Intrinsic Symplectic Structure and Sharp Arithmetic Universality

Abstract

We show that formal eigenvalue equations of analytic one-frequency Schr\"od-inger operators admit intrinsic analytic $Sp(2k,\C)$ structures, where $k=k(E)$ is the T-acceleration in global theory. For trigonometric potentials those structures govern the center dynamics of partially hyperbolic dual cocycles; for general analytic potentials they persist, without loss of analyticity, as an intrinsic object even when the dual operator has infinite range and no cocycles exist. For $k=1$, we also introduce the concept of projectively real cocycles: complex symplectic systems whose projective action is algebraically conjugate, up to a scalar phase, to that of a real $\SL(2,\R)$ cocycle. This allows us to define a rotation pair and establish a rotation--IDS correspondence in the general analytic setting, where standard dynamical methods fail. Using these tools, we solve two spectral arithmetic conjectures: universality of the sharp arithmetic transition in frequency (AAJ) and of the absolute continuity of the integrated density of states for all frequencies, throughout the class of non-critical Type I operators, an open and conjecturally dense set. We also prove universality of sharp $1/2$-H\"older continuity of the integrated density of states for Type I operators with Diophantine frequencies, establishing part of You's conjecture. These results also provide the first duality-based spectral framework for general analytic potentials, overcoming the symmetry and finite-range restrictions present in previous work.