NSR measures on hyperelliptic locus and non-renormalization of 1,2,3-point functions

We demonstrate (under a modest assumption) that the sums over spin-structures of the simplest combinations of fermionic correlators (Szego kernels) and DHP/CDG/Grushevsky NSR measures vanish at least on the hyperelliptic loci in the moduli space of Riemann surfaces-despite the violation of the θe4 hypothesis at g>2. This provides an additional important support to validity of these measures and is also a step towards a proof of the non-renormalization theorems in the NSR approach.