If the internal fiber is an infinite tree $T_5$ or an asymptotic local-limit graph, its internal spectral measure is governed by the Kesten–McKay law.

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Kesten–McKay Density for Degree 5

The adjacency spectral density of $T_5$ is:

$${\mu }_5(x)=\frac{5\sqrt{16-x^2}}{2\pi (25-x^2)}{\mathbf{1}}_{[-4,4]}(x).$$

The corresponding shifted combinatorial Laplacian operator $L_F=5I-A_{T_5}$ has continuous spectral support:

$$\boxed{{\displaystyle \mathrm{supp}({\sigma }_{L_F})=[5-4,5+4]=[1,9].}}$$

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Contrast with Spacetime Dispersion

$$\begin{array}{rl}\text{Spacetime Continuum Spectrum }({\mathrm{\Delta }}_X)& ⟶[0,\mathrm{\infty })\text{ with Weyl growth }N(\lambda )\sim {\lambda }^{d_s/2}\\ \text{Internal Fiber Spectrum }(L_F)& ⟶[1,9]\text{ compact internal spectral band}\end{array}$$

The compact support $[1,9]$ generates a localized internal spectral band at scale ${\mu }_F^2[1,9]$ without contaminating the macroscopic unbounded UV spectrum of spacetime.