spatialDimPlusOne
The definition assigns the number of colors in the four-color theorem to equal the spatial dimension plus one. A researcher deriving graph-coloring results from Recognition Science would cite this to tie the theorem directly to the forced value of D. The definition consists of a single arithmetic expression with no further reduction steps.
claimDefine the number of colors $c$ by $c := D + 1$, where $D$ denotes the spatial dimension.
background
The module develops a Recognition Science account of the four color theorem for planar maps. It notes that four colors suffice because the count equals the spatial dimension plus one when that dimension is three. This identification rests on the recognition lattice at D=3, where the four colors correspond to the elements of the two-dimensional vector space over the field with two elements.
proof idea
The definition is a direct arithmetic assignment of the natural number three plus one. No lemmas are applied and no tactics are used beyond built-in evaluation of addition on natural numbers.
why it matters in Recognition Science
This definition supplies the equality fourColors equals spatial dimension plus one that appears inside the FourColorCert structure. It thereby links the theorem to the forcing-chain step T8 that fixes spatial dimension at three. The downstream theorem fourColors_eq_DplusOne then confirms the equality by decision procedure.
scope and limits
- Does not prove the four color theorem itself.
- Does not derive the spatial dimension from the functional equation.
- Does not address coloring of non-planar graphs.
formal statement (Lean)
27def spatialDimPlusOne : ℕ := 3 + 1
proof body
Definition body.
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