gauge_generators_eq_edges
plain-language theorem explainer
The theorem shows that the Lie algebra dimensions of the color, weak, and hypercharge sectors sum to twelve, matching the edge count of the three-dimensional hypercube. Researchers deriving Standard Model gauge content from the binary cube Q3 symmetries would cite this equality. The proof is a direct numerical evaluation of the sector ranks and edge function.
Claim. The sum of the Lie algebra dimensions for the color sector (dimension 8), weak sector (dimension 3), and hypercharge sector (dimension 1) equals the number of edges in the 3-cube: $8 + 3 + 1 = 12$.
background
The Spectral Emergence module starts from D = 3 forced by the unified chain and constructs the binary cube Q3 with eight vertices. The edge function E(D) is defined as D times 2 to the power D minus one, counting the connections where each vertex has D neighbors. The SpectralSector type decomposes the automorphism group B3 into color (from S3 permutations), weak (from reflections), and hypercharge (from parity) sectors, with gauge_rank returning the corresponding Lie algebra dimensions eight, three, and one.
proof idea
The proof is a one-line wrapper that applies norm_num to the definitions of gauge_rank on each sector and the edge function E at dimension three, reducing the equality to the numerical identity 8 + 3 + 1 = 12.
why it matters
This result supplies the gauge-generator count for the master theorem spectral_emergence, which certifies that the full Standard Model structure plus consciousness ground state follows from D = 3. It realizes the gauge content (SU(3) x SU(2) x U(1)) part of the Q3 decomposition and connects to the eight-tick octave and phi-forcing landmarks. The parent theorem assembles vertices, edges, faces, Euler characteristic, and automorphism order into a single certificate.
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