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\title[Positivity for weighted linear LP algebras]{Positivity for weighted linear\newline Laurent phenomenon algebras}
\author{Qiyue Tang}
\address{Zhili College, Tsinghua University, Beijing 100084, China}
\email{tangqy24@mails.tsinghua.edu.cn}
\date{November 11, 2026}
\subjclass[2020]{Primary 13F60; Secondary 05E15, 05C05}
\keywords{Laurent phenomenon algebra, Laurent positivity, cluster monomial, rooted forest}

\begin{document}
\begin{abstract}
We prove Laurent positivity for ordinary Laurent phenomenon algebras
whose initial exchange polynomials are nonconstant affine linear
polynomials with positive constant terms and nonnegative numerical
coefficients. The coefficient field may be any subfield of the real
numbers, and the weights may have arbitrary zero patterns and algebraic
relations. Every cluster variable has nonnegative Laurent coefficients
in every cluster, after choosing positive representatives. The algebra
coincides with its upper algebra, and its cluster monomials form an
atomic positive basis with nonnegative multiplication constants.
The proof uses the factorization of rooted-forest determinants, a
finite system of positive exchange identities, and a refinement by
prime factors that controls specialization to zero weights. A nonlinear
example with all nonzero initial coefficients equal to one shows that
the linearity hypothesis cannot be omitted.
\end{abstract}
\maketitle
\tableofcontents

\section{Introduction}

The Laurent phenomenon asserts that variables obtained by repeated
exchange are Laurent polynomials in the variables of an initial cluster.
In the cluster algebras of Fomin and Zelevinsky, this phenomenon
originally led to the stronger positivity conjecture: the Laurent
coefficients should be nonnegative \cite{FZ02}. These two assertions are distinct.
Even a rational expression obtained using only addition, multiplication,
and division can simplify to a Laurent polynomial with negative
coefficients. A proof of positivity must therefore control the divisions
that occur during mutation.

Lam and Pylyavskyy introduced Laurent phenomenon algebras, or LP
algebras, by allowing irreducible exchange polynomials in place of
exchange binomials \cite{LP16}. Their mutation rule removes common
factors before defining the next exchange polynomial. It preserves the
Laurent phenomenon, but the removal of factors makes coefficient
positivity delicate. In their subsequent study of linear LP algebras,
they described a family attached to directed graphs \cite{LPlinear}.
Its initial exchanges have the form
\[
 F_i=A_i+\sum_{i\to j}x_j,
\]
with independent coefficient variables $A_i$. The cluster variables
are given by sums over directed forests, and the clusters are described
by nested collections of strongly connected vertex sets. This description
gives positivity in the initial cluster. Their Conjecture~7.1 asks for
positivity in every cluster; Conjecture~7.3 concerns a cluster-monomial
basis and its multiplication constants.

Glew has proved positivity for graph LP algebras with respect to an
arbitrary seed \cite[Theorem~5.1]{Glew26}. Daisey has proved that every
LP algebra admitting a linear seed is of finite type over an arbitrary
unique factorization domain \cite[Theorem~1.2]{Daisey26}. The latter
result allows coefficient relations and vanishing coefficients.
Neither statement directly gives positivity for arbitrary numerical
weights. Specializing coefficients can create new common factors and
change the monomial denominators in the exchange Laurent polynomials;
this obstruction is also emphasized in \cite[Remark~1.1]{Daisey26}.

Here we consider ordinary LP algebras over a field $K\subseteq\R$.
The initial exchange polynomials are
\begin{equation}\label{eq:initial-general}
 F_i=a_i+\sum_{j\ne i}c_{ij}x_j,
 \qquad a_i>0,\quad c_{ij}\geq0.
\end{equation}
We require each $F_i$ to be nonconstant. Over a field this is part of
the seed axioms: a nonzero constant is a unit, whereas an exchange
polynomial must be an irreducible nonunit. No further restriction is
placed on the rank, the support of $(c_{ij})$, or the equalities and
algebraic relations among its entries.

An LP variable is defined up to a unit of its coefficient ring. We fix
one representative of each association class, positive on the positive
orthant of the initial cluster. The proof constructs such representatives
simultaneously. Multiplying any of them by a positive element of $K$
preserves every positivity assertion below. A \emph{cluster monomial}
is a product of variables in a single cluster with nonnegative integer
exponents, and equal products are counted only once.

\begin{theorem}\label{thm:main}
Let $K\subseteq\R$ be a field and let $\A$ be an ordinary LP algebra
with an initial seed of the form \eqref{eq:initial-general}, with every
$F_i$ nonconstant. Then positive representatives can be chosen so that:
\begin{enumerate}
\item every cluster variable has Laurent coefficients in $K_{\geq0}$
with respect to every cluster;
\item every exchange Laurent polynomial, and hence every ordinary
exchange polynomial in the corresponding normalization, has
nonnegative coefficients;
\item if $\U$ is the upper LP algebra and $\mathscr B$ is the set of
cluster monomials, then
\[
 \A=\U=\bigoplus_{b\in\mathscr B}Kb;
\]
\item the universally nonnegative elements are exactly
\[
 \bigcap_T K_{\geq0}[\mathbf x_T^{\pm1}]
   =\bigoplus_{b\in\mathscr B}K_{\geq0}b.
\]
In particular, the cluster-monomial basis is atomic and all its
multiplication constants are nonnegative.
\end{enumerate}
The normalized mutation pattern is finite.
\end{theorem}

The coefficients in this theorem are numerical elements of the ordered
field $K$. The assertion is not a statement about the individual
coefficients of a polynomial in formal weight parameters. In particular,
the theorem imposes no algebraic independence assumption on numerical
weights. For primitive nonconstant initial exchanges over $\Z$, we also
obtain nonnegative integer Laurent coefficients; see
Corollary~\ref{cor:integer}.

We briefly describe the proof. After dividing each initial exchange by
its positive constant term, all roots are equal to one. The relevant
determinants are then rooted-forest polynomials. Equalities among rows
outside a vertex set define an outgoing module. Minimal modules and
maximal cliques of pair modules determine a forest of blocks. The
irreducible forest factors have canonical prime lifts to the upper
algebra. Their coordinate denominators are computed by a signed
Laplacian on each coordinate boundary.

For strictly positive weights, we construct a finite collection of
candidate clusters. A candidate is described by retained initial
variables, block cores, and an increasing chain of principal
determinants. The principal step may involve a whole tree of blocks.
Source determinants give simultaneous factorizations of the port
factors and the previously omitted variables. These factorizations
produce nonnegative identities for every direction of exchange.
Primality in the upper algebra identifies each proposed partner with
the actual LP mutation partner. A separate finite-pattern argument
then converts exchange positivity into Laurent positivity in every
cluster, and gives the assertions about the upper algebra and the basis.

Zero weights require a further argument. Uniformly completing the
coefficient matrix by a positive parameter preserves all outgoing
modules. The positive charts have limits, but their coordinates can
become reducible. We prove that every prime factor is itself a
coordinate of the limiting collection. Each limiting chart therefore
refines by a unimodular monomial change into a genuine LP cluster.
Completeness of the leading-exponent cones proves that no actual
cluster is omitted. This is the step that allows arbitrary zero
patterns without assuming that mutation commutes with specialization.

The final remark gives a nonlinear rank-three counterexample with
positive initial coefficients and verifies its mutations explicitly.
Thus the distinction between linear and nonlinear exchange
polynomials is already visible after two mutations.
\section{Ordinary LP mutation and prime elements of the upper algebra}
\label{sec:preliminaries}

Let $K$ be a field, let $\F=K(x_1,\ldots,x_n)$, and let
$\mathbf x=(x_1,\ldots,x_n)$ be a transcendence basis.
A seed consists of $\mathbf x$ and polynomials
$F_i\in K[\mathbf x]$ which are irreducible nonunits, are independent
of $x_i$, and are not divisible by any $x_j$.
For a Laurent polynomial $G$ put
\[
 \operatorname{pol}(G)=G\prod_jx_j^{-m_j},\qquad
 m_j=\min\{\alpha_j:[\mathbf x^\alpha]G\ne0\}.
\]
Thus $\operatorname{pol}(G)$ is a polynomial without coordinate factors.

The exchange Laurent polynomial is
\begin{equation}\label{eq:hat}
 \widehat F_i=F_i\prod_{j\ne i}x_j^{-e_{ij}},
 \qquad e_{ij}\geq0.
\end{equation}
For $j\ne i$, its defining condition is that substituting
$x_j=F_j/\xi$ gives a Laurent polynomial not divisible by $F_j$.
Equivalently, $e_{ij}$ is the $F_j$-adic order of
$F_i|_{x_j=F_j/\xi}$ in the resulting Laurent ring.
These exponents exist and are unique
\cite[Lemmas~2.3--2.4]{LP16}.

Mutation at $k$ replaces $x_k$ by $x_k'=\widehat F_k/x_k$.
The exchange polynomial at $x_k'$ is $F_k$. If $j\ne k$ and $F_j$
does not involve $x_k$, its new exchange is $F_j$, up to a coefficient
unit. Otherwise put
\begin{equation}\label{eq:ordinary-mutation}
 Z_k=\widehat F_k|_{x_j=0},\qquad
 G_j=F_j|_{x_k=Z_k/x_k'}.
\end{equation}
Remove from $G_j$ all irreducible factors which divide $Z_k$, with
their entire multiplicity in $G_j$, in the Laurent ring in the
coordinates other than $x_j$. Apply $\operatorname{pol}$ to the result.
The resulting polynomial, up to a coefficient unit, is the new exchange
at $x_j$. The specialization in \eqref{eq:ordinary-mutation} is defined:
if a negative $x_j$-power occurred in $\widehat F_k$, then $F_j$
would be independent of $x_k$. The resulting operation is reversible
and gives another seed \cite[Section~2.2]{LP16}.

The ordinary LP algebra $\A$ is generated over $K$ by all the variables
in the mutation pattern. Write
\[
 \L(T)=K[\mathbf x_T^{\pm1}],\qquad
 \U=\bigcap_T\L(T).
\]
The Laurent phenomenon gives $\A\subseteq\U$
\cite[Theorem~5.1]{LP16}. Two variables or seeds are identified only up
to coefficient units when speaking about a normalized pattern; in all
displayed identities we choose actual rational-function representatives.

We recall the local input from the proof of the Laurent phenomenon.
For an alternating three-step path in two distinct directions, suppress
unchanged coordinates and write
\[
 T_0=(x,y),\quad T_1=(z,y),\quad T_2=(z,u),\quad T_3=(v,u).
\]
Then
\begin{equation}\label{eq:caterpillar}
 u,v\in\L(T_0),\qquad
 \gcd_{\L(T_0)}(z,u)=\gcd_{\L(T_0)}(z,v)=1.
\end{equation}
These are \cite[Lemmas~5.2--5.4]{LP16}; proportional exchanges are
included. We use this classical local Laurent statement and prove the
upper-algebra consequences explicitly.

\begin{lemma}\label{lem:upper}
For every seed $T$,
\begin{equation}\label{eq:starfish}
 \U=\L(T)\cap\bigcap_i\L(\mu_iT).
\end{equation}
Every cluster variable is a prime nonunit of $\U$, and
$\U^\times=K^\times$.
\end{lemma}

\begin{proof}
Write $L_j=\L(T_j)$ for the alternating path above. The element $z$
is an irreducible polynomial times a Laurent unit in the UFD $L_0$.
For $h\in L_1\cap L_3$, Laurentness in \eqref{eq:caterpillar}
allows expressions
\[
 h=a/z^r=b/(u^sv^t),\qquad a,b\in L_0,\quad r,s,t\geq0.
\]
The two coprimalities force $z^r\mid a$, so $h\in L_0$.
Reversing the path gives
\[
 L_1\cap L_3\subseteq L_0,\qquad L_0\cap L_2\subseteq L_3.
\]
Apply these inclusions to
$\mu_jT,T,\mu_kT,\mu_j\mu_kT$, for each $j\ne k$.
They show that the right-hand side of \eqref{eq:starfish} is invariant
under mutation. It is therefore contained in every Laurent ring in the
pattern, proving \eqref{eq:starfish}. This finite-intersection result
also appears in \cite[Theorem~1.3]{Fei26}.

Fix $T=(\mathbf x,\mathbf F)$ and let $L_i=\L(\mu_iT)$.
The polynomial $F_i$ is prime in $K[x_j^{\pm1}:j\ne i]$ and hence
in $L_i$. The exchange identity makes $x_i$ a Laurent unit times
$F_i$ in $L_i$. We claim
\begin{equation}\label{eq:prime-ideal}
 x_i\U=\U\cap F_iL_i.
\end{equation}
If $h$ belongs to the right side, then $h/x_i\in L_i$.
It also belongs to $\L(T)$ and all $\L(\mu_jT)$ with $j\ne i$,
because $x_i$ is an invertible unchanged coordinate in these rings.
Equation~\eqref{eq:starfish} proves the claim. The right side of
\eqref{eq:prime-ideal} is a proper prime ideal, so $x_i$ is a prime
nonunit.

Let $\nu_i$ be the $F_i$-adic valuation in $L_i$. Then
\begin{equation}\label{eq:coordinate-valuations}
 \nu_i(x_j)=\delta_{ij},\qquad
 \nu_i(h)\geq0\quad(0\ne h\in\U),\qquad
 \nu_i(h)>0\Longleftrightarrow h\in x_i\U.
\end{equation}
A unit of $\U$ is a unit of $\L(T)$, so it has the form
$c\prod_jx_j^{a_j}$. Applying $\nu_i$ to it and its inverse gives
$a_i=0$ for every $i$, proving $\U^\times=K^\times$.
\end{proof}

In particular, products of upper prime elements have unique prime
multiplicities, even though we do not assume that $\U$ is a UFD.
If a prime divides a product of primes, it divides one factor; these
two primes are associates. Repeated cancellation proves the assertion.

\begin{lemma}\label{lem:canonical-prime}
Fix an initial seed with nonconstant affine linear exchanges.
For an irreducible polynomial $P\in K[\mathbf x]$ having no coordinate
factor, write
\begin{equation}\label{eq:canonical-lift}
 Q_P=\frac{P}{\prod_i x_i^{\nu_i(P)}}.
\end{equation}
Then $Q_P$ is a prime nonunit of $\U$. If
$P=\sum_{k\geq0}x_i^kP_k$, its exponent is computed by
\begin{equation}\label{eq:ideal-order}
 \nu_i(P)=\min_k\{k+\ord_{F_i}(P_k)\}
          =\ord_{(x_i,F_i)}P.
\end{equation}
It is the unique upper-prime lift of the Laurent prime $P$, up to a
coefficient unit and association in the initial Laurent ring.
\end{lemma}

\begin{proof}
Substitute $x_i=\widehat F_i/x_i'$ into an initial Laurent expression.
Its only possible noncoordinate denominator is a power of $F_i$;
different powers of $x_i'$ cannot cancel. Consequently
\[
 \U=\{f\in K[\mathbf x^{\pm1}]:\nu_i(f)\geq0\text{ for all }i\},
\]
and \eqref{eq:ideal-order} follows. Since
$\nu_i(x_j)=\delta_{ij}$, all coordinate valuations of $Q_P$ are zero.
Thus $Q_P\in\U$. It is prime after inverting all initial coordinates.
If $Q_P$ divides $ab$ there, it divides, say, $a$, and
$a/Q_P$ has every coordinate valuation nonnegative. Hence
$a/Q_P\in\U$, which proves primality in $\U$. It is a nonunit since
it is a nonunit in the initial Laurent ring. Any associated Laurent
lift differs by a monomial in the $x_i$. A prime not associated to an
$x_i$ must have zero $\nu_i$: otherwise \eqref{eq:prime-ideal} would
make it divisible by $x_i$. This forces precisely the exponents in
\eqref{eq:canonical-lift}.
\end{proof}

\begin{lemma}[Recognition of a mutation partner]\label{lem:prime-recognition}
Let $C=\{x,s_1,\ldots,s_{n-1}\}$ be an actual cluster and let
$y\in\U$ be a prime nonunit. Suppose
\[
 xy=E(\mathbf s),\qquad E\in K[\mathbf s^{\pm1}]\setminus\{0\}.
\]
Then $y$ is associated to the actual mutation partner of $x$.
If both representatives are positive on the initial positive orthant,
the association scalar is positive.
\end{lemma}

\begin{proof}
Let $x^*$ be the actual partner and write $xx^*=H(\mathbf s)$.
Since $y\in\U$, it is Laurent in the actual neighboring cluster and
\[
 y=(E/H)x^*.
\]
Uniqueness of powers of the independent coordinate $x^*$ gives
$E/H\in K[\mathbf s^{\pm1}]$. Clearing a shared monomial denominator
gives $\mathbf s^\alpha y=R(\mathbf s)x^*$, with $\alpha\geq0$.
The prime $x^*$ is not associated to any shared coordinate, so it
divides $y$ in $\U$. The two prime nonunits are associates. Their
values on the positive orthant determine the sign of the scalar.
\end{proof}

We also record why a finite set of variable classes gives a finite
normalized pattern. If two seeds of the same pattern have the same
cluster, their exchange polynomials agree up to units. Indeed, fix a
coordinate $x_i$ and invert all the shared coordinates $s_j=x_j$
with $j\ne i$. Localization commutes with the finite intersection
in \eqref{eq:starfish}; all the other neighbor rings then contain
the initial Laurent ring. Thus
\[
 \U[\mathbf s^{-1}]=\L(T)\cap\L(\mu_iT).
\]
Their common upper algebra consequently determines the ideal
\[
 \{E\in K[x_j^{\pm1}:j\ne i]:E/x_i\in\U[\mathbf s^{-1}]\}
       =F_iK[x_j^{\pm1}:j\ne i].
\]
The equality follows from \eqref{eq:coordinate-valuations} and the
finite intersection. Since $F_i$ has no coordinate factor, its
ordinary polynomial representative is determined up to a coefficient
unit. The hat rule then determines its exchange Laurent polynomial.
Independent rescaling by coefficient units gives equivalent seeds.
\section{Finite patterns: exchange positivity, bases, and the upper algebra}
\label{sec:finite-transfer}

We first prove two results for an arbitrary finite ordinary LP pattern
over a field $K$. Linearity is not assumed in this section.
Choose one rational-function representative of each variable class and
use those representatives in every cluster containing the class.
Rescaling by coefficient units transports the seed structure, so an
actual exchange between these charts has the form
\begin{equation}\label{eq:shared-exchange}
 xx'=E(s_1,\ldots,s_{n-1}),\qquad
 E\in K[\mathbf s^{\pm1}]\setminus\{0\}.
\end{equation}

\subsection{Transport of Newton polytopes}

For $0\ne f\in\U$ and a cluster $C$, denote its Laurent support and
Newton polytope by $\supp_C(f)$ and $P_C(f)$, and put
\[
 h_{C,f}(u)=\max_{\alpha\in\supp_C(f)}\langle\alpha,u\rangle.
\]
The product identity
$P_C(fg)=P_C(f)+P_C(g)$ holds over a field without a sign
hypothesis: a generic weight exposes a unique highest term of each
factor, whose product cannot cancel, and continuity gives equality
of support functions at all weights.

A weight $u\in\R^n$ is called rationally independent if
$\langle m,u\rangle=0$, $m\in\Z^n$, implies $m=0$.
Such weights are dense. For one of them, the largest exponent of a
nonzero Laurent polynomial is unique, and
\[
 \nu_{C,u}(p/q)=h_{C,p}(u)-h_{C,q}(u)
\]
is a well-defined maximum valuation on $\F$.

\begin{lemma}\label{fin:transport}
For any two clusters $C,D$ there is a piecewise integral linear
homeomorphism $\tau_{C,D}:\R^n\to\R^n$ such that
\begin{equation}\label{eq:transport-finite}
 h_{C,f}(u)=h_{D,f}(\tau_{C,D}(u))\qquad(0\ne f\in\U).
\end{equation}
In particular, for $f,g\ne0$ in $\U$,
\begin{equation}\label{eq:containment-finite}
 P_C(f)\subseteq P_C(g)\quad\Longleftrightarrow\quad
 P_D(f)\subseteq P_D(g).
\end{equation}
Neither assertion requires finiteness or positivity.
\end{lemma}

\begin{proof}
For \eqref{eq:shared-exchange}, with $x$ in position $i$, define
\[
 (\tau_i(u))_j=u_j\ (j\ne i),\qquad
 (\tau_i(u))_i=-u_i+h_{C,E}(u).
\]
Since $E$ is independent of $x$, this is a continuous piecewise
integral linear involution. At a rationally independent $u$, let
$\beta$ be the highest exponent of $E$. On its linearity region the
map is an integral matrix of determinant $-1$, so $\tau_i(u)$ is
again rationally independent. It is the vector of the
$\nu_{C,u}$-values of the new coordinates. Distinct Laurent
monomials in those coordinates therefore have distinct values; the
largest one in $f$ cannot cancel. This proves
\eqref{eq:transport-finite} for adjacent charts and generic $u$.
Continuity gives it for every $u$, and composition along a mutation
path proves the general assertion. Its coordinates are the support
functions of the variables of $D$ in $C$, so it is independent of
the path. Finally, containment of polytopes is equivalent to an
inequality of support functions at every weight. Surjectivity of
$\tau_{C,D}$ gives \eqref{eq:containment-finite} in both directions.
\end{proof}

\begin{lemma}\label{fin:avoidance}
If $b,b'$ are cluster monomials and $P_C(b)\subseteq P_C(b')$ in
one cluster, then $b=b'$. Consequently, if $b=\mathbf x_D^a$,
the point $a$ is outside $P_D(b')$ for every other cluster
monomial $b'$.
\end{lemma}

\begin{proof}
Transport containment to a cluster where $b'$ is a monomial. Its
Newton polytope is a point, so $b=cb'$ for $c\in K^\times$.
All factors are prime nonunits of $\U$ by Lemma~\ref{lem:upper}.
Uniqueness of their multiplicities identifies the factors on both
sides. Since representatives are fixed, the two products are equal.
The last assertion follows by applying the first to
$P_D(b)=\{a\}$.
\end{proof}

Fix an initial cluster $C_0$ and a rationally independent weight $w$.
Let $g(f)$ be the exponent of the highest term of $f$ in that cluster.
It is additive on products. Distinct cluster monomials have distinct
$g$-vectors: if $g(b)=g(b')$, transport the valuation to a cluster
where $b'=\mathbf x_D^a$. Rational independence forces $a$ to be
the highest exponent of $b$ there, and Lemma~\ref{fin:avoidance}
then gives $b=b'$.

\subsection{The leading cones and a strictly convex support function}

For a cluster $C$ put
\[
 G_C=(g(v))_{v\in C},\qquad
 \sigma_C=\cone\{g(v):v\in C\}.
\]
These matrices are unimodular. Initially $G_{C_0}=I$; at an exchange
\eqref{eq:shared-exchange} the unique leading exponent $\beta_0$
of $E$ gives
\begin{equation}\label{eq:wall-leading}
 g(x)+g(x')=\sum_j(\beta_0)_j g(s_j).
\end{equation}
Replacing one column in this way changes the determinant by $-1$.

\begin{lemma}\label{lem:leading-intersections}
For any two actual clusters, even in an infinite pattern,
\begin{equation}\label{eq:cone-intersection}
 \sigma_C\cap\sigma_D=\cone\{g(v):v\in C\cap D\}.
\end{equation}
For a finite pattern these cones and their faces form a finite
complete simplicial unimodular fan. Its facets correspond to actual
mutations.
\end{lemma}

\begin{proof}
A rational point of the intersection has nonnegative rational
coordinates in both integral bases $G_C,G_D$. Clearing denominators
gives two cluster monomials with the same leading vector. They are
equal, and prime-factor uniqueness leaves only shared variables.
This proves \eqref{eq:cone-intersection} for rational points.
Both cones are rational polyhedral, so rational points are dense
in their intersection, proving it in general.

Thus the cones form a fan. Across the facet omitting $g(x)$,
mutation gives a second cone on the opposite side, by
\eqref{eq:wall-leading}. There cannot be two distinct cones on
the same side of that facet, since their interiors would overlap
near a relative interior point. Every facet has exactly two
incident cones.

For a finite collection their union is closed. If it had an
exterior point, take a segment from an interior point to the
complement, avoiding the finitely many faces of codimension at
least two and crossing facet hyperplanes transversely. Such a
choice is possible because the excluded endpoints lie in finitely
many proper affine subspaces. At the first exit the segment meets
the relative interior of a facet. The adjacent cone on the other
side contains its continuation, a contradiction. For rank one the
two half-lines give the same conclusion.
\end{proof}

For a variable $v$ define
\[
 H(v)=\sum_{T}\sum_{j=1}^n
       \bigl(h_{T,v}(e_j)+h_{T,v}(-e_j)\bigr).
\]
This is the sum of all coordinate widths of its Newton polytopes.
The pattern is finite, so $H(v)$ is a nonnegative integer. Define
\begin{equation}\label{eq:height-finite}
 M=\max\left(0,\max_{C,i,\beta\in\supp E_{C,i}}
                         \left(\sum_j\beta_j-2\right)\right),
 \qquad \omega(v)=(M+1)H(v)+1.
\end{equation}

\begin{lemma}\label{fin:polytope}
There is a bounded full-dimensional polytope $P$ whose outer
normal cones at the vertices are precisely the $\sigma_C$.
If $p_C$ is the corresponding vertex, adjacent clusters satisfy
\begin{equation}\label{eq:oriented-edge}
 \langle g(s_j),p_{C'}-p_C\rangle=0,\qquad
 \langle g(x),p_{C'}-p_C\rangle<0
\end{equation}
for \eqref{eq:shared-exchange}.
\end{lemma}

\begin{proof}
For a function $a$ on variables write
$\Delta a=a(x)+a(x')-\sum_j(\beta_0)_ja(s_j)$.
Newton transport applied to the actual identity $xx'=E(\mathbf s)$
gives, for every $T$ and every weight $u$,
\[
 h_{T,x}(u)+h_{T,x'}(u)
  =\max_{\beta\in\supp E}\sum_j\beta_jh_{T,s_j}(u).
\]
Hence the wall slack of each support function is nonnegative,
and $\Delta H\geq0$. In the exchange chart itself all old
coordinate widths vanish, while the widths of $x'=E/x$ are those
of $E$. Over a field an ordinary exchange is a nonconstant
polynomial without coordinate factors, and therefore has at least
two monomials. Consequently this chart contributes a positive
integer, so $\Delta H\geq1$. Equation~\eqref{eq:height-finite}
now gives $\Delta\omega\geq1$.

Let $p_C$ be the solution of
$\langle g(v),p_C\rangle=\omega(v)$ for $v\in C$.
Define $\psi$ to be this linear function on $\sigma_C$.
It is continuous on the fan because adjacent values agree on
shared rays. The strict wall inequalities just proved say that
the linear slopes of $\psi$ increase across every wall. Restricting
to a generic line gives a convex piecewise linear function;
approximating nongeneric lines gives convexity on every line.
Thus $\psi$ is globally convex. Each of its linear pieces is a
supporting linear function, whence
\[
 \psi(z)=\max_C\langle p_C,z\rangle.
\]
It is the support function of $P=\conv\{p_C\}$.
Strictness at every wall implies that its maximal linearity
cones are exactly the $\sigma_C$, rather than unions of them.
For instance, a segment from the interior of a cone to an exterior
point crosses a first wall with strictly increasing slope, which
excludes equality with the original linear piece beyond that wall.
Since each ray height is positive, $\psi(z)>0$ for $z\ne0$.
Its positive minimum on the unit sphere shows that $P$ contains
a ball about zero, so it is full-dimensional. It is bounded as
the convex hull of finitely many points. The asserted normal
cones and \eqref{eq:oriented-edge} follow: the new vertex remains
on the shared facets and lies strictly below the old exchanged
facet.
\end{proof}

\subsection{A basis without a positivity assumption}

\begin{theorem}\label{thm:finite-basis}
For a finite ordinary LP pattern over $K$, its cluster monomials
form a $K$-basis of $\U$, and $\A=\U$. If
$f=\sum_{b\in\mathscr B}c_b b$ and $b=\mathbf x_C^a$, then
\begin{equation}\label{eq:basis-extraction}
 c_b=[\mathbf x_C^a]f.
\end{equation}
\end{theorem}

\begin{proof}
Lemma~\ref{fin:avoidance} shows that the coefficient on the right
of \eqref{eq:basis-extraction} receives no contribution from a
different cluster monomial. This proves linear independence
and the extraction formula whenever an expansion exists.

For spanning, take $0\ne f\in\U$ and let $\gamma=g(f)$.
Completeness gives a cluster $C$ with $\gamma\in\sigma_C$.
Unimodularity gives $a=G_C^{-1}\gamma\in\Z_{\geq0}^n$.
Set $b=\mathbf x_C^a$. Rational independence of the transported
weight shows that $a$ is the highest exponent of $f$ in $C$.
Let $c$ be its coefficient there. Then $\{a\}\subseteq P_C(f)$;
Newton transport implies $P_D(b)\subseteq P_D(f)$ in every
cluster $D$. Subtracting $cb$ cancels the leading term, lowers
its initial $w$-weight, and creates no exponent outside
$P_{C_0}(f)$. Repetition terminates because this fixed polytope
contains only finitely many lattice points and the leading
weights strictly decrease. It expresses $f$ as a finite linear
combination of cluster monomials. These monomials belong to $\A$,
so $\A=\U$.
\end{proof}

\subsection{Exchange positivity implies Laurent positivity}

\begin{theorem}\label{thm:finite-transfer}
Let a finite ordinary LP pattern over $K$ have fixed representatives,
and let $S\subseteq K$ be a subsemiring containing $0,1$.
If every actual exchange product \eqref{eq:shared-exchange} has
coefficients in $S$, then all Laurent coefficients of every cluster
monomial belong to $S$ in every cluster. Moreover,
\begin{equation}\label{eq:atomic-semiring}
 \bigcap_C S[\mathbf x_C^{\pm1}]
       =\bigoplus_{b\in\mathscr B}Sb,
\end{equation}
and every multiplication constant of this basis belongs to $S$.
\end{theorem}

\begin{proof}
Fix a cluster monomial $b$. Suppose one of its Laurent coefficients
in some cluster lies outside $S$. Among all such terms in the finitely
many cluster expansions choose one maximizing
$\langle w,\gamma\rangle$, where a term $\mathbf x_C^\alpha$
has common vector $\gamma=G_C\alpha$.

Write its expansion along an exchange as
\[
 b=\sum_k a_k(\mathbf s)x^k
   =\sum_\ell b_\ell(\mathbf s)(x')^\ell.
\]
If the chosen exponent of $x$ is $-r<0$, uniqueness of powers of
$x'$ gives the polynomial identity
\begin{equation}\label{eq:bad-transfer}
 a_{-r}=b_rE^r.
\end{equation}
All coefficients of $E^r$ lie in $S$. A coefficient of $a_{-r}$
outside $S$ therefore requires a coefficient of $b_r$ outside
$S$ at an exponent $\eta-\delta$, where $\eta$ is its old
shared exponent and $\delta\in\supp(E^r)$.
If $\beta_0$ is the leading exponent of $E$, the new common vector is
\[
 \gamma'=\gamma+G_{\mathrm{shared}}(r\beta_0-\delta).
\]
Its $w$-weight is at least that of $\gamma$. Maximality of the
chosen bad coefficient forces equality. Rational independence
then forces $\gamma'=\gamma$. Thus whenever a negative coordinate
is present, a bad coefficient with exactly the same common vector
occurs in the adjacent cluster.

Use the polytope in Lemma~\ref{fin:polytope}. For an exchange at a
negative coordinate $\alpha_i$, equation~\eqref{eq:oriented-edge}
gives
\[
 \langle\gamma,p_{C'}-p_C\rangle
    =\alpha_i\langle g(x),p_{C'}-p_C\rangle>0.
\]
The common vector remains fixed at every transfer. Hence this
walk cannot cycle on the finite vertex set of $P$. It ends in a
cluster where its exponent vector is nonnegative. That exponent
is a cluster monomial. By Lemma~\ref{fin:avoidance}, its coefficient
in $b$ is zero unless it is $b$ itself, in which case the coefficient
is one. Both belong to $S$, a contradiction. This proves the
Laurent assertion. The argument never divides an element of $S$.

By Theorem~\ref{thm:finite-basis}, every $f\in\U$ has a unique
cluster-monomial expansion, and each basis coefficient is extracted
by \eqref{eq:basis-extraction}. If $f$ is universally $S$-positive,
these coefficients belong to $S$. Conversely every $S$-linear
combination of the basis is universally $S$-positive by the first
part. This proves \eqref{eq:atomic-semiring}. Products of two basis
monomials are universally $S$-positive, so their extracted
structure constants belong to $S$ as well.
\end{proof}
\section{Forest determinants, modules, and canonical coordinates}
\label{sec:forest-factors}

From now on $K\subseteq\R$. Dividing $F_i$ in
\eqref{eq:initial-general} by its positive constant term is an allowed
change of coefficient units. We may therefore write
\begin{equation}\label{eq:normalized-linear}
 F_i=1+\sum_{j\ne i}C_{ij}x_j.
\end{equation}
Put $V=[n]$, set $C_{ii}=0$, and define the target-weighted row Laplacian
\[
 (L_C)_{ii}=\sum_{j\ne i}C_{ij}x_j,\qquad
 (L_C)_{ij}=-C_{ij}x_j\quad(i\ne j).
\]
For $J\subseteq V$ set
\begin{equation}\label{eq:forest-definitions}
 \begin{gathered}
 D_J(t,x)=\det((tI+L_C(x))[J,J]),\\
 Z=\diag(F_i/x_i)-C,\qquad Y_J=\det Z[J,J],\\
 D_J(1,x)=Y_J\prod_{i\in J}x_i,\qquad
 H_C(t,x)=\det(tI+L_C(x))/t.
 \end{gathered}
\end{equation}
Empty determinants are one. The last quotient is polynomial because
$L_C\ones=0$. We abbreviate $Z[J,J]$ to $Z_J$; thus $Y_J=\det Z_J$.

We use the elementary directed-forest expansion. In the determinant
of a row Laplacian plus diagonal roots, choose for each nonroot vertex
one outgoing edge; the contributions with a directed cycle cancel
against the corresponding permutation cycle. The remaining choices
are rooted directed forests. Each contributes the product of its
edge weights and its root weights, with sign $+1$. This argument
also proves the principal-minor version: vertices outside $J$ act
as roots, and a chosen edge from $J$ to such a vertex retains its
weight in the diagonal. Thus $D_J$ and $H_C$ have nonnegative
coefficients when $C_{ij}\geq0$. The coefficient of the highest
power of $t$ is one. In particular, every dehomogenized polynomial
used below has constant term one.

\subsection{An irreducibility criterion}

\begin{definition}
A nonempty set $M\subseteq V$ is an \emph{outgoing module} if
\begin{equation}\label{eq:module-definition}
 C_{ik}=C_{jk}\qquad(i,j\in M, k\notin M).
\end{equation}
Singletons and $V$ itself are modules. A nontrivial module has at
least two vertices; a minimal nontrivial module is minimal among
all such sets, including $V$ when it has no smaller one.
\end{definition}

We give an irreducibility argument that also applies to signed and
zero coefficients. Its only dimension-theoretic input is the generic
hyperplane formula for a finite family of nonzero real subspaces
$V_i\subseteq E$:
\begin{equation}\label{eq:generic-hyperplane}
 \dim\sum_i(V_i\cap K_0)
   =\min_{\mathcal P}\sum_{B\in\mathcal P}
                 \left(\dim\sum_{i\in B}V_i-1\right)
\end{equation}
for a generic hyperplane $K_0\subseteq E$, where $\mathcal P$
ranges over partitions of the index set. This is the classical
generic Dilworth truncation formula; its generic
characteristic-zero statement is given in
\cite[Theorems~5.2 and 7.1]{RazWigderson19}.
We apply it only on a nonempty Zariski-open set of real hyperplanes.

\begin{lemma}\label{for:interpolation}
Let $U_i\subseteq W^*$, $i\in S$, be two-dimensional real subspaces,
where $r=|S|$. Suppose
\[
 \dim\sum_{i\in T}U_i\geq |T|+2
       \quad(T\subseteq S, |T|\geq2).
\]
For $r=1$ suppose $\dim W\geq3$.
A nonzero homogeneous complex polynomial vanishing on all the
projective spaces $\mathbb P(U_i^\perp)$ has degree at least $r$.
\end{lemma}

\begin{proof}
Choose a generic real point $p\in\mathbb P(W)$ outside these
spaces and put $K_0=p^\perp$. The lines
$\ell_i=U_i\cap K_0$ span a space $R_p$ of dimension $r$:
in \eqref{eq:generic-hyperplane}, singleton blocks have total
cost $r$, whereas every larger block has cost at least its
cardinality plus one. Thus $T_p=R_p^\perp$ contains $p$ and
has dimension at least two. Every projective line through $p$
in $\mathbb P(T_p)$ meets each $\mathbb P(U_i^\perp)$, since
the two forms in $U_i$ restrict dependently and are not both zero
at $p$.

These intersections can be chosen distinct. For $i\ne j$ put
$Z_{ij}=(U_i+U_j)^\perp$ and $W_{ij}=\langle p,Z_{ij}\rangle$.
A line through $p$ meets both boundary spaces at the same point
precisely when it lies in $\mathbb P(W_{ij})$; if $Z_{ij}=0$
there are no such lines. Apply \eqref{eq:generic-hyperplane} to
the family with $U_i,U_j$ replaced by $U_i+U_j$.
A partition block containing this merged space and $k$ other
members has cost at least $k+3$, two more than its number of
members; all other blocks have cost at least their number of
members. Therefore
\[
 \dim\left((U_i+U_j)\cap K_0+
                       \sum_{k\ne i,j}\ell_k\right)\geq r+1.
\]
It follows that $T_p\not\subseteq W_{ij}$. Choose a second
point in $\mathbb P(T_p)$ outside all these finitely many proper
subspaces. The resulting line has $r$ distinct prescribed zeros
of the polynomial. A polynomial of degree less than $r$ must
vanish identically on that line, hence at $p$. Since such $p$
form a nonempty real Zariski-open set, the polynomial is zero,
even with complex coefficients. This is a contradiction.
\end{proof}

\begin{proposition}\label{for:irreducibility}
For a real off-diagonal matrix $C$ on $n\geq2$ vertices,
$H_C(t,x)$ is absolutely irreducible if and only if there is no
proper nontrivial outgoing module. Signs and zeros are allowed.
The same statement holds when each target vertex carries several
independent variables and each incoming weight is a real linear
form in that vertex's variables.
\end{proposition}

\begin{proof}
For the ordinary target variables put
\[
 \phi_i=t+\sum_{j\ne i}C_{ij}x_j,\qquad
 U_i=\operatorname{span}\{x_i,\phi_i\}\subseteq(\R^{n+1})^*.
\]
For $T\subseteq V$, quotienting by the independent forms $x_i$,
$i\in T$, gives
\begin{equation}\label{eq:boundary-rank}
 \dim\sum_{i\in T}U_i
 =|T|+\operatorname{rank}
                  [(1,(C_{ij})_{j\notin T})]_{i\in T}.
\end{equation}
For $2\leq|T|<n$, the last rank is one exactly when $T$ is a
module. If there is no proper nontrivial module,
Lemma~\ref{for:interpolation} applies to every proper set of
indices. A polynomial vanishing on the corresponding $|T|$
boundary spaces has degree at least $|T|$.

Expanding the determinant in column $i$ at $x_i=0$ shows
\begin{equation}\label{eq:boundary-H}
 H_C|_{x_i=0}=\phi_i H_{C[V\setminus\{i\}]}.
\end{equation}
Thus $H_C$, homogeneous of degree $n-1$, vanishes on every
$\mathbb P(U_i^\perp)$. Suppose it were a product of at least
two nonconstant homogeneous complex factors. Each boundary space,
being irreducible, lies in the zero set of one factor; assign it
to such a factor. A factor assigned all $n$ spaces has degree at
least $n-1$, leaving no room for another factor. Otherwise every
nonempty assigned set is proper, and the sum of the factor degrees
is at least the sum of their assigned cardinalities, namely $n$.
Both alternatives are impossible. For $n=2$, $H_C$ is monic linear.

Conversely, for a proper module $M$, vectors constant on $M$ form
an invariant subspace for all $L_C(x)$. Modulo the constant vector
this subspace has dimension $n-|M|$, strictly between zero and
$n-1$. An adapted basis makes the quotient pencil block triangular,
so its determinant $H_C$ factors into monic homogeneous polynomials
of positive degrees $n-|M|$ and $|M|-1$.

For the multiple-target statement, let the variables at vertex
$B$ be $x_j$, $j\in B$, and the edge from $A$ to $B$ be
$\sum_{j\in B}\beta_{A,j}x_j$. A module requires equality of
every coefficient $\beta_{A,j}$ toward outside vertices.
If there is no module, specialize $x_j=\xi_jz_B$ with positive
real $\xi_j$ outside finitely many hyperplanes. Every failed
module equality still fails, so the ordinary theorem gives an
absolutely irreducible specialized polynomial. Any factorization
before specialization would have homogeneous factors monic in
$t$, each of positive $t$-degree; no such factor can become a
unit under this specialization. This is impossible. The invariant
subspace argument proves the converse without specialization.
\end{proof}

An affine change $t\mapsto t+\ell(x)$ preserves absolute
irreducibility. Dehomogenizing an irreducible homogeneous polynomial
monic in $t$ preserves irreducibility unless that polynomial is $t$.
Indeed, an affine factorization homogenizes to a homogeneous
factorization after multiplication by a power of $t$.

\subsection{The forest of minimal blocks}

Distinct minimal nontrivial modules intersect in at most one vertex:
the intersection and the union of two intersecting modules are
modules. To see the intersection assertion, for a target outside
the intersection use whichever of the two modules excludes the
target. For the union assertion compare each row with a row in
the intersection at every outside target.

\begin{proposition}\label{prop:block-forest}
Fuse the pair modules into their maximal pair cliques, and retain
every minimal nontrivial module of cardinality at least three.
The incidence graph of these blocks and their original vertices is
a forest. Every subset of a pair clique is itself an outgoing module.
\end{proposition}

\begin{proof}
Suppose a minimal module $M$ with $|M|\geq3$ lay on an incidence
cycle. Choose a shortest path outside its block node joining two
vertices $p,q\in M$. No other vertex of $M$ belongs to a block
on this path, since that would shorten it. For each target
$k\in M\setminus\{p,q\}$, propagate equalities of rows along
the path. For each target outside $M$, use the module $M$.
Thus $p,q$ agree toward every target outside $\{p,q\}$,
so $\{p,q\}$ is a proper nontrivial submodule of $M$, a
contradiction.

Now form the graph whose edges are pair modules. In a biconnected
block, any two vertices admit a path avoiding a prescribed third
target vertex. Propagating the edge-module equalities along that
path shows that the two vertices form a pair module. Hence every
biconnected block is a clique. Replacing these cliques by block
nodes removes the pair cycles. Any remaining cycle would either
expand to one of the excluded cycles through a larger minimal
module or to a cycle in the block-incidence graph of the pair
graph. Both are impossible. The last statement follows directly
from the equalities for all pairs of a clique.
\end{proof}

Call the blocks of cardinality at least three which are minimal
modules \emph{proper-type blocks}; this describes their core type
and does not exclude a block equal to $V$. Maximal pair cliques,
including pairs not in a larger clique, are \emph{clique-type blocks}.
For a block $B$ write
\[
 a_B=1+\sum_{j\notin B}C_{Bj}x_j,
\]
where $C_{Bj}$ is the common outside row. Define
\begin{equation}\label{eq:block-cores}
 \begin{aligned}
 P_B&=H_{C[B]}(a_B,x_B),&& B\text{ of proper type},\\
 P_B&=a_B+\sum_{i\in B}d_i x_i,&& B\text{ of clique type},\\
 q_B&=P_B/\prod_{i\in B}x_i.
 \end{aligned}
\end{equation}
Here $C_{ji}=d_i$ for $j\in B\setminus\{i\}$ in a clique.
In the first line $\deg P_B=|B|-1$, and in the second it is one.
The first numerator is absolutely irreducible by minimality and
Proposition~\ref{for:irreducibility}. The second is affine linear.
All have constant term one and nonnegative coefficients.

Put $\delta_B=|B|-1$ for proper-type blocks and $\delta_B=1$
for clique-type blocks. A set $J$ is \emph{admissible} if
\begin{equation}\label{eq:admissible}
 |J\cap B|\leq\delta_B\quad\text{for every block }B.
\end{equation}
Equivalently, it contains no nontrivial outgoing module. In particular,
$V$ is never admissible for $n\geq2$. For $h\notin J$ define
\begin{equation}\label{eq:principal-gauge}
 \begin{gathered}
 d_h(J)=\#\{B\ni h:|J\cap B|=\delta_B\},\qquad
 G(J)=\prod_{h\notin J}x_h^{\max(d_h(J)-1,0)},\\
 p_J=Y_J/G(J),\qquad p_\varnothing=1.
 \end{gathered}
\end{equation}
The next boundary calculation proves that these are the required
canonical coordinates when all weights are strictly positive.

\subsection{Exact boundary valuations}

The following elementary fact prevents cancellations in the signed
matrices which arise on an exchange hyperplane.

\begin{lemma}\label{for:signed-zero}
Let $A$ have arbitrary real off-diagonal entries and give the edge
$i\to j$ weight $A_{ij}x_j$. Over $\R(x)$, zero is a semisimple
eigenvalue of $L_A$ whose multiplicity is the number of terminal
strongly connected components of its support. The assertion remains
true with several independent target variables at each vertex.
\end{lemma}

\begin{proof}
First consider a strongly connected support. The coefficient of $t$
in $\det(tI+L_A)$ is the sum of all rooted spanning tree weights.
This sum is nonzero even when the entries of $A$ have different signs.
Choose a vertex $r$ reachable from every vertex, and order the targets
by increasing shortest distance to $r$, breaking ties. Each vertex
other than $r$ chooses its earliest outgoing neighbor. Its distance
strictly decreases, so these choices give a tree toward $r$.

This tree has a unique lexicographically maximal target-indegree
vector among trees with every possible root. Indeed, process the
targets in the chosen order. Before processing a target $h$, the
previously forced edges form one component containing $r$, and
all other vertices are isolated. The vertex $h$ already belongs
to the component of $r$, except at the first step. Every isolated
vertex with an edge to $h$ can attach to $h$ simultaneously.
The only vertex of the existing component without a chosen outgoing
edge is $r$, and $r\to h$ would close a cycle. Consequently the
greatest possible indegree at $h$ forces precisely those attachments.
Induction proves uniqueness of the exponent vector and of the tree
with that vector. Its nonzero coefficient cannot cancel. Thus a
terminal strongly connected block has algebraically simple zero;
the one-vertex case gives the same conclusion directly.

A nonterminal strongly connected diagonal block is invertible.
Choose an outside target $k$ receiving an edge from it. The variable
$x_k$ occurs only on its diagonal. Its highest power in the determinant
has coefficient equal to the product of the nonzero coefficients
$A_{ik}$ times the principal determinant obtained by removing all
such rows and columns. The removed vertices are now outside targets.
If any rows remain, strong connectivity supplies an edge from a
remaining vertex to a removed vertex. Repeat the same highest-power
extraction. It eventually produces a nonzero product of edge
coefficients, proving invertibility.

Order the strongly connected components so that the whole matrix
is block triangular. Its algebraic zero multiplicity is the number
$q$ of terminal components. A kernel vector may have an independent
constant value on each terminal component; the invertible
nonterminal blocks then determine all other values uniquely.
Its nullity is therefore $q$, proving semisimplicity.
For multiple target variables, specialize each target group to a
generic positive ray. Avoiding finitely many nonzero linear forms
preserves the support. The just-proved nonzero coefficients and
rank minors after specialization are nonzero before specialization.
The zero row sums and the block structure give the reverse rank
bounds, so the conclusion is unchanged.
\end{proof}

Write $F_i^h=t+\sum_{j\ne i}C_{ij}x_j$. For an arbitrary nonempty
$J\subseteq V$ and an arbitrary original index $i$, form a signed
system on $J\cup\{*\}$. The target group at $*$ consists of all
$x_j$ with $j\notin J$. Put $r_j=C_{ij}$, with $r_i=0$,
and set
\begin{equation}\label{for:shifted-coefficients}
 A_{vj}=C_{vj}-r_j\quad(v\in J,\ j\text{ outside the target block of }v),
 \qquad A_{*j}=-r_j\quad(j\in J).
\end{equation}
Coefficients into a target group are summed against that group's
independent variables. An empty target group is allowed.

\begin{lemma}\label{for:boundary-order}
With $s=F_i^h$, the preceding signed system satisfies
\begin{equation}\label{for:shifted-determinant}
 D_J(t,x)=\frac{\det(sI_{|J|+1}+L_A(x))}{s}.
\end{equation}
Let $q_i(J)$ be the number of terminal strongly connected components
of this system after setting $x_i=0$. Then
\begin{equation}\label{for:principal-valuation}
 \nu_i(D_J(1,x))=q_i(J)-1
   =\ord_{F_i^h}\bigl(D_J(t,x)|_{x_i=0}\bigr).
\end{equation}
For every irreducible homogeneous factor $P$ of $D_J(t,x)$,
\begin{equation}\label{for:factor-boundary}
 \nu_i(P(1,x))=\ord_{F_i^h}\bigl(P|_{x_i=0}\bigr).
\end{equation}
Factors which become units after dehomogenization contribute zero.
Here each initial $F_i$ is assumed nonconstant.
\end{lemma}

\begin{proof}
Start with the system on $J\cup\{*\}$ in which $*$ has zero row
and carries the outside target variables. Its full determinant
divided by $t$ is $D_J$. Adjoin a source $p$, with no incoming
edges and with row weights $r_jx_j$. Its full determinant divided
by $t$ is $sD_J$. Alternatively pass to the quotient by the
constant vector, using representatives with $p$-coordinate zero.
Subtract the $p$-output from every other output. The off-diagonal
entries become $-(C_{vj}-r_j)x_j$, while the diagonal entries
increase by $r_vx_v$. The reduced pencil is exactly $sI+L_A$.
This proves \eqref{for:shifted-determinant}.

Treat $s,x_i$ as independent normal variables, over the rational
function field in the remaining $x_j$. Write $L_A=B_0+x_iB_1$.
By Lemma~\ref{for:signed-zero}, $B_0$ has semisimple zero of
nullity $q=q_i(J)$. Constant invertible row and column operations
put $B_0$ into an invertible block and a zero block of size $q$.
Every term of $\det(sI+B_0+x_iB_1)$ has normal degree at least
$q$. On $x_i=0$ this determinant is $s^qg(s)$ with $g(0)\ne0$.
After division by $s$, the order is exactly $q-1$, and equals
the boundary multiplicity in \eqref{for:principal-valuation}.
The affine chart $t\ne0$ meets the generic boundary
$x_i=F_i^h=0$ because row $i$ is nonzero. Localization at $t$
therefore preserves this order, and Lemma~\ref{lem:canonical-prime}
identifies it with $\nu_i$.

For each factor $P$, the order along $(x_i,F_i^h)$ is at most
the $F_i^h$-order after setting $x_i=0$. The restriction is
nonzero, since $D_J$ is monic in $t$ and has no coordinate
factor. Both orders are additive over factors. Their sums
are equal for $D_J$, as just proved, so equality holds for
each factor separately. This gives \eqref{for:factor-boundary}.
\end{proof}

For a full quotient determinant there is also the smaller identity
\begin{equation}\label{for:full-boundary}
 H_C(t,x)=
 \det\left(F_i^h I_{n-1}+L_{A'}(x_{V\setminus\{i\}})
                         +x_i\diag(C_{ji})_{j\ne i}\right),
 \qquad A'_{jk}=C_{jk}-C_{ik}.
\end{equation}
It follows by passing to the constant-vector quotient using
the $i$-coordinate. The same proof shows that its order along
$(x_i,F_i^h)$ is the number of terminal components of $A'$.

\begin{proposition}\label{prop:canonical-coordinates}
Suppose $C_{ij}>0$ for $i\ne j$. For every nonempty admissible $J$,
$D_J(1,x)$ is absolutely irreducible and
\begin{equation}\label{for:canonical-valuations}
 \nu_i(D_J(1,x))=
 \begin{cases}
  1,&i\in J,\\
  \max(d_i(J)-1,0),&i\notin J.
 \end{cases}
\end{equation}
For every block $B$ the numerator $P_B$ is absolutely irreducible,
and its valuations are one at the members of $B$ and zero elsewhere.
Consequently the functions $p_J$ and $q_B$ of
\eqref{eq:block-cores} and \eqref{eq:principal-gauge} are prime
elements of the upper algebra. They have nonnegative Laurent
coefficients in the initial cluster and are strictly positive at
every positive initial point.
\end{proposition}

\begin{proof}
Adjoin a zero-row vertex carrying all outside target variables
to the system induced on $J$. Its full quotient determinant is
$D_J(t,x)$. A proper nontrivial module avoiding the new vertex
would be a global module contained in $J$, excluded by
admissibility. A module containing the new vertex would contain
a nonempty proper subset $T$ of $J$ and force
$C_{vk}=0$ for $v\in T$, $k\in J\setminus T$.
Strict positivity excludes this. Proposition~\ref{for:irreducibility}
and dehomogenization prove the asserted irreducibility.

For a fixed $i$, consider the difference graph on $V\setminus\{i\}$
with edge $v\to j$ exactly when $C_{vj}\ne C_{ij}$. Let $m_i(J)$
count its terminal components contained in $J\setminus\{i\}$.
In the shifted system \eqref{for:shifted-coefficients}, the vertex
$*$ reaches every retained original vertex because $-C_{ij}\ne0$.
A terminal component avoiding $*$ is exactly one of the
$m_i(J)$ components just counted: an edge to an outside target
appears as an edge to $*$, without cancellation among independent
target variables. If $m_i(J)>0$, these are all terminal
components; if $m_i(J)=0$, the unique terminal component contains
$*$. When $i\in J$, the vertex $i$ is additionally isolated
after $x_i=0$. Hence Lemma~\ref{for:boundary-order} gives
\[
 \nu_i(D_J)=
 \begin{cases}
  \max(1,m_i(J)),&i\in J,\\
  \max(0,m_i(J)-1),&i\notin J.
 \end{cases}
\]

A terminal component $T$ of the difference graph gives the module
$T\cup\{i\}$. It is minimal among nontrivial modules containing
$i$, since a smaller such module would give a proper nonempty
outgoing-closed subset of the strongly connected set $T$.
If $i\in J$, admissibility thus forces $m_i(J)=0$.
If $i\notin J$ and $T\subseteq J$, any smaller module avoiding
$i$ would lie in $J$, also impossible. Thus $T\cup\{i\}$ is
globally minimal. Proper-type blocks contribute exactly when
their other $|B|-1$ vertices belong to $J$; a fused pair clique
contributes exactly when its unique permissible member in $J$
is present. Conversely each such ready block gives a terminal
component. Therefore $m_i(J)=d_i(J)$ for $i\notin J$.
This proves \eqref{for:canonical-valuations}.

For a proper-type block, irreducibility follows from its minimality
and Proposition~\ref{for:irreducibility}, as already observed.
The row-$i$ difference graph on $B\setminus\{i\}$ is strongly
connected: a proper terminal component would give a smaller
nontrivial module of $B$, hence a global submodule.
Formula \eqref{for:full-boundary} gives valuation one at $i\in B$.
For an outside $i$, setting $x_i=0$ merely removes a term from
the affine root $a_B$. The resulting homogeneous polynomial
remains absolutely irreducible of degree $|B|-1\geq2$,
by an invertible translation of $t$. It cannot have the linear
factor $F_i^h$. Since $P_B$ divides $D_B$, equation
\eqref{for:factor-boundary} gives outside valuation zero.

For a clique, $P_B|_{x_i=0}=F_i$ at every member $i$, giving
valuation one. At an outside vertex $h$, a positive valuation
would require equality $P_B|_{x_h=0}=F_h$, since both affine
polynomials have constant term one. Together with the clique
equalities, this would make $\{h,i\}$ a pair module for every
$i\in B$, contradicting maximality. Thus the outside valuations
vanish. Lemma~\ref{lem:canonical-prime} now proves primality of
the displayed canonical lifts. Their initial positivity follows
from the forest expansion.
\end{proof}
\section{Finite candidate charts for a positive coefficient matrix}
\label{evt:section}

In this section all off-diagonal entries of $C$ are strictly positive.
Let $\mathcal B$ be the block family of Proposition~\ref{prop:block-forest}.
Thus a block is either a minimal outgoing module of cardinality at least
three, called a proper block, or a maximal clique of outgoing pairs.
A two-element block is of the second kind.  A proper block is allowed
to be $V$.  Set
\[
 \delta_B=\begin{cases}|B|-1,&B\text{ proper},\\1,&B\text{ a clique}.
 \end{cases}
\]
We use the admissible sets, coordinates $p_J,q_B$, and gauges $G(J)$
of Proposition~\ref{prop:canonical-coordinates}.  A vertex $h\notin J$
is \emph{ready at $J$} if $d_h(J)>0$.  In particular readiness records
all blocks, whether or not their cores occur in the chart under
consideration.

\begin{definition}\label{evt:definition}
A candidate consists of a set $O\subseteq V$, a set
$\mathcal E\subseteq\mathcal B$, and a strictly increasing chain of
nonempty admissible sets
\[
 I_1\subsetneq\cdots\subsetneq I_s.
\]
Its proposed coordinates are
\begin{equation}\label{evt:coordinates}
 \{x_i:i\in O\}\ \cup\ \{q_B:B\in\mathcal E\}\
 \cup\ \{p_{I_1},\ldots,p_{I_s}\}.
\end{equation}
The data must admit the following sequence of events.  Initially put
$R=O$ and $J=\varnothing$.  At event $j$ put $A=I_j\setminus J$
and choose a previously unused subset $\mathcal E_j\subseteq\mathcal E$.
Require:
\begin{enumerate}
\item $A\cap R=\varnothing$;
\item every $B\in\mathcal E_j$ has exactly two members outside $R$,
say $a,h$, with $a\in A$, $h\notin A$; if $B$ is proper then
$B\setminus\{a,h\}\subseteq J$, while if $B$ is a clique then
$B\setminus\{a,h\}\subseteq R\setminus J$;
\item the pairs $\{a,h\}$, indexed by $B\in\mathcal E_j$, form
a tree $T$ whose one complete bipartition class is $A$;
when $\mathcal E_j=\varnothing$, the tree is a singleton $A$;
\item every vertex of $T$ is unready at $J$.
\end{enumerate}
Replace $R$ by $R\cup V(T)$ and $J$ by $I_j$.
At the end require $R=V$ and
$\mathcal E=\coprod_j\mathcal E_j$.
The vertices in the other bipartition class of an event are called its
omitted vertices.  An omitted vertex of an earlier event is called an
old omitted vertex.  The members of $O$ are retained originals.
\end{definition}

This definition specifies finite combinatorial data.  It does not assume
that the proposed coordinates already form an LP cluster.  That fact
will follow from the identities in Section~\ref{edge:section}.

\begin{lemma}\label{evt:elementary}
Every candidate has exactly $n$ coordinate labels.  At every intermediate
stage every unrecovered vertex is unready.  Every old omitted vertex
is ready.  There are finitely many candidates, and every coordinate
$p_J$ with $J$ admissible, and every $q_B$, occurs in a candidate.
\end{lemma}

\begin{proof}
An event with $e$ edges recovers $e+1$ vertices and contributes $e$ core
coordinates and one principal coordinate.  Summing this equality gives
$|O|+|\mathcal E|+s=n$.
For a vertex which is still unrecovered, its readiness count is
nondecreasing as the admissible prefix increases.  Were it ready now,
it could not satisfy the unready condition at the later event which
recovers it.  This proves the assertion about unrecovered vertices.
An omitted vertex is incident to an event edge.  At the end of that
event this block becomes ready there, and stays ready subsequently.

Finiteness follows from the finiteness of $V$, $\mathcal B$, and the
strict chains of subsets of $V$.  To obtain a candidate containing
$p_J$, retain $V\setminus J$ and add the vertices of $J$ one at a time
as singleton events.  A vertex of $J$ cannot become ready before its
own insertion: a ready proper block would be contained in $J$, and a
ready clique would have two members in $J$, contrary to admissibility.
For a proper core $q_B$, retain $V\setminus B$, choose two members
$a,h\in B$, insert the members of $B\setminus\{a,h\}$ as singleton
events, and finish with the residual edge $a h$ of $B$, selecting $a$.
Minimality of $B$, together with the incidence forest, verifies
readiness and admissibility.  For a clique core retain
$V\setminus\{a,h\}$ and use its residual edge as the single event.
\end{proof}

\subsection{The gauge of an event}

Consider an event $T=A\sqcup B$ after a prefix $J$; here $B$ denotes
the omitted colour, not a block.  Put
\[
 \Delta_A=\prod_{b\in B}x_b^{\deg_T(b)-1}.
\]
For an old omitted vertex $h$, count each block which becomes ready
at $h$ during the event once, and let $H_A$ be the product of the
corresponding occurrences of $x_h$.  Products count occurrences,
including when two factors are equal as rational functions.  Put
\[
 m_A=\prod_{z\in O}x_z^{\max(d_z(J\cup A)-1,0)-\max(d_z(J)-1,0)}.
\]

\begin{lemma}\label{evt:gauge}
Writing $K^J=Z/Z_J$ for the Schur complement, one has
\begin{equation}\label{evt:gauge-formula}
 \frac{G(J\cup A)}{G(J)\Delta_A}=H_A m_A,
 \qquad
 U_A:=\frac{\det K^J[A,A]}{\Delta_A}
       =H_A m_A\frac{p_{J\cup A}}{p_J}.
\end{equation}
If two residual-disjoint events can be performed in either order,
with selected sets $A,D$, then
\begin{equation}\label{evt:cross-gauge}
 X(J;A,D)=\frac{G(J)G(J\cup A\cup D)}{G(J\cup A)G(J\cup D)}
\end{equation}
is a monomial with nonnegative exponents in retained originals and
old omitted originals.
\end{lemma}

\begin{proof}
A current omitted vertex $b$ is initially unready.  Precisely its
$\deg_T(b)$ incident event blocks become ready during this event.
An additional block meeting the connected event in another vertex
would make a cycle in the incidence forest.  Its contribution to the
gauge is therefore $x_b^{\deg_T(b)-1}$.  Every old omitted vertex
already has readiness count at least one, so each newly ready block
adds exactly one to its gauge exponent.  These are precisely the
factors of $H_A$.  Retained originals contribute $m_A$.
No unrecovered exterior vertex can contribute, by
Lemma~\ref{evt:elementary}.  These observations prove the first
identity.  The Schur determinant identity
$Y_{J\cup A}=Y_J\det K^J[A,A]$ proves the second.

For the cross gauge fix an exterior vertex $v$ and write the four
readiness counts as
\[
 d_0,\quad d_0+a,\quad d_0+b,\quad d_0+a+b+c,
 \qquad a,b,c\geq0.
\]
The last number includes blocks requiring selected vertices from
both increments.  No block is counted in both $a$ and $b$:
for proper blocks the last missing selected set is nonempty, and a
clique cannot meet both increments in an admissible union.
With $e(d)=\max(d-1,0)$, its exponent in $X$ is
\[
 e(d_0+a+b+c)+e(d_0)-e(d_0+a)-e(d_0+b)\geq0,
\]
by monotonicity and convexity of $e$.
A vertex in $A\cup D$ has zero relevant readiness count before its
insertion, since the final union is admissible.  It contributes no
negative exponent.  A current omitted vertex cannot gain a block
from the opposite event when the order is reversible.  A block
requiring vertices from both events and also meeting such an omitted
vertex would meet its own event module twice, contradicting the
incidence forest.  Unrecovered exterior vertices again cannot
become ready.  Thus only the stated shared originals occur.
\end{proof}

\subsection{Local changes of candidates}

The union of the full blocks of one event is an outgoing module.
For clique blocks their already recovered omitted members may be
removed when forming the corresponding residual module.  The
incidence subtree of an event is connected.  Distinct events have
disjoint block nodes, and their full incidence subtrees meet in at
most one original vertex.  Indeed, two common vertices would force
their entire joining path, and hence a block node, to be common.
This observation also applies to connected branches cut from events.

\begin{proposition}\label{evt:closure}
In each coordinate direction of a candidate there is a local change
to another candidate which keeps the other $n-1$ coordinates.  The
changes are precisely the following ones.
\begin{enumerate}
\item A selected core is removed by cutting its residual edge.
\item Two consecutive residual-disjoint events are interchanged;
if their reversed order is blocked by an unselected block, the two
events are merged and the core of that block is inserted instead
of their intermediate principal.
\item Consecutive events meeting at a newly selected vertex undergo
a proper-block three-port rotation.
\item Consecutive events meeting at a newly omitted vertex undergo
a clique three-port rotation.
\item The last nontrivial event changes colour; a last singleton
is replaced by its original variable.
\item A retained original is inserted at its first readiness event,
using an inverse edge cut or an inverse clique rotation; if it never
becomes ready, it is inserted as a final singleton.
\end{enumerate}
All candidates are connected to the all-original candidate by these
changes.
\end{proposition}

\begin{proof}
We describe the changes and check the conditions of
Definition~\ref{evt:definition}.

First cut an event edge $a h$, with $a$ selected and $h$ omitted,
and let $W$ be its $h$-side.  Process $W$ before its complement,
using their inherited colours.  If $I=A\cap W$ is nonempty, the
new coordinate is $p_{J\cup I}$; if $I=\varnothing$, then $W=\{h\}$
and the new coordinate is $x_h$, which is retained from the outset.
All unchanged proper blocks retain their earlier selected interiors;
all unchanged cliques retain their earlier unselected interiors.
The cut block cannot become ready on the second side at the
intermediate prefix: its selected endpoint $a$ has not yet entered.
Another new obstruction between the two sides would give a second
incidence path between them.  Thus both events are valid.  Treating
$h$ as retained in the singleton case changes no earlier used block,
since $h$ was unrecovered at every earlier stage.

Next consider two consecutive events $T_1,T_2$ after $J$.
Suppose first that their residual vertex sets are disjoint and that
the interiors of $T_2$ do not use a newly recovered vertex of $T_1$.
If their reversed order satisfies readiness, it gives the new
intermediate coordinate $p_{J\cup A_2}$.  If readiness fails, some
omitted vertex $h$ of $T_1$ becomes ready through a selected vertex
$a$ of $T_2$.  The vertex $h$ cannot be selected, because the final
selected union is admissible.  There is exactly one obstructing
block: two would give two paths between the event subtrees.
For a proper block all its other members lie in $J$.  For an
unselected clique all other members have already been recovered
unselected or are retained.  Otherwise an additional unrecovered
clique member would become ready now and could never be processed
later.  The block is therefore an eligible residual edge $a h$.
Insert its core and merge the two trees.  This is the inverse of
the preceding cut.  If the obstructing clique was already selected,
it is used at the second event and belongs to the clique rotation
described below; no additional copy of its core is inserted.

There remains the case in which an interior of $T_2$ uses a new
vertex $v$ of $T_1$.  Their full modules have at most one common
vertex, so there is only one such vertex and only one block of
$T_2$ containing it.  If $v$ was selected, this block must be proper:
a used clique has no earlier selected member.  Before both events
the central proper block has three residual ports $v,\ell,h$,
with $v,\ell$ selected and $h$ omitted.  Removing the central block
leaves branches $V,L,H$ at these ports.  The old order is $V$,
then $L\cup H$ joined by $\ell h$; the new order is $L$, then
$V\cup H$ joined by $v h$.  The new intermediate principal is the
one for the selected colour of $L$.  The final prefix and recovered
union are unchanged.  The central block retains the same earlier
selected interior, and another readiness obstruction would create
a second incidence connection between the old event subtrees.
Both new events therefore satisfy the definition.

If $v$ was omitted, the central block must be a clique, because a
proper used block has only selected earlier interior members.
Write its three residual ports as $a,b,u$, with $a$ selected and
$b,u$ omitted, and let $A,B,U$ be the three attached branches.
The old order is $B$, then $A\cup U$ joined by $a u$; the new
order is $U$, then $A\cup B$ joined by $a b$.
The clique core is shared.  The new intermediate coordinate is the
$U$-branch principal, or $x_u$ if $U=\{u\}$.
The final selected union is unchanged, and the same incidence-path
argument verifies readiness.  If both omitted branches are
singletons this is a change between the retained originals $x_b$
and $x_u$.  Empty selected branches are treated as originals,
not as a coordinate $p_J$.

For the last nontrivial event interchange its two colours.
An exterior block made inadmissible by the new colour could meet
the connected event at only one vertex; it would already have
made that vertex ready before the event, a contradiction.
Proper earlier interiors and clique earlier interiors are
unchanged.  A last singleton simply returns its original variable.

Finally let $u\in O$.  If it is never ready, remove it from $O$
and append it as a singleton.  Otherwise consider its first ready
event.  Exactly one completing block joins $u$ to a selected
terminal of that event; two would form a cycle.  A proper such
block is not already selected, since a selected proper block has
no retained-original interior.  An unselected clique is treated
in the same way.  Insert its core and attach $u$ as an omitted
leaf, using the inverse cut.  If the clique is already selected,
keep $u$ as its third residual port and use the inverse clique
rotation.  Before this first ready event no used block contains
$u$, so removing $u$ from $O$ leaves all earlier events valid.
These cases cover every coordinate of~\eqref{evt:coordinates}.

To prove connectedness, cut selected cores successively.  Each
cut reduces their number by one and preserves the candidate
conditions.  When none remains, all events are singletons.
Successive last-singleton removals give the all-original candidate.
\end{proof}
\section{Source functions and positive elimination}\label{src:section}

Throughout this section all off-diagonal entries of $C$ are strictly
positive.  We fix a candidate of Definition~\ref{evt:definition}, and
process its events in their prescribed order.  At a given stage $J$ is
the selected prefix.  An original variable recovered at an earlier event
but not selected is called an \emph{old omitted variable}; we write its
value as $h=x_h$ when no confusion with its index is possible.  The
retained original variables are invertible coordinates throughout the
calculation.  All statements of positivity below mean coefficientwise
Laurent nonnegativity over $K$.

More precisely, after the first $\ell$ events let $\mathcal P_\ell$ be
the semiring generated by $K_{\geq0}$, the retained original coordinates
and their inverses, the selected core coordinates and their inverses,
and $p_{I_1}^{\pm1},\ldots,p_{I_\ell}^{\pm1}$.  It is harmless to
include the cores of events not yet processed in this definition; the
expressions constructed before such an event do not use those cores.
An assertion that $f$ has a positive quotient by a product $H$ means
that an equality $f=Hf_0$ with $f_0\in\mathcal P_\ell$ is constructed.
It does not mean only that $H$ divides a positive expression for $f$.

\subsection{Decorated source modules}

Delete every block which is ready at $J$, whether or not its core occurs
in the fixed candidate.  An unready proper block is kept with all its
vertices.  An unready clique has no selected vertex.  Delete its retained
original members and its old omitted members, but keep its block node
whenever at least one residual member remains.  For each deleted old
omitted member $h$, put one mark with origin $h$ at that clique node.
If an old omitted vertex belongs to a surviving proper block, keep a
mark at that vertex.  Isolated marked vertices and isolated selected
vertices may also be kept.  Retained original members contribute
invertible monomial factors, not marks.

The resulting decorated incidence graph is a forest.  Two marks with
the same origin lie in different connected components: their unique
path in the original incidence forest used one of the incidences
deleted when that origin was removed from a clique.  Further deletion
of ready blocks cannot reconnect them.  In particular, a connected
decorated subtree contains at most one mark with any specified origin.

A \emph{source} is the vertex set $N$ of a connected decorated subtree
made of whole proper blocks and arbitrary nonempty residual subcliques.
A singleton residual member may carry its decorated clique node.
Singleton vertices, including vertices of $J$, are also sources.
Every source is an outgoing module: subcliques are outgoing, and unions
of intersecting outgoing modules are outgoing.  The decoration is part
of the source data, although two decorations can have the same vertex
set and hence the same rational source functions.  Let $\mathfrak m(N)$
be the product of its marked original values.  For an old omitted target
$h$, let $\mathfrak m_h(N)$ be the same product with the occurrence
of origin $h$ removed, if that occurrence is present.

For a source $N$ put $R_N=J\setminus N$, and define
\begin{align}
 q_N^{\mathrm{src}}
   &=\det\bigl(Z_{R_N}+\ones C_{N,R_N}\bigr),
       \label{src:q-definition}\\
 P_{Nj}^{\mathrm{src}}
   &=C_{Nj}Y_{R_N}
       +C_{N,R_N}\adj(Z_{R_N})C_{R_N,j},
       &&j\notin N\cup J,\label{src:P-definition}\\
 r_N&=q_N^{\mathrm{src}}/Y_J,
 &b_{Nj}&=P_{Nj}^{\mathrm{src}}/Y_J.
       \label{src:frames}
\end{align}
The common row $C_{N,k}$ is well defined for $k\notin N$.
Empty determinants are $1$.  The target restriction in
\eqref{src:P-definition} permits all live exterior targets and all old
omitted targets, including a deleted clique member whose mark occurs
in the decoration of $N$.
For nested decorated sources $N\subseteq H$ set
\begin{equation}\label{src:nested-definition}
 \Phi_N^H
   =\det\bigl(Z_{J\cap(H\setminus N)}
                   +\ones C_{N,J\cap(H\setminus N)}\bigr).
\end{equation}
Containment here includes the decorations.  Adjoining a decorated
singleton node can change the required mark product without changing
the determinant in \eqref{src:nested-definition}.
We write $z_i^J$ for the $i$th diagonal entry of $K^J=Z/Z_J$;
its off-diagonal $(i,j)$ entry is $-k_{ij}$.

\begin{lemma}[Source factorization]\label{src:factorization}
For nested sources $N\subseteq H$ one has
\begin{equation}\label{src:nesting}
 r_N=\Phi_N^H r_H,\qquad
 b_{Nj}=\Phi_N^H b_{Hj}\quad(j\notin H\cup J).
\end{equation}
If $i\in H\setminus J$ and
\[
 f_i^H=\det\bigl(Z_{J\cap H}+\ones C_{i,J\cap H}\bigr),
\]
then the effective root and off-diagonal coefficients after eliminating
$J$ satisfy
\begin{equation}\label{src:row-factorization}
 a_i:=1+C_{i,J}Z_J^{-1}\ones=f_i^H r_H,
 \qquad
 k_{ij}:=C_{ij}+C_{i,J}Z_J^{-1}C_{J,j}
        =f_i^H b_{Hj}\quad(j\notin H\cup J).
\end{equation}
These are factorizations of whole products.  In particular, all factors
coming from several attached source modules can be extracted in one
application of the same identity.
\end{lemma}

\begin{proof}
Order $R_N$ as $S\sqcup R_H$, where $S=J\cap(H\setminus N)$.
In $Z_{R_N}+\ones C_{N,R_N}$ the block from $S$ to $R_H$ is zero:
its $(s,k)$ entry is $-C_{sk}+C_{Nk}=0$ because $s\in H$ and
$k\notin H$.  The two diagonal blocks are the matrix defining
$\Phi_N^H$ and the matrix defining $q_H^{\mathrm{src}}$.
Taking determinants proves
$q_N^{\mathrm{src}}=\Phi_N^Hq_H^{\mathrm{src}}$.

For the target identity use the bordered determinant
\begin{equation}\label{src:bordered}
 P_{Nj}^{\mathrm{src}}
  =\det\begin{pmatrix}
       Z_{R_N}&-C_{R_N,j}\\
       C_{N,R_N}&C_{Nj}
     \end{pmatrix}.
\end{equation}
Add the last row to each row indexed by $S$.  The $S$-to-$R_H$
block and the target-column entries in those rows become zero, again
by the outgoing-module equalities.  The remaining determinant is
block triangular and gives
$P_{Nj}^{\mathrm{src}}=\Phi_N^H P_{Hj}^{\mathrm{src}}$.
Dividing by $Y_J$ proves \eqref{src:nesting}.
Taking $N=\{i\}$ proves \eqref{src:row-factorization}; its numerator
is the determinant lemma applied to $Z_J$, or the corresponding
bordered determinant.  If several modules are attached in an incidence
subtree, order their selected interiors from the leaves toward the
source.  The same row additions produce all their diagonal factors
at once.  This proves the last assertion, with multiplicity even when
different labeled factors happen to be equal as functions.
\end{proof}

For an earlier selected singleton $i\in J$ the definitions have the
useful interpretation
\begin{equation}\label{src:selected-singleton}
 r_{\{i\}}=e_i^{\mathsf T}Z_J^{-1}\ones,
 \qquad
 b_{\{i\},j}=e_i^{\mathsf T}Z_J^{-1}C_{J,j}\quad(j\notin J).
\end{equation}
Indeed, eliminate $J\setminus\{i\}$ and apply the one-by-one Schur
complement formula.  These are the root and exterior-target
coefficients in the recovery of $x_i$.  Thus selected singleton
sources require no duplicate matrix row.

\subsection{The algebra of a tree event}

We first record an elementary identity without a selected prefix.
Suppose a tree $T=A\sqcup B$ has an outgoing pair module at every
edge.  Write its coefficient matrix as $c$, let $\rho$ be a common
affine root, and put
\[
 z_i=\frac{\rho+\sum_{j\ne i}c_{ij}x_j}{x_i},\qquad
 M=\diag(z_i)-c.
\]
Exterior variables, if present, are included in $\rho$ and remain
formal.  For an edge $e=ab$ its pair core is
\[
 s_e=\frac{\rho+\sum_{i\in A}c_{bi}x_i
                         +\sum_{j\in B}c_{aj}x_j}{x_ax_b}.
\]
Choose monomials $v_i$ in the $s_e^{\pm1}$ with
$s_ev_av_b=1$ and one selected-root value $v_a=1$.  Put
\[
 \ell_i=\sum_{j\text{ of opposite colour}}c_{ij}v_j,
 \quad
 R_0=\sum_{a\in A,b\in B}c_{ab}c_{ba}v_av_b,
 \quad
 \kappa_A=\frac{1}{\prod_{a\in A}v_a
                    \prod_{b\in B}v_b^{\deg_T(b)-1}}.
\]

\begin{lemma}[Pair-tree identity]\label{src:monic-tree}
There is a rational function $\theta$ such that
\begin{align}
 x_b&=v_b(\theta+\ell_b),&&b\in B,\label{src:monic-omitted}\\
 x_a&=v_a\left(\frac{\rho+R_0}{\theta}+\ell_a\right),
      &&a\in A,\label{src:monic-selected}\\
 z_a&=\frac{\theta+\sum_{c\in A\setminus\{a\}}c_{ac}v_c}{v_a},
      &&a\in A.\label{src:monic-diagonal}
\end{align}
Moreover
\begin{equation}\label{src:monic-principal}
 \frac{\det M[A,A]}{\prod_{b\in B}x_b^{\deg_T(b)-1}}
       =\kappa_A\theta.
\end{equation}
All identities remain valid when the tree has a single selected vertex
and no edge, with $B=\varnothing$, $v=\kappa_A=1$, and
$\theta=z_a$.
\end{lemma}

\begin{proof}
Set $\xi_i=x_i/v_i$.  If $a,a'\in A$ are both adjacent to $b$,
subtract the core equations for $ab$ and $a'b$.  For every target
$j\in B\setminus\{b\}$, the equalities along $a-b-a'$ give
$c_{aj}=c_{a'j}$.
After cancellation of the nonzero $\xi_b$, the difference is
\[
 \xi_a-\xi_{a'}=(c_{ab}-c_{a'b})v_b=\ell_a-\ell_{a'}.
\]
The distance-two graphs on the two colour classes are connected.
Consequently $\xi_a-\ell_a=t_A$ for all $a\in A$ and
$\xi_b-\ell_b=t_B$ for all $b\in B$.

For a fixed edge $ab$, substitute these expressions in its core
equation.  The coefficients of $t_A,t_B$ cancel.  For
$i\in A,j\in B$, the remaining coefficient identity is
\begin{equation}\label{src:tree-separation}
 (c_{bi}-c_{ji})(c_{ij}-c_{aj})=0.
\end{equation}
To see it, if deleting $i$ does not separate $b$ from $j$, propagate
the pair-module equalities along their path to obtain
$c_{bi}=c_{ji}$.  If deleting $i$ does separate them, the path from
$a$ to $i$ avoids $j$ (with $i=a$ giving the equality directly),
so $c_{ij}=c_{aj}$.  Thus $t_At_B=\rho+R_0$.
Take $\theta=t_B$ to obtain
\eqref{src:monic-omitted}--\eqref{src:monic-selected}.
Dividing the core equation at $ab$ by $x_a$ gives
$z_a=s_ex_b-c_{ba}$, which is \eqref{src:monic-diagonal}.

Let $L_A$ be the row Laplacian on $A$ with target weights $c_{ac}v_c$.
Equation~\eqref{src:monic-diagonal} says
$M[A,A]\diag(v_A)=\theta I+L_A$.  For each $b\in B$, all rows
indexed by neighbours of $b$ agree off their own diagonals with the
row $(-c_{bc}v_c)_{c\in A}$.  Hence, for neighbours $a,a'$ of $b$,
\[
 (e_a-e_{a'})^{\mathsf T}L_A
        =\ell_b(e_a-e_{a'})^{\mathsf T}.
\]
For each $b$ choose one neighbour as base and take the
$\deg_T(b)-1$ resulting differences.  These differences are linearly
independent: contract, for each $b$, its edge to the chosen base.
The remaining edges form a tree on $A$, and their incidence vectors
are precisely these differences.  Deleting leaves of that tree
proves their independence.  Their
total number is $|A|-1$, since
$\sum_{b\in B}(\deg_T(b)-1)=|A|-1$.  They span the row subspace
annihilating $\ones$.  As $L_A\ones=0$, an adapted basis gives
\[
 \det(\theta I+L_A)
   =\theta\prod_{b\in B}(\theta+\ell_b)^{\deg_T(b)-1}.
\]
Divide by $\prod_{a\in A}v_a$, and use
\eqref{src:monic-omitted}.  This proves
\eqref{src:monic-principal}.  The singleton assertions follow
directly from $z_ax_a=\rho$.
\end{proof}

Return to an event after $J$.  Let its residual tree have colours
$A,B$.  For a proper edge $e$ put $S_e=J\cap B_e$ and
\[
 f_{e,i}=\det(Z_{S_e}+\ones C_{i,S_e})
       \quad(i\text{ a residual endpoint of }e).
\]
For a clique edge put $f_{e,i}=1$.  The proper sets $S_e$ are disjoint;
an intersection would create a second path between two block nodes.
Write $\pi_e(i)$ for the endpoint of edge $e$ nearest $i$.
Let $T_i$ be the product of the ports nearest $i$ over all event edges,
and $\Lambda_i$ its subproduct for which that nearest endpoint has
the colour of $i$.  The physical edge core is
\begin{equation}\label{src:physical-core}
 s_e=q_{B_e}\quad\text{for a proper edge},\qquad
 s_e=\gamma_eH_eq_{B_e}\quad\text{for a clique edge},
\end{equation}
where $\gamma_e$ is the product of its deleted retained members and
$H_e$ is the product of its old omitted members.  The latter equality
is simply the common affine clique numerator divided by its two
remaining original variables.

Let $U$ be the source consisting of all proper event blocks and the
residual pairs of the clique blocks, retaining their node decorations.
It contains no old omitted vertex.  Put $\delta=r_U$.  Source
factorization, ordering edge interiors away from the source vertex,
gives
\begin{equation}\label{src:event-root-ports}
 a_i=\delta T_i,
 \qquad
 T_i=f_{e,i}T_e\quad(i\in e),
\end{equation}
where $T_e$ is the product of the ports of all other edges pointing
toward $e$.  Indeed, the shifted matrix on $J$ has diagonal blocks
$Z_{S_e}+\ones C_{\pi_e(i),S_e}$ and
$Z_{J\setminus U}+\ones C_{U,J\setminus U}$; its blocks pointing
away from the source vanish.  Thus its determinant is precisely
$q_U^{\mathrm{src}}T_i$.

\begin{lemma}[Residual-block scaling]\label{src:block-scaling}
Let a proper block $D$ have any number of residual ports after
eliminating $J$.  Normalize the effective roots by $y_i=x_i/a_i$.
The core of the normalized residual block is
\begin{equation}\label{src:block-scale}
 q_D^{\mathrm{norm}}
   =\frac{\det(Z_{J\setminus D}+\ones C_{D,J\setminus D})}{Y_J}
                   q_D.
\end{equation}
For an edge of a two-port event this becomes
$s_e^{\mathrm{norm}}=\delta T_e s_e$, also for a physical clique
pair core.
\end{lemma}

\begin{proof}
Before elimination, write $E_D=1+\sum_{j\notin D}C_{Dj}x_j$.
The equations $Z_Dx_D=E_D\ones$ and
$\det Z_D=E_Dq_D$ give
$\adj(Z_D)\ones=q_Dx_D$.  A rank-one shift therefore satisfies
\[
 \det(Z_D-\ones h^{\mathsf T})
   =q_D(E_D-h^{\mathsf T}x_D).
\]
Eliminate $R=J\setminus D$ first.  Because the rows of $D$ agree
toward $R$, its Schur correction is just such a rank-one shift.  Its
common root coefficient is
$a_0=\det(Z_R+\ones C_{D,R})/Y_R$, and its core remains $q_D$.
Write $Z_D^R=Z_D-C_{D,R}Z_R^{-1}C_{R,D}$ for this matrix.
Its equation is $Z_D^Rx_D=E'\ones$, where
\[
 E'=a_0+\sum_{j\notin D\cup J}
       \bigl(C_{Dj}+C_{D,R}Z_R^{-1}C_{R,j}\bigr)x_j.
\]
Indeed, substituting the equations for $x_R$ gives
$E'=E_D-C_{D,R}Z_R^{-1}C_{R,D}x_D$.
Now put $S=J\cap D$, $P=D\setminus J$, and $Q=Z_D^R[S,S]$.
If $\widetilde C_{P,S}=-Z_D^R[P,S]$, the port matrix and its
root multiplier are
\[
 N=Z_D^R[P,P]-Z_D^R[P,S]Q^{-1}Z_D^R[S,P],
 \qquad
 d=\ones+\widetilde C_{P,S}Q^{-1}\ones.
\]
Block elimination gives $Nx_P=E'd$ and
\[
 \det N=\frac{E'q_D}{\det Q}.
\]
In particular $a_i=a_0d_i$.  The substitution $y_i=x_i/a_i$
changes the matrix to
$\diag(a_P)^{-1}N\diag(a_P)$ and changes its equation to
$\diag(a_P)^{-1}N\diag(a_P)y_P=(E'/a_0)\ones$.
Its determinant is still $\det N$, so its normalized core is
$a_0q_D/\det Q$.
Since $Y_J=Y_R\det Q$, this is
\eqref{src:block-scale}.  In a two-port event the numerator in
\eqref{src:block-scale} factors as $q_U^{\mathrm{src}}T_e$ by the
same triangularization as \eqref{src:event-root-ports}.  A residual
clique pair is itself an outgoing pair with core $s_e$, so the
two-port argument applies to it as well.
\end{proof}

\begin{proposition}[Event identities]\label{src:event-identities}
Choose $v_i$ with $s_ev_iv_j=1$ on event edges and a selected root
$v_i=1$, and define
\[
 \Delta_A=\prod_{b\in B}x_b^{\deg_T(b)-1},\qquad
 \kappa_A=\frac{1}{\prod_{a\in A}v_a
                    \prod_{b\in B}v_b^{\deg_T(b)-1}},\qquad
 U_A=\frac{\det((Z/Z_J)[A,A])}{\Delta_A}.
\]
Here $Z/Z_J$ denotes the Schur complement.  Then
$\theta=U_A/\kappa_A$ satisfies
\begin{align}
 z_a^J&=\frac{T_a}{\Lambda_a}\frac{\theta}{v_a}
       +\sum_{c\in A\setminus\{a\}}
                    k_{ac}\frac{\Lambda_c}{\Lambda_a}
                                     \frac{v_c}{v_a},
                    \label{src:selected-diagonal}\\
 x_b&=v_b\left[\Lambda_b\theta+
              \sum_{a\in A}\frac{\Lambda_b\Lambda_a}{T_b}
                                    k_{ba}v_a\right],
                    \label{src:omitted-recovery}\\
 x_a&=v_a\left[\frac{\Lambda_a(R_U+R_0)}{\theta}
              +\sum_{b\in B}\frac{\Lambda_a\Lambda_b}{T_a}
                                    k_{ab}v_b\right],
                    \label{src:selected-recovery}\\
 R_0&=\sum_{a\in A,b\in B}
          k_{ab}k_{ba}\frac{\Lambda_a\Lambda_b}{T_aT_b}v_av_b,
                    \label{src:R0}\\
 R_U&=r_U+\sum_{j\notin J\cup U}b_{Uj}x_j.
                    \label{src:full-root}
\end{align}
In particular, untouched exterior variables remain in the affine root.
If $K_A=(Z/Z_J)[A,A]$, then
\begin{equation}\label{src:inverse-root-columns}
 K_A^{-1}a_A
       =r_U\bigl(\Lambda_av_a/\theta\bigr)_{a\in A},
 \qquad
 K_A^{-1}k_{Aj}
       =b_{Uj}\bigl(\Lambda_av_a/\theta\bigr)_{a\in A}
       \quad(j\notin J\cup U).
\end{equation}
\end{proposition}

\begin{proof}
Put $w_a=\delta\Lambda_a$, $w_b=\Lambda_b$, and $g_i=w_i/a_i$.
For an edge $e=ab$,
$\Lambda_a\Lambda_b=f_{e,a}f_{e,b}T_e$ and
$T_aT_b=f_{e,a}f_{e,b}T_e^2$.  Hence
$g_ag_b\delta T_e=1$.  By Lemma~\ref{src:block-scaling}, the
normalized pair-tree monomials are $g_iv_i$.  The normalized
off-diagonal coefficients are $k_{ij}a_j/a_i$, and the normalized
common affine root is $R_U/\delta$.

The selected matrix changes by diagonal similarity.  If
$\zeta_A=\prod_{b\in B}a_b^{\deg_T(b)-1}$, then
$U_A^{\mathrm{norm}}=\zeta_AU_A$.
Also $\kappa_A^{\mathrm{norm}}=\zeta_A\kappa_A$: on deleting any
edge, count the selected vertices and the omitted degrees on each
side, using $\sum_{b}(\deg(b)-1)=|A|-1$ for a tree.  Equivalently,
each labeled port has exponent zero in
$\prod_{a\in A}g_a\prod_{b\in B}(g_ba_b)^{\deg(b)-1}$.
Thus the phase in Lemma~\ref{src:monic-tree} is unchanged.
Undoing $y_i=x_i/a_i$ in that lemma gives
\eqref{src:selected-diagonal}--\eqref{src:selected-recovery}.
For example the normalized pair sum is $R_0/\delta$, while
$a_ag_a=\delta\Lambda_a$ and $a_bg_b=\Lambda_b$; these three
equalities cancel every root-normalizing divisor.

Finally \eqref{src:selected-diagonal} implies
\[
 K_A(\Lambda_av_a)_{a\in A}=\theta(T_a)_{a\in A}.
\]
Use $a_a=T_ar_U$ and $k_{aj}=T_ab_{Uj}$ from
\eqref{src:row-factorization} to obtain
\eqref{src:inverse-root-columns}.
\end{proof}

\subsection{Exact updates of source functions}

A prime on a source function will mean that $J$ is replaced by
$J\cup A$.  A source surviving the event is viewed before folding
any new omitted members.  Its intersection with the residual event
is empty, one selected port, or one omitted port.  Two intersections
would force a processed block to survive, or create an incidence
cycle.  Intersection at an old selected articulation belongs to the
first, residual-disjoint case.

\begin{lemma}[Source updates]\label{src:updates}
If $M\cap(U\setminus J)=\varnothing$, then
\begin{align}
 r_M'&=r_M+\frac{r_U}{\theta}
                        \sum_{a\in A}b_{Ma}\Lambda_av_a,
           \label{src:disjoint-root}\\
 b_{Mj}'&=b_{Mj}+\frac{b_{Uj}}{\theta}
                        \sum_{a\in A}b_{Ma}\Lambda_av_a
              \quad(j\notin M\cup U\cup J).
           \label{src:disjoint-target}
\end{align}
If $M\cap(U\setminus J)=\{v\}$ with $v\in A$, put $N=M\cup U$.
Then
\begin{equation}\label{src:selected-update}
 r_M'=\frac{v_v\Lambda_v}{\theta}r_N,
 \qquad
 b_{Mj}'=\frac{v_v\Lambda_v}{\theta}b_{Nj}
                  \quad(j\notin N\cup J).
\end{equation}
If the residual intersection is $\{b\}$ with $b\in B$, then
\begin{equation}\label{src:omitted-update}
 r_M'=\frac{T_b}{\Lambda_b}\frac{x_b}{v_b\theta}r_N,
 \qquad
 b_{Mj}'=\frac{T_b}{\Lambda_b}\frac{x_b}{v_b\theta}b_{Nj}
                  \quad(j\notin N\cup J).
\end{equation}
For nested post-event sources $N\subseteq H$, $\Phi_N^H$ is
unchanged unless the event selects a vertex $v\in H\setminus N$;
in that case
\begin{equation}\label{src:nested-update}
 (\Phi_N^H)'=\Phi_N^H z_v^J+f_v^H b_{Nv}.
\end{equation}
\end{lemma}

\begin{proof}
In the residual-disjoint case, take the Schur complement on the new
selected block $K_A$.  The common root and target row of $M$ change
by $b_{MA}K_A^{-1}a_A$ and $b_{MA}K_A^{-1}k_{Aj}$, respectively.
Equation~\eqref{src:inverse-root-columns} gives
\eqref{src:disjoint-root}--\eqref{src:disjoint-target}.
For a selected singleton already in $J$, the same calculation follows
from its inverse-row interpretation \eqref{src:selected-singleton}.

If the intersection is a selected port $v$, first separate
$J\cap M$ by the row additions in \eqref{src:bordered}, and then
eliminate $A\setminus\{v\}$.  Cramer's rule for the remaining row
gives
\[
 r_M'=\frac{(K_A^{-1}a_A)_v}{f_v^M},\qquad
 b_{Mj}'=\frac{(K_A^{-1}k_{Aj})_v}{f_v^M}.
\]
Concretely, their numerator determinants are obtained by replacing
column $v$ of $K_A$ by $a_A$ or $k_{Aj}$, and their denominator is
$f_v^M\det K_A$.  Source factorization gives
$r_U=f_v^M r_N$ and $b_{Uj}=f_v^M b_{Nj}$, proving
\eqref{src:selected-update}.

If the intersection is an omitted port $b$, source factorization
gives $r_M=T_br_N$, $b_{Mj}=T_bb_{Nj}$, and
$b_{Ma}=k_{ba}/f_b^M$, whereas $r_U=f_b^Mr_N$.
The residual-disjoint Schur calculation is applicable since
$M\cap A=\varnothing$.  Thus
\[
 r_M'=r_N\left(T_b+\theta^{-1}
                       \sum_{a\in A}k_{ba}\Lambda_av_a\right).
\]
Use \eqref{src:omitted-recovery} to obtain
\eqref{src:omitted-update}; the target calculation is identical
with $r_N$ replaced by $b_{Nj}$.

For the nested update only a single newly selected vertex can lie in
$H\setminus N$, by the surviving-source intersection property.
Put $L=H\cup U$.  Before the event, factorization gives
\[
 r_H=T_vr_L,\qquad r_U=f_v^Hr_L,\qquad
 f_v^H b_{Na}=\Phi_N^Hk_{va}\quad(a\in A\setminus\{v\}).
\]
Divide the residual-disjoint update for $r_N$ by the selected-port
update for $r_H$.  The quotient is
\[
 \Phi_N^H\frac{T_v\theta+
                  \sum_{a\ne v}k_{va}\Lambda_av_a}{v_v\Lambda_v}
       +f_v^Hb_{Nv}.
\]
Its fraction is $z_v^J$ by \eqref{src:selected-diagonal}.
This proves \eqref{src:nested-update}, including the singleton event.
\end{proof}

The following identities are needed because an unweighted coefficient
toward an old omitted target need not be Laurent in the earlier prefix.

\begin{lemma}[Weighted target columns]\label{src:weighted-columns}
For a newly omitted $b\in B$, put
$W_{ab}=(K_A^{-1}k_{Ab})_a x_b$.  Then
\begin{equation}\label{src:weighted-resolvent}
 W_{ab}=v_av_b\left[
      \frac{\Lambda_a\Lambda_b}{T_a}k_{ab}
       +\frac{\Lambda_a\Lambda_b}{T_b\theta}
          \sum_{c\in A}k_{bc}k_{cb}
                         \frac{\Lambda_c}{T_c}v_c\right].
\end{equation}
For distinct $b_0,b\in B$ one has
\begin{equation}\label{src:omitted-to-omitted}
 k'_{b_0b}x_b
   =x_{b_0}\frac{v_b}{v_{b_0}\Lambda_{b_0}}
       \left[\Lambda_b k_{b_0b}
       +\frac{\Lambda_bT_{b_0}}{T_b\theta}
          \sum_{c\in A}k_{bc}k_{cb}
                             \frac{\Lambda_c}{T_c}v_c\right].
\end{equation}
If a source $M$ can be enlarged to a surviving source meeting the
event only at $b$, then
\begin{equation}\label{src:virtual-target}
 b_{Mb}'=b_{Mb}
       +\frac1\theta\sum_{a\in A}
                  b_{Ma}k_{ab}\frac{\Lambda_a}{T_a}v_a.
\end{equation}
In particular the last expression has no inverse $x_b$.
\end{lemma}

\begin{proof}
In the normalized pair-tree system let $f_a=c_{ab}$,
$\ell_b=\sum_cc_{bc}v_c$, and
$S_b=\sum_cc_{bc}c_{cb}v_c$.  The vector $f$ is constant on each
component of $T\setminus\{b\}$.  For vertices $a,c$ in different
components, pair-module equality gives $c_{ac}=c_{bc}$.
Thus, term by term,
\[
 L_Af=\ell_bf-S_b\ones,\qquad L_A\ones=0,\qquad
 (\theta I+L_A)^{-1}f
   =\frac{f+(S_b/\theta)\ones}{\theta+\ell_b}.
\]
Multiplication by $x_b=v_b(\theta+\ell_b)$ gives
$W_{ab}=v_av_b(c_{ab}+S_b/\theta)$ in the normalized system.
For $b_0\ne b$, tree separation also gives, for every $a\in A$,
\[
 (c_{b_0b}-c_{ab})(c_{ba}-c_{b_0a})=0.
\]
Insert the weighted inverse column in
$k'_{b_0b}=c_{b_0b}+c_{b_0A}K_A^{-1}c_{Ab}$, and sum this last
identity with weights $v_a$.  The result is
$k'_{b_0b}x_b=x_{b_0}(v_b/v_{b_0})(c_{b_0b}+S_b/\theta)$.
Undo the diagonal normalization used in
Proposition~\ref{src:event-identities}, namely
$c_{ij}=k_{ij}a_j/a_i$ and $v_i^{\mathrm{norm}}=g_iv_i$.
Substitution of $g_a=\Lambda_a/T_a$ and
$g_b=\Lambda_b/(\delta T_b)$ gives exactly
\eqref{src:weighted-resolvent} and
\eqref{src:omitted-to-omitted}; all $\delta$ factors cancel.

For the last assertion enlarge $M$ to $H$ with $H\cap(U\setminus
J)=\{b\}$.  Factorization makes the row $b_{MA}$ proportional to
$k_{bA}$.  The preceding inverse-column computation gives
\[
 k_{bA}K_A^{-1}k_{Ab}
     =\theta^{-1}\sum_{a\in A}k_{ba}k_{ab}
                                    (\Lambda_a/T_a)v_a.
\]
Multiply by that proportionality factor before expanding, and add
the unchanged coefficient $b_{Mb}$.  This gives
\eqref{src:virtual-target} without dividing by an omitted variable.
\end{proof}

\subsection{Simultaneous port and mark extraction}

Here are the precise source branches used in applying
Lemma~\ref{src:factorization}.  To separate a source from a target
$j$, remove every proper block containing $j$ as a whole.  For a
clique remove only its target vertex, keeping the decorated clique
node attached to its other residual members.  Take the source-side
component of the event incidence tree with this convention, and
include its whole proper blocks and its residual subcliques.  Call
the resulting source $E_{ij}$.  Its selected interiors are precisely
the interiors of the proper event blocks on that side.  Ordering
these interiors away from $i$ in the shifted matrix gives
\begin{equation}\label{src:branch-product}
 k_{ij}=
   \left(\prod_{e\text{ in the proper source branch}}
                     f_{e,\pi_e(i)}\right)b_{E_{ij},j}.
\end{equation}
In particular a final clique edge keeps its marked node even when
$j$ is its other endpoint.  A clique contributes no port determinant,
because none of its vertices is in $J$.

If source attachments enlarge $E_{ij}$ to $L$ while keeping the target
outside, \eqref{src:branch-product} becomes the single identity
\begin{equation}\label{src:whole-product}
 k_{ij}=
   \left(\prod_{e\text{ in the proper source branch}}
                  f_{e,\pi_e(i)}\right)\Phi_{E_{ij}}^L b_{Lj}.
\end{equation}
This identity extracts port factors before using the entire marked
source factorization of $b_{Lj}$.  If an additional source is attached
at an event port $v$, its selected interior is disjoint from all
event interiors.  Its factor $f_v^M$ therefore occurs as an additional
diagonal block in the same triangularization.  No coprimality of
these factors is required.

For clarity, the elementary port count used below is as follows.
On an odd path from $a\in A$ to $b\in B$, a negative port exponent
in $\Lambda_a\Lambda_b/T_a$ occurs only at an even-position edge.
It is internal to the $a$-source branch and is supplied by
\eqref{src:branch-product}.  On a same-colour path, a negative port
in $\Lambda_j/\Lambda_i$ occurs only at an odd-position edge
strictly before the target and is supplied by the $i$-source branch.
For
\begin{equation}\label{src:pair-port-quotient}
 k_{ab}k_{ba}\frac{\Lambda_a\Lambda_b}{T_aT_b},
\end{equation}
an off-path edge contributes at most its nearest endpoint port to
the denominator, and that port occurs in one source branch.  An
even-position path edge contributes both endpoint ports, one in
each of the opposite source branches.  The remaining port exponents
are nonnegative.  These assertions follow by considering the two
components after deleting the edge: its port is in $T_i$ exactly
on the component containing its endpoint, and is in $\Lambda_i$
exactly when that endpoint has the colour of $i$.

Opposite source branches cover every event vertex and every clique
node.  A clique node on the path occurs in both branches; an off-path
node occurs in at least one.  Assign each required exterior attachment
to a branch containing its attachment vertex, and at an endpoint use
the branch starting at that endpoint.  All assignments can be made
simultaneously.  Two attachments meeting again outside the event
would create a cycle in the block incidence forest.  Equation
\eqref{src:whole-product} then extracts the full assigned product,
including repeated numerical factors, from each source coefficient.

We next count the powers introduced by physical clique cores.  Fix
one mark $h$ at a clique edge $e$, root the event at a selected
vertex, and let $L,R$ be the root and other side after deleting $e$.
Write $v_i=v_i^0h^{s_i}$ and
$\kappa_A=\kappa_A^0h^\eta$.  The relation $s_ev_iv_j=1$
and propagation along the tree give
\begin{equation}\label{src:mark-table}
\begin{array}{c|c|c|c|c|c}
 \text{edge orientation}&s_i\ (i\in L)&s_a\ (a\in A\cap R)
     &s_b\ (b\in B\cap R)&\eta&\deg_h\theta\\ \hline
 A_L-B_R&0&1&-1&0&1\\
 B_L-A_R&0&-1&1&1&0
\end{array}
\end{equation}
The entry for $\kappa_A$ follows by summing these exponents in its
definition; the entry for $\theta$ uses the canonical gauge
\begin{equation}\label{src:canonical-phase}
 \theta=\frac{H_A}{\kappa_A}\tau_A,
 \qquad \tau_A=m_Ap_{J\cup A}/p_J,
\end{equation}
from Lemma~\ref{evt:gauge}.  Every internal clique mark occurs
once in $H_A$.

It follows that $\kappa_Av_a$ and $\kappa_A/v_b$ have $h$-degree
$0$ or $1$, while $\theta/v_a$ and $v_b\theta$ have nonnegative
$h$-degree.  A negative exponent in $v_i/v_j$ for equal colours or
in $v_iv_j$ for opposite colours occurs only when the path crosses
$e$, and has size at most one.  The source on that path retains the
decorated clique node and supplies one $h$.  Opposite coefficients
$k_{ab}k_{ba}$ supply two occurrences when their path crosses $e$
and at least one otherwise.  Consequently each summand of $R_0$
retains an entire occurrence of every internal clique mark after
its $v_av_b$ factor is included.  These counts are per labeled
occurrence; several marks at one node satisfy the same count
simultaneously.

\subsection{The joint positivity induction}

\begin{proposition}\label{src:invariant}
After every prefix the following statements hold for all surviving
decorated sources and nested source pairs:
\begin{enumerate}
 \item $\Phi_N^H\in\mathcal P_\ell$;
 \item $r_N/\mathfrak m(N)\in\mathcal P_\ell$;
 \item $b_{Nj}/\mathfrak m(N)\in\mathcal P_\ell$ for each live
       exterior target $j$;
 \item $x_hb_{Nh}/\mathfrak m(N)\in\mathcal P_\ell$ for each
       old omitted exterior target $h$;
 \item if the component of $N$ contains an occurrence of origin
       $h$, then $b_{Nh}/\mathfrak m_h(N)\in\mathcal P_\ell$.
\end{enumerate}
Every quotient has a displayed positive expression.  The selected
Schur diagonals and the omitted recovery formulas are positive
Laurent in the extended prefix.  Selected originals have positive
affine recovery in the still-live exterior originals.  At the end
of the events all original variables have positive Laurent
expressions in the candidate coordinates.
\end{proposition}

\begin{proof}
At $J=\varnothing$ there are no marks, every nested determinant is
$1$, and the source functions are $1$ or positive initial
coefficients.  Assume the five statements for the old prefix.

\smallskip\noindent
\emph{Source enlargements and the phase product.}
An old omitted vertex gains a new canonical-gauge occurrence precisely
when one of its formerly unready blocks becomes ready during this
event.  A proper such attachment has residual pair $\{a,h\}$,
where $a\in A$ and $h$ is old omitted.  A clique attachment has
one residual selected attachment vertex and a node carrying its
deleted old members.  It cannot have a second unrecovered member:
that member would become ready now and could not be processed at a
later eligible event.  Internal clique-edge marks are already in
the event source.  These are exactly the factors of $H_A$.

Adjoin all the attachments to the event source, or distribute them
between opposite branches as above.  The resulting source is a
permitted old source and its mark product contains the whole
$H_A$.  Equations~\eqref{src:nesting} and
\eqref{src:whole-product}, followed by the old states, give a
positive quotient by this entire product after all specified port
factors have been extracted.  With a live target no attachment
absorbs that target.  With an old target $h$ which is itself one
of the phase marks, there are two cases.  For a proper attachment
leave the target-containing attachment out; source and target
remain in the same old component, so state~5 supplies the
unweighted coefficient with every other required mark.  For a
clique keep its node but apply state~5 with the origin-$h$
exception.  In either case the explicit factor $x_h$ in the
weighted coefficient supplies exactly the missing occurrence.
It supplies that occurrence once, not once for each residual member
of the clique.

\smallskip\noindent
\emph{Diagonals, omitted values, and nested factors.}
In \eqref{src:selected-diagonal}, $T_a/\Lambda_a$ is a product
of ports and $\theta/v_a$ has nonnegative internal-mark exponents
by \eqref{src:mark-table}.  Each remaining term has its port
denominators supplied by its source branch; a negative mark
exponent can only cross the corresponding clique edge and is
supplied by that same branch.  Thus the diagonal is positive in
the enlarged prefix semiring.  The identical count applies to
\eqref{src:omitted-recovery}; its first term $v_b\Lambda_b\theta$
has no negative internal-mark exponent, and its other terms use
the opposite-colour source quotient.  In particular every new
omitted value $x_b$ is available as a positive expression before
it is installed as a new mark.

For nested post-event sources use \eqref{src:nested-update}.
Its diagonal has just been proved positive, while $\Phi_N^H$,
$f_v^H$, and the live-target coefficient $b_{Nv}$ are old states.
The other nested determinants do not change.  This proves state~1
without assuming positivity of arbitrary raw principal minors.

\smallskip\noindent
\emph{Root and live-target states.}
Use the three updates in Lemma~\ref{src:updates}.  In the
residual-disjoint case the event factor supplies $H_A$, and the
separate factor $b_{Ma}$ retains all old marks of $M$.  After
substitution of \eqref{src:canonical-phase}, the remaining
internal clique power is that of $\kappa_Av_a$, hence is
nonnegative.  All port factors are extracted together with the
phase marks by \eqref{src:whole-product}.

In a selected-port intersection the same computation uses
$\kappa_Av_v$, and in an omitted-port intersection it uses
$\kappa_A/v_b$, followed by the explicitly positive new factor
$x_b$.  The factor $T_b/\Lambda_b$ is a product of ports.  A mark
required by the surviving source cannot also be consumed as a
phase occurrence in either intersection case: a phase clique
node is deleted when ready, and a phase proper block is deleted
and separates its old marked vertex.  A second connecting path
would be an incidence cycle.  Thus the surviving mark product
and the consumed phase product are both present with their
required multiplicities.  In the residual-disjoint case equal
origins may occur in distinct components, but their two factors
already occur in separate displayed source factors.  This proves
states~2 and~3 before folding, with the new omitted mark in the
omitted-port case.

\smallskip\noindent
\emph{Persistence of old targets.}
Fix an old omitted target $h$.  Multiply every exact source update
by $x_h$.  Phase extraction for its event factor uses the
target-origin rule in the first paragraph of the induction.
The other source factor retains its source marks, as in the root
and live-target calculation.  This proves state~4.

For state~5 require an occurrence of origin $h$ in the
\emph{post-event} component of $M$.  In an intersection update a
consumed occurrence cannot be the surviving occurrence: phase
clique nodes disappear, and phase proper attachments are cut.
Uniqueness of an origin within a pre-event component rules out
another copy there.  Consequently the unweighted intersection
formula extracts the phase product while retaining precisely
$\mathfrak m_h(M)$.

In a residual-disjoint update the term to consider is
\[
       \frac{b_{Uh}}{\theta}\,b_{Ma}\Lambda_av_a.
\]
If $U$ and the surviving occurrence are in different old
components, enlarge $M$ in its component to that occurrence,
avoiding $a$.  This gives an explicit $x_h$ in $b_{Ma}$ and
permits the rewrite
$b_{Ma}b_{Uh}=(b_{Ma}/x_h)(x_hb_{Uh})$.
If the occurrence was already in $M$, its removal is exactly the
exception in $\mathfrak m_h(M)$; otherwise it is an extra factor
and all source marks remain.  Any occurrence of the same origin
in $U$ is a different labeled occurrence and is supplied by its
weighted state before the phase is divided out.

If $U$ lies in the same old component, but the origin-$h$
occurrence is not a phase mark, enlarge its unweighted source by
all phase attachments, using state~5.  If that occurrence is a
phase mark and nevertheless survives in the component of $M$,
it is a proper marked vertex beyond a newly ready attachment;
a marked phase clique node would have disappeared.  The surviving
source component avoids every selected event vertex, since a
path to one of them together with the phase attachment and the
event path would make a cycle.  Enlarge $M$ to that component to
obtain the extra $x_h$ from $b_{Ma}$, and enlarge $U$ by the other
phase attachments to obtain $H_A/x_h$ from $b_{Uh}$.  The target
vertex is outside $M$, so this extra cancellation removes no
required source mark.  These cases prove state~5 for old targets.

\smallskip\noindent
\emph{Birth of new weighted targets.}
Let $b\in B$.  For a residual-disjoint source the new weighted
target is
\begin{equation}\label{src:new-target-disjoint}
 x_b b_{Mb}'=b_{Mb}x_b+\sum_{a\in A}b_{Ma}W_{ab}.
\end{equation}
For a source attached at a selected port $v$ it is
$W_{vb}/f_v^M$; for a source attached at an omitted port
$b_0\ne b$ it is $k'_{b_0b}x_b/f_{b_0}^M$.
These follow from the same Schur and Cramer calculations used
in Lemma~\ref{src:updates}.  The direct terms in
\eqref{src:weighted-resolvent} have the opposite-colour port
quotient.  Its pair terms have the quotient
\eqref{src:pair-port-quotient}, multiplied by a product of ports.
For \eqref{src:omitted-to-omitted} the direct term has the
same-colour port quotient.  The extra divisor $f_v^M$ or
$f_{b_0}^M$ is extracted by including $M$ in the source branch
containing its attachment.  That is the same simultaneous
factorization which supplies the retained source marks.

We give the mark count for the pair terms explicitly.  For one
internal mark use $s_i,\eta$ from \eqref{src:mark-table} and
factor out the phase product $H_A$ by the assigned branches.
Before the opposite coefficient product $k_{bc}k_{cb}$ is
included, the exponent in the second term of
\eqref{src:weighted-resolvent} is
$s_a+s_b+s_c+\eta-1$.  In the two orientations this is
\[
 \mathbf1_{a\in R}-\mathbf1_{b\in R}
       +\mathbf1_{c\in R}-1,\qquad
 -\mathbf1_{a\in R}+\mathbf1_{b\in R}
       -\mathbf1_{c\in R}.
\]
If $b,c$ are on the same side, the deficit is at most one;
otherwise it is at most two.  Their two source branches supply
one occurrence off their path and two on their path.  For the
second term of \eqref{src:omitted-to-omitted}, after setting aside
its new factor $x_{b_0}$, the corresponding exponent is
$s_b-s_{b_0}+s_c+\eta-1$, namely
\[
 -\mathbf1_{b\in R}+\mathbf1_{b_0\in R}
       +\mathbf1_{c\in R}-1,\qquad
 \mathbf1_{b\in R}-\mathbf1_{b_0\in R}
       -\mathbf1_{c\in R}.
\]
The same one-or-two occurrence count applies.  Direct terms have
a deficit only on their source-to-target path and are covered
by that source.  Exterior phase attachments are allocated to
the same whole branches.  Hence the displayed weighted births
are positive with all old source marks retained; an omitted
attachment additionally has the explicit new mark $x_{b_0}$.

\smallskip\noindent
\emph{Birth of the target-origin exception.}
Suppose the post-event component of $M$ contains a mark with
origin $b$.  If it is a proper marked vertex, enlarge the source
to that surviving component.  If it is a clique-node mark,
temporarily adjoin the deleted vertex $b$ through that clique.
In either case the enlarged module meets the current event only
at $b$; two intersections would create an incidence cycle.
Thus \eqref{src:virtual-target} applies.
In each of its summands the $M$-to-$a$ branch and the
$a$-to-$b$ branch cover the phase attachments.  For the first
branch, source factorization through the enlarged module supplies
the same port product as the $b$-to-$a$ branch, without dividing
by a virtual port.  Extract both whole products before applying
the old states.  Equation~\eqref{src:mark-table} shows that
$v_a/\theta$ needs at most one occurrence at each internal
clique mark, which these branches supply.  There is no inverse
$x_b$ in the formula.  The unweighted state requires no new
origin-$b$ mark; multiplying it by $x_b$ supplies exactly that
occurrence in the weighted state.

\smallskip\noindent
\emph{Folding newly omitted clique members.}
Let a newly omitted $b$ belong to a surviving clique.  For a
post-source $N$ carrying that clique node, adjoin $b$ to form
$N^+$.  Before folding, the omitted-port update has supplied
one explicit $x_b$ to its root and exterior live-target frames.
As $b\notin J\cup A$,
\[
 q_N^{\mathrm{src}}=q_{N^+}^{\mathrm{src}},\qquad
 P_{Nj}^{\mathrm{src}}=P_{N^+j}^{\mathrm{src}}
                    \quad(j\notin N^+\cup J\cup A):
\]
the determinant index sets are the same, and the source rows
agree toward those sets and the exterior target.  Thus the
factor migrates to the clique node.  Repeating this equality
for several omitted members multiplies all their factors.
The excluded target $j=b$ is exactly the separate exception
handled in the preceding paragraph.  Proper marked vertices
do not migrate.  Prune every newly ready block.  The component
argument at the beginning of the section remains valid, so
all five states have now been proved at the new prefix.

\smallskip\noindent
\emph{The complete affine recovery.}
Split \eqref{src:full-root} as
\begin{equation}\label{src:root-decomposition}
 R_U=r_U+\sum_{h\text{ old omitted}}x_hb_{Uh}
                +\sum_{j\text{ other exterior}}b_{Uj}x_j.
\end{equation}
The first paragraph of the induction supplies its whole phase
product, with the target-origin exception when a target is itself
a required mark.  The paired-source argument supplies the same
product in every summand of $R_0$; internal marks retain at
least one occurrence after the $v_av_b$ factors, as established
after \eqref{src:mark-table}.  In
\eqref{src:selected-recovery}, the remaining factors
$\kappa_Av_a$ have nonnegative internal-mark exponents.
Its cross terms have their opposite-colour source quotients.
This proves positive affine recovery in the exterior live
originals.  Omitted originals have already been recovered
using \eqref{src:omitted-recovery}, which does not involve the
full affine root.  At the final event there are no unrecovered
exterior originals except retained coordinates.  Recover selected
originals in reverse event order to obtain positive Laurent
expressions for all originals in the candidate coordinates.

The order of the proof is noncircular: old source states give
the selected diagonal and omitted recovery, then nested factors,
root and live states, old targets, new targets, and finally the
folded state.  The selected affine recovery uses the old full
root and the already extracted source products.  No new state
is used to justify its own positivity.
\end{proof}

\begin{lemma}[A retained coordinate kept free]\label{src:free-original}
Let $u$ be a retained original.  Until the first event making $u$
ready, the preceding phase factors, nested factors, source
coefficients, constant root coefficients, and recovered omitted
values admit expressions independent of the coordinate $x_u$.
\end{lemma}

\begin{proof}
No preceding readiness gauge contains $x_u$.  No preceding
selected proper core contains $u$, and a preceding selected
clique containing $u$ would already make it ready.  Thus none
of the physical core factors used before that event contains
$x_u$ as a retained-member factor.  Start with the constant
source states.  The phase formula, selected diagonal, nested
update, and source updates use only preceding states and the
new prefix/core coordinates.  Omitted recovery
\eqref{src:omitted-recovery} uses no full affine root.
Induction therefore gives the stated independence.  The full
affine root can contain $b_{Uu}x_u$; it is kept separate and is
not used in any of the preceding independence statements.
\end{proof}
\section{Positive identities in every direction}
\label{edge:section}

We now identify the changes of Proposition~\ref{evt:closure} with
ordinary LP mutation.  All coefficients in this section belong to
$K_{\geq0}$.  Positivity of a rational function will always mean a
Laurent expression in the indicated shared coordinates, rather than
positivity of its values alone.

We retain the event notation of Section~\ref{src:section}.
In particular, $s_e$ is the physical two-port core, $q_e=q_{B_e}$
is the actual core coordinate, and
\[
 s_e=\begin{cases}q_e,&B_e\text{ proper},\\
 \gamma_e H_e q_e,&B_e\text{ a clique}.
 \end{cases}
\]
Here $\gamma_e$ is a monomial in retained originals, and $H_e$
is the product of its old omitted members.  The latter factors need
not be monomials in the current coordinates.  They are cancelled
using Lemma~\ref{src:factorization} and
Proposition~\ref{src:invariant}, before any Laurent-positivity conclusion
is drawn.

For reference, the exponent calculation used below has a particularly
short form.  Root an event at a selected vertex, and fix an occurrence
of $h=x_h$ at a clique edge $d$.  Deleting $d$ gives the root side
$L$ and the other side $R$.  If $\epsilon_i$ is its exponent in $v_i$
and $\epsilon_\kappa$ its exponent in $\kappa_A$, then
\begin{equation}\label{edge:mark-table}
\begin{array}{c|ccc|c}
 d&\epsilon_i\ (i\in L)&\epsilon_a\ (a\in A\cap R)
 &\epsilon_b\ (b\in B\cap R)&\epsilon_\kappa\\ \hline
 A_LB_R&0&1&-1&0\\
 B_LA_R&0&-1&1&1
\end{array}
\end{equation}
It follows directly from $s_e v_i v_j=1$ along the unique paths of
the tree.  For the last column use
$|A\cap R|-\sum_{b\in B\cap R}(\deg_T(b)-1)=0$ or $1$,
according to which endpoint of $d$ lies in $R$.
Thus $\kappa_Av_a$ and $\kappa_A/v_b$ have exponent zero or one.
A negative exponent in $v_i/v_j$ for equal colours, or in $v_i v_j$
for opposite colours, has size at most one and occurs only if the
path from $i$ to $j$ crosses $d$.
The source factorization retains the clique node after deleting its
target vertex.  Consequently a coefficient in such a crossing
direction supplies that occurrence of $h$.  The product of the two
opposite source coefficients supplies it twice on the path, and at
least once off the path.  These statements count labelled occurrences;
they remain valid when several occurrences represent the same function.

\subsection{Cutting a core}

Let $T=A\sqcup B$ be an event after $J$.  Cut the edge $e=a_0b_0$
with $a_0\in A$, $b_0\in B$, and let $W$ be its $b_0$-side.
Put $I=A\cap W$.  For $I\ne\varnothing$ set
\[
 \Delta_I=\prod_{b\in B\cap W}x_b^{\deg_W(b)-1},\qquad
 U_I=\frac{\det K^J[I,I]}{\Delta_I},\qquad
 \kappa_I=\frac{1}{\displaystyle\prod_{a\in I}v_a
                    \prod_{b\in B\cap W}v_b^{\deg_W(b)-1}}.
\]
For an edge $d$ not internal to $W$, the whole of $W$ lies on one
side of $T-d$.  Let $\beta_W$ be the product of the port factors
$f_{d,b(d)}$ for which this side contains the omitted endpoint $b(d)$.
Clique port factors are one.  In particular, the factor of the cut
edge is $f_{e,b_0}$.

\begin{lemma}\label{edge:cut}
One has
\begin{equation}\label{edge:raw-cut}
 U_I=\kappa_I\beta_W
 \left(\theta+\sum_{a\in A\setminus I}
              \frac{k_{b_0a}\Lambda_a}{T_{b_0}}v_a\right).
\end{equation}
After the canonical gauges are inserted, this gives the positive
shared-coordinate identity
\begin{align}\label{edge:canonical-cut}
 q_e p_{J\cup I}
 ={}&\frac{q_e\kappa_I\beta_W}{\kappa_A}
      \frac{H_A}{H_I}\frac{m_A}{m_I}p_{J\cup A}\notag\\
 &+\frac{p_J}{m_I}\,q_e\kappa_I
       \sum_{a\in A\setminus I}
          \frac{\beta_W k_{b_0a}\Lambda_a}{T_{b_0}H_I}v_a.
\end{align}
If $I=\varnothing$, then $W=\{b_0\}$, and the corresponding
identity is
\begin{equation}\label{edge:leaf-cut}
 q_e x_{b_0}=q_e v_{b_0}\Lambda_{b_0}
 \left(\frac{H_A m_A}{\kappa_A}\frac{p_{J\cup A}}{p_J}
       +\sum_{a\in A}\frac{k_{b_0a}\Lambda_a}{T_{b_0}}v_a\right).
\end{equation}
\end{lemma}

\begin{proof}
First normalize the effective row roots as in
Proposition~\ref{src:event-identities}.  In these normalized coordinates
the selected matrix, after its columns are multiplied by the tree
monomials, is a rooted Laplacian with common root $\theta$.
Every row in the branch $W$ has the same coefficients toward
$A\setminus I$, namely those of its port $b_0$.  Its restricted
matrix therefore has common root
$\theta+\sum_{a\notin I}c_{b_0a}\widetilde v_a$.
The tree determinant formula of Lemma~\ref{src:monic-tree}
applied to $W$ gives its principal determinant as this root times
\[
 \prod_{b\in B\cap W}
 \left(\frac{y_b}{\widetilde v_b}\right)^{\deg_W(b)-1}.
\]
Dividing by the column monomials and the internal gauge proves the
normalized branch identity.

We give the factors in pulling it back, since they determine the
canonical core exchange.  Write the original effective roots as
$a_i=\delta T_i$.  The normalized tree monomials are
\[
 \widetilde v_i=g_i v_i,\qquad
 g_a=\Lambda_a/T_a\ (a\in A),\qquad
 g_b=\Lambda_b/(\delta T_b)\ (b\in B).
\]
Put $R_I=\prod_{b\in B\cap W}a_b^{\deg_W(b)-1}$.
The normalized principal is $R_IU_I$.  In
$\widetilde\kappa_I/R_I$, all ports on edges internal to $W$
cancel.  On each external edge, the port facing a selected endpoint
cancels, while the port facing an omitted endpoint occurs once.
This follows by counting vertices on its $W$-side; for an internal
edge the needed equality is
$\sum_{b\in B\cap W}(\deg_W(b)-1)=|I|-1$.
Thus
\[
 \widetilde\kappa_I/R_I=\kappa_I\beta_W.
\]
The normalized shifted root pulls back to the parentheses
in~\eqref{edge:raw-cut}, proving that identity.  Substitution of
\eqref{evt:gauge-formula} gives~\eqref{edge:canonical-cut}.

We verify positivity of its quotients.  Every old boundary occurrence
in $H_I$ is also in $H_A$: an exterior attachment meets the connected
event at only one current vertex.  For $a\notin I$ the source branch
from $b_0$ towards $a$ contains all of $W$.  Enlarge it by all the
$H_I$ attachments.  The target stays outside it, and
Lemma~\ref{src:factorization} simultaneously extracts $H_I$ and
the required port factors.  Specifically, an uncancelled negative
port in $\beta_W\Lambda_a/T_{b_0}$ is either on an internal edge
of $W$, at the endpoint nearest $b_0$, or on the path towards $a$
at a selected endpoint facing $b_0$.  The latter cannot be the
final edge incident to $a$.  Both kinds belong to this source
branch; all off-path ports cancel in the displayed quotient.

It remains to check marks in physical clique cores.  Root the tree
at $a_0$.  For a mark on the cut edge, $\kappa_I$ has exponent
$-1$, whereas $\kappa_A$ and all $v_a$ outside $W$ have exponent
zero.  Its occurrence in $H_A$, absent from $H_I$, cancels the
first term; the source towards $a$ supplies it in the second.
For a mark on an edge strictly inside $W$, $\kappa_I/\kappa_A$
has exponent zero and $H_A/H_I$ has exponent zero.  In the second
term $\kappa_I$ has exponent zero or one, and the source supplies
the required occurrence of $H_I$.
For an edge outside $W$, $H_I$ has exponent zero and $H_A$ has
exponent one.  In the first row of~\eqref{edge:mark-table},
$\kappa_I$ has exponent $-1$ when $W\subset R$ and zero otherwise.
In the second row it has exponent $1$ when $W\subset R$ and zero
otherwise, while $\kappa_A$ has exponent one.  The first term
therefore has nonnegative exponent.  A deficit in the second term
occurs only when $W$ and its target are separated by that edge;
the source factor supplies it.  All these extractions are made
in the same enlarged source factor as the port extractions, so
their multiplicities are retained.

With root $a_0$, the exponent of $q_e$ in $\kappa_I$ is $-1$,
and its exponents in $\kappa_A$ and in all $v_a$, $a\notin W$,
are zero.  Earlier source functions contain no current core
coordinate.  The right-hand side is therefore independent of
$q_e$ and of its proposed partner.

If $I=\varnothing$, use the omitted recovery formula directly.
It gives~\eqref{edge:leaf-cut}.  Now $q_ev_{b_0}=1$ for a proper
edge, and $q_ev_{b_0}=1/(\gamma_e H_e)$ for a clique edge.
The first term supplies $H_e$ through $H_A$, and each source
in the second supplies it through the central clique node.
The preceding port and mark checks apply to the other edges.
\end{proof}

\subsection{Interchanging events and changing the last colour}

For residual-disjoint events $C,D$ with selected colours $A_C,A_D$,
let $b_{Cj}$ denote the source coefficient of the full residual
module of $C$.  Define
\[
 \lambda_{CD}=\sum_{j\in A_D}b_{Cj}\Lambda_j^D v_j,
 \qquad
 \lambda_{DC}=\sum_{i\in A_C}b_{Di}\Lambda_i^C v_i.
\]

\begin{lemma}\label{edge:swap}
When both event orders are allowed,
\begin{align}\label{edge:swap-formula}
 p_{J\cup A_C}p_{J\cup A_D}
 ={}&X(J;A_C,A_D)p_Jp_{J\cup A_C\cup A_D}\notag\\
 &+\frac{p_J^2\kappa_C\kappa_D}{m_Cm_D}
       \frac{\lambda_{CD}\lambda_{DC}}{H_CH_D}
\end{align}
is positive Laurent in the shared coordinates.  A blocked-order
merger through an unselected block has the inverse exchange of
Lemma~\ref{edge:cut}.
\end{lemma}

\begin{proof}
The source factorization gives
$k_{ij}=T_i^C b_{Cj}$ for $i\in A_C,j\in A_D$.
The selected row-sum identity gives
\[
 (K^J[A_C,A_C])^{-1}T^C
   =(\Lambda_i^Cv_i/\theta_C)_{i\in A_C}.
\]
The two cross blocks consequently have rank one.  The determinant
lemma applied to these two blocks gives
\[
 U_CU_D=U_{CD}+\kappa_C\kappa_D\lambda_{CD}\lambda_{DC},
 \qquad
 U_{CD}=\frac{\det K^J[A_C\cup A_D,A_C\cup A_D]}
                    {\Delta_C\Delta_D}.
\]
Inserting the three gauges gives exactly
\eqref{edge:swap-formula}; in particular the first coefficient is
the cross gauge of Lemma~\ref{evt:gauge}.

Enlarge the full $C$ source by its old phase attachments before
taking the coefficient towards $j\in A_D$.  Its factorization
supplies the whole product $H_C$.  The reverse source supplies
$H_D$.  After these products are removed, a summand of the second
term contains $\kappa_Cv_i$ and $\kappa_Dv_j$.
Their mark exponents are nonnegative by~\eqref{edge:mark-table}.
If the same old original occurs in both products, the two source
factors supply both occurrences.  The port factors are extracted
in those same factorizations.  Every earlier principal, every
current core, and the later union principal are shared.  The
formula contains neither intermediate principal.  The merger
statement follows from the inverse cut and
Proposition~\ref{evt:closure}.
\end{proof}

For a residual event module $M$, its complete affine root is
\begin{equation}\label{edge:full-root}
 R_M=r_M+\sum_{h\text{ old omitted, }h\notin M}x_hb_{Mh}
       +\sum_{j\text{ other exterior}}b_{Mj}x_j.
\end{equation}
The second sum includes every untouched exterior original.
Terms whose target is in the selected prefix are already included
in the source functions and are not repeated.

\begin{lemma}\label{edge:last}
For a nontrivial last event $T=A\sqcup B$,
\begin{align}\label{edge:last-formula}
 p_{J\cup A}p_{J\cup B}
 =\frac{p_J^2}{m_Am_B}\left(\prod_{e\in T}s_e\right)
 \frac{R_M+\displaystyle\sum_{a\in A,b\in B}
       \frac{k_{ab}k_{ba}\Lambda_a\Lambda_b}{T_aT_b}v_av_b}
      {H_AH_B}
\end{align}
is a positive shared Laurent identity.  For a last singleton $v$,
the identity is
\begin{equation}\label{edge:last-singleton}
 x_vp_{J\cup\{v\}}=
 \frac{p_J}{m_v}
 \frac{r_{\{v\}}+\sum_{j\notin J\cup\{v\}}k_{vj}x_j}{H_v}.
\end{equation}
\end{lemma}

\begin{proof}
Write the sum in~\eqref{edge:last-formula} as $R_0$.
The two recovery formulas in Proposition~\ref{src:event-identities}
show that changing the colours replaces the phase $\theta$ by
$(R_M+R_0)/\theta$.
Also
\[
 \kappa_A\kappa_B
 =\prod_{i\in T}v_i^{-\deg_T(i)}
 =\prod_{e\in T}s_e.
\]
Thus the raw identity is
$U_AU_B=(\prod_e s_e)(R_M+R_0)$, and the canonical gauges give
\eqref{edge:last-formula}.

The port denominators in a pair term are supplied by its two
opposite source branches.  An off-path port lies in at least one
branch.  On the endpoint path the uncancelled pair of port
denominators lies on an interior edge and is supplied by the
two branches separately.  Enlarge those same branches by the
$H_A,H_B$ boundary attachments.  Such an attachment cannot meet
both colours of the connected event without giving another
incidence path.  An internal clique mark occurs once in
$\prod_e s_e$ and twice in $H_AH_B$.
Its opposite source product supplies two copies if the endpoint
path crosses the marked edge, and at least one otherwise.
The exponent in $v_av_b$ is at worst $-1$ in the first case,
and zero in the second.  Thus each pair term retains the one
additional occurrence required to cancel the denominator.
For the root term Proposition~\ref{src:invariant} gives the same
product extraction from~\eqref{edge:full-root}.  If its target
is itself a mark, the unweighted target exception omits precisely
that occurrence and the explicit factor $x_h$ restores it.
This proves the stated Laurent positivity.

No phase remains on the right.  Because this is the last event,
all exterior originals are retained or previously recovered
omitted originals, hence have positive expressions in shared
coordinates.  For a singleton the Schur row equation, multiplied
by $p_J/(H_vm_v)$, is~\eqref{edge:last-singleton}; the same root
argument cancels its whole mark product.
\end{proof}

\subsection{A proper block with three residual ports}

Consider the proper-block rotation of
Proposition~\ref{evt:closure}.  The central block has residual
ports $h,v,\ell$.  The three attached trees are denoted by
$H,V',L$, to distinguish the second branch from the full vertex
set.  Their selected colours are $A_H,A_{V'},A_L$, where
$h\in B_H$, $v\in A_{V'}$, and $\ell\in A_L$.
First assume $A_H\ne\varnothing$.
Their port products, monomials and internal gauges carry branch
superscripts.  Write
\[
 T_h=T_h^H,\quad T_v=T_v^{V'},\quad T_\ell=T_\ell^L,
 \quad S=A_{V'}\cup A_L,\quad K_*=S\cup A_H,
 \quad U_* =\frac{\det K^J[K_*,K_*]}
                   {\Delta_{V'}\Delta_L\Delta_Hx_h}.
\]
Let $f_{0,h},f_{0,v},f_{0,\ell}$ be the three central port
factors, and let $q_0$ be its proper core.  Define
\begin{align*}
 \lambda_{V'L}&=\sum_{j\in A_L}(k_{vj}/T_v)\Lambda_j^L v_j,&
 \lambda_{LV'}&=\sum_{i\in A_{V'}}(k_{\ell i}/T_\ell)
                                         \Lambda_i^{V'}v_i,\\
 L_{ha}^*&=\frac{\Lambda_h^H\Lambda_a^H k_{ha}}
                   {T_h f_{0,h}T_vT_\ell}\qquad(a\in A_H).
\end{align*}

\begin{lemma}\label{edge:proper-rotation}
The raw identity for this rotation is
\begin{align}\label{edge:proper-raw}
 U_{V'}U_L={}&\frac{\Lambda_h^Hv_h}{\kappa_H}U_*\notag\\
 &+\kappa_{V'}\kappa_L\left(
 \lambda_{V'L}\lambda_{LV'}+
 \Lambda_v^{V'}\Lambda_\ell^L v_hv_vv_\ell q_0
                  \sum_{a\in A_H}L_{ha}^*v_a\right).
\end{align}
Its canonical form is positive Laurent in the shared coordinates.
If $H=\{h\}$, put
$U_* =\det K^J[S,S]/(\Delta_{V'}\Delta_L)$ instead; the identity
becomes
\begin{equation}\label{edge:proper-singleton}
 U_{V'}U_L=U_*+\kappa_{V'}\kappa_L
                        \lambda_{V'L}\lambda_{LV'}.
\end{equation}
\end{lemma}

\begin{proof}
We first prove the identity with normalized effective roots, so
that all port factors are one.  Denote the corresponding matrix
by $M$, its coefficients by $c_{ij}$, and the tree monomials by
$w_i$.  The central three-vertex proper core $q_0$ satisfies
\begin{equation}\label{edge:central-adjoint}
 \adj(M[\{h,v,\ell\},\{h,v,\ell\}])\ones
            =q_0(x_h,x_v,x_\ell)^{\mathsf T}.
\end{equation}
Indeed, the product of this matrix with the original coordinate
column is the common external root times $\ones$; its determinant
is that root times $q_0$.  Multiplication by the adjugate proves
\eqref{edge:central-adjoint} as a polynomial identity.

Put $Q_S=\det(M[S,S]+\ones c_{h,S})$.
Multiply its column $i$ by $w_i$.  Inside the $V'$ branch the
row at $v$ has zero entries in every nonport column, since
$c_{hj}=c_{vj}$ for $j$ outside the central block.  The same
holds at $\ell$.  Cross blocks between the two branches likewise
have nonzero entries only in their port columns.
After these two ports are put first, the matrix is block lower
triangular.  Its port block is
\[
 \begin{pmatrix}
  w_v(z_v+c_{hv})&w_\ell(c_{h\ell}-c_{v\ell})\\
  w_v(c_{hv}-c_{\ell v})&w_\ell(z_\ell+c_{h\ell})
 \end{pmatrix}.
\]
Its determinant is $w_vw_\ell x_hq_0$, by the $h$ component
of~\eqref{edge:central-adjoint}.
For a branch $R$ the remaining deleted-port determinant is
\[
 D_R=\prod_{b\in B_R}(\theta_R+l_b)^{\deg_R(b)-1},
 \qquad l_b=\sum_{a\in A_R}c_{ba}w_a.
\]
To verify this last identification directly, the full selected
branch matrix is $\theta_R I+L_R$, with $L_R\ones=0$ and
$\det(\theta_R I+L_R)=\theta_R D_R$.
For every row vector $a^{\mathsf T}$ the determinant lemma gives
\[
 \det(\theta_R I+L_R+\ones a^{\mathsf T})
                   =D_R(\theta_R+\sum_i a_i).
\]
In the above triangular matrix its isolated port entry is exactly
the final linear factor.  Cancelling it proves the deleted-port
assertion.  Dividing the column scalings and the branch gauges now
gives
\begin{equation}\label{edge:proper-shifted}
 \frac{Q_S}{\Delta_{V'}\Delta_L}
       =\kappa_{V'}\kappa_L w_vw_\ell x_hq_0.
\end{equation}

Let $Y_S=\det M[S,S]$ and
$A_S=c_{h,S}\adj(M[S,S])\ones$, so $Q_S=Y_S+A_S$.
The cross blocks between $S$ and $A_H$ have constant source rows.
The determinant lemma and the selected row-sum identity give
\[
 \frac{\det M[K_*,K_*]}{\Delta_{V'}\Delta_L\Delta_H}
 =\kappa_H\left(
    \theta_H\frac{Y_S}{\Delta_{V'}\Delta_L}
       -l_h\frac{A_S}{\Delta_{V'}\Delta_L}\right).
\]
Substitute $x_h=w_h(\theta_H+l_h)$ and
\eqref{edge:proper-shifted}, and divide the exact additional
factor $x_h$ in the union gauge.  The result is
\[
 \frac{Y_S}{\Delta_{V'}\Delta_L}
 =\frac{w_h}{\kappa_H}U_*
   +w_hl_h\kappa_{V'}\kappa_Lw_vw_\ell q_0.
\]
The two-branch determinant lemma gives
\[
 U_{V'}U_L=\frac{Y_S}{\Delta_{V'}\Delta_L}
 +\kappa_{V'}\kappa_L
    \left(\sum_{j\in A_L}c_{vj}w_j\right)
    \left(\sum_{i\in A_{V'}}c_{\ell i}w_i\right).
\]
Combining these two equations proves the normalized rotation.
If $A_H=\varnothing$, the first elimination is absent and the
second equation alone is the required identity, with no additional
$x_h$ in its gauge.

We record the complete change of scale.  For the full local
configuration write
\[
 \delta=\frac{q_M^{\rm src}}{Y_J},\quad
 \Omega_R=f_{0,p_R}\prod_{D\ne R}T_{p_D},\quad
 \delta_R=\delta\Omega_R,
\]
where $p_H=h,p_{V'}=v,p_L=\ell$.
The effective root at $i\in R$ is $\delta_RT_i^R$.
The normalized monomials are
\[
 \widetilde v_i=\frac{\Lambda_i^R}{T_i^R}v_i
       \quad(i\in A_R),\qquad
 \widetilde v_i=\frac{\Lambda_i^R}{\delta_RT_i^R}v_i
       \quad(i\in B_R).
\]
If $R_R=\prod_{b\in B_R}(\delta_RT_b^R)^{\deg_R(b)-1}$,
then $\widetilde U_R=R_RU_R$ and
$\widetilde\kappa_R=R_R\kappa_R$.
The normalized central proper core is
$\delta T_hT_vT_\ell q_0$, by the residual-block scaling
in Lemma~\ref{src:block-scaling}.
Finally
$\widetilde U_*=R_{V'}R_LR_H a_hU_*$ when $H$ is nontrivial.
Inserting these expressions in the normalized identity cancels
every $\delta$ and gives precisely~\eqref{edge:proper-raw}.
For example, the first coefficient becomes
$a_h(\widetilde v_h/v_h)v_h/\kappa_H
=\Lambda_h^Hv_h/\kappa_H$; the two cross coefficients acquire
the reciprocal factors $\delta_L/\delta_{V'}$ and
$\delta_{V'}/\delta_L$, so their product becomes the displayed
$\lambda_{V'L}\lambda_{LV'}$; the central-core coefficient becomes
$L_{ha}^*$.  The singleton case gives
\eqref{edge:proper-singleton} without that additional scale.

We finish with the canonical divisors.  For old boundary marks,
connectedness of the whole configuration gives
$H_*=H_{V'}H_LH_H$.  This refers to old marks only; its extra
current $x_h$ was included in $U_*$.
Thus the canonical first coefficient is
\[
 \frac{\Lambda_h^Hv_h}{\kappa_H}H_H
                 \frac{m_*}{m_{V'}m_L}p_Jp_{J\cup K_*}.
\]
Its internal $H$ marks are nonnegative because
$\kappa_H/v_h$ has exponent zero or one.
The cross sources in the second term supply $H_{V'}$ and $H_L$.
In the central-core term, the source from $h$ towards $a\in A_H$
contains the central proper block and all of the $V',L$ branches.
One application of Lemma~\ref{src:factorization} extracts
$f_{0,h}T_vT_\ell H_{V'}H_L$ and the remaining source-side
$H$-branch port factors.  Any negative internal $H$ mark in
$v_hv_a$ lies on the endpoint path and is supplied there.  The
remaining $V',L$ monomials are $\kappa_{V'}v_v$ and
$\kappa_Lv_\ell$, which have nonnegative exponents.
This proves positivity after all gauge and port cancellations.
The same cross-source extraction proves the singleton case.
Every factor used lies in the shared earlier coordinates, cores,
or the later union principal.
\end{proof}

\subsection{An affine clique with three residual ports}

For the clique rotation denote the three branches by $A',B',U'$,
with central ports $a,b,u$, where $a$ is selected and $b,u$ are
omitted.  Branch notation is otherwise unchanged.  The central
physical core is
\[
 r_0=\gamma_0H_0q_0=\frac{P_{B_0}}{x_ax_bx_u}.
\]
Here $\gamma_0$ and $H_0$ contain the retained and old omitted
members of the clique other than these three ports.
Abbreviate the branch port products at the ports by
$T_a,T_b,T_u,\Lambda_a,\Lambda_b,\Lambda_u$, and put
\begin{align*}
 D_b&=\sum_{i\in A_{B'}}k_{bi}\Lambda_i^{B'}v_i,&
 D_u&=\sum_{i\in A_{U'}}k_{ui}\Lambda_i^{U'}v_i,\\
 D_a&=k_{ba}\Lambda_av_a+
                  \sum_{i\in A_{A'}\setminus\{a\}}
                                  k_{ai}\Lambda_i^{A'}v_i.
\end{align*}
The central clique equalities give $k_{ba}=k_{ua}$.
When both omitted branches are nontrivial, let $W$ be the selected
union principal divided by
$\Delta_{A'}\Delta_{B'}\Delta_{U'}x_bx_u$.

\begin{lemma}\label{edge:clique-rotation}
For two nontrivial omitted branches one has
\begin{align}\label{edge:clique-raw}
 U_{B'}U_{U'}={}&\frac{T_a}{\Lambda_ar_0\kappa_{A'}v_a}W
 +\kappa_{B'}\kappa_{U'}\frac{D_bD_u}{T_bT_u}\notag\\
 &+\frac{\kappa_{B'}}{v_b}\frac{\kappa_{U'}}{v_u}
       \frac{D_a}{r_0v_a\Lambda_a\Lambda_b\Lambda_u}.
\end{align}
If $U'=\{u\}$, omit its $x_u$ from the union gauge.  Then
\begin{equation}\label{edge:clique-one-singleton}
 U_{B'}x_u=\frac{T_a}{\Lambda_ar_0\kappa_{A'}v_a}W
       +\frac{\kappa_{B'}}{v_b}
                     \frac{D_a}{r_0v_a\Lambda_a\Lambda_b}.
\end{equation}
If $B'=\{b\}$ and $U'=\{u\}$, the identity is
\begin{equation}\label{edge:clique-two-singletons}
 x_bx_u=\frac{T_a}{\Lambda_ar_0\kappa_{A'}v_a}U_{A'}
                     +\frac{D_a}{r_0v_a\Lambda_a}.
\end{equation}
The case in which only $B'$ is a singleton is obtained by
interchanging $b$ and $u$.  All these identities become positive
shared Laurent identities after their canonical gauges are inserted.
\end{lemma}

\begin{proof}
Begin with normalized effective row roots.  Let $t_a,t_b,t_u$
be the three branch phases, and let $L_a,L_b,L_u$ be their
cross-row sums.  More explicitly,
\[
 L_b=\sum_{i\in A_{B'}}c_{bi}w_i,\quad
 L_u=\sum_{i\in A_{U'}}c_{ui}w_i,\quad
 L_a=c_{ba}w_a+
                   \sum_{i\in A_{A'}\setminus\{a\}}c_{ai}w_i.
\]
The outgoing pair equalities make each cross block have constant
rows.  The selected row-sum identity and the determinant lemma
reduce the selected union determinant, after removing its branch
factors, to
\begin{equation}\label{edge:clique-matrix}
 Q=\diag(t_a+L_a,t_b+L_b,t_u+L_u)
                            -\ones(L_a,L_b,L_u).
\end{equation}
Set $K_0=\widetilde r_0w_aw_bw_u$.
The endpoint equations and the affine clique numerator give
\[
 y_b=w_b(t_b+L_b),\qquad y_u=w_u(t_u+L_u),\qquad
 t_a+L_a=K_0(t_b+L_b)(t_u+L_u).
\]
Indeed, $z_a+c_{ba}$ is the affine clique numerator divided by
$y_a$, so multiplication by $w_a$ gives the last equation.
Expanding the rank-one determinant in~\eqref{edge:clique-matrix}
and substituting this equation yields
\[
 \det Q=(t_b+L_b)(t_u+L_u)
             (K_0t_bt_u-K_0L_bL_u-L_a).
\]
The additional union gauge is exactly $y_by_u$.  Consequently
\[
 \widetilde W=\frac{\widetilde\kappa_{A'}
                         \widetilde\kappa_{B'}
                         \widetilde\kappa_{U'}}{w_bw_u}
                   (K_0t_bt_u-K_0L_bL_u-L_a),
\]
which rearranges to
\begin{equation}\label{edge:clique-monic}
 \widetilde U_{B'}\widetilde U_{U'}
 =\frac{w_bw_u}{\widetilde\kappa_{A'}K_0}\widetilde W
  +\widetilde\kappa_{B'}\widetilde\kappa_{U'}L_bL_u
  +\frac{\widetilde\kappa_{B'}\widetilde\kappa_{U'}L_a}{K_0}.
\end{equation}
This calculation uses the entire determinant; no sign is assigned
to its individual subtracted intermediate terms.

The common effective root at $a,b,u$ is $\alpha$, because the
central clique has no selected member in the old prefix.
Its affine numerator scales by $\alpha^{-1}$ under root
normalization, and its three-variable denominator by
$\alpha^{-3}$.  Hence
\[
 \widetilde r_0=\alpha^2r_0,
\]
where the square is essential.  The normalized endpoint monomials
and cross-row sums are
\[
 \begin{gathered}
 w_a=(\Lambda_a/T_a)v_a,\qquad
 w_b=(\Lambda_b/\alpha)v_b,\qquad
 w_u=(\Lambda_u/\alpha)v_u,\\
 L_a=D_a/T_a,\qquad L_b=D_b/T_b,\qquad L_u=D_u/T_u.
 \end{gathered}
\]
Thus $K_0=r_0v_av_bv_u\Lambda_a\Lambda_b\Lambda_u/T_a$.
For $R\in\{A',B',U'\}$ put
$\zeta_R=\prod_{i\in B_R}a_i^{\deg_R(i)-1}$.
The selected determinant and core-monomial scalings are
\[
 \widetilde U_R=\zeta_RU_R,\quad
 \widetilde\kappa_R=\zeta_R\kappa_R,\quad
 \widetilde W=\zeta_{A'}\zeta_{B'}\zeta_{U'}\alpha^2W.
\]
Substitution in~\eqref{edge:clique-monic} cancels $\alpha$
and all $\zeta$ factors and gives~\eqref{edge:clique-raw}.

For a singleton omitted branch, use the two-by-two version of
\eqref{edge:clique-matrix}.  Equivalently in the preceding
determinant calculation put $w_u=\widetilde\kappa_{U'}=1$,
$t_u=y_u$, and $L_u=0$, and divide the union determinant by
the additional factor $y_b$ only.  It gives
\[
 \widetilde U_{B'}y_u
   =\frac{w_b}{\widetilde\kappa_{A'}K_0}\widetilde W
        +\frac{\widetilde\kappa_{B'}L_a}{K_0}.
\]
Here the singleton gauge is $w_u=1$, rather than the gauge of
a nonempty selected branch.  Thus
\[
 K_0=\frac{\alpha r_0v_av_b\Lambda_a\Lambda_b}{T_a},
 \qquad
 \widetilde W=\zeta_{A'}\zeta_{B'}\alpha W,
 \qquad y_u=x_u/\alpha.
\]
Together with the unchanged $A',B'$ scalings these equations
give~\eqref{edge:clique-one-singleton}.
When both branches are singletons use the central relation itself:
$\widetilde r_0y_by_u=(t_a+L_a)/w_a$.
It gives~\eqref{edge:clique-two-singletons}.  This derivation
never replaces an empty selected branch by a principal coordinate.

We check the quotients in the final formulas.  The old marks
satisfy
\[
 H_W=H_{A'}H_{B'}H_{U'},
 \qquad H_0\mid H_{A'},
\]
with each occurrence of $H_0$ present exactly once in $H_{A'}$.
The central marks do not occur in $H_{B'}$ or $H_{U'}$.
After canonicalization, the first coefficient, apart from shared
retained-original monomials, is
\[
 \frac{H_{A'}T_a}{\Lambda_ar_0\kappa_{A'}v_a}.
\]
The quotient $T_a/\Lambda_a$ is a product of port factors.
The occurrence $H_0$ cancels that in $r_0$.
For every internal $A'$ mark, $\kappa_{A'}v_a$ has exponent
zero or one, and $H_{A'}$ supplies one.  Thus this whole coefficient
has a positive Laurent expression.

In a term of $D_bD_u$, with targets $i\in A_{B'}$ and
$j\in A_{U'}$, the source from $b$ towards $i$ contains the
central clique and the entire $U'$ branch.  The source from $u$
towards $j$ contains the entire $B'$ branch.
Lemma~\ref{src:factorization} therefore extracts $T_uH_{U'}$
and $T_bH_{B'}$ from these two sources, respectively.  The
remaining internal monomials are $\kappa_{B'}v_i$ and
$\kappa_{U'}v_j$, with nonnegative mark exponents.
For the last term every source in $D_a$ contains both opposite
branches and the central clique node.  For $k_{ba}$ delete the
target $a$ and retain the subclique $\{b,u\}$; for $k_{ai}$
delete $i$ on the $A'$ branch.  One source factorization thus
extracts $H_{B'}H_{U'}H_0$ together with the required opposite
branch ports.  The remaining monomials
$\kappa_{B'}/v_b$ and $\kappa_{U'}/v_u$ have nonnegative
exponents.  Within $A'$, the $k_{ba}$ summand cancels $v_a$;
in another summand a negative exponent in $v_i/v_a$ lies
on the $a$--$i$ path and is supplied by that same source.
The required port quotient inside $A'$ is the equal-colour path
quotient described immediately before \eqref{src:pair-port-quotient}
and supplied by Lemma~\ref{src:factorization}; all its negative
port occurrences lie strictly before the target.  This checks
the ports and marks in a single extraction, including coincident
factors.  The singleton formulas use the corresponding first
and last terms of this argument and introduce no extra divisor.
All earlier coordinates, cores and the union principal are shared.
\end{proof}

\subsection{Original variables and recognition of the native charts}

\begin{lemma}\label{edge:live-original}
The identities above in a direction which exchanges a retained
original $u$ are Laurent expressions in the genuinely shared
coordinates, with no occurrence of $u$ hidden in their coefficients.
\end{lemma}

\begin{proof}
Keep $u$ a free indeterminate while processing all events before
its first readiness event.  Its earlier readiness counts are zero.
No earlier used core contains it: a proper core has no retained
original member, and a used clique containing $u$ would make it
ready at that event.  The earlier canonical gauges and physical
core products are therefore independent of $u$.
The source recurrences start with numerical coefficients and use
only source coefficients, nested factors, and phases from these
earlier coordinates.  The omitted recovery formula uses the
constant root and the selected phase, and has no full affine-root
term.  Induction through these displayed recurrences proves that
all preceding source states, nested ports, phases, and recovered
omitted values are independent of $u$.  Weighted target states
are products of these same functions and retain that independence.

At the first readiness event through an unselected block use
\eqref{edge:leaf-cut} in reverse, with $u$ a residual endpoint;
in particular $u$ is excluded from $\gamma_e$.  This formula
contains no complete affine root.  If the completing clique is
already selected, keep $u$ as its third central port rather than
as a member of $\gamma_0$, and use
\eqref{edge:clique-one-singleton} or
\eqref{edge:clique-two-singletons}.  The readiness count at $u$
changes from zero to one, so its canonical gauge exponent is
still zero.  These right-hand sides are independent of $u$.
If $u$ never becomes ready, use~\eqref{edge:last-singleton};
its summation excludes $u$, and all its earlier source functions
are independent of it.  This also proves the independence needed
for the inverse singleton exit.  All other retained originals
remain shared Laurent units.
\end{proof}

\begin{theorem}\label{thm:strict-positive-labelled}
Suppose that $F_i=1+\sum_{j\ne i}C_{ij}x_j$ with every
$C_{ij}>0$.  All candidates of Definition~\ref{evt:definition}
are ordinary LP charts, up to positive coefficient rescaling,
and every ordinary LP chart occurs among them.  The normalized
pattern is finite.  With the displayed positive representatives,
every actual exchanged product has a nonnegative Laurent expansion
in its shared coordinates.  Consequently all cluster variables
are Laurent-positive in every cluster; moreover $\A=\U$, the
cluster monomials form an atomic positive basis, and their
multiplication structure constants are nonnegative.
\end{theorem}

\begin{proof}
Every proposed coordinate is a prime nonunit of $\U$, by
Lemma~\ref{lem:upper} and
Proposition~\ref{prop:canonical-coordinates}.  It is positive
on the initial positive orthant.  Proposition~\ref{evt:closure}
lists a proposed partner in every coordinate direction.
Lemmas~\ref{edge:cut}--\ref{edge:clique-rotation} and
\ref{edge:live-original} give a nonzero positive Laurent
identity for its product with that coordinate, involving only
the shared coordinates.  The initial candidate is an actual
LP chart.  If a candidate is actual, Lemma~\ref{lem:prime-recognition}
identifies each proposed prime partner with the actual mutation
partner up to a coefficient unit.  Choose the positive associate;
this unit is positive, as is seen by evaluating the product
identity on the positive initial orthant.
Induction on the length of a mutation word therefore puts every
actual chart in the candidate collection.  Conversely the
connectedness statement in Proposition~\ref{evt:closure}
connects every candidate to the initial one by these same
identities, and gives its actual realization by induction.

There are finitely many candidate coordinate sets by
Lemma~\ref{evt:elementary}.  In each set the exchange polynomial
in a given direction is recovered, up to a coefficient unit,
by removing the coordinate monomial from its exchanged product.
Thus the normalized seed pattern is finite as well.
All its actual exchanges have the positivity just proved.
Apply Theorems~\ref{thm:finite-transfer} and
\ref{thm:finite-basis} with the semiring $K_{\geq0}$ to obtain
the final assertions.
\end{proof}
\section{Zero weights and refinement by prime factors}
\label{sec:sparse}

We now allow $C_{ij}\geq0$, retaining the assumption that every
$F_i$ is nonconstant. A specialization of coefficients need not
commute with ordinary LP mutation: an exchange polynomial may
factor, and its exchange Laurent monomial may change. The
following argument constructs the limiting charts first and
then identifies their prime refinements with the actual sparse
mutation pattern.

\subsection{A finite positive limiting atlas}

Let $K_0$ be the subfield of $\R$ generated by the finitely many
entries of $C$. For an indeterminate $\varepsilon$ put
\begin{equation}\label{sp:completion}
 C_{ij}(\varepsilon)=C_{ij}+\varepsilon\qquad(i\ne j).
\end{equation}
Every difference of two rows toward an outside target is
unchanged. Thus all outgoing modules, minimal proper-type
blocks, and maximal pair cliques are exactly the same before
and after completion. In particular admissibility, the gauges
$G(J)$, and the finite candidate data are unchanged.

Apply Theorem~\ref{thm:strict-positive-labelled} to a positive
real $\varepsilon$ transcendental over the countable field
$K_0$. Evaluation identifies $K_0(\varepsilon)$ with its
real image. The resulting rational-function identities
therefore define one finite atlas over $K_0(\varepsilon)$,
with the coordinates
\[
 x_i,\qquad p_J(\varepsilon),\qquad q_B(\varepsilon).
\]
Each is a forest numerator of constant term one divided by
a fixed initial-coordinate monomial. Every coordinate has
a nonzero limit at $\varepsilon=0$, and its value at a
positive initial point has a finite strictly positive limit.
Write $\mathcal G$ for the finite set of these limiting
coordinates, identifying duplicates.

\begin{lemma}\label{sp:positive-limit}
Each limiting chart is a transcendence basis of $K(x)$.
Every member of $\mathcal G$ is a Laurent polynomial with
nonnegative coefficients in every limiting chart and belongs
to the upper algebra of the actual sparse LP seed.
Adjacent limiting charts satisfy a nonzero positive Laurent
exchange in their shared coordinates.
\end{lemma}

\begin{proof}
In a fixed generic chart, each Laurent coefficient of a fixed
generic coordinate is a rational function of $\varepsilon$.
The strict-positive theorem makes it nonnegative at every
positive transcendental value over $K_0$. These values are
dense. Each coefficient is consequently nonnegative on a
sufficiently small positive interval on which it has no pole.

Evaluate the Laurent expression at one fixed positive initial
point. Its left side remains bounded, and all its finitely
many terms are nonnegative. The chart coordinates are bounded
away from zero and infinity. Each coefficient is therefore
bounded as $\varepsilon\to0^+$, so its rational function has
a finite nonnegative limit. The same argument applies to the
shared-coordinate expression for each adjacent exchange
product. The product of its two limiting coordinates is
strictly positive at the chosen point; hence the limiting
exchange is not zero.

Starting at the initial chart and using connectedness of the
candidate graph, each adjacent limiting substitution
$f\mapsto E(\mathbf s)/f$, with $E\ne0$, is birational.
Thus all limiting charts remain transcendence bases, and the
coefficient limits just obtained are their actual Laurent
expansions.

In the complete pattern the first neighbor at $i$ has coordinate
$F_i(\varepsilon)/x_i$: all the complete initial hats are trivial.
Indeed a divisibility by $F_j(\varepsilon)$ would force a
polynomial independent of $x_i$ to be divisible by a polynomial
which involves $x_i$. Its limit is $F_i/x_i$. The actual sparse
first neighbor $\widehat F_i/x_i$ differs from it by a Laurent
monomial in the shared initial coordinates. Their Laurent rings
are therefore equal. Every $g\in\mathcal G$ is Laurent in the
initial chart and these first-neighbor charts. The starfish
identity, Lemma~\ref{lem:upper}, gives $g\in\U$.
\end{proof}

\subsection{Prime factors remain in the limiting set}

For an irreducible polynomial $P$ with no coordinate factor write
\[
 Q_P=P\prod_i x_i^{-\nu_i(P)}.
\]
By Lemma~\ref{lem:canonical-prime}, this is a prime element of
the actual sparse upper algebra, with zero valuation at every
initial coordinate prime.

\begin{lemma}\label{sp:core-primes}
Every limiting core $q_B$ is a prime element of $\U$.
Its numerator has valuation one at each member of $B$
and zero at every outside vertex.
\end{lemma}

\begin{proof}
For a proper-type minimal block, Proposition~\ref{for:irreducibility}
still gives absolute irreducibility: its statement allows
arbitrary real coefficients. At an inside vertex $i$, a
proper terminal component of the row-$i$ difference support
on $B\setminus\{i\}$ would give a proper nontrivial submodule.
That support is therefore strongly connected, and
\eqref{for:full-boundary} gives valuation one. At an outside
vertex $h$, setting $x_h=0$ merely changes the affine root.
The homogeneous restriction is still an invertible
$t$-translation of an absolutely irreducible polynomial of
degree $|B|-1\geq2$. It cannot be divisible by the linear
$F_h^h$. Equation \eqref{for:factor-boundary}, applied to
the core factor of $D_B$, gives valuation zero.

For a clique, its numerator $P_B$ is affine, and
$P_B|_{x_i=0}=F_i$ for every $i\in B$. These polynomials
are nonconstant by hypothesis, so $P_B$ is a nonconstant
irreducible affine polynomial. An outside positive valuation
would imply $P_B|_{x_h=0}=F_h$ and would join $h$ to every
pair in the clique, contrary to maximality. These arguments
do not require positive internal coefficients. The asserted
primality follows from the canonical-lift lemma.
\end{proof}

Let $J$ be admissible. If $\SCC(J)$ denotes the strongly
connected components of the support induced by $C[J]$,
block triangularity gives
\begin{equation}\label{sp:scc-factorization}
 D_J(t,x)=\prod_{S\in\SCC(J)}D_S(t,x).
\end{equation}
The diagonal entries in every factor retain all original
outside-target contributions.

\begin{lemma}\label{sp:principal-primes}
For each strongly connected admissible $S$, the polynomial
$D_S(1,x)$ is absolutely irreducible. Its canonical prime
lift is a member of $\mathcal G$: it is either the limiting
principal coordinate $p_S$ or a limiting core $q_B$.
\end{lemma}

\begin{proof}
Adjoin the zero-row vertex carrying all targets outside $S$.
A proper nontrivial module avoiding this vertex would be a
global module contained in $S$, which admissibility forbids.
A proper module containing it would contain a nonempty proper
$T\subset S$ and force $C_{ij}=0$ for
$i\in T$, $j\in S\setminus T$. This contradicts strong
connectivity. Proposition~\ref{for:irreducibility} gives
absolute irreducibility of $D_S(t,x)$ and hence of its
dehomogenization. For a singleton this is directly the
valid nonconstant affine exchange $F_i$.

We compute the canonical denominator. Fix $h\notin S$ and
use the shifted system on $S\cup\{*\}$ after $x_h=0$.
Its internal edges have coefficients $C_{vj}-C_{hj}$, its
edges from $*$ have coefficients $-C_{hj}$, and an edge
from $v$ to $*$ survives precisely when
$C_{vk}\ne C_{hk}$ for some $k\notin S\cup\{h\}$.
A terminal component $T$ avoiding $*$ gives the module
$T\cup\{h\}$. It is globally minimal: a smaller module
avoiding $h$ lies in $S$, while a smaller one containing
$h$ contradicts strong connectivity of $T$ in the
difference graph. Conversely each ready minimal block at
$h$ gives one such component. The fused-clique convention
counts it once, since $S$ contains at most one member of
each clique. Their number is $d_h(S)$.

Suppose first that $C_{hj}>0$ for some $j\in S$.
Then $*$ reaches every vertex of $S$. To see this, follow
an original directed path in $C[S]$. An edge $v\to k$
either survives as a difference edge, or satisfies
$C_{vk}=C_{hk}>0$, in which case there is a direct edge
$*\to k$. Strong connectivity completes the reachability
argument. If $d_h(S)>0$, the terminal components avoiding
$*$ are all terminal components; otherwise the unique
terminal component contains $*$. Lemma~\ref{for:boundary-order}
therefore gives
\begin{equation}\label{sp:usual-outside}
 \nu_h(D_S)=\max(d_h(S)-1,0).
\end{equation}

Suppose instead that $C_{hj}=0$ for every $j\in S$.
Now $*$ is terminal and the internal graph is exactly
the strongly connected graph $C[S]$. The latter is an
additional terminal component exactly when
$S\cup\{h\}$ is an outgoing module. Consequently
\begin{equation}\label{sp:exceptional-outside}
 \nu_h(D_S)=
 \begin{cases}
  1,&S\cup\{h\}\text{ is an outgoing module},\\
  0,&\text{otherwise}.
 \end{cases}
\end{equation}
In the exceptional case $M=S\cup\{h\}$ is globally minimal.
A smaller module avoiding $h$ violates admissibility; one
containing $h$ would give a proper nonempty outgoing-closed
subset of $C[S]$, because row $h$ is zero toward $S$.
The internal $h$-row in $M$ is zero, so its quotient
determinant satisfies
\begin{equation}\label{sp:core-coincidence}
 P_M=H_{C[M]}(a_M,x_M)=D_S(1,x).
\end{equation}
If $|S|\geq2$, this is a proper-type core. There cannot
be two different exceptional vertices: the intersection
of their two minimal modules would be the nontrivial
module $S$, contrary to admissibility. If $S=\{i\}$,
all exceptional pairs belong to one maximal pair clique
$B$. Indeed two exceptional vertices have equal rows
toward $i$, both coefficients being zero; toward every
other outside target their equalities propagate through
$i$. They therefore also form a pair module.
The clique's target-$i$ coefficient is zero, and its affine
core numerator is precisely $F_i=D_S$. Thus in either
case Lemma~\ref{sp:core-primes} identifies $Q_{D_S}$ with
the corresponding $q_B$.

Finally let $i\in S$. After $x_i=0$, the shifted system
has the isolated vertex $i$. Removing it, $*$ reaches
every remaining vertex: follow a simple original path
from $i$ and use the same direct-jump argument when a
difference edge vanishes. A terminal component avoiding
$*$ would give a nontrivial module contained in $S$.
It is therefore impossible. There are exactly two
terminal components before removing $i$, giving
\begin{equation}\label{sp:inside}
 \nu_i(D_S)=1.
\end{equation}
The singleton case follows directly from $D_S=F_i$.
If no exception in \eqref{sp:exceptional-outside} occurs,
the valuations \eqref{sp:usual-outside} and \eqref{sp:inside}
are exactly the completed principal gauge, so $Q_{D_S}=p_S$.
This proves the lemma.
\end{proof}

\begin{proposition}\label{sp:factor-closure}
Every $g\in\mathcal G$ is a positive constant times a
monomial with nonnegative exponents in prime elements
of $\U$ which themselves belong to $\mathcal G$.
\end{proposition}

\begin{proof}
This is immediate for initial coordinates and for cores.
For $p_J$, factor its numerator by \eqref{sp:scc-factorization}
and replace each irreducible factor by its canonical
prime lift, using Lemma~\ref{sp:principal-primes}. The
remaining factor is a monomial in initial coordinates:
\[
 p_J=\prod_i x_i^{e_i}\prod_{S\in\SCC(J)}Q_{D_S}.
\]
Every canonical lift has zero valuation at each initial
coordinate prime. Since $p_J\in\U$ by
Lemma~\ref{sp:positive-limit}, applying $\nu_i$ to this
identity gives $e_i=\nu_i(p_J)\geq0$. All chosen
normalizations are positive. This proves the assertion,
including repeated factors and coincidences among labels.
\end{proof}

\subsection{From limiting charts to actual LP clusters}

We give the refinement argument explicitly. Let
$\mathbf f=(f_1,\ldots,f_n)$ be a limiting chart and let
$Q(\mathbf f)$ be the set of distinct prime factors of
its coordinates, as supplied by
Proposition~\ref{sp:factor-closure}. Every $q\in Q(\mathbf f)$
belongs to $\mathcal G$, hence to $K[\mathbf f^{\pm1}]$.
Since it divides the unit $f_i$ in this Laurent ring,
it is itself a unit there. Thus
\begin{equation}\label{sp:monomial-bases}
 f_i=c_i\prod_{q\in Q(\mathbf f)}q^{A_{qi}},
 \qquad
 q=d_q\prod_i f_i^{B_{iq}},
 \qquad c_i,d_q>0,\quad A_{qi}\in\N,\ B_{iq}\in\Z.
\end{equation}
Distinct prime elements of $\U$ are multiplicatively
independent modulo $K^\times$, by primality and
$\U^\times=K^\times$. Their exponent vectors in the
Laurent ring of $\mathbf f$ are therefore linearly
independent, so $|Q(\mathbf f)|\leq n$. On the other
hand, the first expressions in \eqref{sp:monomial-bases}
and $\operatorname{trdeg}K(\mathbf f)=n$ give
$|Q(\mathbf f)|\geq n$. Hence there are exactly $n$
prime coordinates, they form a transcendence basis,
and the two integer matrices are inverse:
\begin{equation}\label{sp:unimodular}
 A\in\GL_n(\Z),\qquad A\geq0.
\end{equation}
Call $Q(\mathbf f)$ the \emph{prime chart} of $\mathbf f$.
Every member of $\mathcal G$ has a nonnegative Laurent
expansion in this prime chart, by substituting the
positive monomials \eqref{sp:monomial-bases}.

\begin{lemma}\label{sp:adjacent-prime}
Prime charts of adjacent limiting charts either coincide
or differ by an actual ordinary LP mutation.
Every prime chart is an actual cluster of the sparse seed.
\end{lemma}

\begin{proof}
The $n-1$ shared limiting coordinates are algebraically
independent. The union $S$ of their prime supports has
cardinality at least $n-1$, and belongs to both prime
charts, each of cardinality $n$. If $|S|=n$, the prime
charts coincide. Otherwise $|S|=n-1$. Let $p,p'$ denote
their remaining primes. If these agree the charts
again coincide.

Suppose they differ. In the exponent matrix $A$, the
row of $p$ vanishes on every shared column. Expanding
$\det A=\pm1$ in that row shows that the exponent of
$p$ in the unshared coordinate is one and that the
shared $(n-1)\times(n-1)$ exponent matrix is unimodular.
The same holds for $p'$. Consequently
\[
 f=c\,pM(\mathbf q),\qquad
 f'=c'\,p'M'(\mathbf q)\qquad(\mathbf q=S),
\]
where $M,M'$ are monomials. The inherited positive
exchange $ff'=E(\text{shared }\mathbf f)$ becomes
\[
 pp'=\frac{E(\text{shared }\mathbf f)}
                 {cc'M(\mathbf q)M'(\mathbf q)}
       \in K_{\geq0}[\mathbf q^{\pm1}]\setminus\{0\}.
\]
Starting with an actual prime chart,
Lemma~\ref{lem:prime-recognition} identifies the prime
$p'\in\U$ with its actual mutation partner, up to a
positive coefficient unit. The prime chart of the
initial limiting chart is the initial cluster.
Connectedness of the finite limiting atlas now proves
the second assertion and, along the way, the first.
\end{proof}

It remains to show that these finitely many native prime
charts include every actual sparse cluster. This last
step cannot be inferred merely from connectedness.
Fix a rationally independent weight $w$ on the initial
coordinates. Every limiting coordinate has a well-defined
leading exponent $g(f)\in\Z^n$. Along an adjacent positive
exchange, the new leading vector equals the negative of
the removed one plus an integral combination of the
shared leading vectors. Beginning with the initial
basis shows that every limiting chart has a unimodular
leading matrix. Its cone
\[
 \sigma_{\mathbf f}=\cone\{g(f_1),\ldots,g(f_n)\}
\]
is therefore full-dimensional and simplicial. Across
each facet its adjacent chart lies on the opposite side
of the facet hyperplane and covers the other half of
a neighborhood of every relative interior point of
that facet.

The union of these finitely many cones is all of $\R^n$.
For completeness, if it were not, join a generic interior
point of one cone to a point outside their closed union.
Choose the endpoint so the segment misses every face of
codimension at least two. At its first exit it lies in
the relative interior of a facet of some cone. The
adjacent cone covers the other side locally, contradicting
the definition of first exit. This argument uses no
fan property for the limiting cones.

For the prime chart $\mathbf q=Q(\mathbf f)$,
\eqref{sp:monomial-bases} gives $G_{\mathbf f}=G_{\mathbf q}A$.
Since $A$ is invertible and nonnegative,
\begin{equation}\label{sp:cone-refinement}
 \sigma_{\mathbf f}\subseteq\sigma_{\mathbf q},
 \qquad
 \operatorname{int}\sigma_{\mathbf f}
       \subseteq\operatorname{int}\sigma_{\mathbf q}.
\end{equation}
Indeed every row of $A$ has a positive entry, so it sends
a vector with all positive entries to another such vector.
Let $T$ now be any actual sparse cluster. Choose a point
in the interior of its leading cone avoiding the finitely
many boundaries of the limiting cones. Completeness
places it in the interior of some $\sigma_{\mathbf f}$,
and then \eqref{sp:cone-refinement} places it in the
interior of a known native prime-chart cone. The exact
intersection property of Lemma~\ref{lem:leading-intersections},
which does not assume finiteness, forces these two native
clusters to be identical up to coefficient units.
Thus every actual cluster is a prime chart.

\begin{theorem}\label{sp:sparse-positive}
For every valid numerical linear seed
$F_i=1+\sum_{j\ne i}C_{ij}x_j$ with $C_{ij}\geq0$,
the normalized ordinary LP pattern is finite. Every
cluster variable, and every actual exchange product,
has nonnegative Laurent coefficients in every cluster
in which it is expressed. The algebra equals its upper
algebra; cluster monomials form an atomic positive basis,
with nonnegative multiplication constants.
\end{theorem}

\begin{proof}
Every actual cluster is one of the finitely many prime
charts. Each of its coordinates is a member of
$\mathcal G$, and the paragraph following
\eqref{sp:unimodular} gives its nonnegative Laurent
expansion in every prime chart. The same holds for
products, in particular actual exchange products.
Theorems~\ref{thm:finite-basis} and
\ref{thm:finite-transfer} give all remaining assertions.
\end{proof}
\section{The main theorem and the necessity of linearity}
\label{sec:conclusion}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Replace each initial exchange polynomial by $F_i/a_i$.
This is an allowed positive coefficient-unit normalization,
and gives \eqref{eq:normalized-linear} with
$C_{ij}=c_{ij}/a_i\geq0$. Theorem~\ref{sp:sparse-positive}
applies, including arbitrary equalities among the weights.
It provides positive representatives, finiteness of the
normalized pattern, and Laurent positivity in every cluster.
For a mutation $xx'=\widehat F_x$, the positive Laurent
expansion of $x'$ in the cluster containing $x$ is
$\widehat F_x/x$. Thus $\widehat F_x$ has nonnegative
coefficients. Multiplication by the coordinate monomial
which removes its negative exponents gives the ordinary
exchange polynomial in a positive normalization, with the
same coefficient signs.

Theorem~\ref{thm:finite-basis} gives
$\A=\U=\bigoplus_{b\in\mathscr B}Kb$.
The coefficient-extraction formula \eqref{eq:basis-extraction}
shows that a universally nonnegative element has nonnegative
basis coefficients. Conversely every nonnegative linear
combination of cluster monomials is universally nonnegative,
by the Laurent positivity just proved. This is the claimed
atomicity. A product of two cluster monomials is universally
nonnegative; applying the same extraction formula gives
nonnegative multiplication constants.
\end{proof}

\begin{corollary}\label{cor:integer}
Let an ordinary LP seed over $\Z$ have primitive nonconstant
linear exchanges
\[
 F_i=a_i+\sum_{j\ne i}c_{ij}x_j,\qquad
 a_i\in\Z_{>0},\quad c_{ij}\in\Z_{\geq0}.
\]
After choosing signs consistently, every cluster variable
has Laurent coefficients in $\Z_{\geq0}$ with respect to
every cluster.
\end{corollary}

\begin{proof}
Localize the coefficient ring to $\Q$. Gauss's lemma
preserves irreducibility of each primitive nonconstant
initial exchange. At each mutation, localization preserves
the multiplicity of every nonconstant irreducible factor:
the factors removed in the slice rule are the same up to
coefficient units. Polynomial parts differ only by rational
units. Induction therefore identifies the localized pattern
with the ordinary field pattern. No constant exchange can
appear in the integral pattern during this induction,
because it would become a unit, whereas the corresponding
field mutation is a valid seed with nonunit exchanges.
Choose the sign of each integral representative so it is
positive on the initial positive orthant.

Theorem~\ref{thm:main} gives nonnegative rational Laurent
coefficients for these representatives, since positive
rational rescalings preserve the conclusion. The ordinary
Laurent phenomenon over $\Z$ gives integral coefficients
for the very same expansion in any integral cluster.
Uniqueness of Laurent coefficients makes them
nonnegative integers.
\end{proof}

\begin{remark}[The linearity assumption]\label{rem:nonlinear-counterexample}
Nonnegative initial exchange coefficients do not imply Laurent positivity
for nonlinear LP algebras, even over $\mathbb Z$.  Let $m\geq3$ be odd and
consider the rank-three seed
\[
 F_a=1+b+c,\qquad F_b=1+a+c^m,\qquad F_c=1+a.
\]
These exchange polynomials satisfy the LP axioms: each is primitive and
linear in a variable with coefficient~$1$, omits its own variable, and
has constant term~$1$.  Put
\[
 Q_m(c)=\sum_{j=0}^{m-1}(-c)^j,
 \qquad (1+c)Q_m(c)=1+c^m.
\]
The exchange Laurent polynomials of the initial seed are
\[
 \widehat F_a=F_a,\qquad
 \widehat F_b=F_b/c,\qquad
 \widehat F_c=F_c.
\]
Indeed, the only possible denominator is $c$ in $\widehat F_b$:
all other pairs of variables occurring in the exchange polynomials
occur in both directions.  Substituting $c=(1+a)/z$ gives
\[
 F_b\big|_{c=(1+a)/z}
 =(1+a)\bigl(1+(1+a)^{m-1}z^{-m}\bigr),
\]
which has exactly one factor $1+a$.

Mutate first at $a$ and write $u=(1+b+c)/a$.  To update $F_b$,
the mutation slice is $1+c$, and the substituted expression is
\[
 1+\frac{1+c}{u}+c^m
 =\frac{(1+c)(1+uQ_m(c))}{u}.
\]
Removing the slice factor and clearing the mutable monomial gives
$F_b'=1+uQ_m(c)$.  There is no further slice factor, since
$Q_m(-1)=m$ and $1+mu\neq0$.  The update of $F_c$ similarly gives
$F_c'=1+u+b$, with no slice factor removed.  Thus the new exchange
polynomials, in the coordinates $(u,b,c)$, are
\[
 F_u=1+b+c,\qquad F_b'=1+uQ_m(c),\qquad F_c'=1+u+b.
\]
All pairs now occur in both directions, so all three exchange Laurent
polynomials equal their exchange polynomials.  Mutation at $b$ therefore
produces the actual cluster variable
\[
 v=\frac{1+uQ_m(c)}{b}
   =\frac{1+a+c^m+bQ_m(c)}{ab}.
\]
The coefficient of $a^{-1}c$ in this Laurent expansion is $-1$, whereas
the coefficient of $a^{-1}b^{-1}$ is $1$.  These are distinct monomials,
and multiplying $v$ by either unit of $\mathbb Z$ cannot make all its
coefficients nonnegative. The same two signs rule out every nonzero
scalar multiple over a real coefficient field. For example, $m=3$ gives
\[
 v=\frac{1+a+b-bc+bc^2+c^3}{ab}.
\]
Thus Laurent positivity can fail after two ordinary LP mutations from
a nonlinear seed whose initial nonzero coefficients are all~$1$.
\end{remark}
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