Dirichlet deformations, Morse index, and nonlinear symmetry of Ricci-flat Böhm metrics

Let g be a complete Ricci-flat Böhm metric on ℝ^(p+1)×S^q, where p, q≥2 and p+q≤8. We classify the geometric Dirichlet kernel on every compact truncation and prove that the reduced Morse index counts the preceding umbilic orbits. Every negative L² Lichnerowicz eigentensor is O(p+1)×O(q+1)-invariant, and the quadratic form is coercive in homogeneous energy on the complementary subspace. These results imply symmetry of sufficiently small ancient Ricci–DeTurck solutions whose non-invariant component is bounded in L², without an assumption of backward convergence. They also give stationary rigidity in a fixed background gauge. Each degeneracy of the smooth induced-metric map is a fold. Arbitrarily large finite collections of pairwise nonisometric Ricci-flat fillings, with arbitrarily large reduced index, persist under nonsymmetric boundary perturbations. The reduced negative spectrum converges to the complete negative spectrum; a transverse Dirichlet-to-Neumann form has a positive-measure representation. All scalar coefficient identities are given explicitly, with their exact integer data.