Mathematics Four Color Theorem From Rs Four Eq F2sq

The arithmetic identity

The four color theorem is a classical result in mathematics: any map drawn on a flat plane can be colored with at most four colors so that no two regions sharing a border have the same color. The number four is the tight bound; five colors would always suffice, but four is the smallest number that works for every possible map. The theorem was proved in 1976 by Kenneth Appel and Wolfgang Haken, and it was the first major theorem proved with substantial computer assistance.

In the Recognition Science framework, the declaration four_eq_F2sq establishes a simple arithmetic identity: the number four equals 2 squared, which is also the number of elements in the two-bit vector space F₂² = {00, 01, 10, 11}. The framework's library proves this identity as a formal theorem: fourColors = f2sq_card, where fourColors is defined as the natural number 4 and f2sq_card is defined as 2 ^ 2. The proof is by computation, using the decide tactic, and the library reports zero axioms and zero unproved assumptions.

The framework's structural observation connects this arithmetic fact to its model of space. In the framework, the number of colors corresponds to the number of elements in F₂², and the framework notes that 4 = 2² = 2^(D-1) at D=3, where D is the spatial dimension. The declaration itself, however, does not prove the four color theorem. It proves only the arithmetic identity that four equals 2 squared. The classical theorem about coloring maps remains a separate result, proved by Appel and Haken, and the framework's declaration does not provide a new proof of that theorem.

theorem four_eq_F2sq : fourColors = f2sq_card := by decide

def fourColorCert : FourColorCert where
  four_eq_Dp1 := fourColors_eq_DplusOne
  four_eq_2sq := fourColors_eq_2sq
  f2sq := four_eq_F2sq

What this page does not claim

The declaration does not prove the four color theorem itself, only the arithmetic identity 4 = 2^2. The framework does not claim that the four color theorem is a consequence of the arithmetic identity alone. The declaration does not establish any connection between the two-bit vector space and actual map coloring beyond the numerical coincidence.

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

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