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

Let $g$ be a complete Ricci-flat B\"ohm metric on $\R^{p+1}\times S^q$,
where $p,q\ge2$ and $p+q\le8$. We classify the geometric Dirichlet
kernel on every compact truncation and prove that the reduced Morse
index counts the preceding umbilic orbits. Every negative $L^2$
Lichnerowicz eigentensor is $O(p+1)\times O(q+1)$-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
$L^2$, 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.