The Osserman (1990) conjecture is fully resolved

The Osserman conjecture says that a Riemannian manifold whose Jacobi operators have the same eigenvalues for all unit tangent vectors at all points is flat or locally rank-one symmetric. Chi and Nikolayevsky proved it in every dimension except one case in dimension sixteen: a Jacobi eigenvalue of multiplicity seven or eight. Nikolayevsky reduced this case to an algebraic statement, the Nikolayevsky (2009) conjecture: an Osserman algebraic curvature tensor on ℝ16 with a Jacobi eigenvalue of multiplicity seven or eight has a Clifford structure or is a linear combination of the constant-curvature tensor and an orthogonal conjugate of the curvature tensor of the Cayley projective plane. We prove this conjecture. Hence the Osserman conjecture is true in every dimension, and so is the conformal Osserman conjecture, which is stated for dimensions greater than four. For two eigenvalues, a Cartan–Münzner quartic built from the Jacobi operators, together with the classification of isoparametric hypersurfaces with four principal curvatures, shows that the tensor is Clifford or comes from the Cayley projective plane. For three or more eigenvalues, polynomial eigenfields settle nineteen multiplicity patterns. In the remaining sixteen patterns, a sums-of-squares argument treats the case where the eigenvalue of multiplicity seven or eight is the smallest or the largest; otherwise we study the poles of a rational factorization of the Jacobi operators: a topological rigidity lemma for quadratic Gram matrices settles the case without poles, and degree bounds, real positivity and homogeneous factorization exclude each pole type or force a Clifford structure.