solid_angle_Q3
The definition computes the total solid angle subtended by the boundary of the 3-cube Q₃ as the product of its eight vertices and the angular deficit at each vertex. Researchers deriving the fine-structure constant from the cubic ledger geometry cite this quantity as the source of the 4π factor in the geometric seed. It is a direct definition that multiplies the vertex count 2^D by the per-vertex deficit 2π minus three times the cube dihedral angle.
claimLet Ω(∂Q₃) denote the total solid angle of the boundary of the 3-cube Q₃. Then Ω(∂Q₃) := 2^D ⋅ (2π − 3 ⋅ (π/2)), where D = 3 and the second factor is the angular deficit at each vertex.
background
The Alpha Derivation module constructs α^{-1} from the geometry of the cubic ledger unit cell Q₃ in D = 3 dimensions. During one recognition tick, a single active edge is traversed while the remaining passive edges couple to the vacuum geometry. The module records cube_vertices(D) = 2^D = 8 and vertex_angular_deficit = 2π − faces_per_vertex ⋅ cube_dihedral, which evaluates to 2π − 3(π/2) for the cube whose dihedral angle is π/2.
proof idea
One-line definition that multiplies the upstream cube_vertices(D) by the upstream vertex_angular_deficit. No tactics or reductions are applied beyond the product.
why it matters in Recognition Science
This supplies the 4π total curvature factor (via Gauss-Bonnet on ∂Q₃ ≅ S²) that enters geometric_seed = solid_angle_Q3 ⋅ 11 and then alpha_seed_structural. It realizes the first main result listed in the module: structural derivation of 4π from vertex deficits of Q₃ with zero imported constants. The quantity anchors the subsequent curvature term 103/102π^5 and the full α derivation chain.
scope and limits
- Does not compute solid angle for D ≠ 3.
- Does not include Planck-scale or PMNS corrections.
- Does not derive the dihedral angle from more primitive axioms.
- Does not distinguish active versus passive edges inside the solid-angle expression.
formal statement (Lean)
129noncomputable def solid_angle_Q3 : ℝ :=
proof body
Definition body.
130 (cube_vertices D : ℝ) * vertex_angular_deficit
131