\section{The cluster-scattering formulation}\label{sec:cluster} In this section $b,c$ denote positive cluster parameters, replacing the polynomial degree bound $b$ used in the preceding construction. Consider the rank-two diagram with initial functions $1+x_1^b$ and $1+x_2^c$, in the tropical vertex convention. For primitive $d,e>0$, write its wall function on the ray $\mathbb R_{\leq0}(db,ec)$ as \[ H_{d,e}(z)=1+\sum_{k\geq1}\tau^{(b,c)}(kd,ke)z^k, \qquad z=x_1^{db}x_2^{ec}. \] This defines the coefficients $\tau^{(b,c)}(kd,ke)$ used here. The support in multiples of $(db,ec)$ follows because the initial monomials lie in $b\ZZ\oplus c\ZZ$, a property preserved by the completed commutator and its homogeneous factorization. Put $g=\gcd(db,ec)$; the primitive lattice vector of this wall is $(db/g,ec/g)$. \begin{proposition}\label{prop:dictionary} Let $G_{d,e}$ correspond to $P_1(X)=(1+X)^c$ and $P_2(Y)=(1+Y)^b$. Then $g\mid bc$ and \begin{equation}\label{eq:dictionary} G_{d,e}(z)=H_{d,e}(z)^{bc/g}. \end{equation} In particular the two series are algebraic simultaneously. \end{proposition} \begin{proof} For every prime $p$, at least one of $d,e$ is not divisible by $p$. Writing $\nu_p(n)$ for the exponent of $p$ in a positive integer $n$, we obtain \[ \nu_p(g)=\min\{\nu_p(db),\nu_p(ec)\}\leq\nu_p(bc). \] Thus $bc/g$ is a positive integer. Restrict the initial automorphisms to $X=x_1^b$, $Y=x_2^c$. They act as \[ (X,Y)\longmapsto (X,Y(1+X)^c),\qquad (X,Y)\longmapsto (X(1+Y)^{-b},Y). \] The general wall with primitive vector $(db/g,ec/g)$ restricts to \[ X\longmapsto XH_{d,e}(X^dY^e)^{-bce/g},\qquad Y\longmapsto YH_{d,e}(X^dY^e)^{bcd/g}. \] The change of lattice preserves slope order. Uniqueness of ordered factorization therefore gives \eqref{eq:dictionary}. Algebraicity is preserved under integral powers. Conversely $H_{d,e}$ satisfies $Z^{bc/g}-G_{d,e}=0$, so transitivity of algebraic extensions proves the other implication. \end{proof} \begin{lemma}\label{lem:monomial} Let $J(z)\in\FF[[z]]$ and let $q\in\ZZ_{\geq0}^r\setminus\{0\}$. For independent formal variables $X_1,\ldots,X_r$, write $X^q=\prod_iX_i^{q_i}$. Then $J(z)$ is algebraic over $\FF(z)$ if and only if $J(X^q)$ is algebraic over $\FF(X_1,\ldots,X_r)$. \end{lemma} \begin{proof} The forward implication follows by substitution. For the converse, clear denominators in a nonzero polynomial relation to obtain \[ P(X,Z)\in\FF[X_1^{\pm1},\ldots,X_r^{\pm1},Z]\setminus\{0\}. \] Group its finitely many $X$-exponents by their cosets in $\ZZ^r/\ZZ q$, choosing a representative $r_C$ for each occurring coset. Then \[ P(X,Z)=\sum_C X^{r_C}P_C(X^q,Z),\qquad P_C(z,Z)\in\FF[z^{\pm1},Z]. \] After substituting $J(X^q)$, different cosets have disjoint monomial supports. Each component therefore vanishes separately. At least one $P_C$ is nonzero, and clearing its negative powers of $z$ gives a nonzero polynomial annihilating $J(z)$. \end{proof} Theorem~\ref{thm:main} and Proposition~\ref{prop:dictionary} prove algebraicity of every $H_{d,e}$. Lemma~\ref{lem:monomial}, applied to $(db,ec)$ and $(d+e,d,e)$ respectively, identifies this assertion with the original multivariable statements for the cluster wall and for $G_{d,e}(t^{d+e}x^dy^e)$. The correspondence of initial parameters is $(\ell_1,\ell_2)=(c,b)$. \section{The diagonal factor in Kontsevich's statement}\label{sec:historical} For a formal automorphism of $\mathbb C[[U,V]]$ tangent to the identity and preserving $dU/U\wedge dV/V$, group its positive-cone factorization into the directions below, on, and above the diagonal. Kontsevich's lecture states that the automorphism is algebraic if and only if all three grouped factors are algebraic~\cite{Kontsevich,Auroux}. Here a map is algebraic when its coordinate series are algebraic over $\mathbb C(U,V)$. The notes give a geometric outline through rational curves in a surface obtained by two blowups of $\mathbb P^1\times\mathbb P^1$; the diagonal factor is identified using marked points and tangent data. \begin{proposition}\label{prop:historical} Kontsevich's diagonal preservation statement implies algebraicity in every primitive positive direction for the two polynomial initial functions over $\mathbb C$, and implies Corollary~\ref{cor:KGP} over $\QQ$. \end{proposition} \begin{proof} Fix a primitive positive direction $(d,e)$ in \eqref{eq:factorization}. Use the finite monomial covering \[ X=U^e,\qquad Y=V^d. \] An automorphism $X\mapsto XA(X,Y)$, $Y\mapsto YB(X,Y)$, where $A,B$ have constant term $1$, lifts uniquely to the branch tangent to the identity: \[ U\longmapsto U A(U^e,V^d)^{1/e},\qquad V\longmapsto V B(U^e,V^d)^{1/d}. \] Fractional powers mean the unique formal roots with constant term $1$. Uniqueness shows that lifting respects composition. The pullback of the logarithmic two-form is $de\,dU/U\wedge dV/V$, so the lift preserves the latter form. The lifts of the polynomial initial maps are \[ (U,V)\longmapsto(U,V P_1(U^e)^{1/d}),\qquad (U,V)\longmapsto(U P_2(V^d)^{-1/e},V). \] Their inverses have the same formulas with the exponents $1/d$ and $-1/e$ negated, since each initial map fixes the variable on which its multiplier depends. All four maps have algebraic coordinate series: the multipliers satisfy their defining root equations. Composition preserves this property. Indeed, substituting an algebraic formal automorphism into a polynomial relation gives a nonzero relation whose coefficients lie in a finite algebraic extension of $\mathbb C(U,V)$; injectivity of the substitution preserves nonzeroness, and transitivity gives algebraicity over $\mathbb C(U,V)$. The lifted commutator is therefore algebraic. The monomial in a direction $(r,s)$ becomes $U^{er}V^{ds}$; the slope is multiplied by $d/e$. Exactly the target direction $(d,e)$ becomes diagonal. Its factor lifts to \begin{equation}\label{eq:lift} U\longmapsto U G_{d,e}((UV)^{de})^{-1},\qquad V\longmapsto V G_{d,e}((UV)^{de}). \end{equation} Ordered-factorization uniqueness identifies \eqref{eq:lift} with the diagonal factor of the lifted commutator. Thus the stated diagonal preservation theorem yields the algebraicity of $G_{d,e}((UV)^{de})$, and Lemma~\ref{lem:monomial} yields that of $G_{d,e}(z)$ over $\mathbb C(z)$. For the original binomial-power inputs, the coefficients of $G_{d,e}$ lie in $\QQ$. A complex polynomial relation then descends to $\QQ$: choose a $\QQ$-basis for the finite-dimensional space spanned by its coefficients and take the component along each basis vector. Since every coefficient of the series is rational, each component is a relation and at least one is nonzero. The monomial substitution of Corollary~\ref{cor:KGP} gives the original formulation. \end{proof} \appendix \section{Nested algebraic approximation}\label{sec:nested} There is a second algebraic closure argument for the system \eqref{eq:proper}. It uses nested algebraic approximation, as in Popescu~\cite[Theorem~1.4]{Popescu} and its combinatorial application in~\cite[Section~5.3]{BBMDP}. We give the reduction because it explains the role of a single catalytic variable. Write $\FF\langle t,u\rangle$ for the subring of $\FF[[t,u]]$ consisting of series algebraic over $\FF(t,u)$, and similarly for $\FF\langle t\rangle$. Nested approximation says that a finite polynomial system with a formal solution whose unknowns depend on prescribed initial segments of an ordered list of variables has an algebraic solution with the same dependence. We use the order $t,u$, so the allowed dependence sets are $\{t\}$ and $\{t,u\}$. \begin{proof}[Alternative proof of algebraicity for \eqref{eq:proper}] Let $n$ be the number of scalar unknowns in \eqref{eq:proper}. For $1\leq i\leq n$, introduce unknowns $D_{i,j}(t,u)$, $0\leq j\leq a$, and $B_{i,j}(t)$, $0\leq j