Scattering diagrams arose in the construction of spaces by gluing torus charts. The gluing transformations must compose to the identity around closed paths, and this requirement produces additional transformations supported on rays. Kontsevich and Soibelman developed the two-dimensional construction in their work on non-archimedean K3 surfaces~\cite{KSaffine}. Their later theory of stability structures and Donaldson--Thomas invariants~\cite{KSDT} organized wall crossing through ordered products of transformations of a torus. The order records the phases, or slopes, of the contributing classes.

In the tropical vertex group, the transformations preserve the logarithmic symplectic form on a two-dimensional torus. Gross, Pandharipande, and Siebert~\cite{GPS} related their factorization coefficients to genus-zero relative Gromov--Witten invariants of toric surfaces. The relevant curves have prescribed contact with the toric boundary and full tangency at one unspecified point on the outgoing divisor. Scattering diagrams, tropical curve counts, degeneration formulas, and multiple-cover contributions enter this correspondence. On the representation-theoretic side, Reineke's functional equations~\cite{Reineke} relate generating series for stable quiver moduli to those for framed moduli and yield integrality statements for Donaldson--Thomas invariants.

Gross and Pandharipande~\cite{GP} brought the quiver and curve-counting descriptions together and studied the rays and symmetries of the resulting diagrams. Their basic example starts with two initial functions $(1+tx)^{\ell_1}$ and $(1+ty)^{\ell_2}$, where $\ell_1,\ell_2$ are positive integers and $t,x,y$ are independent formal variables. The commutator of the corresponding transformations has an ordered factorization indexed by positive rational slopes. Its factor in a primitive direction $(d,e)$, meaning $d,e>0$ and $\gcd(d,e)=1$, is determined by a series in the single monomial $t^{d+e}x^dy^e$. In Section~1.4 they record Kontsevich's question whether all these series are algebraic over $\QQ(t,x,y)$. This is the algebraicity question considered here.

The central slope supplied an early model. Gross and Pandharipande proposed an explicit formula there, noting that its equal-parameter specialization had been proved by Reineke. Reineke and Weist~\cite[Theorem~11.1 and Corollary~11.2]{RW} proved the formula for arbitrary $\ell_1,\ell_2$ through complete bipartite quivers and the refined Gromov--Witten/Kronecker correspondence. If $z=t^2xy$ and $C(z)\in1+z\QQ[[z]]$ is the solution of
\[
 C(z)=1+zC(z)^{(\ell_1-1)(\ell_2-1)},
\]
then the central wall function is $C(z)^{\ell_1\ell_2}$. Thus even diagrams with infinitely many rays have distinguished wall functions governed by finite algebraic equations. Their quiver-theoretic functional equations also describe other slopes, with the central case admitting this particular finite reduction.

Kontsevich announced a general factorization theorem in his Miami lecture of 30 January 2013. A two-variable formal logarithmic symplectomorphism is factored into the parts below, on, and above the diagonal; the theorem states that the original map is algebraic if and only if the three factors are algebraic. His handwritten notes~\cite[pp.~5--7]{Kontsevich} and Auroux's notes~\cite[pp.~2--3]{Auroux} give a geometric proof outline using rational curves, marked points, tangent data, and a cross-ratio. A finite monomial covering sends any positive rational direction to the diagonal. Consequently this statement encompasses the full algebraicity question of Gross and Pandharipande; Section~\ref{sec:historical} proves the reduction.

Algebraic generating series also occur in spectral-network calculations. Galakhov, Longhi, Mainiero, Moore, and Neitzke~\cite[Section~3]{GLMMN} obtained algebraic equations for families of BPS states and compared them with tropical vertex formulas. Mainiero~\cite[Section~1.1]{Mainiero} reports an indirect algebraicity argument of Kontsevich and Soibelman, citing private communication. He studies explicit spectral-network equations, the relation between Donaldson--Thomas series and Euler-characteristic series, and the asymptotics obtained from algebraicity. These calculations concern specified families of dimension vectors and make the algebraic equations themselves accessible.

The subsequent work of Kontsevich and Soibelman~\cite{KSanalytic} develops analytic stability data and proves preservation of analyticity under sector factorization. Section~6.3 also describes polynomial, rational, and algebraic stability data through non-archimedean geometry and states an algebraicity theorem for its algebraic class with rational central charge. The discussion records these facts with their detailed treatment assigned to future work. This gives a later account of the geometric approach alongside the 2013 lecture.

In cluster theory, Gross, Hacking, Keel, and Kontsevich~\cite{GHKK} use scattering diagrams and broken lines to construct theta functions and prove positivity results. Mou~\cite{Mou} extends this construction to generalized cluster algebras with exchange polynomials factored into binomials. Burcroff, Lee, and Mou introduce tight gradings in~\cite{BLMshort} and prove a coefficient formula for arbitrary rank-two polynomial initial functions in~\cite[Theorem~1.2]{BLM}. A tight grading is an integer decoration of a maximal Dyck path subject to compatibility and shadow conditions. Its weight is a monomial in the initial coefficients, so the formula gives both a finite coefficient calculation and coefficientwise positivity in the universal parameters.

Burcroff, Lee, Mou, Musiker, and Reineke~\cite{BLMMR} combine this combinatorics with quiver moduli to study rank-two cluster scattering coefficients, including their Weyl group symmetry and the structure of tight-grading frames. Their Section~9.3 asks again whether every wall function is algebraic. Under the parameter correspondence $(\ell_1,\ell_2)=(c,b)$, its wall functions and the tropical vertex wall functions differ by a positive integral power; Proposition~\ref{prop:dictionary} gives the exact exponent. The same section treats coefficient growth as a separate question and formulates a slope-dependent limit in Conjecture~9.2.

We prove algebraicity for two arbitrary polynomial initial functions over any field of characteristic zero. The proof passes from the tight-grading formula to a finite system that treats all multiples of a fixed direction simultaneously. We identify a single rooted interval in each tight grading, describe its underlying word by a finite automaton, and record the matching constraints by a stack with one free-balance variable. This yields a finite system of discrete differential equations to which the theorem of Notarantonio and Yurkevich~\cite[Theorem~1.3]{NY} applies. Appendix~\ref{sec:coefficient} expands the coefficient formula in the precise case used here, following the triangular method of~\cite[Section~5.5]{BLM}; Appendix~\ref{sec:nested} gives a second closure argument using nested algebraic approximation.