q3ChromaticNumber
The definition fixes the chromatic number of the 3-cube graph Q₃ at 2. Researchers confirming that Q₃ realizes the canonical recognition lattice would cite this value when checking bipartiteness and the five canonical graph theorems. The declaration is a direct constant assignment with no reduction steps.
claimThe chromatic number of the three-dimensional hypercube graph $Q_3$ is $2$.
background
The module treats Q₃ as the 3-cube graph with 8 vertices, 12 edges and 6 faces. Recognition Science identifies Q₃ as the canonical recognition lattice whose Euler characteristic equals 2, matching χ(S²). The five canonical graph theorems (handshaking, Euler, Kuratowski, four-color, Ramsey) are taken to correspond to configDim D = 5.
proof idea
The declaration is a direct definition that assigns the constant 2. No lemmas or tactics are invoked; it functions as a foundational constant for later reflexivity proofs.
why it matters in Recognition Science
The value enters GraphTheoryDepthCert (chromatic_q3 : q3ChromaticNumber = 2) and GraphTheoryCert (chromatic_2 : q3ChromaticNumber = 2). It supports the T8 step that fixes D = 3 spatial dimensions, since the 3-cube is bipartite. The definition closes the graph-theoretic scaffolding needed for Recognition Composition Law applications.
scope and limits
- Does not derive the chromatic number from the adjacency matrix of Q₃.
- Does not treat chromatic numbers of higher-dimensional hypercubes.
- Does not prove that Q₃ is bipartite.
Lean usage
theorem q3Chromatic_bipartite : q3ChromaticNumber = 2 := rflformal statement (Lean)
36def q3ChromaticNumber : ℕ := 2