On irreducible central limit theorems

We consider sequences of homogeneous sums based on independent random variables and satisfying a central limit theorem (CLT). We address the following question: "In which cases is it not possible to reduce such an asymptotic result to the classical Lindeberg-Feller CLT through a restriction of the summation domain?". We provide several sufficient conditions for such irreducibility, expressed both in terms of (hyper)graphs Laplace eigenvalues, and of a certain notion of combinatorial dimension. Our analysis combines Cheeger-type inequalities with fourth moment theorems, showing that the irreducibility of a given CLT for homogeneous sums can be naturally encoded by the connectivity properties of the associated sequence of weighted hypergraphs. Several ad-hoc constructions are provided in the special case of quadratic forms.