
\end{theorem}
\begin{proof}
The initial $x_i$ commute. Define the exchange monomials
\[
 A=\prod_{i\ne k}x_i^{[-b_{ik}]_+},\qquad
 C=\prod_{i\ne k}x_i^{[b_{ik}]_+}.
\]
Skew-symmetrizability and $B_{k,J}\ge0$ imply $B_{J,k}\le0$. Thus $C$ involves only frozen $x_i$ and is a unit in $\U_R(J)$; $A$ is a product of initial cluster variables and also belongs to $\U_R(J)$. Both commute with $x_k$ and $y_k$. In the based normalization,
\[
 x_ky_k=v^{-2d_k}y_kx_k,\qquad
 x'_k=x_k^{-1}(A+v^{-d_k}Cy_k),\qquad
 x'_kx_k=A+v^{d_k}Cy_k.
\]
For a positive integer $t$, put
\begin{equation}\label{eq:monic-divisor}
 P_t(y_k)=\prod_{j=1}^t\bigl(y_k+v^{-(2j-1)d_k}AC^{-1}\bigr)
 =v^{-d_kt^2}C^{-t}(x'_k)^t x_k^t.
\end{equation}
The second equality follows by commuting $x_k$ through the exchange factors. This polynomial is monic of degree $t$ in $y_k$.

Assign degree one to $y_k$ and degree zero to every other initial variable. The row corresponding to $y_k$ in the principal coefficient block vanishes in columns $J$, and remains zero under mutations in $J$. Every resulting cluster variable therefore has $y_k$ degree zero. If $\T_s^0$ is the degree-zero subring of such a seed torus, that torus is
\[
 \T_s^0[y_k^{\pm1};\tau],\qquad \tau(a)=y_kay_k^{-1}.
\]
Here the skew Laurent notation means $y_ka=\tau(a)y_k$, with coefficients on the left. Conjugation by the frozen unit $y_k$ preserves every $\T_s^0$. The common degree decomposition gives
\begin{equation}\label{eq:independent-coefficient}
 E=\bigcap_s\T_s^0,\qquad
 \U_R(J)=E[y_k^{\pm1};\tau].
\end{equation}
Indeed, membership of a finite Laurent sum in each seed torus is equivalent to membership of each of its degree components. We have $A,C,C^{-1}\in E$.

First suppose $h\in\U_R(J)$ and $\nu_k(h)\ge0$. For a finite theta function of the frozen seed, its support has exponents $b+p_Ja$, $a\ge0$. Thus $B_{k,J}\ge0$ and its leading coefficient one give
\[
 \nu_k(\Theta_b^J)=b_k.
\]
Expand $h$ in its finite $R$-basis of frozen theta functions. Valuative independence shows that every participating $b$ has $b_k\ge0$. Lemma~\ref{lem:frozen-face}, applied to the single added vertex, identifies these functions with theta functions for $J\cup\{k\}$. Consequently
\begin{equation}\label{eq:no-pole}
 h\in\U_R(J),\quad \nu_k(h)\ge0
 \quad\Longrightarrow\quad h\in\Mt(J\cup\{k\}).
\end{equation}

Take $f\in\U_R(J\cup\{k\})\subseteq\U_R(J)$. Right multiplication by a sufficiently large power of $y_k$ makes $f$ polynomial in $y_k$ and preserves $\nu_k(f)$. Suppose $\nu_k(f)=-t<0$. Monic left division in $E[y_k;\tau]$ gives
\[
 f=P_tQ+R_t,\qquad \deg_{y_k}R_t<t.
\]
Here $\deg_{y_k}$ is the largest exponent with nonzero coefficient, with $\deg_{y_k}0=-\infty$. A highest term $a_sy_k^s$ is removed by subtracting
$P_t\tau^{-t}(a_s)y_k^{s-t}$. These operations take place in $E$ and require only addition and multiplication. The coefficients of $P_t$ have initial $x_k$ degree zero, and $\tau$ scales initial based monomials by powers of $v$. Hence
\[
 \nu_k(Q),\nu_k(R_t)\ge-t.
\]

Perform the same monic division in the initial skew Laurent ring in the generators other than $y_k$, with $y_k$ adjoined polynomially. Decompose its coefficients by their initial $x_k$ degree. Each coefficient of $P_t$ has $x_k$ degree zero, and $\tau$ preserves $x_k$ degree. Thus every subtraction in the division preserves the homogeneous degree, and taking an $x_k$ component commutes with the remainder map. 

The adjacent $k$-mutated Laurent expansion has the form
\[
 f=\sum_j(x'_k)^j a_j,
\]
where each $a_j$ is Laurent in the other initial generators and has $x_k$ degree zero. The nonzero summands have distinct $x_k$ degrees $-j$. Therefore the largest occurring $j$ is $t$, and the lowest component is $(x'_k)^t a_t$. By \eqref{eq:monic-divisor} it equals
\[
 P_t\bigl(v^{d_kt^2}x_k^{-t}C^t a_t\bigr).
\]
The parenthesized expression is initially Laurent in $y_k$. Its lowest $y_k$ degree is nonnegative: otherwise multiplication by the nonzero constant coefficient of $P_t$ would leave a nonzero negative-degree term, contradicting polynomiality of the component of $f$. Thus its monic-division remainder is zero. Commutation of the remainder map with the ambient homogeneous projection gives
\[
 \nu_k(R_t)>-t.
\]

Now set $h=v^{-d_kt^2}C^{-t}x_k^tQ$. This element belongs to $\U_R(J)$, since $C$ and $x_k$ are frozen units there, and has $\nu_k(h)\ge0$. Equation~\eqref{eq:no-pole} gives $h\in\Mt(J\cup\{k\})$. Since $C$ commutes with $x'_k$, equation~\eqref{eq:monic-divisor} yields
\[
 P_tQ=(x'_k)^th\in\Mt(J\cup\{k\}).
\]
It follows that $R_t=f-(x'_k)^th$ remains in $\U_R(J\cup\{k\})$ and has smaller pole order. Induction terminates after at most $t$ steps, with \eqref{eq:no-pole} handling the final remainder. Right multiplication by the original compensating power of $y_k^{-1}$ finishes the expansion of the original $f$.
\end{proof}

\begin{corollary}\label{cor:block-spanning}
Suppose $B_{F,I}\ge0$ and $B_{F,F}$ is acyclic. If the frozen core has $\Mt=\U$ for one compatible realization with symmetrizer $D_I$, then the full matrix has $\Mt=\U$ for every compatible realization with symmetrizer $D$.
\end{corollary}
\begin{proof}
First use the standard principal realization of the full matrix, freezing $F$. Proposition~\ref{prop:twist} and Theorem~\ref{thm:central}, applied to the core matrix and $D_I$, give its theta upper basis in this induced compatible realization. Order $F=\{k_1,\ldots,k_s\}$ so that $b_{k_jk_i}\ge0$ for $i<j$. Theorem~\ref{thm:pole-descent} applies successively with mutable sets $I$, $I\cup\{k_1\}$, and so on. This proves the equality for the full standard principal seed. Apply Proposition~\ref{prop:twist} and Theorem~\ref{thm:central} again, now to the full matrix and $D$, to obtain every compatible realization.
\end{proof}

\subsection{Large cluster complexes}\label{sec:large-complex}
In a fixed seed chart of the standard principal realization, project the labels of normalized cluster monomials to their mutable coordinates $\Z^n$. We say that the cluster complex is large when these projected labels are contained in no closed half-space $\{g:\ell(g)\ge0\}$ for a nonzero real linear functional $\ell$. This is the condition of \cite[Definition 8.27]{GHKK}, expressed in the seed basis for the label lattice. The projection sends a principal cluster-monomial label to its usual $g$-vector: both are obtained from the initial exponent by the same tropical mutation maps. The full principal labels include every frozen translate of each such label.

\begin{corollary}\label{cor:large-complex}
If the cluster complex is large, every quantum theta label is finite and the theta functions form an $R$-basis of $\U$ for every compatible realization of the same $B,D$.
\end{corollary}
\begin{proof}
Consider first the full principal label lattice $M=\Z^{2n}$. A functional nonnegative on all cluster-monomial labels and their frozen translates must vanish on the frozen subspace, because both signs of each frozen unit occur. It then descends to a functional on the mutable label space. Largeness implies that this functional is zero.

For each functional on the unit sphere in $M_\R^*$, choose a cluster-monomial label on which it is negative. These strict inequalities give an open cover of the sphere. Compactness provides finitely many labels $g_1,\ldots,g_q$ with the same property. Their finitely generated cone is closed and has zero dual cone, hence equals $M_\R$. For any $m\in M$, the rational system $\sum_i\lambda_i g_i=m$, $\lambda_i\ge0$, has a nonnegative rational solution. Clearing denominators gives $km=\sum_i n_i g_i$ with $k>0$ and $n_i\ge0$ integral. Cluster monomials are finite theta functions, so additivity and saturation of finite labels imply $m\in\Theta_{\rm fin}$.

Under the projection from the principal label lattice to the mutable label lattice, the classical cluster complex is the inverse image of the original one, as in the proof of \cite[Proposition 8.28]{GHKK}. That proposition supplies the classical full theta basis of $\U_{\rm cl}$ for this principal $\mathcal A$-type object. The functions are defined over $\Q$; their basis property descends to $\Q$ by finite coefficient comparison in each unique expansion. The finite valuative independence used in Theorem~\ref{thm:deform} is given by \cite[Theorems 3.8 and 3.10]{CMMM}. The finiteness just proved verifies the quantum-lift hypothesis, so Theorem~\ref{thm:deform} gives the principal quantum upper basis. Proposition~\ref{prop:twist} and Theorem~\ref{thm:central} transfer the result to the other compatible realizations.
\end{proof}

\subsection{Classical integral bases under unfreezing}\label{sec:classical-unfreezing}
In the standard principal model, write $x_i,y_i$ for the classical initial variables. For $J\subseteq\{1,\ldots,n\}$ and $b\in M$, write $\theta_b^J$ for the classical theta function with label $b$ and mutation restricted to $J$. Let $\U_{\mathrm{cl},\Z}(J)$ be the integer classical upper algebra with mutation restricted to $J$, and let $\mathcal M_{\mathrm{cl},\Z}(J)$ be the $\Z$-span of its finite classical theta functions. Classical finite theta multiplication \cite{GHKK} makes the latter an algebra. We use finite classical valuative independence over $\Q$ from \cite[Theorems 3.8 and 3.10]{CMMM}. Define $\nu_k(f)$ as the least initial $x_k$ exponent of a nonzero Laurent polynomial and set $\nu_k(0)=+\infty$.

\begin{theorem}\label{thm:classical-pole}
If $k\notin J$, $B_{k,J}\ge0$ entrywise and
$\mathcal M_{\mathrm{cl},\Z}(J)=\U_{\mathrm{cl},\Z}(J)$, then
\[
 \mathcal M_{\mathrm{cl},\Z}(J\cup\{k\})=\U_{\mathrm{cl},\Z}(J\cup\{k\}).
\]
Every nonzero $f\in\U_{\mathrm{cl},\Z}(J\cup\{k\})$ has an integral theta expansion obtained in at most $\max(0,-\nu_k(f))$ strict pole-order reductions.
\end{theorem}
\begin{proof}
Define
\[
 A=\prod_{i\ne k}x_i^{[-b_{ik}]_+},\qquad
 C=\prod_{i\ne k}x_i^{[b_{ik}]_+},\qquad x'_k=x_k^{-1}(A+Cy_k).
\]
Skew-symmetrizability gives $B_{J,k}\le0$, so $C$ uses only frozen variables of the core and is a unit there. The principal row of $y_k$ has zero entries in columns $J$ and remains zero under mutations in $J$. Consequently all resulting cluster variables have $y_k$-degree zero. If $\T^0_{s,\mathrm{cl},\Z}$ is the degree-zero subring of the corresponding Laurent ring, its common degree decomposition gives
\[
 E=\bigcap_s\T^0_{s,\mathrm{cl},\Z},\qquad
 \U_{\mathrm{cl},\Z}(J)=E[y_k^{\pm1}],\qquad A,C,C^{-1}\in E.
\]
The intersections here range over mutations in $J$.

For a core theta function $\theta_b^J$, its exponents lie in $b+p_J\N^J$. The sign hypothesis and unit leading term give $\nu_k(\theta_b^J)=b_k$. If $h\in\U_{\mathrm{cl},\Z}(J)$ has $\nu_k(h)\ge0$, its finite integer theta expansion therefore uses only labels $b_k\ge0$, by valuative independence applied over $\Q$. For such labels the full classical theta function equals the core function. Indeed, the one-step charge inequality \eqref{eq:onehop} gives $a_k\le[-b_k]_+=0$, and the zero-$k$-charge quotient removes precisely the $k$th initial wall. These are the coefficientwise classical specializations of the support bound and frozen-face identity in Lemma~\ref{lem:frozen-face}; the identities hold through every fixed height. Thus $h\in\mathcal M_{\mathrm{cl},\Z}(J\cup\{k\})$.

Now take $f\in\U_{\mathrm{cl},\Z}(J\cup\{k\})$. Multiply by a power of $y_k$ to make $f$ polynomial in $y_k$; this leaves $\nu_k(f)$ unchanged. If $\nu_k(f)=-t<0$, divide in $E[y_k]$ by the monic polynomial
\[
 P_t(y_k)=(y_k+AC^{-1})^t=C^{-t}(x'_k)^t x_k^t,
 \qquad f=P_tQ+R_t,\quad\deg_{y_k}R_t<t.
\]
Every coefficient of $P_t$ has initial $x_k$-degree zero. Each subtraction in monic division therefore preserves the lower bound $-t$, so $\nu_k(Q),\nu_k(R_t)\ge-t$.

In the adjacent torus write $f=\sum_j(x'_k)^j a_j$, where $a_j$ is Laurent in the other initial variables. The nonzero summands have distinct $x_k$-degrees $-j$. The lowest component is consequently $(x'_k)^t a_t=P_t x_k^{-t}C^t a_t$. The quotient on the right is polynomial in $y_k$: if it had a negative lowest power, multiplication by the nonzero constant term $(AC^{-1})^t$ of $P_t$ would preserve that power in the integer Laurent domain, contradicting polynomiality of this component of $f$.

Performing the same monic division in the initial Laurent coefficient ring commutes with taking an $x_k$-degree component, because the coefficients of $P_t$ have degree zero. The degree-$-t$ remainder is therefore zero, and $\nu_k(R_t)>-t$. Set $h=C^{-t}x_k^tQ$. Both $C$ and $x_k$ are core units, so $h\in\U_{\mathrm{cl},\Z}(J)$, and $\nu_k(h)\ge0$. The first part of the proof gives
\[
 P_tQ=(x'_k)^th\in\mathcal M_{\mathrm{cl},\Z}(J\cup\{k\}).
\]
Hence $R_t$ remains in the full upper algebra with strictly smaller pole order. Induction, followed by multiplication by the compensating power of $y_k^{-1}$, gives a finite integer theta expansion. Classical theta independence makes these functions a basis.
\end{proof}

\begin{lemma}\label{lem:classical-coefficients}
For the principal enlargement and map $\varphi$ of Section~\ref{sec:coeff}, the monomial map $\Phi(x^{m^+})=x^{\varphi m^+}$ induces
\[
 \U^+_{\mathrm{cl},\Z}/(x^\kappa-1:\kappa\in K_0)
 \simeq\U_{\mathrm{cl},\Z}.
\]
The property that the finite classical theta functions form an integer upper-algebra basis passes in both directions between these two realizations.
\end{lemma}
\begin{proof}
Full column rank holds in every seed. By \cite[Proposition 1.8, Corollary 1.9 and Lemma 4.1]{BFZ}, the classical upper algebra is the finite intersection of the adjacent algebras over $\Z$. In that argument the exchange binomials are coprime in the integer Laurent ring: full rank excludes proportional columns, and their unit coefficients have no common integer prime factor.

Use the exact sequence $0\to K_0\to G^+\to G\to0$ from Theorem~\ref{thm:central}. Choose a degree $\alpha\in G^+$ over $\gamma\in G$ and a representative $a^+\in M^+$ of $\alpha$. The monomial map is an integer-module isomorphism of the two torus components, since their exponents are parametrized by the same $b\in N$ as $a^++p^+b$ and $\varphi(a^+)+pb$. Each adjacent generator lifts: the exchange variable uses the frozen correction $x^{(0,0,[-C_{\bullet k}]_+)}$ from that theorem. Lift a finite sum of homogeneous generator words and move each word to degree $\alpha$ by its unique kernel monomial. Uniqueness in the torus component makes this lift independent of the adjacent chart. It therefore lies in their intersection.

For the kernel, group a finite zero-image sum by target degree and move its terms to one chosen source degree. Their images sum to zero, so their lifts sum to zero by the component isomorphism. Subtracting these zero sums writes the original element as a sum of $(x^\kappa-1)F_\kappa$, with $\kappa\in K_0$ and $F_\kappa\in\U^+_{\mathrm{cl},\Z}$. This proves the quotient statement.

Fixed-label theta correspondence and frozen translation are the coefficientwise classical versions of Theorem~\ref{thm:central}. Each label in a target degree has a unique lift to a prescribed source degree, because $\varphi p^+=p$ and $p$ is injective. Thus finite homogeneous theta expansions transfer in both directions through the component isomorphism. Their coefficients remain integers, and independence follows from their unit leading terms.
\end{proof}

\begin{corollary}\label{cor:integer-classes}
For reddening classes, large cluster complexes, the Markov matrix, and their successive one-way acyclic extensions satisfying \eqref{eq:oneway}, every compatible coefficient realization has
\[
 \U_R/(v-1)\U_R\simeq\U_{\mathrm{cl},\Z}.
\]
\end{corollary}
\begin{proof}
In a reddening class, Theorem~\ref{thm:box} gives finite quantum theta functions for all labels, with integral classical specializations in every seed. Their leading coefficients remain one, and Proposition~\ref{prop:mutation} supplies compatible pointedness. Compatibility makes $p$ full rank, and the reddening word gives injective reachability. The classical criterion \cite[Theorem 1.2.1(1)]{Qin}, with charge map $p$ and order $g'=g+pa$, therefore gives an integer classical basis indexed by all labels.

For a large cluster complex, \cite[Proposition 8.28]{GHKK} gives the principal classical basis used in Corollary~\ref{cor:large-complex}. For the Markov matrix, use \cite[Theorem 5.1]{Zhou}. These functions are integral by Proposition~\ref{prop:integral}; Lemma~\ref{lem:matching-integral} descends the rational bases to $\Z$. Lemma~\ref{lem:classical-coefficients} transfers them to each compatible realization. Extra frozen variables absent from the exchange columns are handled by their independent Laurent degrees.

For an extension $I\sqcup F$, order $F=\{k_1,\ldots,k_s\}$ so that $b_{k_jk_i}\ge0$ for $i<j$. In the standard principal realization, apply Theorem~\ref{thm:classical-pole} successively, starting with the frozen core; its basis is supplied by Lemma~\ref{lem:classical-coefficients} applied to the core matrix. All classical labels remain finite: the construction in Theorem~\ref{thm:unfreezing} uses a finite core theta function and powers of one-step cluster variables with leading product label $m$. Classical positive theta multiplication puts the support of its leading output $\theta_m$ inside that finite product. This verifies all-label finiteness at each step. Lemma~\ref{lem:classical-coefficients} then gives every compatible realization.

On the quantum side the corresponding bases, also indexed by all labels, are given by Theorem~\ref{thm:box}, Corollary~\ref{cor:large-complex}, the Markov argument in Section~\ref{sec:markov}, and Theorem~\ref{thm:unfreezing} with Corollary~\ref{cor:block-spanning}. Their classical specializations have the same labels. Lemma~\ref{lem:matching-integral} proves the isomorphism.
\end{proof}

\section{Low-rank applications}\label{sec:examples}
\subsection{Non-cluster families in rank three}
Consider principal coefficients and the standard compatible form for
\[
 B=\begin{pmatrix}0&-2&0\\1&0&-1\\0&2&0\end{pmatrix},
 \qquad D=\diag(1,2,1).
\]
The matrix is connected, skew-symmetrizable, singular, and acyclic. For
\[
 m_r=(r,0,-r;0,0,0),\qquad r\in\Z_{\ge1},
\]
the acyclic bound gives
\[
 \supp\Theta_{m_r}\subseteq\{m_r+pa:0\le a\le(2r,r,r)\}.
\]
These are not cluster-monomial labels. To verify this, use the acyclic valued representation description \cite[Theorem 7.3 and Corollaries 7.4--7.5]{Rupel}. With $H=[-B]_+$, a cluster monomial has mutable leading degree $(H-I)d+s$, where $d$ is a rigid representation dimension vector, $s\ge0$ records initial-variable factors, and $s_i d_i=0$. In the quotient-dimension convention for the cluster character, the leading degree is $(H-I)d$. The counting-polynomial result for rigid representations in the same reference gives this formula over the formal parameter $v$.

Here $(H-I)(1,1,1)^T=(1,0,-1)^T$. The assumed label would force
\[
 d=r(1,1,1)^T+(I-H)^{-1}s.
\]
The inverse is $I+H+H^2\ge0$ with diagonal ones. If any $s_i>0$ then $d_i>0$, a contradiction. Thus $s=0$, but the Euler quadratic form at $r(1,1,1)$ is zero, whereas a nonzero rigid object has Euler value $\dim\End>0$.

Likewise, for
\[
 B=\begin{pmatrix}0&1&0\\-2&0&3\\0&-1&0\end{pmatrix},
 \quad D=\diag(2,1,3),\quad m_r=(-r,0,r;0,0,0),
\]
the support box is $0\le a\le(r,2r,2r)$. The same calculation forces rigid dimension $r(1,2,1)$, whose Euler value is $-r^2$, again impossible. After mutation at vertex two the matrix is
\[
 B_c=\begin{pmatrix}0&-1&3\\2&0&-3\\-2&1&0\end{pmatrix}.
\]
For endpoint anchor $(11,17,23)$ the words $21,23,31$, executed left to right, have rows
\[
 e_1^TA_{21}=(-1,0,0),\quad e_3^TA_{23}=(0,-1,2),\quad
 e_1^TA_{31}=(5,0,-3).
\]
Their weighted sum with weights $6,1,1$ is $(-1,-1,-1)$. All selected second-step endpoint coordinates are positive: after the first step they are read from $(11,-17,40)$ or $(80,17,-23)$. Formula~\eqref{eq:charge-rec} gives offsets $r,0,0$ on these rows for $m_r$. Thus the cyclic-chart support satisfies $a_1+a_2+a_3\le6r$. The mutated original principal coefficient block is $\left(\begin{smallmatrix}1&0&0\\0&-1&3\\0&0&1\end{smallmatrix}\right)$; the charge bound also applies to a new principal realization at this seed.

\subsection{A rank-four class with no acyclic seed}
Take
\[
 B^{(0)}=\begin{pmatrix}0&2&-1&-1\\-2&0&1&1\\2&-2&0&0\\2&-2&0&0\end{pmatrix},
 \qquad D=\diag(2,2,1,1),
\]
with initial principal coefficients at seed $r$. The mutation class, up to relabeling, is closed on the following additional representatives:
\[
 B^{(1)}=\begin{pmatrix}0&0&1&-1\\0&0&-1&1\\-2&2&0&0\\2&-2&0&0\end{pmatrix},
 \quad
 B^{(2)}=\begin{pmatrix}0&0&-1&1\\0&0&-1&1\\2&2&0&-2\\-2&-2&2&0\end{pmatrix}.
\]
In the table, $(j,\pi)$ means $\mu_k(B^{(i)})_{ab}=B^{(j)}_{\pi(a),\pi(b)}$; permutation strings list the four images.
\begin{center}
\begin{tabular}{c|cccc}\toprule
 $i\backslash k$&1&2&3&4\\\midrule
 0&$(0,2134)$&$(0,2134)$&$(1,1234)$&$(1,1243)$\\
 1&$(2,1234)$&$(2,1243)$&$(0,1234)$&$(0,2134)$\\
 2&$(1,1234)$&$(1,1243)$&$(2,1243)$&$(2,1243)$\\\bottomrule
\end{tabular}
\end{center}
Each identity follows by the matrix mutation formula, and all permutations preserve $D$. Each representative contains a directed cycle: respectively $1231$, $13241$, and $1431$. Mutation commutes with relabeling, so induction on arbitrary word length proves that no seed is acyclic.

Nevertheless the word $(3,2,1,3,4,2,1,4)$ is green and ends at $B^{(0)}$ with principal $c$-matrix $-I$. Its successive selected $c$-columns are
\[
 e_3,\ e_2,\ e_1+2e_3,\ e_1+e_3,\ e_2+e_4,\ e_2+2e_4,\ e_1,\ e_4.
\]
These columns are nonnegative and nonzero. At the final seed $s$, $p_s=\binom{B^{(0)}}{-I}$ and the inverse positive-chamber map is $L=\diag(-I,I)$. Thus Theorem~\ref{thm:box} gives an upper theta basis in this non-mutation-acyclic class.

For a label with mutable coordinates $x=(x_1,x_2,x_3,x_4)$, write $d_i(x)$ for the $i$th component of $d_s(m)$; the frozen coordinates do not affect it. The coordinate bounds of the support box are
\begin{align*}
 d_1(x)=\max\{&0,-x_1,-x_1-x_3,-x_1-x_4,-x_1-x_3-x_4,\\
 &-x_1-2x_2-x_3-x_4,-2x_1-2x_2-x_3-x_4\},\\
 d_2(x)=\max\{&0,-x_2,-2x_1-x_2,-2x_1-x_2-x_3,-2x_1-x_2-x_4,\\
 &-2x_1-x_2-x_3-x_4,-2x_1-2x_2-x_3-x_4\},\\
 d_3(x)=\max\{&0,-x_3,-2x_2-x_3,-2x_1-2x_2-x_3,-2x_1-2x_2-2x_3\},\\
 d_4(x)=\max\{&0,-x_4,-2x_2-x_4,-2x_1-2x_2-x_4,-2x_1-2x_2-2x_4\}.
\end{align*}
Apply \eqref{eq:charge-rec} along $(4,1,2,4,3,1,2,3)$ on all $2^8$ sign branches. Seventy-two branches have strict rational witnesses. On each of the other 184 branches a nonzero nonnegative combination of the strict sign rows is zero, proving infeasibility. On each feasible branch, the active gradient minus every displayed competing gradient is a nonnegative rational combination of its sign rows. The 1440 resulting equalities prove the maxima on the closures of the branches; continuity covers their boundaries. The companion certificate lists these rational vectors and verifies the equalities after reconstructing each mutation branch. For example $x=(-r,0,0,0)$ gives $d_s=2r(1,1,1,1)$ for every $r>0$.

\subsection{The Markov matrix}\label{sec:markov}
Let
\[
 B=\begin{pmatrix}0&-2&2\\2&0&-2\\-2&2&0\end{pmatrix}.
\]
Every positive diagonal skew-symmetrizer is $D=dI$. For every compatible integral $p=\binom BC$ and $\Lambda$, all theta labels are finite and their theta functions form an $R$-basis of $\U$.

Indeed, Zhou \cite[Theorem 5.1]{Zhou} proves the full classical theta-basis statement for the principal-coefficient cluster variety of the once-punctured torus. His row pairing matrix is $B^T$ in our column convention, so its principal model is the one used here. Davison--Mandel positivity identifies the full quantum and classical supports in the standard principal model for $d=1$; replacing the formal parameter by $v^d$ gives every positive $d$. Theorem~\ref{thm:deform}, with classical valuative independence from \cite{CMMM}, gives its quantum upper basis. Proposition~\ref{prop:twist} and Theorem~\ref{thm:central} transfer it to every compatible realization.

There is no reddening sequence. Every principal matrix in the mutation history is $B$ or $-B$. By sign coherence, if the selected $c$-column has sign $\eps$, the column-sum update is
\[
 \sum_jc'_j=\sum_jc_j+
 \left(-2+\sum_{j\ne k}[\eps b_{kj}]_+\right)c_k=\sum_jc_j.
\]
It remains $(1,1,1)^T$ from the initial identity matrix. A terminal matrix with all columns nonpositive is therefore impossible. In the standard principal model the degree assigning one to mutable variables and zero to frozen variables is also preserved by mutation. A global monomial is an element of the upper algebra that is a monomial in some seed. Every global monomial has nonnegative mutable exponents in that seed: a negative exponent would require a Laurent unit to be divisible by a nonunit adjacent exchange polynomial. Its degree is consequently nonnegative. The enough-global-monomials condition requires, at every nonzero tropical point, a global monomial with negative tropical value; see \cite[Definition 4.17]{DM}. At the tropical point defined by this degree, the value of a monomial is its degree. Hence the condition fails for this principal-coefficient cluster variety.

A compatible nonsaturated example has one frozen variable:
\[
 p=\begin{pmatrix}0&-2&2\\2&0&-2\\-2&2&0\\1&1&1\end{pmatrix},\qquad
 \Lambda=\begin{pmatrix}0&-1&1&-2\\1&0&-1&-2\\-1&1&0&-2\\2&2&2&0\end{pmatrix}.
\]
Here $p^T\Lambda=(6I,0)$ and the Smith invariants are $(1,2,6)$, so $M/pN\simeq\Z\oplus\Z/2\oplus\Z/6$.

For arbitrary compatible Markov data, take $e_1$ with zero frozen part and list charges
\[
 a_0=000,\ a_1=100,\ a_2=101,\ a_3=102,\ a_4=112,\ a_5=122,\ a_6=222,
\]
where each three-digit string denotes its coordinate triple. Then
\begin{equation}\label{eq:markov}
 \Theta_{-e_1}=\sum_{j\in\{0,1,3,5,6\}}X^{-e_1+pa_j}
 +(v^d+v^{-d})(X^{-e_1+pa_2}+X^{-e_1+pa_4}).
\end{equation}
Zhou's negative-basic theta formula gives these seven classical support positions, and the quantum support is the same. On a slice of $k$th exponent $-r<0$, use a formal variable $z$ to record increments by $pe_k$ in its based-monomial coefficient sequence. Membership in $S_k$ requires this Laurent polynomial in $z$ to be divisible by
\[
 \prod_{s=1}^r(1+v^{(r+1-2s)d}z).
\]
This follows by expanding $(X'_k)^r$ in based monomials; multiplying by an exponent with $k$th coordinate zero gives the same phase in all orbit positions. For the seven positions, the negative slices are $\{a_0,a_1\}$ and $\{a_5,a_6\}$ at $k=1,r=1$, $\{a_3,a_4,a_5\}$ at $k=2,r=2$, and $\{a_1,a_2,a_3\}$ at $k=3,r=2$. Writing $c_j=[X^{-e_1+pa_j}]\Theta_{-e_1}$, the coefficients must satisfy
\[
 c_0=c_1=c_3=c_5=c_6,\qquad
 c_2=(v^d+v^{-d})c_1,\quad c_4=(v^d+v^{-d})c_3.
\]
Conversely these conditions put the polynomial in every $S_k$. There is exactly one with $c_0=1$, proving \eqref{eq:markov}. In the four-variable example it reads
\begin{align*}
 \Theta_{-e_1}={}&X^{(-1,0,0,0)}+X^{(-1,2,-2,1)}
 +(v^6+v^{-6})X^{(1,0,-2,2)}+X^{(3,-2,-2,3)}\\
 &+(v^6+v^{-6})X^{(1,-2,0,4)}+X^{(-1,-2,2,5)}+X^{(-1,0,0,6)}.
\end{align*}

\subsection{One-way Markov extensions}\label{sec:markov-extension}
Write $B_M$ for the preceding Markov matrix. For an integer $r\ge2$ and a nonzero column $c\in\N^3$, define
\[
 B_{r,c}=\begin{pmatrix}B_M&-rc\\c^T&0\end{pmatrix},
 \qquad D_{r,c}=\diag(1,1,1,r).
\]
The matrices $B_{r,c}$ are connected, skew-symmetrizable, and not skew-symmetric. A compatible principal realization is
\[
 p=\binom{B_{r,c}}{I_4},\qquad
 \Lambda=\begin{pmatrix}0&-D_{r,c}\\D_{r,c}&-D_{r,c}B_{r,c}\end{pmatrix}.
\]

\begin{corollary}\label{cor:markov-extension}
For every compatible realization with symmetrizer $D_{r,c}$, all theta labels for $B_{r,c}$ are finite and their theta functions form an $R$-basis of $\U$. For each label $m$, every support charge satisfies
\begin{equation}\label{eq:markov-extension-box}
 0\le a_4\le[-m_4]_+,\qquad
 0\le a_i\le2\sum_{j=1}^3[rc_j[-m_4]_+-m_j]_+\quad(1\le i\le3).
\end{equation}
The matrix $B_{r,c}$ admits no reddening sequence and is not mutation-acyclic.
\end{corollary}
\begin{proof}
Direct multiplication gives $p^T\Lambda=(D_{r,c},0)$. Freeze vertex four. The core is a compatible Markov seed with symmetrizer $I_3$ and additional frozen coefficients, so all its theta labels are finite by the Markov result above. Equation~\eqref{eq:oneway} holds with $F=\{4\}$ and $H=0$. Theorem~\ref{thm:unfreezing} gives finiteness, and Corollary~\ref{cor:block-spanning} gives the upper basis. For the support bound, $t_4(m)=[-m_4]_+$ and the core coordinates are
\[
 b_i=m_i-rc_i[-m_4]_+\quad(1\le i\le3).
\]
Each negative basic Markov theta function has charges with every coordinate at most two, by \eqref{eq:markov} and cyclic permutation. Positive basic theta functions and frozen monomials have charge zero. A product of these functions with leading label $b$ therefore has core charge coordinates at most $2\sum_j[-b_j]_+$. Proposition~\ref{prop:finmult} places the entire support of $\Theta_b$ in this product. Equation~\eqref{eq:unfreezing-bound} gives the stated box.

Cao--Li \cite[Lemma 4.9]{CaoLi} prove that existence of a green-to-red sequence for a skew-symmetrizable matrix passes to its principal submatrices. This is the valued version of the scattering deletion and projection argument in \cite[Section 5.1]{Muller}. The Markov principal submatrix has no such sequence, so $B_{r,c}$ has no reddening sequence. Since mutation-acyclic matrices admit reddening sequences, it is not mutation-acyclic.
\end{proof}

The bound gives charge height at most
\[
 6\sum_{j=1}^3[rc_j[-m_4]_+-m_j]_++[-m_4]_+.
\]
For $r=2$, $c=e_1$, and $m=-e_4$, this is $13$. Put $I=\{1,2,3\}$ and let $p_I$ be the corresponding columns of the full extended matrix. For $c=e_1$ and $h\in\Z_{\ge0}$, Lemma~\ref{lem:frozen-face} gives
\begin{align*}
 \Theta_{-e_1+he_4}
 ={}&\sum_{a\in\{000,100,102,122,222\}}X^{-e_1+he_4+p_Ia}\\
 &+(v+v^{-1})\bigl(X^{-e_1+he_4+p_I(101)}+X^{-e_1+he_4+p_I(112)}\bigr).
\end{align*}
The three-digit strings denote charges in the Markov core. The core symmetrizer is $I_3$, which gives the parameter $v$ in these coefficients.

\begin{proposition}\label{prop:negative-fourth}
For $B_{2,e_1}$ with standard principal coefficients, let $e_1,\ldots,e_8$ be the standard basis of the exponent lattice, so $y_4=X^{e_8}$. Let $\Theta^I_{-e_1}$ be the seven-term Markov theta function for the seed obtained by freezing vertex four, embedded in the same based quantum torus. Then
\[
 \Theta_{-e_4}=X^{-e_4}
 +X^{-e_4+e_8}\bigl(\Theta^I_{-e_1}\bigr)^2.
\]
The right side has 26 nonzero based-monomial terms.
\end{proposition}
\begin{proof}
Equation~\eqref{eq:markov-extension-box} places the support in the 250 charge positions $0\le a_1,a_2,a_3\le4$, $0\le a_4\le1$. The charge-face quotient in Lemma~\ref{lem:frozen-face} makes its $a_4=0$ part equal to $X^{-e_4}$, because the frozen label has zero mutable core coordinates. Membership in each adjacent torus imposes the centered Gaussian divisibility conditions used in \eqref{eq:markov}.

These conditions form a linear system over $R$ in the 250 charge coefficients. The accompanying exact certificate constructs this system directly from $B_{2,e_1}$ and $D=\diag(1,1,1,2)$. It verifies all 90 nontrivial Laurent-polynomial remainder tests for the displayed candidate. After specialization $v=2$ over $\mathbb F_{1000003}$, the homogeneous system, including the fixed zero-charge face, has column rank 250. Since its entries are Laurent polynomials and $2$ is nonzero in that field, a nonzero specialized maximal minor proves full column rank over $\Q(v)$. Thus there is at most one candidate in the stated box with the required face and adjacent divisibilities. The theta function satisfies these conditions, which identifies the candidate.

To obtain the compact expression, expand the seven-term core formula by based-monomial multiplication using the original eight-dimensional $\Lambda$. The square has 25 terms. The vector $-e_4+e_8$ pairs to zero with $-2e_1$ and every $p_Ia$, and $pe_4=-2e_1+e_8$. Multiplying this square on the left by $X^{-e_4+e_8}$ therefore puts these terms on the charge face $a_4=1$, with their unchanged coefficients. Adding $X^{-e_4}$ gives exactly the verified 26-term candidate. \end{proof}

\paragraph{Supplementary data.}
The companion archive \nolinkurl{quantum_theta_supplement_20261012.zip}, supplied with this manuscript, contains the 26-term coefficient table at \nolinkurl{spanning/incoming/theta_minus_e4.json}, the divisibility and uniqueness check \nolinkurl{spanning/incoming/verify_explicit.py}, and the multiplication check \nolinkurl{spanning/check_compact_formula.py}. Its \nolinkurl{README.md} gives the commands and explains the certificate's dependence on the support bound, frozen face and mutation results used in Proposition~\ref{prop:negative-fourth}. The archive also contains the rational rank-four support certificate and its checker.

For comparison with Proposition~\ref{prop:cert}, every elementary Markov charge map satisfies
\[
 J_{k,\sigma}(\pm B_M)\mathbf1=\mathbf1,
 \qquad\mathbf1=(1,1,1)^T,
\]
because the changed coordinate is $-1+2=1$. Since every mutated matrix is $\pm B_M$, one has $A_w\mathbf1=\mathbf1$ for every word. The transported leading term has charge $b_w(m)\ge0$. Hence all vectors $t\mathbf1$, $t\in\N$, satisfy every word constraint $A_wa+b_w(m)\ge0$. Even the full collection of these constraints is unbounded, although the theta functions are finite.

\begin{proof}[Proof of Theorem~\ref{thm:applications}]
The first assertion is Theorem~\ref{thm:box}, the second is Theorem~\ref{thm:central}, and the third is Theorem~\ref{thm:deform}. The Markov calculation and transfer in Section~\ref{sec:examples} prove the fourth assertion, including the absence of a reddening sequence. Theorem~\ref{thm:unfreezing} and Corollaries~\ref{cor:block-spanning} and \ref{cor:markov-extension} give the fifth assertion. Corollary~\ref{cor:large-complex} gives the sixth.
\end{proof}


\subsection{A mixed-sign extension and its Chebyshev subalgebra}\label{sec:mixed-chebyshev}
Consider the compatible seed on $M=\Z^8$ given by
\[
 B=\begin{pmatrix}0&2&2&-3\\-2&0&-2&3\\-1&1&0&3\\1&-1&-2&0\end{pmatrix},
 \quad D_0=\diag(1,1,2,3),\quad D=4D_0,
\]
\[
 p=\binom B{2I_4},\qquad
 \Lambda=\begin{pmatrix}0&-2D_0\\2D_0&-D_0B\end{pmatrix}.
\]
Direct multiplication gives $p^T\Lambda=(D,0)$. Freeze indices $3,4$, retaining $I=\{1,2\}$. The block $B_{\{3,4\},I}=\left(\begin{smallmatrix}-1&1\\1&-1\end{smallmatrix}\right)$ has mixed signs. Moreover
\[
 \tfrac12(pe_1+pe_2)=(1,-1,0,0;1,1,0,0)\in M\setminus pN,
\]
since its bottom coordinates would force charge $(1/2,1/2,0,0)$.
Write $p_i=pe_i$ in this subsection, and set
\[
 m_0=(-1,1,0,0;0,0,0,0),\quad
 m=m_0+e_3+e_4,\quad
 \eta=2m+p_1+p_2=(0,0,2,2;2,2,0,0).
\]
For commuting indeterminates $U,V$, define $\mathcal T_k(U,V)\in\Z[U,V]$ for $k\ge0$ by
\[
 \mathcal T_0=2,\qquad \mathcal T_1=U,\qquad
 \mathcal T_{k+1}=U\mathcal T_k-V\mathcal T_{k-1}\quad(k\ge1).
\]

\begin{proposition}\label{prop:mixed-chebyshev}
Put $F=\Theta_m$ and $Z=X^\eta$. Then
\[
 F=X^m+X^{m+p_1}+X^{m+p_1+p_2}
   =X'_1X'_2-v^{-8}X^{m+p_2},\qquad FZ=ZF,
\]
where $X'_i$ is the variable obtained by one mutation at $i$ in the initial full seed.
For integers $b\ge0$ and $k\ge1$,
\[
 \Theta_{b\eta}=Z^b,\qquad
 \Theta_{km+b\eta}=Z^b\mathcal T_k(F,Z).
\]
These theta functions form an $R$-basis of the polynomial subalgebra $R[F,Z]\simeq R[U,V]$. Writing $H_{k,b}=\Theta_{km+b\eta}$ for $k,b\ge0$, their multiplication is
\[
 H_{0,b}H_{k,c}=H_{k,b+c},\qquad
 H_{k,b}H_{\ell,c}=
 \begin{cases}
 H_{k+\ell,b+c}+H_{|k-\ell|,b+c+\min(k,\ell)},&k\ne\ell,\\
 H_{2k,b+c}+2H_{0,b+c+k},&k=\ell,
 \end{cases}
\]
for $k,\ell\ge1$ and $b,c\ge0$. No $H_{k,b}$ with $k>0$ is a normalized cluster monomial.
\end{proposition}
\begin{proof}
We first determine $\Theta^I_{m_0}$. For any nonzero support charge $(a_1,a_2)$, the mutations $(1,2)$ transform it successively into $(1-a_1,a_2)$ and $(1-a_1,1-a_2)$ by Proposition~\ref{prop:mutation}. At a positive endpoint with first coordinates $(q_1,q_2)$, the corresponding endpoint coordinates are $(-q_1,q_2)$ and $(-q_1,-q_2)$, so each mutated coordinate is positive before its mutation. Nonnegativity of transformed charges implies $0\le a_1,a_2\le1$.

Charge zero contributes one. Charge $e_1$ has a single bend at wall 1 with Gaussian coefficient one; its bend point in the first two coordinates is $(0,q_2-q_1)$. Charge $e_2$ cannot occur, because its final exponent has first two coordinates $(1,1)$ and its reverse ray stays in the positive quadrant. For charge $e_1+e_2$, a single bend has zero initial pairing and is impossible. A last bend at wall 1 is also impossible, since the final first coordinate is positive. The remaining order, wall 1 followed by wall 2, has bend points $(0,-q_1-q_2)$ and $(q_1+q_2,0)$, with both Gaussian coefficients one. Thus
\[
 \Theta^I_{m_0}=X^{m_0}+X^{m_0+p_1}+X^{m_0+p_1+p_2}.
\]
The walls are cylindrical in the other coordinates; those coordinates are uniquely recovered from the endpoint and the segment exponents. Lattice generality excludes seams. Frozen translation in the seed with mutable set $I$ gives the same three coefficients at label $m$.

The one-step exchange variables have exponents
\[
 g_1=(-1,2,1,0;0,0,0,0),\quad
 g_2=(0,-1,0,1;0,0,0,0),
 \qquad X'_i=X^{g_i}+X^{g_i+p_i}.
\]
The four pairing values are
\[
 \begin{pmatrix}\Lambda(g_1,g_2)&\Lambda(g_1,g_2+p_2)\\
 \Lambda(g_1+p_1,g_2)&\Lambda(g_1+p_1,g_2+p_2)\end{pmatrix}
 =\begin{pmatrix}0&-8\\0&0\end{pmatrix}.
\]
Since $g_1+g_2=m$ and $m+p_2=(1,1,2,0;0,2,0,0)$ has nonnegative mutable coordinates, the asserted expression for $F$ first proves $\Theta_m^I\in\A$. Theorem~\ref{thm:finite-extension} then identifies it with $\Theta_m$.

For all multiples of $m$, use the quantum Kronecker identity of \cite[Example 5.8]{MQ}, together with the coefficient comparison in \cite[Definition 6.6 and Lemma 6.7]{MQ}. After interchanging its two mutable indices and setting $\Lambda_{\rm MQ}=-\Lambda_*$ and $t=v^{-1}$, its source data have
\[
 m_*=(-1,1,0,0),\quad p_{1*}=(0,-2,1,1),\quad
 p_{2*}=(2,0,-1,-1),\quad
 \Lambda_*=\begin{pmatrix}0&2&0&0\\-2&0&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}.
\]
Writing $p_*=(p_{1*}\ p_{2*})$, we have $p_*^T\Lambda_*=(4I_2,0)$ and $2m_*+p_{1*}+p_{2*}=0$.
The parameter change inverts the initial dilogarithm, while the change of pairing reverses its crossing exponent; the resulting action agrees with ours. Both old mutable multipliers are four, so the similarity comparison introduces no further parameter rescaling. Its conclusion is equality of the charge coefficients of $\Theta_{km}^I$ and $\Theta_{km_*}$ for every $k\ge1$.

For $a=(a_1,a_2)\in\Z^2$, write $pa=a_1p_1+a_2p_2$, and use the analogous convention for $p_*a$. For charges $a,b\in\Z^2$ and integers $i,j$, compatibility gives
\[
 \Lambda(im+pa,jm+pb)
 =4i(b_1-b_2)-4j(a_1-a_2)+8(a_1b_2-a_2b_1).
\]
The same formula holds with $m,p,\Lambda$ replaced by $m_*,p_*,\Lambda_*$. Also $\Lambda(\eta,m)=\Lambda(\eta,p_1)=\Lambda(\eta,p_2)=0$. Consequently
\[
 X^\eta X^{(k-1)m+pa}=X^{(k+1)m+p(a+(1,1))}.
\]
In the source the corresponding shift is zero. Comparing coefficients in the source Chebyshev identity therefore gives
\[
 \Theta_{2m}^I=F^2-2Z,\qquad
 F\Theta_{km}^I=\Theta_{(k+1)m}^I+Z\Theta_{(k-1)m}^I\quad(k\ge2).
\]
This proves $\Theta_{km}^I=\mathcal T_k(F,Z)$ for $k\ge1$. Since $F,Z\in\A$, each right side has a finite full-theta expansion. Theorem~\ref{thm:finite-extension} identifies its full label. In the seed with mutable set $I$, $\eta$ is frozen and has zero pairing with this family, so the same argument applies to $Z^b\Theta_{km}^I$. For $k=0$, $Z^b$ is an initial cluster monomial and $H_{0,b}=Z^b$, whereas $\mathcal T_0=2$.

For independent commuting indeterminates $u,w$, the substitution $U=u+w$, $V=uw$ is injective and one has $\mathcal T_k(U,V)=u^k+w^k$ for $k\ge1$. This proves all displayed product identities. The labels $km+b\eta$ are distinct because $m,\eta$ are linearly independent. Pointed independence and the fact that $\mathcal T_k$ is monic in $U$ prove the basis and polynomial-algebra assertions. Finally,
\[
 H_{k,b}^2=H_{2k,2b}+2H_{0,2b+k}.
\]
A normalized cluster monomial squares to the normalized monomial with doubled label in its seed. Homogeneity of tropical mutation transports that doubled label to $2km+2b\eta$, contradicting this identity over $R$.
\end{proof}

The unshifted $\Theta^I_{m_0}$ is not in the full upper algebra. Its initial $X_3$-degrees are $0,-1,0$; the unique negative layer has a unit monomial coefficient. The adjacent-torus divisibility condition would require this unit to be divisible by the nonunit exchange binomial at 3. The same obstruction applies to every nonnegative shift in directions $3,4$ whose third coordinate is zero. Corollary~\ref{cor:eventual-shift} gives equality with the full theta for every shift at least $e_3+e_4$.


\section{Further questions}
Does $\Mt=\U$ hold for every integral compatible skew-symmetrizable seed? Proposition~\ref{prop:twist} reduces the question to standard principal coefficients. Theorem~\ref{thm:deform} applies when a classical theta basis has finite quantum lifts. Theorem~\ref{thm:pole-descent} applies to one-way extensions of a quantum theta upper basis. For mixed-sign couplings, its exchange monomial $C$ can contain mutable variables and need not be invertible in the frozen-core upper algebra, so the monic division requires a different argument.

A second problem is to characterize $\Theta_{\rm fin}$ from the mutation data. It is the set of lattice points in its conical hull by Corollary~\ref{cor:finite-cone}. The Markov example shows that the finite-word inequalities of Proposition~\ref{prop:cert} need not bound the support even when every theta label is finite.

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\bigskip
\noindent Zhili College, Tsinghua University, Beijing 100084, China\\
\noindent E-mail: \href{mailto:tangqy24@mails.tsinghua.edu.cn}{tangqy24@mails.tsinghua.edu.cn}
\end{document}