  Let $U$ be an affine log Calabi-Yau variety containing an open algebraic
torus. We show that the naive counts of rational curves in $U$ uniquely
determine a commutative associative algebra equipped with a compatible
multilinear form. This proves a variant of the Frobenius structure conjecture
by Gross-Hacking-Keel in mirror symmetry, and the spectrum of this algebra is
supposed to give the hypothetical mirror family. Although the statement of our
theorem involves only elementary algebraic geometry, our proof employs
Berkovich non-archimedean analytic methods. We construct the structure
constants of the algebra via counting non-archimedean analytic disks in the
analytification of $U$. We establish various properties of the counting,
notably deformation invariance, symmetry, gluing formula and convexity. In the
special case when $U$ is a Fock-Goncharov skew-symmetric X-cluster variety, we
prove that our algebra generalizes, and gives a direct geometric construction
of, the mirror algebra of Gross-Hacking-Keel-Kontsevich. The comparison is
proved via a canonical scattering diagram constructed from counts of
infinitesimal non-archimedean analytic cylinders, without using the
Kontsevich-Soibelman algorithm. Several combinatorial conjectures of GHKK, as
well as the positivity in the Laurent phenomenon, follow readily from the
geometric description.
