The optimal eigenvalue count in Aubry's spectral sphere theorem for dimensions at least four

For every n≥4, we determine the least number of positive Laplace eigenvalues whose pinching to n, under Ric_g≥(n−1)g, forces a closed connected n-manifold to be homeomorphic or diffeomorphic to Sⁿ. This number is n, also within the simply connected class. The examples are defined on a fixed simply connected manifold Mₙ with H₂(Mₙ;ℤ)≅ℤ². At every sufficiently small prescribed volume V>0, it admits a smooth family with λ₁,…,λₙ₋₁→n and λₙ≥n+cₙ, where cₙ>0 is an Aubry constant. The proof uses the fixed-volume degeneration theorem for core-admissible doubles from a companion preprint. Applying that theorem also to Sⁿ gives two families, on nonhomeomorphic manifolds, with the same volume, Ricci lower bound, and metric-measure limit S^(n−2)*S¹_(2πq), where q=V/Vol(Sⁿ,g_(Sⁿ)). Spectral convergence gives the same limit for every fixed eigenvalue index. For q=1/m, the known spectrum of the corresponding cyclic spherical quotient makes these limits explicit. In particular, for all sufficiently large integers m≥3, the nth positive eigenvalue of both families tends to 2(n+1). Each family has constant volume density and, for fixed V, a uniform positive lower bound for the volumes of unit balls.