E
plain-language theorem explainer
E(D) computes the number of edges in the D-dimensional hypercube via the formula D times 2 to the power of D minus one. Recognition Science modelers cite it when enumerating connections in Q3 that generate the gauge group dimensions and 48 fermionic states. The definition is a direct algebraic implementation of the handshaking lemma on the hypercube graph.
Claim. $E(D) = D · 2^{D-1}$, the number of edges in the D-dimensional hypercube.
background
The Spectral Emergence module starts from T8 forcing D=3, yielding the binary cube Q₃ with 8 vertices. E supplies the edge count for this graph and its generalizations. The module doc states that this structure simultaneously forces SU(3)×SU(2)×U(1) gauge content, three generations from face pairs, and |Aut(Q₃)|=48 matching chiral fermion states in the Standard Model.
proof idea
Direct definition. It encodes the combinatorial count by multiplying dimension D by the per-direction edge multiplicity 2^{D-1}, which follows from each of the D coordinates contributing a binary choice across the remaining D-1 bits.
why it matters
E supplies the edge cardinality that downstream results in Action.EulerLagrange and Action.Hamiltonian use to construct geodesic and energy functionals on the recognition metric. It participates in the module's self-consistency loop linking T8 (D=3) to the 48-state fermion count and the phi-ladder on cube edges. It touches the uniqueness claim that D≠3 fails to produce the observed gauge and generation structure.
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