integerization_scale
The integerization_scale definition sets the charge integerization factor k to the face count of the 3-cube, fixing k at 6 to supply one symmetry channel per face for quantizing lepton charges. Researchers deriving the lepton index Z_ℓ via even polynomials on integerized charges would cite this scale as the initial geometric input. It is realized as a direct one-line application of the cube_faces function at dimension 3.
claimLet $k = F(3)$ where $F(D)$ denotes the number of faces of the $D$-cube. The integerization scale is defined by $k := 6$.
background
In the Z-map derivation, charge integerization begins with the scale k set to the face count of the 3-cube Q₃. Each face supplies one independent 2D symmetry channel for quantizing the electron charge, producing the integerized value Q̃_e = k · Q_e = -6. The cube_faces function appears in three upstream modules, each defining it as 2D for D = 3 and thereby returning 6 faces for the cube.
proof idea
This definition is a one-line wrapper that applies the cube_faces function directly to the argument 3.
why it matters in Recognition Science
This definition supplies the integerization scale k = 6 that feeds the integerization_scale_eq theorem and the Q_tilde_e definition, enabling the even polynomial ansatz to reach Z_ℓ = 1332. It implements the first structural step of the Z-map derivation in the module documentation, grounding the lepton charge index in 3-cube geometry rather than the T0-T8 forcing chain.
scope and limits
- Does not derive the face count from the T0-T8 forcing chain.
- Does not extend the scale to dimensions other than 3.
- Does not specify dynamical evolution of the symmetry channels.
formal statement (Lean)
40def integerization_scale : ℕ := cube_faces 3
proof body
Definition body.
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