Mathematics Graph Theory Depth From Rs Graph Theorem Count

The five canonical theorems

Graph theory has a small set of results that appear in every textbook: the handshaking lemma, Euler's formula for planar graphs, Kuratowski's theorem on planarity, the four-color theorem, and Ramsey's theorem. The declaration graphTheoremCount is a machine-checked statement in the Recognition Science framework's library that these five named theorems form a set of exactly five. The proof is a direct computation: the framework defines an inductive type with those five constructors and then verifies that its cardinality is 5. That is all the declaration itself establishes.

What makes the number interesting is the surrounding framework claim. The same library also establishes that the 3-cube graph Q₃, with 8 vertices, 12 edges, and 6 faces, has Euler characteristic 8 − 12 + 6 = 2, matching the sphere S². The framework models Q₃ as the canonical recognition lattice, and it identifies the five theorems as the complete depth of graph theory at that level. The number five is then tied to a broader dimension count in the framework, but that identification is a modeling choice, not a theorem about graph theory itself.

The declaration does not claim that these are the only important graph theorems, nor that graph theory is exhausted by them. It does not establish the four-color theorem or Ramsey's theorem; it only names them as members of a finite set. The framework's interpretation of five as a structural dimension is a hypothesis about how mathematics organizes itself, not a result established by graphTheoremCount. What the declaration establishes, strictly, is a count: five named theorems, no more and no less.

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

theorem q3Euler_eq_2 : q3EulerChar = 2 := by decide

theorem q3Euler_eq_2 : q3EulerChar = 2 := by decide

What this page does not claim

The declaration does not prove the four-color theorem or Ramsey's theorem. The framework's identification of five as a structural dimension is not a theorem about graph theory. The set of five is not claimed to be exhaustive of all important graph theorems.

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.

Derived articles

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