Mathematics Graph Theory From Rs Q3 Vertices Eq

The cube's vertex count

A cube has eight corners. This is one of the first facts a child learns about solid geometry, and it follows from the simplest counting: each of the three dimensions contributes a factor of two, one for each side along that axis, so the total is 2 × 2 × 2 = 8. The Recognition Science framework encodes this same object as the recognition lattice, a discrete record of events arranged on the three axes, and its machine-checked library of formal theorems contains a declaration named q3Vertices_eq that certifies the count.

The declaration is a theorem in the formal sense: it states that the number of vertices in this three-dimensional lattice, written q3Vertices, equals 8. The proof is by direct computation, marked with the command "decide", which means the Lean kernel checks the arithmetic without any external assumptions. The library also certifies the companion facts: the lattice has 12 edges, computed as 3 × 2^(3-1), and its chromatic number is 2, meaning the vertices can be colored with two colors so that no edge connects two vertices of the same color. These properties match the ordinary cube: 8 corners, 12 edges, and a two-coloring by even and odd parity.

The theorem does not claim that the cube is the only graph with eight vertices, nor that the recognition lattice is the only way to build a cube. It does not assert that the number 8 arises from any physical law or that the framework's derivation of three dimensions is complete. The declaration simply verifies the arithmetic identity q3Vertices = 8 within the framework's own definitions, a small but exact step in a larger chain of formal results.

What the theorem changes is the status of a trivial fact: it moves the cube's vertex count from an observation to a certified statement inside a formal system. For a reader, the practical consequence is that the framework's graph-theoretic vocabulary is grounded in checkable arithmetic, not in hand-waving. The eight vertices, twelve edges, and two-coloring are not asserted but proved, and the proof is available for anyone to inspect.

theorem q3Vertices_eq : q3Vertices = 8 := by decide

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 theorem does not claim the cube is the only graph with eight vertices. The theorem does not claim the number 8 arises from any physical law. The theorem does not claim the framework's derivation of three dimensions is complete.

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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