graphTheoryDepthCert
plain-language theorem explainer
The depth certificate for graph theory asserts that the three-cube graph Q3 satisfies exactly five canonical theorems, has Euler characteristic 2, and chromatic number 2. Researchers in Recognition Science cite it to confirm that Q3 meets the lattice depth for spatial recognition in the framework. The construction is a direct record that pulls the theorem count from a decision procedure and the two graph invariants from prior equalities.
Claim. The graph theory depth certificate is the structure with three fields: the cardinality of the set of canonical graph theorems equals 5, the Euler characteristic of the 3-cube graph equals 2, and the chromatic number of the 3-cube graph equals 2.
background
The module presents the 3-cube graph Q3 as the canonical recognition lattice with 8 vertices, 12 edges, and 6 faces. Five canonical graph theorems (handshaking, Euler, Kuratowski, four-color, Ramsey) are identified with configuration dimension 5. The Euler characteristic is V minus E plus F, which equals 2 for Q3 and matches the sphere topology. The referenced structure requires exactly these three fields to be populated. Upstream results establish the theorem count by decision and the two Q3 invariants by decision and reflexivity.
proof idea
The definition constructs the certificate by direct field assignment. It sets the theorem count field to the established value of 5, the Euler characteristic field to the decided equality of 2, and the chromatic number field to the reflexive equality of 2. No further tactics or reductions are applied.
why it matters
This definition supplies the concrete witness that Q3 realizes the five graph theorems required for configuration dimension 5 in the Recognition Science framework. It packages the Euler and chromatic results into the required structure, closing the module on graph theory depth. The module documentation positions Q3 as the canonical recognition lattice, linking to the eight-tick octave and three spatial dimensions.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.