We prove that Boolean matrices with bounded γ2-norm or bounded normalized trace norm must contain a linear-sized all-ones or all-zeros submatrix, verifying a conjecture of Hambardzumyan, Hatami, and Hatami. We also present further structural results about Boolean matrices of bounded γ2-norm and discuss applications in communication complexity, operator theory, spectral graph theory, and extremal combinatorics. As a key application, we establish an inverse theorem for MaxCut. A celebrated result of Edwards states that every graph G with m edges has a cut of size at least m2+8m+1-18, with equality achieved by complete graphs with an odd number of vertices. To contrast this, we prove that if the MaxCut of G is at most m2+O(m), then G must contain a clique of size Ω(m).