Q3_euler_characteristic
plain-language theorem explainer
The Euler characteristic for the 3-cube satisfies V(3) + F₂(3) = E(3) + 2. Researchers deriving the Standard Model gauge content and fermion counts from D = 3 in Recognition Science cite this as the topological confirmation that Q₃ is sphere-like. The proof is a direct numerical check of the closed-form counts via native_decide.
Claim. For the three-dimensional binary cube, the vertex count plus the two-face count equals the edge count plus two: $V(3) + F_2(3) = E(3) + 2$, where $V(D) = 2^D$, $E(D) = D · 2^{D-1}$, and $F_2(D) = (D(D-1)/2) · 2^{D-2}$.
background
The Spectral Emergence module starts from the forced dimension D = 3 and constructs the binary cube Q₃ with 8 vertices. Vertex count is defined as V(D) := 2^D. Edge count is E(D) := D * 2^(D-1). Two-face count is F₂(D) := (D*(D-1)/2) * 2^(D-2). These supply the raw numbers for extracting |Aut(Q₃)| = 48 and the gauge dimensions 3 + 2 + 1.
proof idea
The proof is a one-line wrapper that applies native_decide to evaluate the three explicit arithmetic expressions and confirm the equality.
why it matters
This theorem is invoked inside the master result spectral_emergence to populate the euler_2 field of SpectralEmergenceCert. It supplies the sphere topology step in the chain from T8 (D = 3) through the eight-tick octave and phi-ladder to the emergence of SU(3) × SU(2) × U(1) and three generations. The relation V + F = E + 2 is the standard Euler characteristic for a 2-sphere realized combinatorially on the self-dual Q₃.
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