Q3_edges
plain-language theorem explainer
The theorem establishes that the three-dimensional cube has exactly twelve edges. Physicists deriving Standard Model gauge content and fermion counts from forced D=3 in Recognition Science cite this as the edge count in the numerological summary. The proof is a direct numerical normalization from the general edge formula E(D) = D * 2^(D-1).
Claim. Let $E(D)$ be the number of edges in the $D$-cube, defined by $E(D) = D · 2^{D-1}$. Then $E(3) = 12$.
background
The Spectral Emergence module derives Standard Model structure and consciousness from the forced dimension D=3 (T8), yielding the binary cube Q₃ with 8 vertices. The edge function E counts edges via the formula D times 2 to the power D-1, since each vertex has D neighbors and each edge is shared by two vertices. This supplies the combinatorial input for gauge group dimensions (3+2+1=6) and the 48 fermionic states from |Aut(Q₃)| = 48.
proof idea
The proof is a one-line wrapper that applies norm_num to unfold the definition of E at argument 3.
why it matters
This populates the SpectralEmergenceCert (edges_12 field) and numerological_summary theorem, confirming 12 edges match the gauge generators. It closes part of the self-consistency loop from T8 (D=3) through the eight-tick octave and phi-ladder to the observed particle content with three generations. Downstream AlphaFrameworkCert and Q3Cert structures depend on this exact count.
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