A doubled Gordon threshold for palindromic quasiperiodic Schrödinger operators

We consider one-frequency quasiperiodic Schr\"odinger operators $$(H_{v,\alpha,\theta}u)(n) = u(n+1)+u(n-1) + v(\theta+n\alpha)u(n)$$ acting on $\ell^2(\mathbb Z)$, where $\alpha\notin\mathbb Q$ and $v\in C^2(\mathbb T,\mathbb R)$ is an even function. We develop a new Gordon-type method that exploits approximate repetitions and palindromic symmetries simultaneously. Denote by $L(E)$ the Lyapunov exponent and let $$\beta(\alpha) = \limsup_{|k|\to\infty} -\frac{\log\|k\alpha\|_{\mathbb R/\mathbb Z}}{|k|}.$$ We prove that, for every completely resonant phase $2\theta\in\alpha\mathbb Z+\mathbb Z$, $E$ cannot be an eigenvalue if $L(E)<2\beta(\alpha)$. As an application, consider the almost Mathieu operator $$(H_{\lambda,\alpha,\theta}u)(n) = u(n+1)+u(n-1) + 2\lambda\cos\bigl(2\pi(\theta+n\alpha)\bigr)u(n).$$ We show that if $2\theta\in\alpha\mathbb Z+\mathbb Z$ and $1<|\lambda|<e^{2\beta(\alpha)}$, then $H_{\lambda,\alpha,\theta}$ has purely singular continuous spectrum. This resolves the remaining absence-of-eigenvalues part of a conjecture of Avila and Jitomirskaya concerning the sharp spectral transition for completely resonant phases.