Q3_simplex_vertices
plain-language theorem explainer
Q3_simplex_vertices supplies the simplex vertex count D + 1 for the three-dimensional cube graph. Recognition Science researchers cite the value when inserting the η₁ correction into scaling exponents. The definition is a direct one-line arithmetic extension of the graph degree.
Claim. The critical simplex vertex count is defined by $D + 1$ where $D = 3$, so the value equals 4. This integer enters the first-order correction term in the Recognition Science mass formula on the phi-ladder.
background
The module formalizes combinatorial and spectral properties of the 3-dimensional hypercube Q₃, the unit cell of ℤ³. It records 8 vertices, 12 edges, 6 faces, and Laplacian eigenvalues {0, 2, 2, 2, 4, 4, 4, 6}. Q3_degree is the sibling definition that fixes this dimension at 3, matching the eight-tick octave that forces D = 3 in the unified forcing chain.
proof idea
The definition is a one-line wrapper that adds one to Q3_degree.
why it matters
The definition supplies the integer 4 required by the Q3Cert structure that certifies cube properties and spectral ratios. Those ratios determine correction-to-scaling terms in Recognition Science. The construction aligns with T8 of the forcing chain, where three spatial dimensions are forced, and feeds the η₁ correction referenced in the module doc-comment.
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