A central problem in complex dynamics is the MLC conjecture which states that the Mandelbrot set is locally connected. It is equivalent to the rigidity conjecture which states that any two infinitely renormalizable quadratic polynomials with the same combinatorics are conformally conjugate. In the bounded combinatorics case, this follows from a priori bounds which was proven by Kahn, Dudko and Lyubich using analysis of near-degenerate surfaces. I will explain a new proof of this theorem via "totally degenerate regime". The idea is inspired by Thurston's compactification of Teichmuller space: assuming lack of a priori bounds, renormalizations converge to a limiting modular lamination on an infinite type surface that is invariant under a limiting topological transcendental map.