A conjecture of Elgin, Reading and Stella on rank-two scattering coefficients

For the rank-two cluster scattering diagram with initial walls $1+x$ and $1+y^5$, we prove $\tau^{1,5}(2j,j)=\frac1j\sum_{\ell}\binom{\ell}{j-\ell+1}\binom{j+\ell-1}{\ell}$ for every $j\geq1$, in the convention of Burcroff and coauthors and of Elgin, Reading and Stella's table and code. This proves Elgin, Reading and Stella's Conjecture 13. Elgin, Reading and Stella provided extensive computations and stated eighteen conjectures on rank-two cluster scattering coefficients. Burcroff, Lee, Mou, Musiker and Reineke proved Conjectures 1–3, 5–10 and 14–18, leaving Conjecture 13 unresolved and "mysterious from both the quiver moduli and tight grading points of view." Kontsevich and Soibelman provided ordered factorization, and Gross, Hacking, Keel and Kontsevich provided the change of lattice used here. The binomial sum was already recorded by Vladimir Kruchinin, with functional equations by Paul D. Hanna; Burcroff and coauthors identified its algebraicity and growth rate conditionally on the conjecture. We identify these numbers with the scattering coefficients. We observe that OpenAI's closely related quantum scattering framework admits the same five-and-one incoming multiplicities; whether it implies the coefficient formula or cubic remains unresolved. After a change of lattice and a monomial substitution, the fifth power of the wall function becomes the diagonal factor of an explicit Poisson automorphism. A grading separates that factor from the other walls and yields a differential equation with a unique algebraic solution. The coefficient identity is formalized in Lean from the definitions. This work has not yet been reviewed by experts. The proofs, the Lean formalization and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction.

Checked in Lean from the definitions; not yet reviewed by experts. Produced by AI models under the author's direction.