cube_vertices
plain-language theorem explainer
The definition states that a D-dimensional hypercube has exactly 2^D vertices. Researchers deriving discrete curvature on the 3-cube or matching Planck lengths via ledger-curvature extrema cite this count when summing vertex deficits or enumerating Gray-cycle ticks. The declaration is introduced by direct equating to the exponential formula with no additional lemmas.
Claim. The number of vertices $V(D)$ of the $D$-dimensional hypercube is given by $V(D) = 2^D$.
background
In the Planck-Scale Matching module the derivation of the recognition wavelength from the ledger-curvature extremum begins with the unique cost functional $J(x) = ½(x + x^{-1}) - 1$ evaluated at the self-similar fixed point φ, yielding $J_{bit} = cosh(ln φ) - 1$. A ±4 curvature packet is distributed over the eight faces of the 3-cube Q₃, producing the equilibrium condition $J_{bit} = J_{curv}(λ_{rec})$ that determines λ_rec after restoring SI dimensions with the averaging factor 1/π. The vertex count enters the discrete Gauss-Bonnet relation on ∂Q₃, where the total curvature equals 4π.
proof idea
The declaration is a direct definition that sets the vertex count equal to the exponential 2^D. No lemmas are applied; the body is the closed-form combinatorial expression for the number of vertices in a D-hypercube.
why it matters
This definition supplies the vertex count required for the discrete Gauss-Bonnet theorem on Q₃ and the solid-angle calculation in AlphaDerivation. It anchors the eight-tick octave in the Recognition Science forcing chain (T7), where the 3-cube has 8 vertices corresponding to the period 2^3. The count also feeds the octave offset in BaselineDerivation for mass rungs and the Q3_vertices theorem that equates the count to 8 for D = 3, closing the combinatorial step in Conjecture C8 for the Planck ratio.
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