Nondegeneracy of the spectrum of the twisted cocycle for interval exchange transformations

We prove the positivity of the top Lyapunov exponent of the twisted (spectral) cocycle, associated with IETs, with respect to a family of natural invariant measures. The proof relies on relating the top exponent to limits of exponents along families of affine invariant submanifolds of genus tending to infinity. Applications include an observation about a conjecture of Kontsevich and Zorich, a discrepancy estimate, and a formula for the lower local dimension of spectral measures.