Spectral-Dimension Obstructions for Operators with Superlinear Counting Laws

We show that single-valuation exponential kernels, under mild regularity assumptions, converge in the continuum limit to a fourth-order operator with heat asymptotics $\mathrm{\Theta }(t)\sim t^{-1\mathrm{/}4}$ and hence spectral dimension $d_s=\frac{1}{2}$. Independently, a Tauberian analysis implies that any self-adjoint operator with superlinear eigenvalue counting $N(\lambda )\sim \lambda L(\lambda )$ must satisfy $\mathrm{\Theta }(t)\sim t^{-1}L(1\mathrm{/}t)$ and therefore has spectral dimension $d_s=2$. Since spectral dimension is invariant under unitary equivalence and compact perturbations, these exponents are incompatible, yielding a structural obstruction that separates single-valuation kernel limits from operators with accelerated spectral growth.