A Graph-Fourier Companion to Paper #53: The 6/7 Dark-Energy Factor as the Interior-Spectrum Projector Trace

Paper #53 derived the dark-energy correction factor (F − χ)/F = 6/7 by identifying the two topologically inert face-modes of the Kelvin-cell face Laplacian as the A₁g zero mode (λ = 0, forced by connectivity) and the A₂u maximum mode (λ = 9, forced by bipartiteness of the hexagonal subgraph Q₃). The Tier 1 sub-theorems there — A₁g uniqueness, A₂u uniqueness, and inert count equal to χ(S²) = 2 — settle the counting, while the physical identification linking the inert subspace to the absence of pressure-wave energy rested on the A₂u ↔ Higgs identification of Paper #45 (Tier 2). This companion paper gives an independent route. We show that the 6/7 factor equals the trace over F = 14 of the spectral projector of the face Laplacian onto the open interval (λ_min, λ_max) = (0, 9). This identity is purely algebraic — a consequence of the spectrum multiplicities 1 + 3 + 2 + 3 + 4 + 1 = 14 with the two 1-dimensional irreps living at the band edges. A band-edge zero-group-velocity lemma from graph Fourier analysis then provides a physical complement: for the hyperbolic wave equation ∂²ψ/∂t² + c²Lψ = 0, the unique eigenvectors at λ_min and λ_max carry zero group velocity and therefore transport no wave energy. Together these establish the 6/7 factor as the fraction of propagating modes, without invoking Higgs absorption. Consequence: the algebraic identity tr(P_int)/F = 6/7 is Tier 1. The physical link to the cosmological pressure wave remains Tier 2 — shifted from the Paper-#45 A₂u=Higgs identification to the Part-X wave-equation identification, but not promoted. Full Tier 1 for ρ_Λ requires closure of Paper #48.