Spectral resolution and rigidity of spherical space forms

Let n≥5 be odd. We prove quantitative scalar spectral rigidity for homogeneous spherical space forms of even order and scalar lens spaces of odd order at least five. At fixed sufficiently small normalized frequency tolerance, an explicit bound on the lower limit of unmatched multiplicity divided by R^((n+1)/2), along prescribed cutoffs, forces isometry to the reference space. The exponent is sharp in general. A Besse homotopy n-sphere admitting a constant-displacement isometry of finite odd order at least five is round. For self-adjoint classical elliptic operators of order one with positive scalar principal symbol on Hermitian bundles over closed connected d-manifolds, d≥2, concentration modulo one near s fixed phases, with vanishing tolerance and o(R^d) exceptional copies along a sequence of cutoffs, forces the time-2π Hamiltonian map to have order at most s and gives a commuting quantizer A. Finer resolutions bound the order of P−A and, when it is not smoothing, determine the Liouville eigenvalue distribution of its normalized leading symbol. For finite free isometric quotients of closed connected Zoll manifolds of dimension at least two, the Poisson relation is an equality; the scalar spectrum determines maximal returning-component dimensions. In odd dimension n≥5, scalar multiplicities m_j=cj^(n−1)+O(j^(n−2)) have precisely the positive coefficients c=2^n/((n−1)!N), N a positive integer. The largest characterizes round real projective space up to scale.