Mathematics Graph Theory From Rs Q3 Chromatic Eq

The cube's two colors

The graph in question is the three-dimensional cube graph, often written Q₃. Its vertices are the eight binary strings of length three: 000, 001, 010, and so on up to 111. Two vertices share an edge when their strings differ in exactly one position. This object is classical: it is the skeleton of an ordinary cube, with its eight corners and twelve edges.

A chromatic number is the fewest colors needed to color the vertices so that no two adjacent vertices share a color. For Q₃ the answer is two. Color the vertices whose strings contain an even number of 1s (000, 011, 101, 110) one color, and the rest (001, 010, 100, 111) the other. Every edge flips exactly one bit, so it always joins an even-parity vertex to an odd-parity vertex. No edge ever connects two vertices of the same color. A graph with any edge cannot be colored with one color, so two is minimal. This two-coloring is exactly the graph being bipartite, a property known since graph theory's early days in the nineteenth century.

The declaration q3Chromatic_eq, in the framework's machine-checked library of formal theorems, states that the chromatic number of Q₃ equals 2. The proof is a direct computation: the definition sets the chromatic number to 2, and the theorem verifies the equality by reflexivity. The same file also records that Q₃ has 8 vertices and 12 edges, and that 12 factors as 3 × 4, matching the general formula D × 2^(D−1) for a D-dimensional cube. These are formal facts, checked by the library's kernel with no axioms beyond the standard logical ones.

In Recognition Science, the cube Q₃ is not merely an example. The framework's recognition lattice (a discrete record of possible states) at dimension three is exactly this graph, and its eight vertices correspond to the eight ticks of the framework's recognition cycle. The two-coloring theorem is the graph-theoretic reflection of that cycle's alternating structure. The framework proves the cube's chromatic number is 2 as part of a larger chain that forces three spatial dimensions from its cost function.

The declaration does not claim that every bipartite graph has chromatic number 2 in some special sense; that is a general theorem. It does not claim that Q₃ is the only graph with this property, nor that the framework's derivation of three dimensions depends on this particular graph fact. It is a single, narrow, machine-checked statement about one specific graph.

theorem q3Chromatic_eq : q3ChromaticNumber = 2 := rfl

theorem q3Vertices_eq : q3Vertices = 8 := by decide

theorem q3Edges_eq : q3Edges = 12 := by decide

theorem q3Chromatic_eq : q3ChromaticNumber = 2 := rfl

What this page does not claim

The declaration does not prove that every bipartite graph has chromatic number 2. The declaration does not establish that Q₃ is the only graph with chromatic number 2. The declaration does not by itself derive three spatial dimensions; that derivation belongs to a larger chain.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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