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\hypersetup{pdftitle={Another proof of Laurent positivity for skew-symmetric quantum cluster algebras},pdfauthor={Qiyue Tang},pdfsubject={An algebraic proof and finite graded coefficient spaces},pdfkeywords={quantum cluster algebras, Laurent positivity, shuffle algebras}}
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\title[Another proof of quantum Laurent positivity]{Another proof of Laurent positivity for skew-symmetric quantum cluster algebras}
\author{Qiyue Tang}
\address{Zhili College, Tsinghua University, Beijing 100084, China}
\email{tangqy24@mails.tsinghua.edu.cn}
\date{November 9, 2026}
\subjclass[2020]{Primary 13F60; Secondary 16G20, 16T30}
\keywords{Quantum cluster algebra, Laurent positivity, polynomial shuffle algebra, primitive element, scattering diagram}

\begin{document}
\begin{abstract}
We give another proof of Laurent positivity for skew-symmetric quantum
cluster algebras.
\end{abstract}
\maketitle
\tableofcontents

\section{Introduction}

Fomin and Zelevinsky introduced cluster algebras to study the
coordinate changes underlying total positivity and canonical bases
\cite{FZ02}. A seed consists of a set of algebraically independent
variables and exchange data; mutation replaces one variable by the
quotient of a sum of two monomials by that variable. Their Laurent
phenomenon \cite[Theorem~3.1]{FZ02} states that every variable obtained
by repeated mutation is a Laurent polynomial in every seed. Immediately
after that theorem they conjectured that the Laurent coefficients are
nonnegative. Repeated exchange relations give rational expressions
involving sums, products and inverses. The Laurent phenomenon reduces
the denominators to monomials, while the positivity conjecture concerns
the coefficients in the resulting Laurent expansions.

Berenstein and Zelevinsky introduced quantum cluster algebras in
\cite{BZ05}. Their compatible pairs couple an exchange matrix with an
integral alternating form. The variables in a quantum seed commute up
to powers of a parameter, and centered monomials form a distinguished
basis of its quantum torus. They proved the quantum Laurent phenomenon
\cite[Corollary~5.2]{BZ05} and constructed the bar involution
\cite[Section~6]{BZ05}. Write $\N=\{0,1,2,\ldots\}$ and $v=q^{1/2}$.
Quantum positivity asks that the coefficients in the centered basis
belong to $\N[v^{\pm1}]$. Specialization at $v=1$ gives classical positivity.

Early results connected positivity with explicit bases and expansion
formulas. Sherman and Zelevinsky constructed positive canonical bases
for rank-two cluster algebras of finite and affine types \cite{SZ04}.
Musiker, Schiffler and Williams gave Laurent expansion formulas for
cluster algebras from triangulated surfaces, including punctured
surfaces, and proved positivity in that setting \cite{MSW11}.
Lee and Schiffler proved positivity for every skew-symmetric cluster
algebra \cite{LS15}. Gross, Hacking, Keel and Kontsevich constructed
classical theta functions by broken lines in scattering diagrams and
proved positivity in their theory of canonical bases \cite{GHKK18}.
In this description, a Laurent coefficient is obtained by recording the
monomial changes along a broken line, and positive wall functions give
positive contributions to the expansion.

Geometric methods also gave graded interpretations of quantum
coefficients. Nakajima related cluster monomials to simple modules and
perverse sheaves on graded quiver varieties, obtaining classical
positivity for cluster algebras with a bipartite seed \cite{Nak11}.
Qin expressed quantum cluster monomials and quantum $F$-polynomials in
terms of Serre polynomials of quiver Grassmannians of rigid modules;
this proved quantum positivity for expansions in acyclic seeds
\cite{Qin12}. Kimura and Qin subsequently used graded quiver varieties
and deformed Grothendieck rings to prove quantum positivity throughout
the mutation class of an acyclic seed \cite{KQ14}.

Donaldson--Thomas theory supplied a way to study arbitrary mutation
sequences. Nagao related compositions of cluster transformations to
wall crossing and quiver Grassmannians \cite{Nag13}. Efimov expressed
quantum cluster variables through vanishing-cycle cohomology and
reduced positivity to a purity statement; he proved positivity and a
Lefschetz property when the initial or final seed is acyclic
\cite{Efi11}. Davison, Maulik, Sch\"urmann and Szendr\H oi proved the
required purity for positively graded nondegenerate potentials
\cite{DMSS15}. Davison then proved quantum Laurent positivity for
every skew-symmetric quantum cluster algebra
\cite[Theorem~2.4]{Dav18}. His purity theorem for vanishing-cycle
cohomology supplies the nonnegative coefficients
\cite[Theorem~5.2 and Section~6]{Dav18}. The result also gives a
Lefschetz property and a parity constraint on the coefficients.

Davison and Mandel developed quantum scattering diagrams and quantum
theta bases with positive multiplication \cite{DM21}. Their positivity
for local wall factors \cite[Theorem~2.15]{DM21} follows from
cohomological DT integrality and positivity of DT invariants
\cite[Theorems~6.6 and~6.15]{DM21}. Cluster monomials occur among the
theta functions, so this also proves quantum Laurent positivity.
Kontsevich and Soibelman gave a polynomial presentation of cohomological
Hall algebras \cite{KS11}. For symmetric quivers with zero potential,
Efimov proved that a sign twist makes the Hall algebra free
supercommutative and obtained a quantum-dilogarithm factorization with
nonnegative multiplicities
\cite[Theorem~1.1 and Corollary~4.1]{Efi12}.
Semistable Hall quotients \cite{FR18} and the PBW and integrality
theorems of cohomological Donaldson--Thomas theory \cite{DM20}
describe related structures behind these factorizations.

Huang gave combinatorial quantum Laurent expansion formulas for
orbifolds without orbifold points of weight two, including punctured
cases \cite[Section~6]{Hua24}.

The dependence on frozen coefficients and quantization has also been
studied separately. Tran introduced quantum $F$-polynomials and their
recursion \cite{Tra11}. Qin used these polynomials to compare compatible
quantizations and coefficient patterns. In particular,
\cite[Theorem~3.0.1]{Qin14} transfers expansions from compatibility
constant one to an arbitrary positive constant by a change of the
quantum parameter; the correction technique of \cite[Section~9]{Qin14}
compares more general compatible realizations. Ye used a central lattice
extension to separate
exchange-polynomial roots in generalized quantum scattering diagrams,
and combined it with Davison--Mandel positivity and a positive cylinder
projection \cite{Ye26}. That construction gives positivity in the
adjoined polynomial roots.

The proof below applies the polynomial shuffle and primitive-string
construction of OpenAI \cite[Section~3]{OAI26} to quantum cluster
mutation. Independent-center restrictions on polynomial shuffle
quotients define a regular signed coproduct. A direct PBW argument and
polynomial-string decomposition express the elementary factor
multiplicities as dimensions of primitive starting spaces. The formal wall
transport, including the change of roots, is developed in
\cite[Section~4]{OAI26}; finite energy slices and decorated histories
appear in \cite[Section~6]{OAI26}. We reconstruct the local and transport
arguments, give their comparison with arbitrary integral compatible BZ
seeds, and construct finite graded spaces for all Laurent coefficients.
The comparison uses a central coefficient extension and an integral
auxiliary form agreeing with the original quantization on root actions.

\subsection{The positivity theorem}

Let $\widetilde B$ be an $m\times n$
integer matrix whose principal $n\times n$ part $B$ is skew-symmetric.
Let $\la$ be an integer alternating $m\times m$ matrix satisfying
\begin{equation}\label{intro:compatibility}
 \widetilde B^T\la=(D\ \ 0),
 \qquad D=\operatorname{diag}(d_1,\ldots,d_n),\qquad d_i>0.
\end{equation}
The first $n$ variables are mutable and the remaining variables are
frozen. We invert the frozen variables. If $M$ is a toric frame, our
normalization is
\begin{equation}\label{intro:torus}
 M(a)M(b)=v^{\la(a,b)}M(a+b).
\end{equation}
A normalized cluster monomial in a seed $M'$ means $M'(a)$ with
$a_i\geq0$ for $i\leq n$; the frozen exponents may be any integers.

\begin{samepage}
\begin{theorem}[Quantum Laurent positivity]\label{thm:main}
For every compatible seed satisfying \eqref{intro:compatibility}, every
seed $M'$ obtained by a finite sequence of mutations, and every
normalized cluster monomial $M'(a)$, the expansion
\[
 M'(a)=\sum_{b\in\Z^m}c_b(v)M(b)
\]
is finite and satisfies $c_b(v)\in\N[v^{\pm1}]$ for every $b$.
The same assertion holds with any seed in the mutation class as the
expansion seed.
\end{theorem}
\end{samepage}

Theorem~\ref{thm:main} proves Laurent positivity for all skew-symmetric
mutation classes, with the general positive diagonal matrix $D$ of the
BZ definition. Davison uses $\widetilde B^T\la=(I_n\ \ 0)$ in
\cite[Section~2.1]{Dav18}. Our comparison treats the compatibility
constant on each mutable component and retains all frozen pairings.

The quantum Laurent phenomenon \cite[Corollary~5.2]{BZ05}, recalled in
Theorem~\ref{pre:laurent}, supplies finite Laurent support and integral
coefficients. The polynomial reordering and wall-transport statements
used to prove nonnegativity are established below.

\subsection{Finite coefficient spaces}

The construction also gives an interpretation of individual Laurent
coefficients by finite-dimensional vector spaces. These spaces are
defined by polynomial maps, kernels, quotients and finite choices of
wall-crossing terms.

\begin{samepage}
\begin{theorem}[Finite coefficient spaces]\label{thm:finite}
Fix the integer seed data, a finite mutation word, and the exponent of a
normalized cluster monomial at the last seed. There is a terminating
construction, using rational linear algebra and integer arithmetic, of
finite-dimensional $\Z$-graded rational vector spaces
$\mathcal V_b=\bigoplus_E\mathcal V_{b,E}$ such that
\begin{equation}\label{intro:dimension-formula}
 M'(a)=\sum_{b\in\Z^m}\sum_{E\in\Z}
       \dim_{\Q}\mathcal V_{b,E}\;v^E M(b).
\end{equation}
Only finitely many $\mathcal V_{b,E}$ are nonzero. The construction
produces bounds on all exponents $b$ and all energies $E$ before the
coefficient spaces are computed.
\end{theorem}
\end{samepage}

The spaces depend on graded complements, rational bases and generic
paths; their graded dimensions are determined by
\eqref{intro:dimension-formula}.

\subsection{The proof}

Sections~\ref{loc:shuffle-section}--\ref{loc:bounds-section} prove
positive reordering of elementary quantum products. An ordered
Harder--Narasimhan decomposition identifies each ray factor with the
Hilbert series of a polynomial shuffle quotient. The primitive
decomposition gives its elementary factors with nonnegative integral
multiplicities.

Sections~\ref{tr:section}--\ref{tr:mutation-section} pass from positive
factors to Laurent coefficients. Incoming coefficients determine a
section by a triangular recurrence with positive terms. Under mutation,
a bound on each non-pure wall in the mutated-root direction makes the
change of completion a comparison of finite Laurent fibers. A terminal
normalized cluster monomial has one incoming coefficient, equal to one;
the change of roots transports this datum to the expansion seed.

Section~\ref{bz:section} compares this construction with BZ mutation.
Adjoining central frozen variables makes the exchange-root lattice an
integral direct summand. On a component of compatibility constant $d$,
an integral auxiliary form gives the same left and right root actions
after $t=v^d$. The centered two-term exchange relation identifies the
formal and BZ mutations. Specializing the auxiliary variables to one in
the finite Laurent algebra proves Theorem~\ref{thm:main}.

Section~\ref{eff:section} proves Theorem~\ref{thm:finite}. Exact
auxiliary-variable valuations give a finite box of exponents. Local
parameter bounds give an upper quantum energy, and bar invariance gives
the lower energy. Backward parameter bounds then reduce the primitive
spaces and the incoming recurrence to finite rational matrices and
finite sets of decorated histories.


% ===== sections/01_preliminaries.tex =====
\section{Quantum seeds and Laurent expressions}\label{pre:section}

\subsection{Based tori and normalization}

For an integer alternating form $\la$ on $\Z^m$, the based quantum
torus is the free $\Z[v^{\pm1}]$-module
\[
 \mathcal T_\la=\bigoplus_{a\in\Z^m}\Z[v^{\pm1}]X^a,
 \qquad X^aX^b=v^{\la(a,b)}X^{a+b}.
\]
Bilinearity proves associativity. Its identity is $X^0$, and
$(X^a)^{-1}=X^{-a}$. Let $e_1,\ldots,e_m$ be the standard basis of
$\Z^m$ and put $X_i=X^{e_i}$. Then
\begin{equation}\label{pre:toric}
 X_iX_j=v^{2\la_{ij}}X_jX_i,
 \qquad
 X^a=v^{-\sum_{i<j}a_i a_j\la_{ij}}
      X_1^{a_1}\cdots X_m^{a_m}.
\end{equation}
The second identity is valid for arbitrary integer exponents. We call
these basis elements \emph{centered monomials}.

Over $\Q(v)$ this torus is an iterated skew Laurent polynomial ring, so
it has an Ore division ring of fractions. A toric frame $M$ in such a
division ring is a copy of a based torus with basis $M(a)$ satisfying
\eqref{intro:torus}; its images generate the ambient division ring.
Whenever a seed is fixed, we identify $M(a)$ with $X^a$.

\subsection{Compatibility and mutation}

A quantum seed consists of a toric frame $M$ and an integer matrix
$\widetilde B$ satisfying \eqref{intro:compatibility} for its form
$\la$. Write $\widetilde b_k$ for the $k$th column. For a scalar $r$,
write $[r]_+=\max(r,0)$; positive parts of vectors are taken
coordinatewise. Mutation in $k\leq n$ replaces the generators by
\begin{equation}\label{pre:mutation}
 \begin{split}
 M'(e_j)&=M(e_j)\quad(j\ne k),\\
 M'(e_k)&=M(-e_k+[\widetilde b_k]_+)
              +M(-e_k+[-\widetilde b_k]_+).
 \end{split}
\end{equation}
The exchange matrix changes according to
\begin{equation}\label{pre:matrix-mutation}
 \widetilde b'_{ij}=
 \begin{cases}
  -\widetilde b_{ij},&i=k\text{ or }j=k,\\
  \widetilde b_{ij}+[\widetilde b_{ik}]_+b_{kj}
             +\widetilde b_{ik}[-b_{kj}]_+,&i\ne k,\ j\ne k.
 \end{cases}
\end{equation}
The form of the new frame is $\la'=E^T\la E$, where
\begin{equation}\label{pre:E}
 Ee_j=e_j\ (j\ne k),\qquad
 Ee_k=-e_k+[\widetilde b_k]_+.
\end{equation}
Let $E_0$ be the principal $n\times n$ block of $E$. Skew-symmetry
and \eqref{pre:matrix-mutation} give
\[
 \widetilde B'=E\widetilde B E_0^T,
 \qquad B'=E_0BE_0^T,
\]
so $B'$ is skew-symmetric. The matrix
$\widetilde B^T\la\widetilde B=DB$ is alternating; taking transposes
gives $DB=BD$, or $(d_i-d_j)b_{ij}=0$. The definition of $E$ then gives
$E_0D=DE_0$, $E^2=I_m$ and $E_0^2=I_n$. Hence
\[
 \begin{aligned}
  (\widetilde B')^T\la'
   &=E_0\widetilde B^T(E^T)^2\la E\\
   &=E_0(D\ \ 0)E=(E_0DE_0\ \ 0)=(D\ \ 0).
 \end{aligned}
\]
The normalized monomial $M'(a)$ is defined from these generators by
\eqref{pre:toric}, using $\la'$.

Let $g_+=-e_k+[\widetilde b_k]_+$ and
$g_-=-e_k+[-\widetilde b_k]_+$. For $j\ne k$,
\[
 \la(g_+,e_j)-\la(g_-,e_j)
  =\la(\widetilde b_k,e_j)=d_k\delta_{kj}=0.
\]
Thus the two summands in $M'(e_k)$ have the same commutation factor
with $M(e_j)$. That factor is the one prescribed by $E^T\la E$.
Let $\mathcal D$ be the division subring generated over $\Q(v)$ by
the other coordinates, and put $x=M(e_k)$, $x'=M'(e_k)$.
The exchange relation has the form $x'=P x^{-1}$ with
$0\ne P\in\mathcal D$. Thus $x=(x')^{-1}P$, so the new coordinates
generate the same division ring. For every $r\in\Z$,
$(x')^r=P_r x^{-r}$ with $P_r\in\mathcal D^\times$.
The distinct powers of $x$ are linearly independent over
$\mathcal D$ in the skew Laurent polynomial ring. A finite relation
between the new monomials, grouped by the exponent $r$ of $x'$, therefore
has zero coefficient in each group. Linear independence of the based
monomials in the other coordinates makes every scalar coefficient
zero. This proves that the normalized monomials define a toric frame.

Mutation is involutive. To verify the scalar factors, write
$b_+=[\widetilde b_k]_+$ and $b_-=[-\widetilde b_k]_+$.
Their $k$th entries vanish, and compatibility gives
\[
 \la(b_+,b_-)=\la(\widetilde b_k,b_-)=0,\qquad
 \la'(e_k,b_\pm)=\la(g_+,b_\pm)=-\la(e_k,b_\pm).
\]
The second mutation reverses column $k$. If its new coordinate is
$x''$, its exchange relation, multiplied on the right by $x'$, is
\[
 x''x'=\sum_{\epsilon\in\{+,-\}}
 v^{-\la'(e_k,b_\epsilon)}M'(b_\epsilon)
 =\sum_{\epsilon\in\{+,-\}}
 v^{\la(e_k,b_\epsilon)}M(b_\epsilon)=xx'.
\]
Here $M'(b_\epsilon)=M(b_\epsilon)$ because these exponents have
zero $k$th coordinate. Cancelling $x'$ gives $x''=x$.
These are the mutation conventions of \cite[Section~4]{BZ05}.

All frozen generators remain fixed. A normalized cluster monomial has
nonnegative exponents on the mutable generators and arbitrary exponents
on these fixed frozen generators. An unnormalized ordered monomial
differs from it by a power of $v$.

\begin{theorem}[Quantum Laurent phenomenon, Berenstein--Zelevinsky]
\label{pre:laurent}
For any two seeds in the same quantum mutation class, every cluster
variable of either seed belongs to the based quantum torus of the
other. In particular, every normalized cluster monomial has a finite
expansion with coefficients in $\Z[v^{\pm1}]$ in the based torus of
every seed.
\end{theorem}

The first assertion is \cite[Corollary~5.2]{BZ05}. The assertion for
monomials follows by multiplication, and the frozen inverse powers are
already basis monomials in every seed. We use the theorem for integrality
and finite Laurent support.

The mutable graph has vertices $1,\ldots,n$ and an edge between
distinct $i,j$ exactly when $b_{ij}\ne0$.

\begin{lemma}\label{pre:components}
The compatibility condition has the following consequences.
\begin{enumerate}
 \item The columns of $\widetilde B$ are linearly independent over $\Q$.
 \item $DB=BD$. Consequently $d_i=d_j$ whenever $i$ and $j$ belong
 to the same connected component of the mutable graph.
 \item Distinct mutable components stay disconnected under mutation,
 and a mutation in one component does not change the exchange columns
 of another component.
\end{enumerate}
\end{lemma}

\begin{proof}
The right side of \eqref{intro:compatibility} has rank $n$, proving the
first assertion. The relation $DB=BD$ proved above says
$(d_i-d_j)b_{ij}=0$ for all $i,j$. Equality of the
constants then propagates along every edge. For the last assertion,
if $k$ lies in one component and $j$ in another, $b_{kj}=0$.
Every correction to column $j$ in \eqref{pre:matrix-mutation} is
therefore zero, including all its frozen entries. The mutable
cross-component entries remain zero, so induction proves the assertion
for an arbitrary mutation word.
\end{proof}

The following seed illustrates the integral lattice and frozen pairings
treated by the central extension in Section~\ref{bz:section}.

\begin{example}\label{pre:nonsaturated-example}
With one mutable variable and three frozen variables, take
\[
 \widetilde B=\begin{pmatrix}0\\2\\0\\0\end{pmatrix},
 \qquad
 \la=\begin{pmatrix}
  0&-1&0&0\\
  1&0&0&0\\
  0&0&0&1\\
  0&0&-1&0
 \end{pmatrix}.
\]
Then $\widetilde B^T\la=(2\ 0\ 0\ 0)$. The column is nonprimitive
and generates the nonsaturated sublattice $2\Z e_2$, while the
frozen--frozen pairing $\la(e_3,e_4)=1$ is not divisible by the
compatibility constant $2$. The exchange relation is
$M'(e_1)=M(-e_1+2e_2)+M(-e_1)$.
\end{example}

\subsection{The abstract parameter}

The polynomial and wall arguments will use an integral alternating form
$\Omega$ on a lattice $L$ and an independent symbol $t$. Their
coefficient field is
\[
 \K=\Q((t^{-1})),
 \qquad X^aX^b=t^{\Omega(a,b)}X^{a+b}.
\]
An element of $\K$ has a bounded upper $t$-degree, and may have an
infinite negative tail. Write $\N((t^{-1}))$ for the subsemiring whose
coefficients are nonnegative integers. An integer exponent of $t$ will
be called its \emph{energy}. All limits will be taken first at bounded
positive root degree, with explicit energy bounds for any remaining
infinite products. On a mutable component with compatibility constant
$d$, the comparison with the original quantum seed will set $t=v^d$.


% ===== sections/02_shuffle.tex =====
\section{Polynomial shuffle algebras and ordered decomposition}
\label{loc:shuffle-section}

We give the polynomial construction of the positive reordering theorem
of \cite[Section~3]{OAI26}, whose multiplicities will be used at
two-dimensional intersections of walls.

Let $L$ be a lattice with an integral alternating form $\Omega$, and
write $\K=\Q((t^{-1}))$.  Let $\mathcal C\subset L$ be an additive
monoid contained in a strictly convex cone in a real plane.  Choose an
additive integral charge $c:\mathcal C\to\N$ which is positive on
$\mathcal C\setminus\{0\}$ and has finite bounded sublevel sets.
We use the multiplication
\[
 X^aX^b=t^{\Omega(a,b)}X^{a+b}
\]
and complete with respect to charge.  For $p\in\mathcal C\setminus\{0\}$
and $k\in\Z$, put
\begin{equation}\label{loc:units}
 E_{p,k}=\prod_{j\geq0}(1+t^{k-2j}X^p),\qquad
 H_{p,k}=\prod_{j\geq0}(1-t^{k-2j}X^p)^{-1}.
\end{equation}
An occurrence of a unit consists of its root $p$, parameter $k$, and
type $E$ or $H$.  Repeated occurrences are distinguished.

\begin{definition}\label{loc:admissible}
A list of unit occurrences is \emph{admissible} if, for every charge
bound $B$, the parameters of its roots of charge at most $B$ are bounded
above, and only finitely many such occurrences have parameter at least
$J$, for every $J\in\Z$.
\end{definition}

\begin{theorem}[Positive reordering]\label{loc:reordering}
Suppose an admissible product of the units \eqref{loc:units} is ordered
by rays, with $\Omega(p_i,p_j)>0$ whenever the ray of $p_i$ precedes
the distinct ray of $p_j$.  Its factorization in the opposite ray order
has, on every ray, an admissible product of $E$ and $H$ units with
nonnegative integral multiplicities.  At each exponent, the coefficients
belong to $\N((t^{-1}))$.  Every coefficient at a fixed exponent and
fixed power of $t$ is a finite sum.

The assertion holds after any charge truncation.  If the input list
above any parameter threshold and below any charge bound is effectively
given, and a bound on the relevant pairings is computable, every bounded
part of the output unit list is computable by finite rational linear
algebra.
\end{theorem}

Ray factorization itself is unique over $\K$: at the next charge, the
coefficient of $X^p$ not already supplied by smaller charges is assigned
to the ray of $p$.  A product of this new term with a positive-charge
term has larger charge.

\subsection{The shuffle product and its quotients}

First take a finite list, indexed by $1,\ldots,r$ in input ray order.
Indices on the same ray may occur in any order.  Define
\begin{equation}\label{loc:arrow-data}
 a_{ii}=\begin{cases}0&\text{for type }E,\\1&\text{for type }H,
 \end{cases}
 \qquad a_{ji}=[\Omega(p_i,p_j)]_+\quad(i\ne j).
\end{equation}
For dimension vectors $d,e\in\N^r$, set
\begin{align}
 p(d)&=\sum_i d_ip_i,&
 \chi(d,e)&=\sum_i d_ie_i-\sum_{i,j}a_{ij}d_ie_j,\label{loc:euler}\\
 s(d)&=\tfrac12\chi(d,d),&
 u(d)&=\sum_i d_i(k_i+1-a_{ii}).\label{loc:shifts}
\end{align}
Directly from \eqref{loc:arrow-data},
\begin{equation}\label{loc:euler-skew}
 \chi(d,e)-\chi(e,d)=\Omega(p(d),p(e)).
\end{equation}
Extend $c$ linearly to the span of the cone. Choose a rational linear
functional $\eta$ on the root span such that
$\eta(p)/c(p)$ increases in the input ray order, and define
$\mu(d)=\eta(p(d))/c(p(d))$ for $d\ne0$.  Such an $\eta$ exists by
choosing a rational affine coordinate on the line $c=1$ with the
prescribed orientation; for a single ray take $\eta=0$.  In particular,
$\chi(d,e)=\chi(e,d)$ when $\mu(d)=\mu(e)$.

Introduce a pack of variables of size $d_i$ for each index $i$, and let
\[
 S_d=\Q[x_{i,\alpha}:1\leq i\leq r,\ 1\leq\alpha\leq d_i]
       ^{\prod_i\mathfrak S_{d_i}},\qquad S_0=\Q.
\]
Every variable has ordinary degree one.  For $f\in S_d$ and $g\in S_e$
define
\begin{equation}\label{loc:shuffle}
 f*g=\sum_{\text{pack shuffles}}f(x)g(y)K_{d,e}(x,y),\qquad
 K_{d,e}(x,y)=
 \prod_{i,j}\prod_{\substack{\alpha\leq d_i\\\beta\leq e_j}}
 (y_{j,\beta}-x_{i,\alpha})^{a_{ij}-\delta_{ij}}.
\end{equation}
Each shuffle selects the variables belonging to the first input in
each output pack, and each pack shuffle occurs once.  This is the
polynomial shuffle presentation of the Hall product
\cite[Section~2.4]{KS11}.

\begin{lemma}\label{loc:polynomial-ideal}
The product \eqref{loc:shuffle} is polynomial and associative.  It
changes ordinary degree by $-\chi(d,e)$.  For $d\ne0$, the subspace
\begin{equation}\label{loc:quotient}
 I_d=\sum_{\substack{a+b=d, a,b\ne0\\\mu(a)>\mu(d)}}S_a*S_b
 \quad\text{is an ordinary homogeneous ideal of }S_d.
\end{equation}
Put $B_d=S_d/I_d$ and $B_0=\Q$.  The shuffle product induces
$B_d\otimes B_e\to B_{d+e}$ on each common slope.  It preserves the sum
of the shifted degrees
\begin{equation}\label{loc:shifted-degree}
 \deg_{\mathrm{sh}}f=\deg_{\mathrm{pol}}f+s(d).
\end{equation}
Common translation and common differentiation preserve $I_d$.
\end{lemma}

\begin{proof}
Only differences between variables of the same pack can occur in a
denominator, and each has exponent at most one.  Along a fixed collision
hyperplane, interchange the two colliding variables in a shuffle.  The
corresponding residues are opposite: all other factors have equal
restrictions to that hyperplane.  Thus the common numerator is divisible
by its defining difference.  The distinct differences are relatively
prime irreducible polynomials, so all possible poles cancel.  Both
parenthesizations of a threefold product enumerate the same final
splittings and use each of the three cross-kernels once.  This proves
associativity.  The degree of $K_{d,e}$ is $-\chi(d,e)$.

Let $h\in S_d$, and restrict it to packs of dimensions $a,b$.  This
restriction is separately symmetric, hence is a finite sum
$\sum_\nu h_{\nu,1}\otimes h_{\nu,2}\in S_a\otimes S_b$.
Its invariance under output shuffles gives
\[
 h(f*g)=\sum_\nu(h_{\nu,1}f)*(h_{\nu,2}g).
\]
Every summand $S_a*S_b$ of \eqref{loc:quotient} is therefore an ordinary
$S_d$-submodule.  The degree formula gives homogeneity.

Suppose $d,e$ have slope $\theta$.  A destabilizing first factor in
$S_d$ remains the first factor after multiplication by $S_e$.  If
$e=a+b$ with $\mu(a)>\theta$, associativity rewrites
$S_d*(S_a*S_b)=(S_d*S_a)*S_b$, whose first dimension $d+a$ also has
slope greater than $\theta$.  This proves descent in both arguments.
On that slope,
$s(d+e)=s(d)+s(e)+\chi(d,e)$, which compensates the shuffle degree.
Finally, every kernel depends on differences of variables.  Simultaneous
translation consequently preserves each destabilizing image and its
sum.  Differentiating the translation identity proves the assertion for
common differentiation.
\end{proof}

\subsection{Separated restrictions}

For an ordered list $P=(r_1,\ldots,r_q)$ of nonzero vectors with
$\sum_jr_j=d$, divide each pack into labeled target groups of these
dimensions.  Add an independent center $z_j$ to every variable in
target group $j$, and take the quotient in each group.  This defines
\begin{equation}\label{loc:raw-restriction}
 \widetilde R_P^z:S_d\longrightarrow
       \left(\bigotimes_{j=1}^qB_{r_j}\right)[z_1,\ldots,z_q].
\end{equation}
Pack symmetry makes the choice of variables within each pack irrelevant.
Translation is an automorphism of $B_d[z]$, with inverse translation by
$-z$.

To restrict a shuffle expression, divide every source group among the
target groups.  Over
$\Q(z_1,\ldots,z_q)$, expand kernels between distinct targets in the
internal variables.  In each fixed internal degree their coefficients
are finite sums of polynomial tensors, separately symmetric in the
source--target intersection packs.  Absorb these tensors into the
source factors before taking the target shuffles.  Kernels within a
target are left unchanged.  If a source prefix has nonzero dimension
$b$ in a target of dimension $a$, with $\mu(b)>\mu(a)$, that target
product belongs to $I_a$ and vanishes.  The homogeneous quotient ideals make
these Taylor calculations valid in the internal-degree completion.
Extension to the independent-center field is faithful; a zero Taylor
restriction therefore implies a zero original polynomial restriction.

\begin{lemma}[Ordered decomposition]\label{loc:hn}
Choose homogeneous vector-space sections $B_a\to S_a$ of the quotient
maps.  Ordered shuffle multiplication induces an isomorphism
\begin{equation}\label{loc:hn-map}
 \bigoplus_{\substack{d^1+\cdots+d^q=d,\ d^j\ne0\\
              \mu(d^1)>\cdots>\mu(d^q)}}
 B_{d^1}\otimes\cdots\otimes B_{d^q}\longrightarrow S_d,
\end{equation}
with the ordinary degree shifts supplied by the shuffle kernels.
For $d=0$ the source is the empty tensor $\Q$.
\end{lemma}

\begin{proof}
We first prove surjectivity by induction on $|d|_1=\sum_i d_i$.
After removing the chosen lift of $B_d$, an element of $S_d$ is a sum
of destabilizing twofold products.  Each factor has smaller dimension
and may be expanded by induction.  Associate to a list of factors the
polygon whose successive vertices are
\[
 \left(c(p(d^1+\cdots+d^j)),
       \eta(p(d^1+\cdots+d^j))\right).
\]
If two adjacent slopes are nondecreasing, merge their factors and
expand the product by induction.  The old two-segment polygon lies
below its chord, whereas every replacement has decreasing slopes and
lies above that chord.  Hence the polygon weakly rises.  Equality
requires a single collinear replacement, so the number of segments
decreases.  The initial destabilizing cut puts the polygon strictly
above the total chord somewhere; all subsequent replacements preserve
this property.  Consequently the merged pair cannot be the whole
list: a two-segment list with nondecreasing slopes lies below the total
chord.  Thus each application of induction really has smaller total
dimension.  There are only finitely many ordered lists of nonzero
subvectors of $d$, so the process terminates with strictly decreasing
slopes.

For injectivity, restrict a source product to a target list
$R=(r_1,\ldots,r_q)$ with strictly decreasing slopes.  In a surviving
assignment, every nonempty part of a source prefix inside target $j$
has slope at most $\mu(r_j)$.  If that part has charge $b_j$, its height
is at most $b_j\mu(r_j)$; an empty part has charge and height zero.
At fixed total charge, the sum of these bounds
is maximized by filling the targets in decreasing slope order.
The source polygon therefore lies below the target polygon at each
source corner, and hence everywhere by concavity of the target polygon.
If the polygons coincide, equality at their successive corners forces
complete filling of the first target, then the second, and so on.
Positive charge excludes a nonzero unfilled subdimension of charge
zero.  Thus the complete dimension-vector lists agree, even if distinct
input occurrences have the same projected root.

In a finite relation among \eqref{loc:hn-map}, choose a nonzero summand
whose polygon is minimal for pointwise comparison, and use its list as
the target.  A surviving source polygon must be smaller, or equal with
the same complete list.  Minimality leaves only the chosen list.  Its
restriction is the tensor of the translated quotient classes multiplied
by the cross-target kernels.  These kernels are separately symmetric in
the complete target packs and have nonzero scalar constant term over
the center field.  They are invertible in the internal-degree
completion of the quotient tensor.  Canceling them, and then undoing
translation, shows that the chosen tensor is zero, a contradiction.
\end{proof}

\begin{lemma}\label{loc:slope-restrictions}
If all entries of $P$ have the slope of $d$, then
\eqref{loc:raw-restriction} descends to a map $R_P^z$ on $B_d$.
These maps compose under refinement, with successive centers added.
When a product on a common slope is restricted, only assignments
whose nonempty source--target subgroups have that slope survive.
\end{lemma}

\begin{proof}
A destabilizing first source group has slope greater than the common
target slope.  Its nonempty pieces in the targets cannot all have slope
at most that slope, since their charge-weighted average is the source slope.
At least one target is killed by its instability ideal.  This proves
descent.  Refinement is composition of substitutions, and therefore
still holds after descent. For the first source factor already on the common
slope, a nonempty off-slope piece forces another piece above the slope by the
same weighted-average argument.  That target vanishes.  In all
remaining terms the products descend in their input subgroups by
Lemma~\ref{loc:polynomial-ideal}; the Taylor coefficients can be
absorbed because the quotient ideals are ordinary ideals.
\end{proof}

\subsection{The Hilbert-series identity}

For an ordinarily graded space $V=\bigoplus_{\ell\ge0}V_\ell$ with
finite-dimensional pieces, write
\[
 \Hilb_V(z)=\sum_{\ell\ge0}\dim_\Q V_\ell\,z^\ell.
\]
The pack-symmetric polynomial ring has Hilbert series
\[
 \Hilb_{S_d}(z)=\prod_i\prod_{j=1}^{d_i}(1-z^j)^{-1}.
\]
Indeed, monomial orbits in a pack of size $b$ are indexed by
$\lambda_1\ge\cdots\ge\lambda_b\ge0$.  With $\lambda_{b+1}=0$,
their differences $\nu_j=\lambda_j-\lambda_{j+1}$ satisfy
$\sum_j\lambda_j=\sum_j j\nu_j$ and range independently over $\N$.
From
\eqref{loc:units}, the coefficient of $X^{bp}$ in $E_{p,k}$ and
$H_{p,k}$ is, respectively,
\[
 \frac{t^{bk-b(b-1)}}{\prod_{j=1}^b(1-t^{-2j})},
 \qquad
 \frac{t^{bk}}{\prod_{j=1}^b(1-t^{-2j})}.
\]
For $E$, subtract $0,1,\ldots,b-1$ from the strictly increasing
chosen indices; for $H$ the index list is already weakly increasing.
Thus the full ordered input product equals
\begin{equation}\label{loc:input-hilbert}
 \sum_{d\in\N^r}X^{p(d)}
       t^{u(d)-\chi(d,d)}\Hilb_{S_d}(t^{-2}).
\end{equation}
The exponent is
\[
 u(d)-\chi(d,d)
 =\sum_i\bigl(d_ik_i-(1-a_{ii})d_i(d_i-1)\bigr)
   +\sum_{i<j}d_id_j\Omega(p_i,p_j).
\]
The diagonal summands give the two single-occurrence formulas, and the
second sum is the quantum multiplication shift in the input order.
Occurrences on the same ray commute.  The identity holds for every
complete dimension vector; at bounded charge there are finitely many
such vectors, whose contributions are then collected at each projected
root.

For two factors of dimensions $a,b$ in \eqref{loc:hn-map}, the
ordinary degree changes by $-\chi(a,b)$.  Their energy relative to
separate normalization consequently changes by
\[
 2\chi(a,b)-\chi(a,b)-\chi(b,a)=\Omega(p(a),p(b)).
\]
Lemma~\ref{loc:hn} therefore identifies the factor on a ray of slope
$\theta$ in the reversed product as
\begin{equation}\label{loc:slope-hilbert}
 1+\sum_{\mu(d)=\theta}X^{p(d)}
       t^{u(d)-\chi(d,d)}\Hilb_{B_d}(t^{-2}).
\end{equation}
An element of original shifted degree $h$ contributes energy
$u(d)-2h$.

\section{A regular coproduct and primitive strings}
\label{loc:primitive-section}

Fix a slope $\theta$.  All nonzero dimensions in the remainder of
this section have slope $\theta$.  In particular $\chi$ is symmetric
on these dimensions.  For a list $P=(r_1,\ldots,r_q)$ write
$s(P)=\sum_j s(r_j)$.

\subsection{The separated-center filtration}

Give the centers auxiliary degree zero.  Define a decreasing filtration
of $B_d$, indexed by $w\in\tfrac12\Z$, by
\begin{equation}\label{loc:filtration}
 f\in F^wB_d
 \quad\Longleftrightarrow\quad
 R_P^zf\text{ has internal degree at least }w-s(P)
 \text{ for every }P.
\end{equation}
Each restriction preserves total polynomial degree in the internal
variables and centers. Components of distinct total degree cannot cancel
in the internal-degree tests, so $F^wB_d$ is homogeneous for the original
polynomial degree.
The one-entry list is included.  For fixed $d$, there are finitely many
lists; their shift sums give a lower bound for the filtration.  In
ordinary degree $\ell$, the one-entry test gives upper bound
$\ell+s(d)$.  Thus the filtration is finite in each ordinary degree.
Set $\gr^wB_d=F^wB_d/F^{w+1/2}B_d$ and give $B_0=\Q$ weight zero.
Passing to $\gr B_d$ preserves every original shifted graded dimension.

For $[f]\in\gr^wB_d$ define
\begin{equation}\label{loc:symbol}
 L_P[f]=\prod_{i<j}(z_j-z_i)^{\chi(r_i,r_j)}(R_P^zf)_w,
\end{equation}
where the subscript selects internal degree $w-s(P)$; a negative or
nonintegral degree selects zero.  These symbols initially have rational
center coefficients.

\begin{lemma}\label{loc:polynomial-symbols}
All symbols \eqref{loc:symbol} are polynomial in their centers, and
together they detect $\gr^wB_d$.  For the tensor filtration
\begin{equation}\label{loc:tensor-filtration}
 F^w(B_a\otimes B_b)=
     \sum_{u+v\geq w}F^uB_a\otimes F^vB_b,
\end{equation}
an element $f\in B_a\otimes B_b$ belongs to $F^w(B_a\otimes B_b)$
if and only if, for every pair of lists $R,S$ summing to $a,b$,
$(R_R^{z^R}\otimes R_S^{z^S})f$ has total internal degree at least
$w-s(R)-s(S)$, where all centers are mutually independent.
For a zero dimension take the empty list, with $s(\varnothing)=0$
and $R_\varnothing=L_\varnothing=\id_\Q$.
The symbols $L_R\otimes L_S$ detect the associated graded.
\end{lemma}

\begin{proof}
Merge two target groups $a,b$ in $P$.  The shift sum increases by
$\chi(a,b)$.  Refine that merged group again, counting the relative
center $\delta=z_b-z_a$ as an internal variable of degree one.
The coarse filtration test gives degree at least
$w-s(P)-\chi(a,b)$.  To contribute fine internal degree $w-s(P)$,
a term must therefore contain $\delta$ to degree at least
$-\chi(a,b)$.  When $\chi(a,b)<0$, this is precisely the divisibility
needed for the corresponding denominator in \eqref{loc:symbol}.
Perform this comparison for every pair.  In a rational vector-space
basis of the target quotient tensor, each scalar polynomial coefficient
is divisible by each required diagonal power.  The differences for
distinct unordered pairs are relatively prime irreducibles in the
center polynomial ring, so their product divides that coefficient.
This proves polynomiality.

If every leading restriction is zero, \eqref{loc:filtration} places the
representative in $F^{w+1/2}B_d$.  Multiplication by the nonzero scalar
prefactors cannot alter this conclusion after extension to the
independent-center field.  This proves detection.

The tensor tests hold on the right-hand side of
\eqref{loc:tensor-filtration}.  For the converse, choose filtration
complements in the finite-dimensional spaces of each original degree.
If a tensor violated the proposed bound, take its lowest nonzero total
weight below $w$.  The tensor product of the detecting families on its
two factors is injective.  Different factor weights $(u,v)$ have
different internal bidegrees $(u-s(R),v-s(S))$ and cannot cancel.
Some tensor restriction thus detects the violating term.  The same
argument proves joint detection on every associated tensor grade.
Whenever scalar parameters are adjoined, all testing centers are taken
independent of those parameters.  Accordingly the detection applies
coefficientwise to actual rational vectors and to their scalar
extensions.
\end{proof}

For example, with one particle $B_d=\Q[x]$ and
$R_{(d)}^zf=f(x+z)$.  Every $x^\ell$ has auxiliary weight $s(d)$,
but its original shifted degree is $\ell+s(d)$; its symbol is $z^\ell$.
Thus the auxiliary weight and the degree recording energy are distinct.

\subsection{Product symbols}

For a target list $R=(r_1,\ldots,r_q)$, let $*^{\otimes R}$ mean:
bring the two factors assigned to each target together, with no
additional sign, and use its internal shuffle product.  Zero entries
in a symbol list are omitted, with units reinserted in their target
positions.

\begin{lemma}\label{loc:product-symbols}
The common-slope product preserves the sum of filtration weights.
For $f\in\gr^uB_d$, $g\in\gr^vB_e$, its symbols are
\begin{equation}\label{loc:product-symbol-formula}
 L_R(f*g)=
 \sum_{\substack{d_i+e_i=r_i\ (1\leq i\leq q)\\
                  \sum_i d_i=d,\ \sum_i e_i=e}}
 (-1)^{\sum_{i>j}\chi(d_i,e_j)}
 *^{\otimes R}\bigl(L_{(d_i)}f\otimes L_{(e_i)}g\bigr).
\end{equation}
Only dimensions on the fixed slope occur in nonzero terms.
\end{lemma}

\begin{proof}
Lemma~\ref{loc:slope-restrictions} leaves only the indicated assignments.
Expand their cross-target kernels at separated centers.  In target $i$
the shuffle lowers polynomial degree by $\chi(d_i,e_i)$, which is
compensated by
$s(r_i)-s(d_i)-s(e_i)=\chi(d_i,e_i)$.
Every nonconstant internal Taylor term raises internal degree.  This
proves preservation of the filtration bound.

On the leading grade, the cross-target kernel for $i<j$ contributes
\[
 (z_j-z_i)^{-\chi(d_i,e_j)}
 (z_i-z_j)^{-\chi(d_j,e_i)}.
\]
Multiplication by the target prefactor in \eqref{loc:symbol} cancels
the cross terms in $\chi(r_i,r_j)$.  What remains is the product of
the two source prefactors and the sign $(-1)^{\chi(d_j,e_i)}$.
Multiplying over all $i<j$ gives \eqref{loc:product-symbol-formula}.
\end{proof}

\subsection{Construction of the coproduct}

Let $d=a+b$ be a proper splitting on the slope, and let $f\in F^wB_d$
have ordinary degree $\ell$.  At $Z=\infty$ expand
\begin{equation}\label{loc:Cj}
 K_{a,b}(x+Z,y)^{-1}f(x+Z,y)=\sum_j C_j(f)Z^j.
\end{equation}
Every $C_j$ is an actual separately symmetric polynomial tensor: a
specified power of $Z$ involves only finitely many terms from the
finitely many integer powers of linear factors.  The restrictions
descend, and their target ideals are ordinary ideals, so $C_j$ induces
a map from $B_d$ to $B_a\otimes B_b$.  Homogeneity gives
\begin{equation}\label{loc:Cj-degree}
 \deg_{\mathrm{pol}}C_j(f)=\ell+\chi(a,b)-j,
 \qquad C_j(f)=0\quad\text{if }j>\ell+\chi(a,b).
\end{equation}

\begin{lemma}\label{loc:coproduct-lemma}
Each $C_j$ preserves the tensor filtration.  On the associated graded,
all its negative-$j$ components vanish.  The assignment
\begin{equation}\label{loc:coproduct}
 \Delta_{a,b}[f]=[C_0(f)]
\end{equation}
is therefore a well-defined map
$\gr B_d\to\gr B_a\otimes\gr B_b$, preserving the original shifted
degree and the total auxiliary weight.  It is characterized by
\begin{equation}\label{loc:coproduct-symbol}
 (L_R\otimes L_S)\Delta_{a,b}[f]=L_{R,S}[f],
\end{equation}
where $R,S$ are arbitrary refinements of $a,b$, with independent
centers.  The components with a zero dimension are the identity
tensored with the unit.
\end{lemma}

\begin{proof}
Refine $a,b$ into lists $R=(r_i)$ and $S=(s_j)$.  The internal-degree-zero
term of the inverse kernel is
\[
 \prod_{i,j}(z_j^S-z_i^R-Z)^{\chi(r_i,s_j)}.
\]
All its other internal terms have positive degree.  The restriction of
$f$ to the concatenated list satisfies the lower weight bound $w$.
Thus every refined restriction of each $C_j(f)$ satisfies the same
tensor lower bound.  Lemma~\ref{loc:polynomial-symbols} implies
$C_j(f)\in F^w(B_a\otimes B_b)$ in the actual tensor space.
For fixed $j$, \eqref{loc:Cj-degree} bounds the internal degrees
contributing to extraction, so coefficient extraction commutes with
these refinements and quotient maps.

On weight $w$, only the displayed zero-degree inverse kernel remains.
Multiplying by the two separate symbol prefactors supplies exactly the
cross-list factors, and gives
\begin{equation}\label{loc:separation-symbol}
 (L_R\otimes L_S)\left(\sum_j[C_j(f)]Z^j\right)
       =L_{R,S}[f](z^R+Z,z^S).
\end{equation}
The right side is a polynomial in the independent centers and $Z$.
For $j<0$, tensor detection therefore gives $[C_j(f)]=0$ before any
specialization of $Z$.  Equation~\eqref{loc:Cj-degree} leaves only
finitely many nonnegative powers on this grade.  Their polynomial may
now be evaluated at $Z=0$, defining \eqref{loc:coproduct} and yielding
\eqref{loc:coproduct-symbol}.  Replacing $f$ by a representative in
$F^{w+1/2}$ raises every tensor coefficient by that weight, so the map
is independent of its representative.  Finally,
\[
 \ell+\chi(a,b)+s(a)+s(b)=\ell+s(d)
\]
proves preservation of the original shifted degree.
\end{proof}

Set
\[
 \mathcal H=\Q\oplus\bigoplus_{\mu(d)=\theta}\gr B_d,
 \qquad \varepsilon(a,b)=(-1)^{\chi(a,b)}.
\]
On the chosen slope, $\varepsilon$ is a symmetric bilinear interchange
sign.  On tensor products we use
\begin{equation}\label{loc:braided-product}
 (x\otimes y)(x'\otimes y')
  =\varepsilon(\deg y,\deg x')
      (x*x')\otimes(y*y'),
\end{equation}
where these degrees are dimension vectors.

\begin{proposition}\label{loc:bialgebra}
With $\Delta=\sum_{a+b=d}\Delta_{a,b}$, the space $\mathcal H$ is a
connected dimension-graded bialgebra.  Its coproduct is cocommutative
with interchange sign $\varepsilon$.
\end{proposition}

\begin{proof}
Both iterated coproducts, tested against three lists of symbols, give
the symbol of their concatenation by \eqref{loc:coproduct-symbol}.
Joint tensor detection proves coassociativity.  Interchanging two lists
changes their scalar prefactor by $(-1)^{\chi(a,b)}$ and permutes the
underlying restriction.  Detection proves signed cocommutativity.
The empty-list restrictions give the counit and unit identities.

For multiplicativity use \eqref{loc:product-symbol-formula}.  If the
two inputs contribute dimensions $d_R,d_S$ and $e_R,e_S$ to two test
lists, the signs for the concatenated list differ from the two separate
signs by
$(-1)^{\chi(d_S,e_R)}$.
This is exactly the factor in \eqref{loc:braided-product}.  Both sides
of multiplicativity consequently have identical symbols, hence are
equal.  Lastly $\mathcal H_0=\Q$, and every proper splitting strictly
reduces $|d|_1$, proving connectedness.
\end{proof}

\subsection{Signed PBW and strings}

Let $\mathcal P=\Prim(\mathcal H)$ denote the subspace of positive
dimension annihilated by all proper coproducts.  Equation
\eqref{loc:coproduct-symbol} and tensor detection identify it with the
common kernel of all symbols for lists having at least two nonzero
entries.  Indeed such a list can be divided into two nonempty lists;
the converse follows by testing each proper coproduct against all its
refinements.

\begin{lemma}[Signed vector-space PBW]\label{loc:pbw}
Symmetrized multiplication gives an isomorphism of vector spaces
preserving dimension, original shifted degree, and auxiliary weight:
\begin{equation}\label{loc:pbw-map}
 \Sym_{\varepsilon}(\mathcal P)\xrightarrow{\sim}\mathcal H.
\end{equation}
Here exchanging homogeneous vectors of dimensions $a,b$ in the signed
symmetric space multiplies the tensor by $\varepsilon(a,b)$.
\end{lemma}

\begin{proof}
For maps $f,g:\mathcal H\to A$ into an algebra with multiplication
$m_A$, their convolution is
$f\star g=m_A(f\otimes g)\Delta$.  The augmentation is projection to
$\mathcal H_0=\Q$.  Write $e$ for the augmentation followed by the
unit, $J=\id-e$, and
\[
 \mathcal E=\log_\star(\id)
      =\sum_{q\geq1}\frac{(-1)^{q-1}}qJ^{\star q}.
\]
On dimension $d$, a nonzero term has at most $|d|_1$ positive-dimensional
factors, so this is a finite operator.  In the braided tensor target
put $\alpha(h)=h\otimes1$ and $\beta(h)=1\otimes h$.  Then
$\alpha\star\beta=\Delta$, and signed cocommutativity also gives
$\beta\star\alpha=\Delta$.  Since $\Delta$ is multiplicative,
\[
 \Delta\mathcal E=\log_\star(\Delta)
    =\log_\star(\alpha)+\log_\star(\beta)
    =\mathcal E\otimes1+1\otimes\mathcal E.
\]
Thus $\mathcal E$ has primitive image.  The convolution exponential
identity
\[
 \id=e+\sum_{q\geq1}\frac1{q!}\mathcal E^{\star q}
\]
and the signed permutation invariance of the iterated coproduct show
that signed symmetrized products of primitives span $\mathcal H$.

In characteristic zero the signed symmetric tensors are identified
with invariant tensors by averaging over signed permutations.  In a
finite relation among their products, choose the largest tensor length
$q$.  Apply the iterated coproduct with $q$ factors, projecting each
factor to positive dimension.  Products of fewer than $q$ primitives
vanish.  For a product of $q$ primitives the result is the sum of its
$q!$ signed permutations, which on the invariant tensor is $q!$ times
that tensor.  The length-$q$ term of the relation is therefore zero.
Descending on $q$ proves injectivity.  All maps used preserve the
three stated gradings.
\end{proof}

For a homogeneous basis of $\mathcal P$, a basis vector can be repeated
in a signed symmetric tensor exactly when its self-sign is $+1$.
Thus \eqref{loc:pbw-map} describes polynomial generators for even
self-sign and exterior generators for odd self-sign.  In the original
algebra, an odd primitive $x$ satisfies
\[
 \Delta(x*x)=(x*x)\otimes1+1\otimes(x*x),
\]
because its two mixed coproduct terms cancel.  Thus its square is an
even primitive.

\begin{lemma}[Primitive strings]\label{loc:strings}
For every nonzero $d$, the space $\mathcal P_d$ is a direct sum of
polynomial strings with original shifted degrees
\[
 h,h+1,h+2,\ldots,\qquad h-s(d)\in\N.
\]
Each graded piece is finite dimensional, and every vector in these
strings has self-sign $(-1)^{\chi(d,d)}$.
\end{lemma}

\begin{proof}
On $S_d$ consider the operators
\begin{equation}\label{loc:weyl-operators}
 D_d=\sum_{i,\alpha}\frac{\partial}{\partial x_{i,\alpha}},
 \qquad T_df=\left(\frac1{|d|_1}\sum_{i,\alpha}x_{i,\alpha}\right)f.
\end{equation}
They descend to $B_d$ by translation invariance and ordinary ideality,
respectively.  We use the same symbols for the induced operators on
$B_d$ and its associated graded.  On a restriction to
$P=(r_1,\ldots,r_q)$,
\begin{align*}
 R_P^z(D_df)&=\sum_j\partial_{z_j}R_P^zf,\\
 R_P^z(T_df)&=
 \left(\frac{\sum_j|r_j|_1z_j}{|d|_1}
       +\frac{\sum_{j,i,\alpha}x_{j,i,\alpha}}{|d|_1}\right)R_P^zf.
\end{align*}
Both preserve the filtration lower bound.  In the second formula the
internal-variable summand raises internal degree and disappears on
the given associated grade.  The symbol prefactor depends only on
center differences, so its common center derivative is zero.  Hence
\begin{equation}\label{loc:weyl-symbols}
 L_P(D_df)=\sum_j\partial_{z_j}L_P(f),\qquad
 L_P(T_df)=\frac{\sum_j|r_j|_1z_j}{|d|_1}L_P(f).
\end{equation}
These operators preserve the common kernel of all proper symbols,
and therefore preserve $\mathcal P_d$.

Their commutator is $[D_d,T_d]=1$.  Moreover $D_d$ is locally
nilpotent and $T_d$ raises the original shifted degree by one.  Define
the finite-on-each-vector operator
\[
 \pi_d=\sum_{j\geq0}\frac{(-1)^j}{j!}T_d^jD_d^j.
\]
The commutator gives
\begin{align*}
 D_d\pi_d
 &=\sum_{j\ge0}\frac{(-1)^j}{j!}T_d^jD_d^{j+1}
   +\sum_{j\ge1}\frac{(-1)^j}{(j-1)!}T_d^{j-1}D_d^j=0,\\
 \sum_{r\ge0}\frac{T_d^r}{r!}\pi_dD_d^r
 &=\sum_{m\ge0}T_d^mD_d^m
       \sum_{j=0}^m\frac{(-1)^j}{(m-j)!j!}=\id.
\end{align*}
All sums are finite on each vector; the last equality is the binomial
identity.  Hence
\[
 f=\sum_{j\geq0}\frac{T_d^j}{j!}\pi_d(D_d^jf).
\]
A homogeneous basis of $\mathcal P_d\cap\ker D_d$ and its $T_d$-iterates
thus span $\mathcal P_d$.  They are independent: in a finite relation
$\sum_{j=0}^qT_d^jb_j=0$ with $D_db_j=0$, applying $D_d^q$ gives
$q!b_q=0$, and downward induction eliminates the other terms.  Therefore
\begin{equation}\label{loc:string-decomposition}
 \mathcal P_d\simeq
 \Q[T_d]\otimes(\mathcal P_d\cap\ker D_d).
\end{equation}
Every component comes from nonnegative ordinary polynomial degree
plus $s(d)$.  The starting degrees have the claimed form.  Finite
ordinary polynomial spaces and the finite filtration give finite
graded dimensions.  The self-sign depends only on $d$.
\end{proof}

\section{Positive factors and effective bounds}
\label{loc:bounds-section}

\subsection{Completion of the finite-input proof}

For a primitive string starting in ordinary degree $\ell$ and auxiliary
weight $w$, its original shifted degree is $h=\ell+s(d)$.
Its successive energies in \eqref{loc:slope-hilbert} are
$u(d)-2h,u(d)-2h-2,\ldots$.
Lemma~\ref{loc:pbw} and Lemma~\ref{loc:strings} consequently make this
string contribute
\begin{equation}\label{loc:string-unit}
 \begin{cases}
 E_{p(d),\,u(d)-2h},&\chi(d,d)\text{ odd},\\
 H_{p(d),\,u(d)-2h},&\chi(d,d)\text{ even}.
 \end{cases}
\end{equation}
The integer multiplicity of this occurrence is
\begin{equation}\label{loc:multiplicity}
 \dim_\Q(\mathcal P_d\cap\ker D_d)_{\ell,w}.
\end{equation}
Summing over auxiliary weights recovers the original Hilbert series.
Thus \eqref{loc:slope-hilbert} is a positive product
of the required units.  Combining this with Lemma~\ref{loc:hn} proves
Theorem~\ref{loc:reordering} for finite input lists.

\subsection{Admissible infinite inputs}

For an infinite list give every occurrence its own index and use
dimension vectors of finite support.  All constructions for one fixed
$d$ involve only its support: the polynomial rings, instability ideals,
restrictions, filtrations, primitive spaces, and differentiation kernels
are unchanged by adding unused indices.

Fix a charge bound $B\geq1$.  Choose $K\geq0$ bounding above all input
parameters with root charge at most $B$, and choose $A\geq1$ bounding
the absolute pairings of roots of charge at most $B$.  Such roots form
a finite set.  If $c(p(d))\leq B$, then $|d|_1\leq B$ and
\[
 \chi(d,d)\geq-A B^2,\qquad
 u(d)\leq\sum_i d_ik_i+B.
\]
Since $h\geq s(d)$, the output parameter $k=u(d)-2h$ satisfies
\begin{equation}\label{loc:parameter-upper}
 k\leq\sum_i d_ik_i+B+AB^2\leq BK+B+AB^2.
\end{equation}
If $k\geq J$, single out one of the active copies of an occurrence
$i$.  There are at most $B-1$ remaining copies, each with parameter
at most $K$.  Hence every active occurrence satisfies
\begin{equation}\label{loc:parameter-lower}
 k_i\geq J-(B-1)K-B-AB^2.
\end{equation}
Admissibility leaves finitely many indices in this interval and thus
finitely many possible $d$.  For each such vector,
$h\leq(u(d)-J)/2$ bounds the ordinary starting degree.  The spaces
in \eqref{loc:multiplicity} are finite dimensional, and only finitely
many auxiliary weights occur in a fixed ordinary degree.  This proves
admissibility of the output list.

Consider now a coefficient with root charge at most $B$ and energy
$e$.  For either the input or the output list, let $K'\geq0$ bound
the parameters and let $A'\geq0$ bound the absolute root pairings at
this charge. List the $D\leq B$ selected monomial copies in their
multiplication order, with roots $p_\nu$, parameters $k_\nu$, and string
indices $j_\nu\geq0$. Their energy is
\begin{equation}\label{loc:energy-count}
 e=\sum_{\nu=1}^D(k_\nu-2j_\nu)
      +\sum_{1\leq\mu<\nu\leq D}\Omega(p_\mu,p_\nu).
\end{equation}
The multiplication shift satisfies
\[
 \left|\sum_{\mu<\nu}\Omega(p_\mu,p_\nu)\right|
      \leq A'\binom D2\leq A'B^2.
\]
The $j_\nu$ belonging to one $E$ occurrence are distinct; for $H$
repetitions are allowed.  In either case,
\begin{align}
 e&\leq BK'+A'B^2,\label{loc:energy-upper}\\
 k_\nu&\geq e-(B-1)K'-A'B^2,
 &\sum_\nu j_\nu&\leq\frac{BK'+A'B^2-e}{2}.
 \label{loc:energy-finite}
\end{align}
Only finitely many unit occurrences and index choices contribute.
Each coefficient is therefore a finite nonnegative integer, and its
upper energy is bounded.  The finite-list identities pass
coefficientwise to the infinite list, proving the remaining algebraic
assertions of Theorem~\ref{loc:reordering}.

\subsection{Finite linear-algebra realization}

Suppose the
bounded-charge input list above a specified parameter threshold can be
listed, and a bound $A$ is supplied.  The maximum of zero and the
parameters in the list above threshold zero gives $K$.  For an output
threshold $J$, equation~\eqref{loc:parameter-lower} gives a finite list
of possible indices.  Enumerate the dimension vectors supported there
with charge at most $B$.  The equation
$k=u(d)-\chi(d,d)-2\ell$ bounds every relevant ordinary degree.

In a fixed degree use monomial symmetric functions as bases of $S_d$.
The instability test is the rational comparison
$\eta(p(a))c(p(d))>\eta(p(d))c(p(a))$.
To form the instability ideal in output degree $\ell$, only pairs of
input degrees with sum $\ell+\chi(a,b)$ occur; these are nonnegative
integers and hence give a finite computation.  Shuffle multiplication
and quotient maps are rational matrices.  Each separated restriction
is a polynomial substitution with finite ordinary degree, so the
filtration tests and their associated graded spaces are kernels and
quotients of finite rational matrices.  The common kernels of the
proper symbols, followed by the kernel of $D_d$, give
\eqref{loc:multiplicity}.  Equation
\eqref{loc:energy-finite} makes the expansion of every requested product
coefficient finite.

In the later wall applications all roots lie in a free positive cone
with known generators $c_1,\ldots,c_n$, and charge is the sum of their
coordinates.  One can then take
\[
 A_B=\max\left\{1,\,B^2\max_{i,j}|\Omega(c_i,c_j)|\right\}.
\]
This also bounds pairings in the monoids cut out by two-dimensional
joint planes.


% ===== sections/03_transport.tex =====
\section{Finite wall systems and positive coefficients}
\label{tr:section}

We use the wall construction and change of roots of
\cite[Section~4]{OAI26}.

Let $L$ be a lattice with an integral alternating form $\Omega$. Write
$L^\vee=\operatorname{Hom}(L,\Z)$, $L_\R=L\otimes_\Z\R$ and
$L_\R^\vee=\operatorname{Hom}_\R(L_\R,\R)$, and put
$\K=\Q((t^{-1}))$. Multiplication is
$X^aX^b=t^{\Omega(a,b)}X^{a+b}$.  Fix a finite set $C\subset L$ contained
in a lattice basis, and write
\begin{equation}\label{tr:cone}
 C^+=\sum_{c\in C}\N c,\qquad
 \left|\sum_{c\in C}r_c c\right|_C=\sum_{c\in C}r_c,\qquad
 a\le_C b\ \Longleftrightarrow\ b-a\in C^+.
\end{equation}
The nonzero elements of $C^+$ are called roots.
Every interval for $\le_C$ is finite. Our completed modules
consist of series whose support is contained in a finite union of translates
of $C^+$.  Multiplication of two such series is defined coefficientwise:
for a fixed output exponent, only finitely many pairs of exponents from
their prescribed translates can contribute.

We may enlarge $L$ to $L\oplus L^\vee$, with form
\[
 \Omega((a,f),(b,g))=\Omega(a,b)+f(b)-g(a).
\]
Pairing with all $(0,g)$ forces $a=0$ for a vector $(a,f)$ in the
radical; pairing then with all $(b,0)$ forces $f=0$.
Thus the form is nondegenerate, and its restriction to $L$ is the given form.
When the original form is degenerate, we perform the wall construction
in this extension. Wall factors have exponents in $C^+$, so a section
starting in the original lattice stays there. Pairings between original
exponents are unchanged, and all resulting identities restrict to the
original torus. In this construction we continue to write $L$ and
$\Omega$ for the enlarged lattice and form when the extension is used.
Set
\begin{equation}\label{tr:V}
 V(a)=\Omega(-,a)\in L_\R^\vee,
 \qquad D_p=E_{p,-1}
       =\prod_{j\ge0}(1+t^{-1-2j}X^p).
\end{equation}

\subsection{The total element and its incoming prescription}

The group of positive units is
\[
 G_C=1+\prod_{0\ne r\in C^+}\K X^r.
\]
Inversion, logarithms and exponentials are well-defined in root degree.
For $h\in L_\R^\vee$, let $G_{C,+}(h)$, $G_{C,0}(h)$ and
$G_{C,-}(h)$ be the subgroups supported respectively on roots with
positive, zero and negative $h$-value.

\begin{lemma}\label{tr:factorization}
Every $g\in G_C$ has a unique factorization
\begin{equation}\label{tr:three-factors}
 g=g_+(h)g_0(h)g_-(h),\qquad g_\sigma(h)\in G_{C,\sigma}(h).
\end{equation}
At any bounded root degree the factors are constant on the cells of a
finite hyperplane arrangement.  On perturbing $h$, they are refined by
factoring the middle element according to the perturbation.
\end{lemma}

\begin{proof}
Suppose the coefficients of all three factors below degree $d$ are known.
For a root $r$ of degree $d$, precisely one of the three factors may contain
$X^r$.  Its unknown coefficient occurs in the product with coefficient
one; all other contributions use smaller positive roots.  Subtracting
these known contributions determines it.  Induction proves existence and
uniqueness.  There are finitely many roots of degree at most $d$; their
hyperplanes therefore give the stated finite arrangement.  Near a fixed
$h$, strict signs do not change at that order.  Factoring its middle
element by the new signs and invoking uniqueness proves refinement.
\end{proof}

At a fixed root order, a \emph{chamber} is a connected component of
the complement of the relevant hyperplanes. A \emph{wall} is a
codimension-one cell on which the middle factor is nonidentity, and a
\emph{joint} lies on the intersection of at least two independent
wall hyperplanes. We write $\Ad(u)(f)=ufu^{-1}$ for conjugation.
The all-positive and all-negative chambers are given respectively by
$h(c)>0$ and $h(c)<0$ for every $c\in C$.

Write $r^\perp=\{h\in L_\R^\vee:h(r)=0\}$. At a generic point of
$r^\perp$, the middle factor is supported on the
positive ray through $r$.  We regard it as the wall element there.  A
crossing from positive to negative evaluation applies $\Ad(g_0)$, and a
crossing in the other direction applies its inverse.  Earlier crossing
elements occur on the right in an ordered product.  If $I$ and $II$ are
the chambers before and after a positive-to-negative crossing, refinement
gives
\[
 g_-(II)=g_0g_-(I).
\]
Thus transport from an all-positive chamber to $h$ is $\Ad(g_-(h))$.
It depends only on the endpoint.  If the value $F_-$ is specified in the
all-negative chamber, its transported value is consequently
\begin{equation}\label{tr:section-value}
 F(h)=\Ad(g_-(h)g^{-1})F_-.
\end{equation}
All these statements are interpreted at the finite root order needed for
a requested coefficient.

Conversely, a consistent system of wall elements with this positive
support comes from a total element.  Take its ordered product $g$ from an
all-positive to an all-negative chamber.  At the chosen finite order,
join an all-positive point to a generic wall point $h_0$, and $h_0$ to an
all-negative point, by generic straight segments.  Replace their common
endpoint by a short transverse crossing of the wall.  The first segment
crosses only roots negative at $h_0$, the short crossing contributes the
wall element, and the last segment crosses only roots positive at $h_0$.
Its product is a
factorization of the form \eqref{tr:three-factors}.  Uniqueness identifies
its middle factor with the prescribed wall element.

For every $0\ne r\in C^+$, prescribe
\begin{equation}\label{tr:incoming-prescription}
 [X^r]\log g_0(V(r))=
 \begin{cases}
 [X^r]\log D_p,&r=jp,\ p\in C,\ j\ge1,\\
 0,&\text{otherwise}.
 \end{cases}
\end{equation}
These equations also have a unique solution.  Indeed, in
$\log g=\sum_r\ell_rX^r$, the coefficient $\ell_r$ contributes linearly
with coefficient one to the middle logarithm at $V(r)$, because
$V(r)(r)=0$.  All its other contributions involve roots of smaller
degree.  The equations therefore determine the $\ell_r$ successively.

The incoming end of the ray $r$ is the positive $V(r)$ direction.
On the subalgebra supported in $\ker V(r)$ the
functional $\tau_r(B)=[X^r]B$ is a trace.  In fact, if $a+b=r$ and
$\Omega(a,r)=0$, then $\Omega(a,b)=0$, so the two orders of multiplication
have the same $X^r$-coefficient. For positive units $U,W$ in this subalgebra,
\begin{equation}\label{tr:trace-log}
 \tau_r\log(UW)=\tau_r\log U+\tau_r\log W.
\end{equation}
To prove this, introduce a central variable $z$ by replacing $X^a$ with
$z^{|a|_C}X^a$. At each fixed root order, expansion of $\log(1+A)$
and cyclicity combine the differentiated terms of each power of $A$:
\[
 \tau_r\frac{d}{dz}\log U(z)
 =\tau_r\bigl(U'(z)U(z)^{-1}\bigr).
\]
The product rule and cyclicity then give
\[
 \tau_r\bigl((UW)'(UW)^{-1}\bigr)
 =\tau_r(U'U^{-1})+\tau_r(W'W^{-1}).
\]
The logarithms have zero constant term in $z$. Integrating and setting
$z=1$ proves \eqref{tr:trace-log} coefficientwise in root degree.
Apply this to a small generic perturbation of $V(r)$ within $r^\perp$.
In the refined factorization, a strictly positive or strictly negative
factor cannot contain $X^r$ in its logarithm.  Hence the middle factor
retains its prescribed coefficient.  Repeating this for every positive
multiple on the ray shows that its incoming wall element is $D_p$ on a
simple ray and the identity on every other ray.

\subsection{Positive crossing formulas}

\begin{lemma}\label{tr:positive-crossing}
Let $U_{a,k}$ be either $E_{a,k}$ or $H_{a,k}$, and let
$b=\Omega(a,m)\in\Z$.  If $b\ne0$, the coefficients of
$\Ad(U_{a,k})^{\sgn b}X^m$ are nonnegative.  Its coefficient at $X^m$ is
one.  In centered notation its factors have slots of energies
\begin{equation}\label{tr:slots}
 k-|b|+2h,\qquad 1\le h\le |b|.
\end{equation}
An $E$-slot can be chosen at most once; an $H$-slot can be chosen any
nonnegative number of times.  If $b=0$, the adjoint action is the identity.
For the simple unit, more precisely,
\begin{equation}\label{tr:simple-crossing}
 \Ad(D_p)^{\sgn\Omega(p,m)}X^m
 =\sum_{j=0}^{q}
 e_j(t^{1-q},t^{3-q},\ldots,t^{q-1})X^{m+jp},
 \qquad q=|\Omega(p,m)|.
\end{equation}
Here $e_j$ denotes the elementary symmetric polynomial of degree $j$,
with $e_0=1$.
\end{lemma}

\begin{proof}
Write $U_{a,k}(Z)=\prod_{j\ge0}f(t^{k-2j}Z)$, where
$f(Z)=1+Z$ or $f(Z)=(1-Z)^{-1}$.  The commutation relation gives
\begin{equation}\label{tr:unit-ratio}
 \Ad(U_{a,k})X^m
 =X^m\frac{U_{a,k}(t^{2b}X^a)}{U_{a,k}(X^a)}.
\end{equation}
For $b>0$ the quotient is
$\prod_{h=1}^{b}f(t^{k+2h}X^a)$.  For $b<0$, its reciprocal is
$\prod_{h=b+1}^{0}f(t^{k+2h}X^a)$.  Multiplying a selected copy of
$X^a$ on the right of $X^m$ contributes the centering factor $t^{-b}$.
In either case this gives exactly the list \eqref{tr:slots}, up to its
order.  All coefficients are nonnegative, and the constant selection
has coefficient one.  Taking $k=-1$ and $f(Z)=1+Z$ gives
\eqref{tr:simple-crossing}.  In particular, pairing one gives precisely
$X^m+X^{m+p}$.
\end{proof}

\begin{theorem}\label{tr:wall-positive}
At every generic point of a wall, and to every specified root order,
its element is an admissible product of $E/H$ units on that ray with
nonnegative integral multiplicities.  A monomial crossing in the direction
$-V(m)$ has coefficients in $\N((t^{-1}))$.
\end{theorem}

\begin{proof}
Fix a degree bound $d$ and proceed by induction in $d$.  For a ray
$\R_{>0}r$, follow a generic line in $r^\perp$ parallel to $V(r)$ from its
incoming end to the desired cell.  At this finite order it meets finitely
many joints. If $s$ is not proportional to $r$ and $\Omega(s,r)=0$,
then $s^\perp$ is parallel to that line
and can be avoided by its offset.  The offset also avoids intersections
of three independent normals.  Every joint where the factor changes is
therefore governed by a two-dimensional normal plane on which $\Omega$
is nonzero.

Order its positive rays so that the pairing of an earlier ray with a
later ray is positive.  At the incoming direction $V(a)$ of one of these
rays, earlier normals evaluate positively and later ones negatively.
Refinement in Lemma~\ref{tr:factorization} consequently identifies the
incoming local product with the product in this order, and the outgoing
factors with its factorization in reverse order.  The monoid obtained by
intersecting this plane with $C^+$ is pointed, with positive integral
charge $a\mapsto|a|_C$ and finitely many roots of bounded charge.
Theorem~\ref{loc:reordering} applies.

At degree $d$, the factor on the ray being followed is known on the
preceding segment, and other rays are needed only through degree $d-1$.
A degree-$d$ term contributes at that degree only linearly, on its own
ray; every interaction with another positive root has larger degree.
Choose any admissible extensions of
the known factorizations, apply positive reordering, and use uniqueness
of the truncated ray factorization.  This determines the required factor
through degree $d$.  Starting with the incoming factors supplied by
\eqref{tr:incoming-prescription} proves the assertion successively along
the line.

For a crossing in direction $-V(m)$, positive pairing with its positive
normal gives a positive-to-negative crossing and hence an adjoint;
negative pairing gives an inverse adjoint.  Every unit argument is a
positive multiple of that normal, so Lemma~\ref{tr:positive-crossing}
proves nonnegativity.  At bounded root degree, only a bounded number of
nonconstant copies can be selected, and their interaction pairings are
bounded.  The admissible unit parameters have a common upper bound.
For a fixed output energy, none of the selected parameters can tend to
$-\infty$, since the remaining bounded number of parameters and pairing
shifts cannot compensate for it.  Admissibility then leaves only finitely
many occurrences; their string indices are bounded as well.  Thus every
energy coefficient is a finite nonnegative integer, and the upper energy
is bounded.  This is precisely membership in $\N((t^{-1}))$.
\end{proof}

\subsection{Incoming coefficients and a decreasing recursion}

A \emph{section} is a family of chamber values related by the consistent
wall operations.  We will use sections obtained by transporting a finite
Laurent polynomial.  More generally the same construction works with a
value supported in a finite union
$U=\bigcup_{j=1}^s(m_j+C^+)$.  Transport and inverse transport add only
positive roots, so every chamber value retains this support bound.
Define
\begin{equation}\label{tr:incoming-coordinate}
 A_w(F)=[X^w]F(V(w)+\epsilon),
\end{equation}
where $\epsilon\in L_\R^\vee$ is a sufficiently small generic
perturbation in the finite arrangement needed for this coefficient.
The coefficient is independent of this choice: a separating wall has
normal $a$ with $\Omega(a,w)=0$,
and every potential predecessor $w-ja$ also pairs to zero with $a$.
Its wall action therefore fixes all monomials that could contribute to
$X^w$.

Fix a generic target point $h$.  At the needed finite order, follow
$h+sV(w)$ from large $s$ down to zero.  Write its successive crossing
maps as $W_i:I_i\longrightarrow I_{i+1}$ and let $a_i$ be the primitive
positive normal of the $i$th wall.  Put
\[
 J_i(u,j)=[X^{u+ja_i}]W_i(X^u).
\]
The initial coefficient on this path is $A_w(F)$.  Every crossing has
the direction of $-V(w)$; also
$\Omega(a_i,w-ja_i)=\Omega(a_i,w)$.  The preceding theorem gives
$J_i(w-ja_i,j)\in\N((t^{-1}))$, with $J_i(u,0)=1$.
Subtracting successive $X^w$-coefficients and telescoping gives
\begin{equation}\label{tr:positive-recursion}
 [X^w]F(h)=A_w(F)+
 \sum_i\ \sum_{\substack{j>0\\w-ja_i\in U}}
 J_i(w-ja_i,j)[X^{w-ja_i}]F(I_i).
\end{equation}
The inner exponents lie in the finite set $U\cap(w-C^+)$.  Only finitely
many roots can join two members of this set, so the outer sum is finite
as well.

\begin{theorem}\label{tr:incoming-positive}
If all incoming coefficients of a section belong to $\N((t^{-1}))$,
then every coefficient of every chamber value belongs to
$\N((t^{-1}))$.
\end{theorem}

\begin{proof}
For a target $w$, induct on the finite partially ordered set
$U\cap(w-C^+)$, simultaneously in all chambers.  In
\eqref{tr:positive-recursion}, every predecessor is strictly smaller than
the target.  The incoming term is nonnegative by assumption, and all
other terms are products of a nonnegative jump coefficient with a
coefficient already treated by the induction.  The finiteness and energy
bounds established above justify the sums and products in $\K$.
\end{proof}

The recursion is also effective.  For example,
\begin{equation}\label{tr:simple-coefficient}
 [X^{jp}]D_p=
 \frac{t^{-j^2}}{\prod_{i=1}^j(1-t^{-2i})}.
\end{equation}
This follows by selecting $j$ distinct string positions and subtracting
$0,1,\ldots,j-1$ from them.  Thus the incoming coefficients of the wall
elements are rational functions of $t$.  Finite triangular factorization
computes any bounded-order wall data.  Alternatively, positive
factorizations are computed using the finite shuffle constructions of
Theorem~\ref{loc:reordering}.  Once a support interval and an energy are
specified, \eqref{tr:positive-recursion} has finitely many strictly
decreasing dependencies.

\section{Change of roots and equality of Laurent coefficients}
\label{tr:mutation-section}

Fix $p\in C$ and put
\begin{equation}\label{tr:mutated-roots}
 \alpha_c=\Omega(p,c),\qquad
 c'=c+[-\alpha_c]_+p\quad(c\ne p),\qquad
 C'=\{-p\}\cup\{c':c\in C\setminus\{p\}\}.
\end{equation}
The roots $C'$ are again part of a lattice basis.  The symplectic shear
and its action on covectors are
\begin{equation}\label{tr:shear}
 S(w)=w-\Omega(p,w)p,\qquad S^\vee h=h\circ S^{-1}.
\end{equation}
Indeed, expanding $\Omega(Sa,Sb)$ makes the two cross terms cancel.
Since $\Omega(p,S(w))=\Omega(p,w)$, the inverse is
$S^{-1}(w)=w+\Omega(p,w)p$.
Let $L_+=\id$, $L_-=S$, and define the piecewise linear covector map
\begin{equation}\label{tr:covector-change}
 T(h)=
 \begin{cases}h,&h(p)\ge0,\\S^\vee h,&h(p)\le0.
 \end{cases}
\end{equation}
The two definitions agree on $p^\perp$.

\subsection{A uniform bound along the mutation direction}

Write a non-pure positive root uniquely as
\[
 r=kp+u,\qquad
 u=\sum_{c\ne p}u_c c\ne0,\qquad k,u_c\in\N,
\]
and set $|u|=\sum_{c\ne p}u_c$.  For $\sigma\in\{+1,-1\}$ define
\begin{equation}\label{tr:M-bound}
 M_\sigma(u)=\sum_{c\ne p}[-\sigma\alpha_c]_+u_c.
\end{equation}

\begin{lemma}\label{tr:fiber}
On the open side $\sgn h(p)=\sigma$, every non-pure root $kp+u$ in
the support of a wall element or its logarithm satisfies
\begin{equation}\label{tr:fiber-inequality}
 0\le k\le M_\sigma(u).
\end{equation}
The pure wall on $p^\perp$ has element $D_p$ wherever no other wall
direction is present.
\end{lemma}

\begin{proof}
Projection onto the face $\N p$ is an algebra homomorphism: a sum
involving a nonzero non-$p$ component cannot be pure.  At $p^\perp$,
the pure part of the middle factor is therefore the pure part of the
total element.  The incoming prescription identifies it with $D_p$.

We induct on $|u|$. For each fixed $k$, work in a finite full-root
truncation containing $kp+u$; the bound below is independent of that
truncation.
Let $b=\Omega(p,u)$.  If $b\ne0$, call the side
$\sigma=\sgn b$ upstream.  The path in $(kp+u)^\perp$ from its incoming
end to a point on that side stays off the pure cut, since its forward
tail has evaluation $h(p)+sb$ on $p$.  If $b=0$, the path is parallel
to the cut and the same reasoning applies on either side.  At the
incoming end, the only nonidentity data are simple, and their non-pure
roots have $k=0$.

At a joint away from the cut, all normals have positive non-$p$ degree.
A new nonlinear term of non-$p$ degree $|u|$ uses only terms of strictly
smaller non-$p$ degree.  A term of the current degree can enter linearly
only on its own ray and is carried from the preceding segment.
By induction all smaller terms have the stated bound.  Since
$M_\sigma$ is additive, the support set
\[
 \{kp+u:0\le k\le M_\sigma(u)\}
\]
is closed under addition.  The coefficient recursion for ray
factorization therefore preserves it.  This proves the upstream bound
for each fixed $k$, through finitely many joints at the necessary full
order.  Its numerical bound is independent of that order.  We may now
conclude that the entire upstream $p$-fiber is finite.

For $b\ne0$, continue across a generic joint on $p^\perp\cap u^\perp$.
In its two-dimensional normal plane, the non-pure projections are
positive multiples of the same ray modulo $p$.  Their pairings with
$p$ all have sign $\sigma$.  The upstream bounds already proved allow
this genericity to be imposed in a finite full-root arrangement for all
non-$p$ degrees currently needed.  Separate the pure factor from the
incoming product.  For $\sigma=+1$ it is first, so passage across it
conjugates the non-pure product by $D_p$; for $\sigma=-1$ it is last,
so conjugation is by $D_p^{-1}$.

Formula~\eqref{tr:simple-crossing} adds between zero and
$|\Omega(p,w)|$ copies of $p$ to a monomial with non-$p$ part $w$.
On the projected ray under consideration,
\[
 M_{-\sigma}(w)=M_\sigma(w)+|\Omega(p,w)|.
\]
The conjugated product thus satisfies the downstream bound.
Refactorization preserves that additive support set: for a first
putative exponent outside it, the product coefficient is zero and all
subtracted terms are sums of already allowed exponents, so its factor
coefficient must also be zero. At the current non-$p$ degree, the cut
calculation uses smaller degrees and current upstream coefficients.
Subsequent off-cut joints use the same smaller-degree induction.
Finally, logarithms and exponentials preserve
an additive positive support set, proving both assertions about support.
\end{proof}

In particular, non-cut wall elements have constant term one in non-$p$
degree, and their inverses have finite $p$-fibers too.  The same is true
after either shear, with finite Laurent fibers.
Furthermore,
\begin{equation}\label{tr:pushed-support}
 L_\sigma(kp+u)=\sum_{c\ne p}u_c c'
                    +(M_\sigma(u)-k)(-p).
\end{equation}
For $\sigma=+1$ this follows by substituting the definitions of $c'$;
for $\sigma=-1$ use
$M_+(u)+\Omega(p,u)=M_-(u)$.  Thus supported roots map into $C'^+$.
Put
\[
 B=\max\bigl(\{0\}\cup\{|\alpha_c|:c\ne p\}\bigr).
\]
If a pushed root has new degree at most $d$, then $|u|\le d$ and
$k\le B|u|$, so its old degree is at most $(B+1)d$.  Every bounded
new-order operation is consequently determined by finite old-order data.

\subsection{Comparison in a common rational ring}

\begin{theorem}\label{tr:mutation}
The covector map \eqref{tr:covector-change}, with exponent map $L_\sigma$
on the open side of sign $\sigma$, carries the wall system for $C$ to
that for $C'$, after replacing the pure wall by $D_{-p}$.

Let $F$ and $F'$ be sections specified by finite Laurent polynomials in
the respective all-negative chambers.  Suppose their initial values
satisfy the rational identity
\begin{equation}\label{tr:initial-identity}
 F_-=\Ad(D_p)F'_-.
\end{equation}
Their actual transported coefficients satisfy
\begin{equation}\label{tr:section-mutation}
 F'(T(h))=L_\sigma F(h),\qquad \sigma=\sgn h(p),
\end{equation}
where the equality is taken in each finite Laurent $p$-fiber.  In particular,
\begin{equation}\label{tr:incoming-mutation}
 A_w(F)=A_{\mu_p(w)}(F'),\qquad
 \mu_p(w)=w+[-\Omega(p,w)]_+p,
\end{equation}
and $\mu_p$ is a bijection of $L$.
\end{theorem}

\begin{proof}
First we prove the identity at the pure cut:
\begin{equation}\label{tr:pure-cut}
 S\circ\Ad(D_p)=\Ad(D_{-p})^{-1}.
\end{equation}
For $q=\Omega(p,m)\ge0$, put
$b_{q,j}=e_j(t^{1-q},t^{3-q},\ldots,t^{q-1})$.
The letters are invariant under inversion and have product one, giving
$b_{q,j}=b_{q,q-j}$. Formula~\eqref{tr:simple-crossing}, followed by
$S$, sends $X^m$ to
$\sum_{j=0}^q b_{q,j}X^{m-(q-j)p}$.
This equals $\Ad(D_{-p})^{-1}X^m$.  For negative pairing, apply the
proved equality to $X^{-m}$ and take inverses.  Both sides act by
algebra homomorphisms on rational monomial expressions, so
\eqref{tr:pure-cut} follows.

Set $Z=X^p$ and choose a lattice complement
$L=\Z p\oplus L_0$ containing $C\setminus\{p\}$.
Allow coefficients in $\K(Z)$ on the monomials $X^\ell$, $\ell\in L_0$,
and complete only in the free positive monoid generated by the images
of $C\setminus\{p\}$.  Section values may be supported in finitely many
translates of this monoid.  The multiplication rule, in addition to the
centered torus multiplication, is
\begin{equation}\label{tr:rational-ring}
 X^\ell f(Z)=f(t^{-2\Omega(p,\ell)}Z)X^\ell.
\end{equation}
Every specified non-$p$ coefficient is therefore a rational function
of $Z$, and its products use finitely many decompositions of that
non-$p$ exponent.

Lemma~\ref{tr:fiber} places each non-cut wall element and its inverse
in this ring, with finite Laurent support in $Z$ at every non-$p$
degree.  Pure conjugation fixes $Z$ and multiplies each $X^\ell$ by
a rational function of $Z$, by \eqref{tr:unit-ratio}.  The shear is
the identity modulo $p$.  Hence all operations in the comparison act
on the same ring.

Old consistency holds there as well.  The expansion map
$\K(Z)\hookrightarrow\K((Z))$ at zero is injective.  On a monomial,
the expansion of an old crossing agrees with its action in the old
positive completion.  At bounded non-$p$ degree, the non-cut operations
are determined by finitely many old roots, by the fiber bound.  To
compare a further coefficient in $Z$, choose the additional finite
full-root order needed for it.  Old path independence applies at that
order.  Thus a difference of two rational path operations has zero
expansion at zero in every non-$p$ coefficient, and injectivity makes
the difference zero in the common ring. The path maps are continuous
algebra homomorphisms; their equality on monomials extends through the
localization and then the non-$p$ completion.

Push the non-cut elements by $L_\sigma$ and insert $D_{-p}$ on the pure
cut.  Equation~\eqref{tr:pushed-support} and the following degree bound
place them in the new completion, with finite data at every new order.
In a loop, the linear identifications on successive open sides cancel;
at a cut crossing the remaining discrepancy is precisely corrected by
\eqref{tr:pure-cut}.  The pushed actions are therefore consistent in
the common rational ring.  Expansion at $Z^{-1}=0$ gives the new-cone
actions.  Indeed, replacing $c$ by $c'$ changes a fixed projected
coefficient only by a Laurent monomial in $Z$ and a power of $t$, and
\eqref{tr:pushed-support} ensures that every increment is new-cone
positive.  The finite-order bound makes this expansion legitimate.
Consistency of actions implies consistency of elements after the
nondegenerate enlargement: if a logarithm has a first nonzero term
$\lambda X^r$, choose $m$ with $\Omega(r,m)\ne0$; its commutator with
$X^m$ is nonzero at that first degree.

We next identify the incoming prescription.  A symplectic map $A$
satisfies $V(Ar)=A^\vee V(r)$.  Thus, on an eventual open side, the
incoming direction of a pushed ray is the image of its old incoming
direction.  For a simple root $c\ne p$, the relevant side has sign
$\alpha_c$, and its image is exactly
$c+[-\alpha_c]_+p=c'$; its unit becomes $D_{c'}$.  Other non-pure
incoming logarithmic coefficients are zero.  If the incoming direction
is parallel to the cut, its exponent pairs to zero with $p$, and both
linear pieces agree there.  Either generic offset gives the same
coefficient by \eqref{tr:trace-log}.  A portion with no old wall
preimage has identity element.  The remaining pure incoming element
is $D_{-p}$.  The uniqueness of \eqref{tr:incoming-prescription}
identifies the pushed system with the system for $C'$.

Choose a point on the cut with $h(c)<-1$ for every $c\ne p$, and
take $h(p)$ small on either side, with $|h(p)|<(B+1)^{-1}$.
Such covectors exist since $C$ is part of a basis.  For every non-pure
supported root,
\[
 h(kp+u)\le B|u|\,|h(p)|-|u|<0.
\]
The short crossing between these points meets only the pure wall.
The negative side is in the old all-negative chamber, and the positive
side is in the new all-negative chamber, since $h(-p)<0$ and
\[
 h(c')=h(c)+[-\alpha_c]_+h(p)<-1+\frac{B}{B+1}<0.
\]
On the positive side the old section takes the value
$\Ad(D_p)^{-1}F_-=F'_-$, a finite Laurent polynomial by hypothesis.
On the negative side the pushed old value is $SF_-$, while
\eqref{tr:pure-cut} and \eqref{tr:initial-identity} give
\[
 SF_-=\Ad(D_{-p})^{-1}F'_-.
\]
This value is also finite.

To reach any other chamber on one open side, transport entirely within
that half-space.  Only non-cut walls are crossed.  Their elements and
inverses have finite Laurent $p$-fibers.  For a fixed non-$p$ root-degree
bound $N$, Lemma~\ref{tr:fiber} leaves only roots of old degree at most
$(B+1)N$.  Thus a specified non-$p$ coefficient requires only finitely
many decompositions and crossings.
Starting from either of the finite values just obtained therefore
preserves finite Laurent support in each fiber.  Applying $L_\sigma$
preserves it as well.  Each compared fiber is the same finite Laurent
polynomial, so its actual coefficients agree independently of whether
we expand at zero or infinity.

Finally, $V(w)$ lies on the side of sign $\Omega(p,w)$.  For positive
pairing the exponent map is the identity, and for negative pairing it
is $S(w)=w-\Omega(p,w)p$.  For zero pairing both pieces fix $w$, and
incoming coefficients are independent of the generic offset.
This proves \eqref{tr:incoming-mutation}.  Its label map preserves
$\Omega(p,w)$, and hence its inverse subtracts
$[-\Omega(p,w)]_+p$.  It is a bijection.
\end{proof}

\subsection{A monomial with one incoming coefficient}

\begin{lemma}\label{tr:singleton}
Suppose $\Omega(c,m)\le0$ for every $c\in C$.  The section whose
all-negative value is $X^m$ satisfies
\begin{equation}\label{tr:singleton-formula}
 A_w(F)=\begin{cases}1,&w=m,\\0,&w\ne m.\end{cases}
\end{equation}
\end{lemma}

\begin{proof}
Every wall action and its inverse add only positive roots.  Thus the
support is contained in $m+C^+$, and the coefficient at $X^m$ is one
in every chamber.  Consider $w=m+r$ with $0\ne r\in C^+$.
In the finite truncation needed for this coefficient, start from a
generic all-negative point $h_-$ and follow $h_-+sV(w)$ for $s\ge0$.
At every crossed wall its positive normal $a$ satisfies
$\Omega(a,w)>0$: its evaluation starts negative and increases to zero.
Any nonidentity contribution from $X^m$ to $X^w$ would therefore give
\[
 r=\sum_j q_ja_j,\qquad q_j>0,\qquad \Omega(a_j,w)>0,
\]
so $\Omega(r,w)>0$.  Alternatingness and the hypothesis instead give
\[
 \Omega(r,w)=\Omega(r,m+r)=\Omega(r,m)\le0.
\]
No such contribution exists.  At the far end the target coefficient
is the incoming coefficient, proving \eqref{tr:singleton-formula}.
\end{proof}


% ===== sections/04_bz.tex =====
\section{From root transport to arbitrary quantum seeds}\label{bz:section}

For the seed data of Theorem~\ref{thm:main}, adjoin central coefficients
so that the exchange-root lattice is an integral direct summand.

\subsection{Central coefficients and an integral root model}

For the moment suppose that all current mutable vertices have the same
compatibility constant $d$. This includes a single mutable component
with the other components held fixed and regarded as frozen. Add $n$ frozen variables
$Y_1,\ldots,Y_n$ which commute with every original variable and with
each other. On $\widehat L=\Z^m\oplus\Z^n$ put
\begin{equation}\label{bz:central-extension}
 \widehat B=\begin{pmatrix}\widetilde B\\ I_n\end{pmatrix},
 \qquad
 \widehat\la=\begin{pmatrix}\la&0\\0&0\end{pmatrix},
 \qquad p_i=-\widehat b_i.
\end{equation}
Thus $p_i=(-\widetilde b_i,-e_i)$, where the second coordinate is in
the new frozen lattice. The extended pair is compatible with the same
$d$. The original coordinate basis together with $p_1,\ldots,p_n$ is
a basis of $\widehat L$: the change-of-basis matrix has diagonal blocks
$I_m,-I_n$. In particular
\[
 \mathsf N=\bigoplus_{i=1}^n\Z p_i
\]
is a direct summand. All subsequent root mutations are integral basis
changes within this same lattice $\mathsf N$.

Let $\Pi:\Z^m\longrightarrow\Z^n$ be the projection to the mutable
coordinates. Define the integral alternating form
\begin{equation}\label{bz:integral-model}
 \Omega_0=
 \begin{pmatrix}0&-\Pi^T\\ \Pi&-B\end{pmatrix}
 \quad\text{on }\widehat L.
\end{equation}
For an ambient vector $a=(a^{\rm old},a^Y)$, compatibility and
skew-symmetry of $B$ give
\begin{equation}\label{bz:root-pairings}
 \begin{aligned}
 \widehat\la(p_i,a)&=-d\,a^{\rm old}_i,
 &\widehat\la(p_i,p_j)&=d b_{ij},\\
 \Omega_0(p_i,a)&=-a^{\rm old}_i,
 &\Omega_0(p_i,p_j)&=b_{ij}.
 \end{aligned}
\end{equation}
Indeed,
$p_i^T\Omega_0=(-e_i^T\Pi,
\widetilde b_i^T\Pi^T+e_i^TB)$, and the second block is zero because
the transpose of column $i$ of $B$ is the negative of its row $i$.
Consequently
\begin{equation}\label{bz:radical}
 (\widehat\la-d\Omega_0)(r,a)=0
 \qquad(r\in\mathsf N, a\in\widehat L).
\end{equation}

\begin{lemma}[Comparison of root actions]\label{bz:root-bimodule}
Set $t=v^d$ and extend scalars from $\Q((t^{-1}))$ to
$\mathbb K_v=\Q((v^{-1}))$. Matching centered monomials by their
labels intertwines left and right multiplication by every root series,
and hence every root-wall conjugation, between the model with form
$\Omega_0$ and the actual model with form $\widehat\la$.
It also intertwines the rational pure-wall operations used in
Theorem~\ref{tr:mutation}.
\end{lemma}

\begin{proof}
For $r\in\mathsf N$ and $a\in\widehat L$, equation
\eqref{bz:radical} gives
\[
 t^{\Omega_0(r,a)}=v^{\widehat\la(r,a)},
 \qquad
 t^{\Omega_0(a,r)}=v^{\widehat\la(a,r)}.
\]
Thus the label matching is an isomorphism of bimodules for the two
identified root subalgebras, on each coset of $\mathsf N$. It
intertwines finite products and all coefficientwise root multiplication,
inverses of root units and conjugations in the positive root completion.
We make this comparison over $\mathbb K_v$, which contains the
coefficients of every actual BZ Laurent expression.

For the rational operation at a simple root $p$, choose
$\widehat L=\Z p\oplus L'$; this is possible because $p$ is a member
of a basis of the direct summand $\mathsf N$. Write $Z=X^p$.
Localizing the nonzero elements of $\mathbb K_v[Z^{\pm1}]$ gives a
module with summands $\mathbb K_v(Z)X^\ell$, $\ell\in L'$. The
commutation rule in both models is
\[
 ZX^\ell
 =v^{2\widehat\la(p,\ell)}X^\ell Z
 =t^{2\Omega_0(p,\ell)}X^\ell Z.
\]
The localization is legitimate: conjugating a nonzero Laurent
polynomial in $Z$ by any monomial scales $Z$ by a nonzero scalar and
therefore preserves the multiplicative set. For a finite sum of
monomials, the product of the finitely many scaled denominators gives
a common left or right denominator, proving the Ore condition.
Multiplication by a root
monomial is likewise intertwined after this localization. The pure
adjoint and its inverse send each monomial to a rational function of
$Z$ times that monomial, by the finite-ratio calculation in
Lemma~\ref{tr:positive-crossing}. Their formulas consequently agree
under the label matching. The same comparison applies after completion
in the remaining positive root directions.
\end{proof}

The wall identities for $\Omega_0$ restrict to $\widehat L$ from the
nondegenerate enlargement of Section~\ref{tr:section}. We then apply
this root bimodule map and specialize in the actual central torus
\eqref{bz:central-extension}.

\subsection{The exact exchange relation}

At the initial extended seed, define $E$ by the convention
\eqref{pre:E}, with $\widehat B$ in place of $\widetilde B$.
Thus $E^2=1$. If $g=Ee_k$, then
$\Omega_0(p_k,g)=1$ and $\Omega_0(p_k,Ee_j)=0$ for $j\ne k$.
The simple wall $D_{p_k}=E_{p_k,-1}$ therefore satisfies
\begin{equation}\label{bz:exchange-adjoint}
 \begin{split}
 \Ad(D_{p_k})X^g
 &=X^g+X^{g+p_k}\\
 &=X^{-e_k+[\widehat b_k]_+}
       +X^{-e_k+[-\widehat b_k]_+}.
 \end{split}
\end{equation}
Both coefficients are exactly one. In the uncentered ratio the single
factor has coefficient $t$; multiplication of $X^g$ by $X^{p_k}$
has coefficient $t^{-1}$, so these powers cancel. All other coordinate
generators are fixed by this adjoint. In the actual extended torus,
Lemma~\ref{bz:root-bimodule} proves the identical assertion after
$t=v^d$. Since the new generators have form $E^T\widehat\la E$,
the identity extends from coordinate generators to every centered
monomial, including those with negative exponents.

Let $E_0$ be the $n\times n$ mutable block of $E$. Matrix mutation is
\begin{equation}\label{bz:matrix-factorization}
 \widehat B'=E\widehat B E_0^T,
 \qquad \widehat\la'=E^T\widehat\la E.
\end{equation}
Identifying the new exponent lattice with the old one by $E$, its
negative exchange columns become
\begin{equation}\label{bz:root-mutation}
 -E\widehat b'_k=-p_k,
 \qquad
 -E\widehat b'_j=p_j+[-b_{kj}]_+p_k\quad(j\ne k).
\end{equation}
This is the root change in Theorem~\ref{tr:mutation}, since
$\Omega_0(p_k,p_j)=b_{kj}$, and is an integral basis change in
$\mathsf N$.

Let $\Pi_0:\widehat L\longrightarrow\Z^n$ be the
mutable-coordinate projection. Since $\Pi_0E=E_0\Pi_0$ and $E_0^2=1$,
\begin{equation}\label{bz:frozen-compatibility}
 (\widehat B')^T\widehat\la'
  =d E_0\Pi_0E=d\Pi_0.
\end{equation}

\subsection{Transport of an actual cluster monomial}

Fix a mutation word of length $s$ and a normalized cluster monomial
at its last seed. Let $E^{(j)}$ be the full mutation matrix from the
exponent lattice of seed $j+1$ to that of seed $j$, and put
$J_0=1$, $J_{j+1}=J_jE^{(j)}$. The map $J_j$ places
the exponent lattice of seed $j$ in the fixed initial lattice. Write
$\widehat B_j$ for its exchange matrix and set
\[
 \begin{aligned}
 \widehat\la_j&=J_j^T\widehat\la J_j,
 &\Omega_j&=J_j^T\Omega_0J_j,\\
 p_{j,i}&=-J_j\widehat b_{j,i},
 &C_j&=\{p_{j,i}:1\le i\le n\}.
 \end{aligned}
\]
Applying \eqref{bz:matrix-factorization} and
\eqref{bz:frozen-compatibility} successively in seed coordinates gives
$\widehat B_j^T\widehat\la_j=d\Pi_0$ and makes $C_j$ a basis of
$\mathsf N$. Since $p_{j,i}\in\mathsf N$, \eqref{bz:radical} yields
\begin{equation}\label{bz:moving-pairings}
 \Omega_0(p_{j,i},J_ja)
 =d^{-1}\widehat\la(p_{j,i},J_ja)
 =d^{-1}\widehat\la_j(-\widehat b_{j,i},a)
 =-a_i.
\end{equation}
Thus the exchange and root-change calculations apply at every stage
with the current forms $\Omega_j$ and $\widehat\la_j$.

Write the chosen cluster monomial in seed $j$ and transport its
monomial labels by $J_j$; call the result $P_j$. Each $P_j$ is an
actual finite based Laurent polynomial over $\Z[v^{\pm1}]$, by
Theorem~\ref{pre:laurent}. Equation~\eqref{bz:exchange-adjoint}
gives the rational identity
\begin{equation}\label{bz:actual-adjacent}
 P_j=\Ad(D_{p})P_{j+1},
 \qquad p\in C_j\text{ the root mutated at step }j.
\end{equation}
The two finite values are the all-negative initial values for the
respective wall systems. Lemma~\ref{bz:root-bimodule} and
Theorem~\ref{tr:mutation} thus identify their actual incoming
coefficients by a bijection of labels.

At the last seed, write the chosen normalized monomial as $M_s(a)$.
Then $P_s=X^{J_sa}$ and
$\Omega_0(p_{s,i},J_sa)=-a_i\le0$ by \eqref{bz:moving-pairings}.
Lemma~\ref{tr:singleton} therefore says that its incoming data consist
of a single coefficient equal to one. Applying the label bijections
backwards proves the same assertion for $P_0$. By
Theorem~\ref{tr:incoming-positive}, all coefficients of $P_0$ belong
to $\N((v^{-1}))$. They were already elements of
$\Z[v^{\pm1}]$; hence
\begin{equation}\label{bz:positive-intersection}
 \Z[v^{\pm1}]\cap\N((v^{-1}))=\N[v^{\pm1}]
\end{equation}
proves positivity in the central extension.

\subsection{Specialization and several components}

The actual initial Laurent torus has the homomorphism
\begin{equation}\label{bz:central-specialization}
 \operatorname{ev}_Y:
 \mathcal T_{\widehat\la}\longrightarrow\mathcal T_{\la},
 \qquad X^{(b,c)}\longmapsto X^b,
 \qquad Y_i\longmapsto1.
\end{equation}
It is a homomorphism because the added directions are in the radical
of $\widehat\la$. Its application to a finite positive Laurent
polynomial preserves positivity.

Deletion of the auxiliary exchange rows commutes with matrix mutation.
If $E$ is the mutation matrix for the original current seed and
its form is $\la$, the extended mutation matrix has the block form
\[
 \widehat E=\begin{pmatrix}E&0\\ R&I_n\end{pmatrix},\qquad
 \widehat E^T
 \begin{pmatrix}\la&0\\0&0\end{pmatrix}\widehat E
 =\begin{pmatrix}E^T\la E&0\\0&0\end{pmatrix},
\]
where $R$ is the $n\times m$ matrix with
$R_{ij}=\delta_{jk}[\widehat b_{m+i,k}]_+$. Thus at every seed the extended
form has the original form as its upper-left block and is zero on the
auxiliary directions.

We prove by induction that evaluation sends every extended cluster
variable to its original counterpart. Let $\widehat M$ and $M$ be
the current frames, with forms $\widehat\la$ and $\la$ in this
calculation, and write
$g_\pm=-e_k+[\pm\widehat b_k]_+$. Multiplying the extended exchange
relation on the right by $\widehat M(e_k)$ gives
\begin{equation}\label{bz:finite-exchange}
 \begin{split}
 \widehat M'(e_k)\widehat M(e_k)
 ={}&v^{\widehat\la(g_+,e_k)}\widehat M([\widehat b_k]_+)\\
 &+v^{\widehat\la(g_-,e_k)}\widehat M([-\widehat b_k]_+).
 \end{split}
\end{equation}
All factors lie in the finite Laurent algebra of the initial extended
seed. The right side uses nonnegative mutable powers and frozen units.
By the induction hypothesis and the block-form calculation, its
evaluation is the original exchange expression multiplied on the right
by $M(e_k)$. Hence
\[
 \operatorname{ev}_Y(\widehat M'(e_k))M(e_k)=M'(e_k)M(e_k).
\]
Right cancellation of the nonzero element $M(e_k)$ proves the induction.
The same block-form calculation preserves the centering factors, so
evaluation also sends every normalized cluster monomial to its
original counterpart.

\begin{proof}[Proof of Theorem~\ref{thm:main}]
If $n=0$, every cluster monomial is already an initial based monomial,
and the assertion is immediate. When the mutable compatibility
constants are equal, the preceding
argument proves the theorem. In general they are constant on each
mutable connected component, by Lemma~\ref{pre:components}. Fix one
component and hold the other mutable variables frozen. The preceding
proof applies to this smaller mutable set. A mutation in another
component changes neither its exchange columns nor the variables that
occur in its exchange relation. Every cluster variable from this
component in an arbitrary mutation word is therefore the same rational
expression as the one obtained from the subsequence of mutations in
that component. All cluster variables are consequently positive in the
chosen initial seed. A normalized cluster monomial is a power of $v$
times an ordered product of these variables and frozen powers. Based
torus multiplication multiplies coefficients and introduces only powers
of $v$, so it preserves positivity. This proves the assertion for
monomials. Finally the initial seed was arbitrary, which proves the
assertion relative to every seed.
\end{proof}

For the finite construction, add one central variable for every mutable
vertex and use the single
form \eqref{bz:integral-model} for the full principal matrix $B$.
For a component $c$, let $\mathsf N_c$ be the span of its roots and
$d_c$ its compatibility constant. Then
\begin{equation}\label{bz:component-model}
 \widehat\la(r,a)=d_c\Omega_0(r,a)
 \quad(r\in\mathsf N_c, a\in\widehat L),
 \qquad
 \Omega_0(\mathsf N_c,\mathsf N_{c'})=0\quad(c\ne c').
\end{equation}
Run the root system of component $c$ with parameter $t_c=v^{d_c}$.
The root subalgebras of distinct components commute, and their wall
actions commute on every ambient monomial. Their product therefore
defines the simultaneous system, with coefficients in
$\Q((v^{-1}))$ and with the same cone and incoming-label rules.
Local refactorization leaves commuting component factors unchanged.
The simple cut in
one component also commutes with every other component action.
Thus the proofs of the incoming recurrence and of change of chart
apply component by component to this product system. Apply it directly
to the whole normalized terminal monomial, whose incoming coefficient
is one. All scalar factors are then those of the original centered
monomial and its root actions.


% ===== sections/05_effective.tex =====
\section{Finite graded spaces for all coefficients}\label{eff:section}

We retain the central extension of Section~\ref{bz:section} and the
finite matrix construction of \cite[Sections~3 and~6]{OAI26}.
We compute all exponent and energy bounds from the seed, mutation word
and terminal monomial exponent.

\subsection{The incoming label and exact auxiliary valuations}

Let $\widehat F$ be the lifted normalized cluster monomial. At a
mutation in direction $k$ from seed $j$, use the current coordinates
$p_k=-\widehat b_{j,k}$ and the pulled-back form
$\Omega_j=J_j^T\Omega_0J_j$. If $w$ is the incoming label in the
new coordinates, the old label is
\begin{equation}\label{eff:inverse-label}
 g_{\rm old}=Ew-[-w_k]_+p_k.
\end{equation}
By \eqref{bz:moving-pairings}, $\Omega_j(p_k,Ew)=w_k$.
The map $z\mapsto z+[-\Omega_j(p_k,z)]_+p_k$ preserves the
pairing with $p_k$, so its inverse subtracts this same multiple of
$p_k$. Applying \eqref{eff:inverse-label} backwards to the terminal
exponent computes $g\in\widehat L$ in the initial coordinates.
The formula holds componentwise in \eqref{bz:component-model}.
From now on, $p_i$ denotes the initial root.

Write
\[
 \iota:\Z^n\longrightarrow\widehat L,\qquad
 \iota(k)=\sum_i k_i p_i,\qquad C^+=\iota(\N^n).
\]
The incoming recurrence gives
\begin{equation}\label{eff:cone-support}
 \supp\widehat F\subset g+C^+.
\end{equation}
By Laurent finiteness, the section with initial value $\widehat F$
is supported in a finite union $\mathcal U$ of translates of $C^+$.
If $w\notin g+C^+$,
every predecessor $w-ja$ in the incoming recurrence is also outside
$g+C^+$. Its incoming datum is zero. Induction over the finite
partially ordered set $\mathcal U\cap(w-C^+)$, simultaneously in all chambers,
therefore makes every such coefficient zero. This proves
\eqref{eff:cone-support}.

The smallest auxiliary exponents are computed using an ordered
series division ring.

\begin{lemma}[Positive leading coefficients]\label{eff:ordered-series}
Let $\mathcal D$ be an ordered division ring and let $\sigma$ be an
order-preserving automorphism. The skew Laurent-series division ring
$\mathcal D((Z;\sigma))$, with $Za=\sigma(a)Z$ and exponents
bounded below, is ordered by its first nonzero coefficient. An integral based
quantum torus over $\Q(v)$ embeds into an iterated division ring of
this kind in which every centered monomial is positive.
For any specified central coordinate $Y$, the last series variable
can be chosen to be $Y$.
\end{lemma}

\begin{proof}
If two positive series have first terms $a_pZ^p$ and $b_qZ^q$,
the first coefficient of their product is
$a_p\sigma^p(b_q)>0$. Addition preserves positivity: at different
first exponents the earlier coefficient survives, and at equal
exponents the two positive coefficients add. These rules also give
trichotomy. A nonzero series is its invertible first monomial times
$1+h$, where $h$ has strictly positive $Z$-valuation. The geometric
series for $(1+h)^{-1}$ is defined coefficientwise and has constant
term one. This proves both the division assertion and the positivity
of inverses of positive series.

Order $\Q(v)$ by the sign at positive infinity. In particular,
every integer power of $v$ is positive. Adjoin the coordinate
variables successively as skew Laurent-series variables. At each
step the required automorphism scales earlier coordinate variables
by central powers of $v$. It extends coefficientwise to
their series division rings and preserves the first exponent. By
induction it preserves the sign of the recursively leading coefficient,
since each scaling factor is a power of $v$. It is thus order-preserving.
The quantum commutation relations
give the embedding. It extends to the Ore division ring of fractions, since
every nonzero torus element has become invertible. Centered monomials
differ from ordered products of coordinate powers by a power of $v$,
which is positive in this order. A central coordinate may be adjoined
last, with identity automorphism.
\end{proof}

Fix $j$ and place $Y_j$ last in this construction. The resulting
division ring is $\mathcal D((Y_j))$, where $\mathcal D$ is ordered.
For a nonzero series, define
\[
 \nu_{Y_j}\!\left(\sum_r f_rY_j^r\right)
   =\min\{r:f_r\ne0\}.
\]
Every cluster variable is positive in this ring: mutation uses sums, products,
inverses and integer powers of $v$. Each of the two exchange
summands is positive. If their $Y_j$-valuations agree, their leading
coefficients in $\mathcal D$ are positive and their sum is nonzero.

Consequently the valuations $\ell_i=\nu_{Y_j}(X_i)$ of the current
frame coordinates satisfy the exact recurrence
\begin{equation}\label{eff:tropical}
 \ell'_k=-\ell_k+
 \min\left\{\sum_i[\widehat b_{ik}]_+\ell_i,\
             \sum_i[-\widehat b_{ik}]_+\ell_i\right\},
 \qquad \ell'_i=\ell_i\quad(i\ne k).
\end{equation}
Initially the old coordinate valuations are zero and
$\nu_{Y_j}(Y_h)=\delta_{jh}$. If the terminal exponent is $a$, set
\begin{equation}\label{eff:aux-minimum}
 L_j=\sum_i a_i\ell_i^{\rm terminal}.
\end{equation}
The centering scalar has valuation zero, so
$L_j=\nu_{Y_j}(\widehat F)$, also when frozen exponents are negative.
Since $\widehat F$ is an actual finite Laurent polynomial, $L_j$ is
exactly the smallest $Y_j$-exponent among its nonzero terms. At any
fixed $Y_j$-exponent the remaining based monomials are linearly
independent in the residue quantum torus, which proves this last
assertion.

For $w=g+\iota(k)$, the auxiliary coordinates are
$w_{Y_i}=g_{Y_i}-k_i$. Combining
\eqref{eff:cone-support}--\eqref{eff:aux-minimum} gives a complete
finite box:
\begin{equation}\label{eff:box}
 0\le k_i\le K_i:=g_{Y_i}-L_i,\qquad
 N=\sum_iK_i,\qquad
 \mathcal I=\{g+\iota(k):0\le k_i\le K_i\}.
\end{equation}
All $K_i$ are nonnegative, since $\widehat F\ne0$. Both their values
and the uniqueness of the label $k$ follow before any coefficients
are computed. If $N=0$, the incoming recurrence gives
$\widehat F=X^g$; its only coefficient space is $\Q$ in degree
zero. We henceforth suppose $N\ge1$.

\subsection{A finite graph of local reorderings}

At positive root degree at most $N$ there are finitely many roots,
wall hyperplanes and cells. The proof of
Theorem~\ref{tr:wall-positive} constructs the required factorization
on each ray and cell by following a generic line and applying
Theorem~\ref{loc:reordering} at its joints. Retain each chosen line
traversal as an ordered list of nodes and join each query to its
predecessor computation. These applications form a finite directed
acyclic graph, ordered first by root degree and then by position
along the line on the ray being constructed.
Queries to other rays use smaller root degree. Indeed, a term of the
current degree on a different ray could affect the target ray only
through a nonlinear product, which has larger degree. Queries on
the current ray refer only to earlier points of that same line.
This proves acyclicity.

All these choices are effective over $\Q$. Enumerate rational
points and lines with the prescribed rational directions until they
avoid the finitely many forbidden
hyperplanes and intersections in the truncated arrangement.
Each choice is tested by integer linear algebra. The input leaves
are the simple units $E_{p_i,-1}$. For a component, choose an integer
\[
 A\ge\max\left(1,\
 N^2\max_{i,j}|\Omega_0(p_i,p_j)|\right).
\]
It bounds the pairings between all relevant roots. If the parameters
of the elementary units entering a node are at most $K\ge0$, the
effective bound in Theorem~\ref{loc:reordering} is
\begin{equation}\label{eff:upper-node}
 k_{\rm out}\le NK+N+AN^2.
\end{equation}
Propagating this bound from $K=0$ at the leaves gives an integer
$K_{*,c}\ge0$ bounding every output parameter in component $c$.

There is a complementary finite procedure for any lower output
threshold $L$. If a node is asked for parameters $k_{\rm out}\ge L$,
an input occurrence that participates in such an output must have
\begin{equation}\label{eff:lower-node}
 k_i\ge L-(N-1)K-N-AN^2.
\end{equation}
The ordinary polynomial degree $\ell$ of the start of any required
primitive string satisfies
\begin{equation}\label{eff:degree-node}
 0\le\ell\le
 \left\lfloor\frac{N(K+1)+AN^2-L}{2}\right\rfloor.
\end{equation}
The degree estimate follows from the exact output
parameter $u(d)-\chi(d,d)-2\ell$ and the inequalities
$u(d)\le N(K+1)$ and $\chi(d,d)\ge-AN^2$. To obtain
\eqref{eff:lower-node}, isolate one active input copy in the
inequality
$k_{\rm out}\le\sum_i d_i k_i+N+AN^2$; at most $N-1$
other copies remain. These are the estimates proved with
Theorem~\ref{loc:reordering}. A negative upper bound in
\eqref{eff:degree-node} means that the node contributes nothing.
Apply \eqref{eff:lower-node} recursively to predecessor nodes.
Acyclicity makes the recursion terminate, and each retained input
list is finite. Each occurrence keeps its own index even if several
occurrences have the same root, parameter and type.

\subsection{A global energy bound}

In the simultaneous component system, set
\begin{equation}\label{eff:energy-constants}
 \kappa_*=\max_c d_cK_{*,c},\qquad
 T_*=\max_{\substack{a\in C^+\cap\mathsf N_c,\ 0<|a|\le N\\
                    w\in\mathcal I,\ c}}
             |\widehat\la(a,w)|,
 \qquad U=N(\kappa_*+T_*).
\end{equation}
Here $|a|$ is the sum of its coefficients in the initial positive
root basis. The maximum is over finite sets. For example, it is
enough to use the explicit upper bound
\[
 T_*\le
 N\max_i|\widehat\la(p_i,g)|
 +N^2\max_{i,j}|\widehat\la(p_i,p_j)|.
\]
The local parameter $k$ in component $c$ is measured in powers of
$t_c=v^{d_c}$. Lemma~\ref{tr:positive-crossing} gives the exact
$v$-energies of its centered slots at a source exponent $w$:
\begin{equation}\label{eff:slots}
 d_ck-|\widehat\la(a,w)|+2d_ch,\qquad
 1\le h\le |\widehat\la(a,w)|/d_c.
\end{equation}
The slot number is an integer by
\eqref{bz:component-model}. At zero pairing the crossing contributes
no nonconstant term. For an $E$ unit each slot is chosen zero or
one times; for an $H$ unit its multiplicity is any nonnegative
integer. Root-degree truncation bounds the latter multiplicities.
Roots on one ray pair to zero, so the energies in
\eqref{eff:slots} add under slot selection.

Expand the incoming recurrence as histories of positive jumps from
$g$. Suppose a history ending at $w\in\mathcal I$ selects $q$
nonconstant copies, with roots $a_\nu$, parameters $k_\nu$ and
components $c_\nu$. Its prefix labels
$w_\nu=g+\sum_{\mu=1}^{\nu}a_\mu$ lie in $\mathcal I$. Since
each $|a_\nu|\ge1$,
\[
 q\le\sum_{\nu=1}^q|a_\nu|=|w-g|\le N.
\]
Let $\delta_\nu$ be the slot shift in \eqref{eff:slots}, evaluated
at $w_{\nu-1}$. Then $|\delta_\nu|\le T_*$ and
$d_{c_\nu}k_\nu\le\kappa_*$. The total energy satisfies
\begin{equation}\label{eff:history-energy}
 E=\sum_{\nu=1}^q(d_{c_\nu}k_\nu+\delta_\nu)
 \le q(\kappa_*+T_*)\le U.
\end{equation}
The exact crossing and incoming identities express each Laurent
coefficient of $\widehat F$ as the sum of these histories.
Admissibility makes each specified energy coefficient finite, so
\eqref{eff:history-energy} bounds every coefficient above by $U$.

Bar invariance supplies the other half of the bound. On the based
quantum torus the anti-involution
\begin{equation}\label{eff:bar}
 \overline v=v^{-1},\qquad \overline{X^a}=X^a,\qquad
 \overline{PQ}=\overline Q\,\overline P
\end{equation}
extends to its division ring. If the coordinates of a toric frame
are fixed, every centered monomial of that frame is fixed: reversing
the ordered product contributes the inverse of the change in the
centering power. The statement includes negative powers. Each
mutated variable is a sum of two centered monomials, so induction
along the mutation word gives $\overline{\widehat F}=\widehat F$.
Write its finite initial Laurent expansion as
$\widehat F=\sum_w c_w(v)X^w$. Linear independence of the based
monomials yields
\[
 c_w(v)=c_w(v^{-1}).
\]
Thus
\begin{equation}\label{eff:bar-window}
 [v^E]c_w(v)=0\qquad\text{unless }-U\le E\le U.
\end{equation}
In a history of final energy $E\ge-U$, isolate a selected copy with
parameter $k$ in component $c$. There are at most $N-1$ other
parameters, each with $v$-weight at most $\kappa_*$, and the sum of
all slot shifts is at most $NT_*$. Thus
\[
 -U\le E\le d_ck+(N-1)\kappa_*+NT_*,
\]
which gives
\begin{equation}\label{eff:global-lower}
 d_ck\ge L_*:=-U-(N-1)\kappa_*-NT_*.
\end{equation}
Query component $c$ only for
$k\ge\lceil L_*/d_c\rceil$, and propagate these queries backwards
using \eqref{eff:lower-node}. Equations
\eqref{eff:degree-node} then give finite bounds for every local
matrix calculation.

Every retained prefix has energy at least
\begin{equation}\label{eff:prefix-window}
 N(L_*-T_*).
\end{equation}
Indeed, each selected copy has energy at least $L_*-T_*$, this
number is nonpositive, and there are at most $N$ copies. Thus
\eqref{eff:prefix-window} and $U$ bound all intermediate energies
used in the enumeration.

\subsection{The finite polynomial matrices}

At a local node, use its finite occurrence list. For each dimension
vector $d$ of root charge at most $N$, use monomial symmetric
functions in each variable pack as a basis of the space $S_d$.
In the required ordinary degrees,
the unstable ideal is the span of the explicit shuffle images
\[
 S_a*S_b,\qquad a+b=d,\quad a,b\ne0,\quad \mu(a)>\mu(d).
\]
In output degree $\ell$, the input degrees sum to
$\ell+\chi(a,b)$. Since
$\chi(a,b)\le\sum_i a_i b_i\le N^2$, the necessary matrix domains
have ordinary degrees at most the bound
\eqref{eff:degree-node} plus $N^2$. Taking the quotient of these
rational matrices gives $B_d$ in all required degrees.

Write $\omega$ for the auxiliary weight. The independently centered
restrictions define the finite filtration $F^\omega B_d$. In ordinary degree $\ell$, the possible weights lie
between
\[
 \min_{P}\sum_{r\in P}s(r)
 \quad\text{and}\quad \ell+s(d),
\]
in half-integer steps, where $P$ runs through the finitely many
ordered decompositions of $d$ on the slope. Restriction does not
increase internal polynomial degree. Thus the filtration and its
associated graded require only finite matrix operations.

The primitive subspace $\mathcal P_d$ is the common kernel of the
proper coproduct symbols. To compute this kernel, one may test the
leading restrictions before multiplying them by their nonzero
rational prefactors in the independent centers. Such a prefactor is
invertible over the rational function field of the centers and does
not change a kernel. Expanding the remaining expression in center
monomials and comparing its quotient-vector coefficients gives
finite rational matrices in the stated degree range. The centers
remain independent throughout this calculation.

Finally impose the kernel of
$D_d=\sum_{i,\alpha}\partial/\partial x_{i,\alpha}$, whose target
has ordinary degree $\ell-1$. The resulting actual multiplicity
spaces are
\begin{equation}\label{eff:primitive-spaces}
 W_{d,\ell,\omega}=
       (\mathcal P_d\cap\ker D_d)_{\ell,\omega}.
\end{equation}
By Lemma~\ref{loc:strings}, a basis vector of this space gives one
output unit with root $p(d)$ and parameter
\[
 k=u(d)-\chi(d,d)-2\ell.
\]
Its type is $E$ if $\chi(d,d)$ is odd and $H$ if it is even.
Choose an ordered basis by rational row reduction, with a fixed
ordering of all input monomials. This gives the individual occurrence
labels needed at subsequent nodes. The full dimension vector and
the separate occurrence labels are retained after projection to roots.

\subsection{Decorated histories and specialization}

Fix the paths used in \eqref{tr:positive-recursion} for every target
exponent $w\in\mathcal I$ and chamber $I$ of the truncated arrangement.
The indices $i$, normals $a_i$ and source chambers $I_i$ below refer
to the fixed path for $(w,I)$.
Retain the unit parameters in \eqref{eff:global-lower} and the
propagated predecessor ranges. For a summand $(i,j)$ of that
recurrence, put $u=w-ja_i$ and let $\mathsf S_i(u,j;e)$ be the set
of nonempty slot selections contributing exponent $ja_i$ and energy
$e$ to the jump coefficient $J_i(u,j)$. Each selected occurrence is
labeled by the chosen basis of \eqref{eff:primitive-spaces}. An $E$
slot permits a subset of these labels and an $H$ slot permits a
multiset. Their energies are given by \eqref{eff:slots}.

For $w\in\mathcal I$ and $N(L_*-T_*)\le E\le U$, define the
history sets recursively by
\begin{equation}\label{eff:histories}
 \mathsf H(w,I;E)=\mathsf H_0(w;E)\ \sqcup
 \bigsqcup_{\substack{i,\ j>0,\ e\in\Z\\u=w-ja_i\in\mathcal I}}
 \mathsf H(u,I_i;E-e)\times\mathsf S_i(u,j;e),
\end{equation}
where $I_i$ is the source chamber of the crossing and
\[
 \mathsf H_0(w;E)=
 \begin{cases}
  \{\varnothing\},&w=g,\ E=0,\\
  \varnothing,&\text{otherwise}.
 \end{cases}
\]
The element $\varnothing$ in the first case denotes the empty history.
Each alternative in \eqref{eff:histories} retains its label $(i,j)$;
the product concatenates the earlier history and its slot selection.
Set $\mathsf H(w,I;E)=\varnothing$ if $w\notin\mathcal I$ or
$E\notin[N(L_*-T_*),U]$. Since $j>0$, each
recursive call strictly decreases root degree. The recursion is
therefore finite. The exponent and chamber sets, unit lists and slot
sets are finite, and every history contains at most $N$ copies.

Let $I_-$ be the all-negative chamber. Define
$\widehat{\mathcal V}_{k,E}$ as the free rational vector space on
$\mathsf H(g+\iota(k),I_-;E)$. Thus the empty history gives
$\widehat{\mathcal V}_{0,0}=\Q$. Products and disjoint unions in
\eqref{eff:histories} give tensor products and direct sums of these
spaces. The slot choices give the exterior and symmetric powers
of the multiplicity spaces for $E$ and $H$, respectively.

At each local node, the primitive-string factorization identifies
the dimensions of \eqref{eff:primitive-spaces} with the elementary
factor multiplicities. At each edge,
Lemma~\ref{tr:positive-crossing} identifies slot enumeration with
the exact centered jump coefficient. Induction in root degree in the
incoming recurrence expresses the actual coefficient as the sum over
decorated histories. Equations~\eqref{eff:global-lower} and
\eqref{eff:prefix-window} retain every history in the final window, so
\begin{equation}\label{eff:lifted-dimensions}
 \widehat F=
 \sum_{\substack{k\in\N^n\\k_i\le K_i\ (1\le i\le n)}}\ \sum_{E=-U}^{U}
     \dim_\Q\widehat{\mathcal V}_{k,E}\,
     v^E X^{g+\iota(k)}.
\end{equation}
Equation~\eqref{eff:bar-window} gives vanishing outside this window.
The component actions in \eqref{bz:component-model} act on the whole
centered monomial, retaining its original normalization in
\eqref{eff:lifted-dimensions}.

\begin{proof}[Proof of Theorem~\ref{thm:finite}]
Let $\operatorname{pr}:\widehat L\longrightarrow\Z^m$ delete the
auxiliary coordinates. For $-U\le E\le U$, define
\begin{equation}\label{eff:final-spaces}
 \mathcal V_{b,E}
   =\bigoplus_{\substack{k\in\N^n,\ k_i\le K_i\ \forall i\\
                \operatorname{pr}(g+\iota(k))=b}}
          \widehat{\mathcal V}_{k,E},
 \qquad
 \mathcal V_b=\bigoplus_{E=-U}^{U}\mathcal V_{b,E}.
\end{equation}
Set $\mathcal V_{b,E}=0$ for $|E|>U$.
The sums are finite and their summands are constructed by rational
linear algebra and finite choices. Apply the
homomorphism \eqref{bz:central-specialization} to
\eqref{eff:lifted-dimensions}. The specialization argument in
Section~\ref{bz:section} identifies the left side with the required
original cluster monomial. The right side is exactly
\eqref{intro:dimension-formula}. The label recursion, tropical
valuations, finite reordering graph and energy bounds are computed
before the coefficient spaces, and every subsequent operation
terminates. If the mutable set is empty, take the single initial
monomial space $\Q$ in energy zero. This proves the theorem.
\end{proof}

\begin{example}\label{eff:one-mutation-example}
For the seed of Example~\ref{pre:nonsaturated-example}, adjoin the
single central coordinate $Y$ and mutate the mutable variable once.
In the extended exponent lattice,
\[
 p=-2e_2-e_Y,\qquad g=-e_1+2e_2+e_Y.
\]
The inverse incoming-label rule gives $g$, and the tropical recurrence
gives $L_Y=0$. Hence $K_Y=1$ and the full exponent box is
$\{g,g+p\}$. The component constant is $d=2$ and
$\widehat\la(p,g)=2$. The simple unit has parameter $k=-1$,
so its sole slot in \eqref{eff:slots} has energy
$2(-1)-2+2\cdot2=0$. The empty history and the history selecting
this slot give, respectively,
\[
 \widehat{\mathcal V}_{0,0}=\Q,\qquad
 \widehat{\mathcal V}_{1,0}=\Q,
 \qquad
 \widehat F=X^g+X^{g+p}.
\]
All other spaces vanish. Specializing $Y=1$ gives the two
coefficient spaces for $M(-e_1+2e_2)+M(-e_1)$. The original
nonprimitive exchange column and the frozen pairing
$\la(e_3,e_4)=1$ have both been retained.
\end{example}


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\end{thebibliography}
\end{document}