The warp factor of supersymmetric $D=11$ near-horizon geometries: single-point rigidity, the $Spin(7)$ perfect square, and global constraints

On a compact connected section ${\cal S}$ of a supersymmetric $M$-horizon the Killing-spinor bilinears give a pointwise identity relating the warp factor $Δ$, the rotation one-form $h$ and the two spinor norms, with nothing assumed about the norm of the Killing spinor. We derive three consequences o