Spectral identities and closed-geodesic periods

We construct closed real-analytic locally symmetric nine-manifolds with equal scalar Laplace spectra and explicitly determine the infinite difference of their closed-geodesic period sets. Their common half-wave trace is real analytic near every period in this difference, although both geodesic flows are clean. In dimension eight, all finite-rank unitary flat connection spectra agree under a fixed Z/7 Z-marking, while the period sets differ. On a fixed closed simply connected eighteen-manifold, one of two real-analytic scalar-isospectral metrics has infinitely many primitive periods absent from the period set of the other. These spectral equalities follow from intrinsic equivariant identities for U(2n+1) and Sp(n)(2n+2), n≥1. For every central extension of a finite group with finitely generated kernel, prescribed finite families of central quotients occur as fundamental groups of closed homogeneous manifolds. Their canonical rough connection spectra agree for every finite-dimensional unitary representation of a common compact isotropy group. Simple connectivity varies in dimension eight, and rational formality in dimension seven. For a closed connected Riemannian manifold, marked finite-holonomy spectra determine the profinite conjugacy envelope of least lengths. Among compact connected Lie groups with bi-invariant metrics and a U(n)(1) marking, prescribed finite jets of the rate and logarithmic amplitude of the normalized equivariant heat trace, dimension and a volume lower bound characterize exactly these two models. The ambient weighted identity gives scalar-isospectral left-invariant normal homogeneous metrics on every SU(m), m≥6, which are not locally isometric, answering Gordon–Sutton’s uniqueness question negatively.