  An idea that became unavoidable to study zero entropy symbolic dynamics is
that the dynamical properties of a system induce in it a combinatorial
structure. An old problem addressing this intuition is finding a structure
theorem for linear-growth complexity subshifts using the S-adic formalism. It
is known as the S-adic conjecture and motivated several influential results in
the theory. In this article, we completely solve the conjecture by providing an
S-adic structure for this class. Our methods extend to
nonsuperlinear-complexity subshifts. An important consequence of our main
results is that these complexity classes gain access to the S-adic machinery.
We show how this provides a unified framework and simplified proofs of several
known results, including the pioneering 1996 Cassaigne's Theorem.
