V
plain-language theorem explainer
The definition V(D) counts vertices of the D-dimensional hypercube as 2^D. Recognition Science researchers cite it when enumerating the 8 vertices of Q3 to derive the 48 chiral fermion states and gauge group dimensions from D=3. It is a direct power definition with no lemmas or tactics beyond built-in natural-number exponentiation.
Claim. Let $V(D)$ denote the number of vertices in the $D$-dimensional hypercube, defined by $V(D) := 2^D$.
background
The Spectral Emergence module starts from T8 forcing D=3, yielding the binary cube Q3 with exactly 8 vertices. V(D) supplies the base count for 24 chiral fermion flavors (= D × 2^D) and the automorphism group of order 48 that matches Standard Model fermion degrees of freedom. The module DOC states: 'From the single forced datum D = 3 (Theorem T8), the binary cube Q3 = {0,1}3 has 8 = 2^3 vertices.' Upstream results such as InflatonPotentialFromJCost.V and PhiForcingDerived.of supply the J-cost and phi-ladder used on the cube edges for mass hierarchies.
proof idea
One-line definition that directly applies the exponentiation operator on natural numbers to produce 2^D. No upstream lemmas are invoked; the body is the primitive power construction.
why it matters
V supplies the vertex count that feeds the parent derivations of SU(3)×SU(2)×U(1) content, three generations from face pairs, and |Aut(Q3)|=48 fermion states. It closes the self-consistency loop T8 (D=3) → Q3 vertices → B3 symmetry group in the module DOC. Downstream uses appear in Hamiltonian energy conservation and conjugate momentum constructions that rely on the same combinatorial counts.
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