GraphTheoryDepthCert
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.
claimLet $Q_3$ 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 in Recognition Science
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.
scope and limits
- Does not establish the validity of the five graph theorems.
- Does not derive the Euler characteristic or chromatic number from first principles.
- Does not address higher-dimensional graphs or other lattices.
- Does not connect to physical constants or mass formulas.
formal statement (Lean)
39structure GraphTheoryDepthCert where
40 five_theorems : Fintype.card GraphTheorem = 5
41 euler_q3 : q3EulerChar = 2
42 chromatic_q3 : q3ChromaticNumber = 2
43