Assume the global graph Laplacians $L_n^X$ converge to ${\mathrm{\Delta }}_X$ in the declared norm-resolvent / quasi-unitary sense. Let $L_F$ be a fixed finite self-adjoint non-negative fiber operator with eigenpairs $({\nu }_a,e_a)$.

Then the combined operator

$$L_n^{LG}=L_n^X\otimes I+{\mu }_F^{2}I\otimes L_F$$

decomposes as the finite orthogonal direct sum:

$$\boxed{{\displaystyle L_n^{LG}\cong \underset{a}{⨁}(L_n^X+{\mu }_F^2{\nu }_a).}}$$

The continuum limit is

$$\boxed{{\displaystyle L^{LG}={\mathrm{\Delta }}_X\otimes I+{\mu }_F^{2}I\otimes L_F.}}$$

Because resolvent convergence is stable under a fixed scalar spectral shift and only finitely many fiber channels occur:

$$\boxed{{\displaystyle L_n^{LG}⟶L^{LG}}}$$

in the corresponding finite-fiber norm-resolvent sense.

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Exact Spectral Consequence

If ${\lambda }_{j,n}$ are base eigenvalues, then

$$\boxed{{\displaystyle \mathrm{Spec}{L}_n^{LG}=\{{\lambda }_{j,n}+{\mu }_F^2{\nu }_a{\}}_{j,a}.}}$$

Likewise, the heat semigroup factorizes:

$$\boxed{{\displaystyle e^{-t{L}_n^{LG}}=e^{-t{L}_n^X}\otimes e^{-t{\mu }_F^2{L}_F}.}}$$

Thus the internal channels are spectrally visible while the continuum limit remains strictly governed by the global Dirichlet convergence.

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Exact Finite Profile: $K_6$

The complete graph $K_6$ is 5-regular on 6 vertices. Its adjacency spectrum is:

$$5,(-1{)}^{\times 5},$$

so its nontrivial adjacency eigenvalues obey the degree-five Ramanujan bound $|\lambda |\le 2\sqrt{4}=4$. Its combinatorial Laplacian spectrum is:

$$\boxed{{\displaystyle 0,6^{\times 5}.}}$$

This gives a six-state finite internal profile with one singlet ground state and one five-dimensional excited multiplet. It does not reproduce the continuous Kesten–McKay density; that belongs to the infinite tree/local-limit profile.