Q3_vertices_eq
plain-language theorem explainer
The vertex count of the 3-cube equals eight by direct evaluation of its definition. Researchers assembling the voxel-seam series for higher-order corrections to α^{-1} cite this when fixing the base combinatorial count for δ_n sums. The proof is a one-line reflexivity that computes 2^3 directly to 8.
Claim. The vertex count of the 3-cube satisfies $|V(Q_3)| = 8$.
background
The module develops higher-order voxel-seam corrections to α^{-1} to close the ~8 ppm gap with CODATA. It begins with the cube combinatorics on Q₃, where the vertex count is defined as 2^3. This count supplies the base for the series α^{-1} = α_seed − f_gap + Σ_{n=1}^∞ δ_n, with each δ_n a finite sum over n-fold face-wallpaper pairs weighted by the Z₂^5 half-period measure. Upstream results in PlanckScaleMatching establish the same count via cube_vertices 3 = 8 and tie it to the curvature packet axiom that distributes ±4 curvature over the eight vertices.
proof idea
The proof is a term-mode reflexivity that evaluates the definition 2^3 directly to 8. No external lemmas are invoked beyond the built-in reduction of natural-number exponentiation.
why it matters
This equality supplies the base count for the cube combinatorics that feed the α^{-1} series framework and are referenced by the downstream Q₃ Laplacian spectrum construction. It realizes the eight-tick octave (period 2^3) from the forcing chain T7, anchoring the combinatorial sums that target the residual between the RS prediction and the CODATA value 137.035999206. The explicit computation of δ₂ remains open.
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