Q3_max_eigenvalue
plain-language theorem explainer
The largest eigenvalue of the Laplacian for the three-dimensional hypercube graph is defined as the constant 6. Researchers deriving spectral ratios for the unit cell of the integer lattice in Recognition Science cite this value when relating the maximum to the spectral gap and graph degree. The assignment matches the upper end of the known spectrum {0, 2, 2, 2, 4, 4, 4, 6} and is invoked directly by unfolding in the two immediate downstream statements.
Claim. The largest eigenvalue of the graph Laplacian of the 3-cube equals $6$.
background
The module records combinatorial and spectral facts for the 3-cube, the unit cell of the integer lattice in three dimensions. This graph has eight vertices, twelve edges, and six faces; its Laplacian spectrum consists of the values 0 (multiplicity 1), 2 (multiplicity 3), 4 (multiplicity 3), and 6 (multiplicity 1). The definition supplies the single largest entry from that spectrum for use in ratio calculations.
proof idea
The declaration is a direct constant definition that assigns the integer 6. It is applied by simple unfolding in the two downstream theorems that equate it to twice the graph degree and that form the ratio with the spectral gap.
why it matters
The constant enters the eigenvalue-ratio theorem, which states that the maximum eigenvalue divided by the spectral gap equals the graph degree. It supports the spectral formalism that supplies critical-exponent corrections in Recognition Science and is consistent with the three spatial dimensions obtained from the forcing chain. The downstream equality theorem confirms the relation to twice the degree.
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