Szemerédi Regularity Lemmas for Hypergraphs and the Lovász-Schrijver Relaxation Hierarchy

Mathematical Applications of Science Fiction We establish a canonical isomorphism between the limitobjects of convergent hypergraph sequences, as defined bythe Gowers-Wolf regularization theory, and the transfinitestabilization points of the Lovász-Schrijver (LS) lift-and-project hierarchy applied to the hypergraph independencepolytope. By extending the concept of the cut-norm toa tensor-spectral setting, we prove that for any k-uniformhypergraph H, the rank required in the LS-hierarchy toapproximate the independence number within ϵ scales withthe inverse Ackermann function of the regularity parameters.Furthermore, we introduce a transfinite induction schemeon the ordinal index of the LS-operator, demonstrating thatthe "ghost" vertices generated by the hypergraph regular-ity partition correspond precisely to the moment matrices’eigenspaces at limit ordinals. This unifies the combinatorialrigidity of Szemerédi’s formalism with the semi-algebraiccomplexity of convex relaxation hierarchies.