solid_angle_per_octant
solid_angle_per_octant assigns the value π/2 steradians to the solid angle of one octant in three-dimensional space. Researchers deriving the recognition wavelength λ_rec from the J-bit and curvature-cost extremum cite this constant when averaging over the eight faces of the Q3 hypercube to recover the factor 1/π. The definition is a direct numerical assignment obtained by dividing the sphere's total solid angle by eight.
claimThe solid angle subtended by one octant equals $π/2$ steradians.
background
The PlanckScaleMatching module derives λ_rec ≈ 0.564 ℓ_P by equating the bit cost J(φ) = cosh(ln φ) - 1 to the curvature cost 2λ² and restoring SI dimensions through geometric averaging over the eight faces of the Q3 hypercube. The solid angle per octant supplies the measure for each face in this averaging step, which introduces the 1/π prefactor in the final Planck ratio. Upstream, the J functional is taken from PhiForcing and the eight-octant partition follows from the three-dimensional space fixed by the forcing chain.
proof idea
The definition is a direct assignment solid_angle_per_octant := Real.pi / 2 that encodes the standard partition of the unit sphere into eight equal octants.
why it matters in Recognition Science
This definition supplies the geometric prefactor used by the theorem octants_cover_sphere, which verifies that eight times the value recovers the full 4π steradians. It closes the face-averaging step in the derivation chain of Conjecture C8, linking the eight-tick octave structure to the continuous curvature cost that yields λ_rec = ℓ_P / √π.
scope and limits
- Does not derive the numerical value of π from Recognition Science axioms.
- Does not generalize the solid-angle partition to dimensions other than three.
- Does not incorporate curvature-induced deformations of octant boundaries.
Lean usage
theorem check : 8 * solid_angle_per_octant = 4 * Real.pi := by simp [solid_angle_per_octant]formal statement (Lean)
175noncomputable def solid_angle_per_octant : ℝ := Real.pi / 2
proof body
Definition body.
176
177/-- There are 8 octants in 3D space. -/