  We exploit the structure of dual cocycles to develop several new concepts and
tools for the study of one-frequency quasiperiodic operators with analytic
potentials. As applications we solve two long-standing arithmetic conjectures:
universality of sharp phase transition in frequency and absolute continuity of
the integrated density of states for all frequencies for the entire class of
(non-critical) type I operators, a large open set including the analytic
neighborhood of several popular explicit models. Additionally, we prove the
universality of $1/2$-H\"older continuity of the integrated density of states
for all type I operators with Diophantine frequencies, partially solving You's
conjecture \cite{You}.
  A key dynamical finding: the special symplectic structure of the center of
the dual cocycle that, remarkably, holds also in the limit where the cocycle
does not even exist, allows, among other things, to pass for the first time
from trigonometric polynomials to general analytic potentials, in the
duality-based arguments. It is also applicable in other outstanding problems
where the results so far have only been obtained for the trigonometric
polynomial case. The hidden subcriticality uncovered by this structure allows
to define the rotation numbers for these complex-symplectic cocycles and
reveals the dynamical nature of the special place of even potentials.
