Positivity-preserving extensions of sum-of-squares pseudomoments over the hypercube

We introduce a new method for building higher-degree sum-of-squares lower bounds over the hypercube $\mathbf{x} \in \{\pm 1\}^N$ from a given degree 2 lower bound. Our method constructs pseudoexpectations that are positive semidefinite by design, lightening some of the technical challenges common to other approaches to SOS lower bounds, such as pseudocalibration. We give general "incoherence" conditions under which degree 2 pseudomoments can be extended to higher degrees. As an application, we extend previous lower bounds for the Sherrington-Kirkpatrick Hamiltonian from degree 4 to degree 6. (This is subsumed, however, in the stronger results of the parallel work of Ghosh et al.) This amounts to extending degree 2 pseudomoments given by a random low-rank projection matrix. As evidence in favor of our construction for higher degrees, we also show that random high-rank projection matrices (an easier case) can be extended to degree $ω(1)$. We identify the main obstacle to achieving the same in the low-rank case, and conjecture that while our construction remains correct to leading order, it also requires a next-order adjustment. Our technical argument involves the interplay of two ideas of independent interest. First, our pseudomoment matrix factorizes in terms of certain multiharmonic polynomials. This observation guides our proof of positivity. Second, our pseudomoment values are described graphically by sums over forests, with coefficients given by the Möbius function of a partial ordering of those forests. This connection guides our proof that the pseudomoments satisfy the hypercube constraints. We trace the reason that our pseudomoments can satisfy both the hypercube and positivity constraints simultaneously to a combinatorial relationship between multiharmonic polynomials and this Möbius function.