angular_contribution_per_dim

formal source

 131∫₀^π dθ = π
 132
 133Total: π⁵ -/
 134noncomputable def angular_contribution_per_dim : ℝ := Real.pi
 135
 136/-- The total angular factor is π^(config_space_dim) = π⁵. -/
 137noncomputable def total_angular_factor : ℝ := Real.pi ^ configSpaceDim
 138
 139theorem total_angular_is_pi5 : total_angular_factor = Real.pi ^ 5 := by
 140  unfold total_angular_factor configSpaceDim
 141  rfl
 142
 143/-! ## Part 4: The Curvature Integral Structure -/
 144
 145/-- The curvature integrand is the seam mismatch 103/102.
 146
 147The seam count arises from:
 148- 102 = 6 faces × 17 wallpaper groups (the denominator)
 149- 103 = 102 + 1 (Euler closure)
 150
 151The ratio 103/102 measures the topological "stress" between spherical
 152and cubic boundaries per unit of configuration space volume. -/
 153noncomputable def seam_ratio : ℝ := 103 / 102
 154
 155theorem seam_ratio_from_topology :
 156    seam_ratio = (seam_numerator D : ℝ) / (seam_denominator D : ℝ) := by
 157  unfold seam_ratio
 158  rw [seam_numerator_at_D3, seam_denominator_at_D3]
 159  norm_cast
 160
 161/-- The curvature correction is the seam ratio divided by the phase space volume.
 162
 163The phase space volume is π⁵ (from the angular integrations).
 164The full curvature correction is: