Consider the family of quadratic polynomials with a neutral fixed point. In this family, bounded-type Siegel maps are dense; their qc (closed) Siegel disks degenerate by developing parabolic fjords. By appropriately filling-in such fjords, we obtain "almost-invariant'' pseudo-Siegel disks with uniformly bounded qc geometry. As a consequence, almost-invariant uniformly-qc pseudo-Siegel disks exist for all neutral quadratic polynomials. After describing this structure of pseudo-Siegel disks, we will discuss how to obtain uniform a priori bounds for Sector Renormalization from uniform bounds for pseudo-Siegel disks.

Joint work with Misha Lyubich.