Mathematics Graph Theory Depth From Rs Graph Theory Depth Cert

A certificate, not a theory

Graph theory studies networks of points and the lines connecting them. Its classic results include the handshaking lemma, Euler's formula for polyhedra, Kuratowski's planarity theorem, the four-color theorem, and Ramsey's theorem. A certificate in the Recognition Science framework's machine-checked library of formal theorems packages three verified statements about a specific graph, the cube graph Q₃, which has 8 vertices, 12 edges, and 6 faces. The certificate records that these five classic theorems form a set of exactly five, that the cube's Euler characteristic V − E + F equals 2, and that its chromatic number is 2.

The Euler characteristic of the cube graph is 8 − 12 + 6 = 2, a number that matches the Euler characteristic of a sphere. This is not a coincidence: the cube graph is topologically a sphere, and the certificate's proof of V − E + F = 2 is a direct computation. The chromatic number being 2 means the cube's vertices can be colored with two colors so that no edge connects two vertices of the same color; this follows because the cube graph is bipartite. Each of these three facts is proved in the library with no unproved assumptions, meaning the certificate's contents are fully checked.

In Recognition Science, the framework models the cube graph as its canonical recognition lattice, a term for a discrete structure of connected points. The framework's library proves the three statements above as part of a larger claim that these five graph theorems represent a certain depth, denoted configDim D = 5. The certificate itself, however, does not prove that claim. It only bundles the three smaller proofs into one named object, so other parts of the library can refer to them together.

The certificate's value is organizational: it gives a single name to a small bundle of checked facts. A reader who wants to know why five graph theorems should correspond to a dimension, or why the cube graph deserves the label canonical, must look elsewhere. The certificate itself is a record, not an argument.

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

theorem q3Euler_eq_2 : q3EulerChar = 2 := by decide

theorem q3Chromatic_bipartite : q3ChromaticNumber = 2 := rfl

What this page does not claim

The certificate does not prove that the five graph theorems represent a depth or dimension of five. The certificate does not prove that the cube graph is the canonical recognition lattice. The certificate does not prove any of the five named graph theorems themselves; it only counts them.

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)

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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