GraphTheoryDepthCert
plain-language theorem explainer
The GraphTheoryDepthCert structure certifies that the 3-cube graph Q₃ satisfies three key properties in the Recognition Science setting: exactly five canonical graph theorems are recognized, its Euler characteristic equals 2, and its chromatic number is 2. Researchers tracing the graph-theoretic foundations of RS would cite this certificate to confirm that Q₃ matches the expected sphere topology and bipartiteness. The declaration is a plain structure that directly records these three equalities.
Claim. Let $Q_3$ be the 3-cube graph. The certificate asserts that exactly five canonical graph theorems exist (handshaking lemma, Euler formula, Kuratowski theorem, four-color theorem, Ramsey theory), that the Euler characteristic satisfies $V - E + F = 2$, and that the chromatic number equals 2.
background
In the module on Graph Theory Depth from RS, the 3-cube graph $Q_3$ is introduced as the canonical recognition lattice with 8 vertices, 12 edges, and 6 faces. The inductive type GraphTheorem enumerates five canonical results: handshaking lemma, Euler's formula, Kuratowski's theorem, the four-color theorem, and Ramsey theory. The definitions q3EulerChar and q3ChromaticNumber are set to the computed Euler characteristic $V - E + F$ and to 2, respectively. This local setting establishes that $Q_3$ is topologically equivalent to a sphere since its Euler characteristic matches that of $S^2$.
proof idea
The declaration is a structure definition that bundles the three required equalities: the cardinality of GraphTheorem equals 5, q3EulerChar equals 2, and q3ChromaticNumber equals 2. No tactics or lemmas are applied; it serves as a direct packaging of the pre-established facts from the sibling definitions.
why it matters
This certificate supplies the concrete data for the downstream definition graphTheoryDepthCert, which constructs an explicit instance using graphTheoremCount, q3Euler_eq_2, and q3Chromatic_bipartite. It fills the graph-theoretic component of the Recognition Science framework by confirming that the five theorems align with configDim D = 5 and that $Q_3$ realizes the sphere Euler characteristic, consistent with the eight-tick octave and D = 3 spatial dimensions.
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