gauge_generators_eq_edges
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.
claimThe 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 in Recognition Science
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.
scope and limits
- Does not derive the explicit gauge group representations acting on fermions.
- Does not compute numerical values of coupling constants or their running.
- Does not address the phi-ladder mass spectrum or rung assignments.
- Does not prove that these dimensions are the only possible gauge content for D = 3.
Lean usage
example : SpectralSector.gauge_rank .color + SpectralSector.gauge_rank .weak + SpectralSector.gauge_rank .hypercharge = E 3 := gauge_generators_eq_edgesformal statement (Lean)
191theorem gauge_generators_eq_edges :
192 SpectralSector.gauge_rank .color +
193 SpectralSector.gauge_rank .weak +
194 SpectralSector.gauge_rank .hypercharge = E 3 := by
proof body
Term-mode proof.
195 norm_num [SpectralSector.gauge_rank, E]
196
197/-! ## Part 3: Generation Structure — Three Families from Face Pairs
198
199The Q₃ cube has 6 faces organized into 3 opposite pairs. Each face pair
200corresponds to a normal axis. The 3 face pairs force exactly 3 generations
201of fermions. -/
202
203/-- **THEOREM**: D = 3 forces exactly 3 generations. -/