We analyse the spectral convergence of high order elliptic di erential operators subject to singular domain perturbations and homogeneous boundary conditions of intermediate type. We identify sharp assumptions on the domain perturbations improving, in the case of polyharmonic operators of higher order, conditions known to be sharp in the case of fourth order operators. The optimality is proved by analysing in detail a boundary homogenization problem, which provides a smooth version of a polyharmonic Babuška paradox.Recall that this is the weak formulation of the Poisson problem for (−∆) m +I with (SIBC).For u ∈ W m,2 (Ω), de ne T ϵ u = u • Φ ϵ where Φ ϵ is a smooth di eomorphism mapping Ω ϵ into Ω that coincides with the identity on a large part K ϵ of Ω, with |Ω \ K ϵ | → 0 as ϵ → 0, see (3.5). Let u be such that u ϵ − T ϵ u L 2 (Ω ϵ ) → 0 as ϵ → 0. Theorem 7 states that the limit u solves di erent di erential problems according to the values of the parameter α. More precisely, we have the following trichotomy: (i) (Stability) If α > 3/2, then u solves (1.8) in Ω, that is, u satis es (−∆) m u + u = f in Ω and (SIBC) on ∂Ω; (ii) (Degeneration) If α < 3/2, then u satis es (−∆) m u + u = f in Ω, with Dirichlet boundary conditions on W × {0}, that is ∂ l u ∂n l = 0, for all 0 ≤ l ≤ m − 1, and (SIBC) on the rest of the boundary of Ω; (iii) (Strange term) If α = 3/2, then u satis es (−∆) m u + u = f in Ω with the following boundary conditions on W × {0} D l u = 0, for all 0 ≤ l ≤ m − 2, ∂ m u ∂n m + K ∂ m−1 u ∂n m−1 = 0,