Mathematics Graph Theory Depth From Rs Q3 Chromatic Bipartite

The cube's two-color proof

The 3-cube graph, written Q₃, is the shape of a cube's corners and edges: 8 vertices, 12 edges, and 6 square faces. A graph is bipartite when its vertices can be split into two groups so that every edge connects one group to the other; equivalently, it can be colored with two colors so that no two adjacent vertices share a color. The cube is the classic example: color the bottom four corners red and the top four blue, and every edge runs between bottom and top. The declaration q3Chromatic_bipartite states this fact formally: the chromatic number of Q₃ equals 2.

The proof is not a long argument; it is a definitional identity. The framework defines the chromatic number of Q₃ to be 2, and the theorem confirms that the definition reduces by reflexivity. In the machine-checked library of formal theorems, this is recorded with zero unproved assumptions. The same library also proves the Euler characteristic of the cube, V minus E plus F, equals 2, matching the sphere's value, and counts five canonical graph theorems (handshaking, Euler, Kuratowski, four-color, Ramsey) as a distinct set of size five.

In Recognition Science, the cube is a canonical recognition lattice, a discrete grid of events that the framework uses to model how a system tracks its own states. The two-color property is part of that role: it shows the lattice's structure is simple enough to separate cleanly into two alternating classes. But the declaration itself does not assert any deeper physical law. It does not claim the cube is the only bipartite graph, nor that every recognition process must be bipartite. It only certifies a single combinatorial fact about one specific graph.

What the reader gains is a precise, machine-checked statement of a familiar property, plus a concrete example of how the framework records such facts. The cube's two-colorability is a small but real anchor: it connects an everyday geometric object to a formal system that aims to derive physical structure from recognition costs. The declaration is a building block, not a grand conclusion.

theorem q3Chromatic_bipartite : q3ChromaticNumber = 2 := rfl

theorem q3Chromatic_bipartite : q3ChromaticNumber = 2 := rfl

theorem q3Euler_eq_2 : q3EulerChar = 2 := by decide

theorem graphTheoremCount : Fintype.card GraphTheorem = 5 := by decide

What this page does not claim

The declaration does not claim that every bipartite graph is a cube. It does not claim that recognition processes must be bipartite. It does not establish any physical law beyond the combinatorial fact.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/GraphTheoryDepthFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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