Before revision:  Our main conceptual contribution is to distill two simple analytic properties - (certifiable) hypercontractivity and  anti-concentration  - that are necessary and sufficient for mixture models to be (efficiently) clusterable.
After revision:  Our main conceptual contribution is to distill two simple analytic properties - (certifiable) hypercontractivity and bounded variance of degree 2 polynomials and anti-concentration - that are necessary and sufficient for mixture models to be (efficiently) clusterable.