The exact Laplacian spectrum of the Berker−Ostlund diamond lattice

We derive, in closed form, the complete spectrum of the combinatorial Laplacian $L^{(L)}$ on the level-$L$ Berker--Ostlund diamond hierarchical lattice $\mathcal{D}_L^{(b)}$ with branching parameter $b\ge 2$. Because the vertex degree grows exponentially under inflation, the renormalized eigenvalue equation is not a Laplacian at the coarser scale but a two-parameter matrix pencil $\alpha D-A$, whose coefficients evolve under a quadratic renormalization map of the parameter plane. Iterating a Schur-complement block recursion yields an exact product formula for the characteristic polynomial $\chi_L(\lambda)$ over the singular polynomials $S_\ell=2\alpha_\ell-\beta_\ell$ and two boundary polynomials $P_L^\pm$, together with the exact multiplicity of every localized eigenvalue; the multiplicity contains a topological correction given by the cyclomatic number of the ancestral graph, and a dimension audit confirms $\deg\chi_L=V_L$. As a consequence, the empirical spectral measure is proved to converge, as $L\to\infty$, to an explicit purely atomic probability measure carried by the singular ladder, whose dominant atom sits at the molecular eigenvalue $\lambda=2$ with weight $(b-1)/b$. The relation between the accumulation set of the spectrum and the complex dynamics of the planar renormalization map is formulated as a set of precise conjectures.