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 beta₂ = C (2k+1, k) (k³-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 beta₂=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.