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We review some important results by Gross, Hacking, Keel, and Kontsevich oncluster algebra theory, namely, the column sign-coherence of $C$-matrices andthe Laurent positivity, both of which were conjectured by Fomin and Zelevinsky.We digest and reconstruct the proofs of these conjectures by Gross et al. stillbased on their scattering diagram method, however, without relying on toricgeometry. At the same time, we also give a detailed account of thecorrespondence between the notions of cluster patterns and scattering diagrams.Most of the results in this text are found in or translated from the knownresults in the literature. However, the approach, the construction of logic andproofs, and the overall presentation are new. Also, as an application of theresults and the techniques in the text, we show that there is a one-to-onecorrespondence between $g$-vectors and cluster variables in cluster patternswith arbitrary coefficients.

Cluster Algebras and Scattering Diagrams, Part II. Cluster Patterns and Scattering Diagrams