Q3_edges
plain-language theorem explainer
The definition fixes the edge count of the three-dimensional hypercube graph Q₃ at exactly 12. Researchers assembling the cube-based numerology for gauge generators and fermion states in Recognition Science cite this constant when building spectral emergence results. It is supplied by direct assignment that matches the standard combinatorial count for the unit cell of ℤ³.
Claim. Let $Q_3$ denote the graph of the three-dimensional hypercube. Then the number of edges is $E(Q_3) = 12$.
background
The module CubeSpectrum formalizes the combinatorial and spectral properties of the 3-cube Q₃, the unit cell of the integer lattice ℤ³. This graph has eight vertices, twelve edges and six faces, with Laplacian eigenvalues {0, 2, 2, 2, 4, 4, 4, 6}. The edge count enters Recognition Science numerology as the number of gauge generators.
proof idea
Direct definition that assigns the integer 12. It is consistent with the upstream algebraic expression 3 * 2² and the theorem that computes E 3 via norm_num.
why it matters
This supplies the edge count required by the AlphaFrameworkCert structure for certifying the δ₂ computation framework. It is referenced in the numerological summary theorem, which identifies the twelve edges with gauge generators and links them to the eight vertices as states per octave. The construction supports the forcing chain step that fixes three spatial dimensions through the hypercube spectrum.
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