  We establish a list of characterizations of bounded twin-width for
hereditary, totally ordered binary structures. This has several consequences.
First, it allows us to show that a (hereditary) class of matrices over a finite
alphabet either contains at least $n!$ matrices of size $n \times n$, or at
most $c^n$ for some constant $c$. This generalizes the celebrated Stanley-Wilf
conjecture/Marcus-Tardos theorem from permutation classes to any matrix class
over a finite alphabet, answers our small conjecture [SODA '21] in the case of
ordered graphs, and with more work, settles a question first asked by Balogh,
Bollob\'as, and Morris [Eur. J. Comb. '06] on the growth of hereditary classes
of ordered graphs. Second, it gives a fixed-parameter approximation algorithm
for twin-width on ordered graphs. Third, it yields a full classification of
fixed-parameter tractable first-order model checking on hereditary classes of
ordered binary structures. Fourth, it provides a model-theoretic
characterization of classes with bounded twin-width.
