unitSphereSurface_D2
The declaration shows that the surface measure of the unit circle equals 2π. Researchers deriving the fine-structure constant from isotropic coupling cite this base case to isolate the D=3 result. The tactic proof unfolds the general surface-area formula and simplifies via the gamma function at one together with real-power rules.
claimThe surface area of the unit circle satisfies $S_1 = 2π$, where the general formula is $S_{D-1} = 2π^{D/2}/Γ(D/2)$.
background
The module establishes why the factor 4π appears in the α derivation α^{-1} = 4π·11 - f_gap - δ_κ. It requires isotropic coupling, normalization to the passive-edge count, and D=3 dimensionality. The supporting definition is unitSphereSurface(D) := 2 * π^{D/2} / Γ(D/2), which recovers the circumference 2π when D=2 and the solid angle 4π when D=3.
proof idea
The proof unfolds unitSphereSurface, introduces the normalization (2/2)=1 by norm_num, and applies simp using Real.Gamma_one together with Real.rpow_one.
why it matters in Recognition Science
This base case supports the module's uniqueness argument that only the D=3 solid angle supplies the required 4π factor. It contrasts with the sibling unitSphereSurface_D3 and feeds the broader claim that isotropic measure is forced by the Recognition Composition Law and the eight-tick octave. The result touches the alpha-band interval (137.030, 137.039) but leaves open the explicit gap term f_gap.
scope and limits
- Does not prove uniqueness of the isotropic measure.
- Does not extend to non-integer dimensions.
- Does not derive the physical coupling interpretation.
- Does not address the gap or defect terms in the α formula.
formal statement (Lean)
91theorem unitSphereSurface_D2 : unitSphereSurface 2 = 2 * Real.pi := by
proof body
Tactic-mode proof.
92 unfold unitSphereSurface
93 -- Γ(1) = 1, π^1 = π
94 -- 2 * π^1 / 1 = 2π
95 have h : ((2 : ℝ) / 2) = (1 : ℝ) := by norm_num
96 -- `simp` knows Γ(1)=1 and π^1=π for `Real.rpow`.
97 simp [h, Real.Gamma_one, Real.rpow_one]
98
99/-! ## Part 2: Uniqueness of Isotropic Measure -/
100
101/-- **Uniqueness Theorem**: For any dimension D, the uniform measure on S^{D-1}
102 is unique up to scaling.
103
104 This is a consequence of the uniqueness of Haar measure on compact groups:
105 SO(D) acts transitively on S^{D-1}, so any SO(D)-invariant measure is
106 proportional to the uniform surface measure. -/