Adjoint Closures of Singular Pairs of Quadratic Forms and a Pfister-Type Criterion

Let $K$ be a field of characteristic different from $2$. First proved that, for a nonsingular pair of quadratic forms over a number field or a real closed field, weak hyperbolicity is equivalent to vanishing of the total signature on the adjoint closure, and he asked whether nonsingularity is necessary. We show that it is. Over every formally real field, we construct a singular pair on $K^7$ whose adjoint closure is exactly its two-dimensional pencil and consists entirely of hyperbolic forms, although the pair is not weakly hyperbolic. We then identify the mechanism. For every symmetric singular Kronecker block $M_{\epsilon }$ of positive minimal index, its adjoint closure is its pencil; moreover, adjoining $M_{\epsilon }$ to a nonsingular pair collapses the visible regular closure to the original pencil. Minimal-index-zero blocks, by contrast, preserve the full regular closure. This yields an exact visibility dichotomy and a general family of counterexamples. Finally, over number fields and real closed fields, we prove that dimension $7$ is minimal. For the minimal example we also compute the full adjoint algebra and its involution-trace form.