Does the perfect matching polytope have a polynomial-size semidefinite lift?

Theorem 1.1: there is $c>0$ such that for every fixed $0<\rho<1$ the odd-cut matrix with entries $|M\cap\delta(U)|-1+\rho$ has real PSD rank at least $2^{cn}$ for large even $n$, with no symmetry or precision restriction on the factors. Corollary 1.2: every exact semidefinite lift of the perfect matching polytope has size $2^{\Omega(n)}$. The method follows Lee-Raghavendra-Steurer with a Grigoriev-type parity functional and new averaging lemmas. The endpoint $\rho=1$ has a polynomial factorization, and approximate lifts are not excluded (Kaniewski-Lee-de Wolf give subexponential approximations).

Theorem 1.1: there is $c>0$ such that for every fixed $0<\rho<1$ the odd-cut matrix with entries $|M\cap\delta(U)|-1+\rho$ has real PSD rank at least $2^{cn}$ for large even $n$, with no symmetry or precision restriction on the factors. Corollary 1.2: every exact semidefinite lift of the perfect matching polytope has size $2^{\Omega(n)}$. The method follows Lee-Raghavendra-Steurer with a Grigoriev-type parity functional and new averaging lemmas. The endpoint $\rho=1$ has a polynomial factorization, and approximate lifts are not excluded (Kaniewski-Lee-de Wolf give subexponential approximations).