The exact second Betti number of the neighborhood complex of the Kneser graph KG(2k+2,k)

We prove that the second Betti number of the neighborhood complex of the Kneser graph KG(2k+2,k) is given by the exact formula beta_2 = C(2k+1,k)(k^3-k+2)/(k+2) - 1 for all k >= 2. The proof proceeds via an explicit computation of the Euler characteristic using a combinatorial characterization of faces, together with Björner's theorem on the homotopy type. Previously conjectured and verified computationally for k=2,...,7 (including a 6.8-hour C-accelerated computation for KG(16,7) yielding beta_2=241,669), this formula is now a theorem. The result is new for k >= 4, extending the work of Nilakantan-Singh (2018) who covered k=2,3 by different methods.