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