unitSphereSurface_D3
Unit sphere surface area in three dimensions equals four pi. Derivations of the fine structure constant from isotropic recognition coupling cite this to fix the geometric seed. The tactic proof unfolds the general hypersurface formula, evaluates the gamma function at three halves via its recurrence relation, and reduces the resulting algebraic expression to four pi with field simplification and ring tactics.
claimThe surface area of the unit sphere in three-dimensional Euclidean space satisfies $S_2 = 4π$.
background
The SolidAngleExclusivity module proves that the geometric seed in the fine structure constant derivation must be four pi because it equals the surface area of the unit sphere in three dimensions. The general definition computes this area as two pi to the D over two divided by gamma of D over two. For D set to three this yields exactly four pi, as required by isotropy in the recognition ledger.
proof idea
The proof first unfolds unitSphereSurface. It then proves the gamma identity at three halves by applying the recurrence Gamma(s+1) equals s times Gamma(s) at s equals one half, together with the known value Gamma one half equals square root of pi. A separate calculation shows pi to the three halves equals pi times square root of pi. The main simplification applies these equalities, coerces the natural number three, clears the denominator with field_simp, and finishes with ring.
why it matters in Recognition Science
This theorem supplies the concrete value needed by four_pi_unique_for_D3 to conclude that four pi is the unique factor for D equals three. It likewise discharges the rewrite in solidAngle_is_sphere_area. Within the framework it confirms the solid angle term in the alpha seed expression alpha inverse equals four pi times eleven minus gap terms, aligning with the D equals three forcing from the unified chain and the requirement of isotropic coupling.
scope and limits
- Does not establish the formula for arbitrary dimension D.
- Does not derive the numerical value of pi from recognition principles.
- Does not address non-spherical geometries or curved spaces.
- Does not connect directly to the phi ladder or mass formulas.
- Does not prove exclusivity without the companion uniqueness theorem.
formal statement (Lean)
51theorem unitSphereSurface_D3 : unitSphereSurface 3 = 4 * Real.pi := by
proof body
Tactic-mode proof.
52 unfold unitSphereSurface
53 -- Γ(3/2) = (1/2)·Γ(1/2) = √π/2
54 have hGamma : Real.Gamma ((3 : ℝ) / 2) = (Real.sqrt Real.pi) / 2 := by
55 -- Use Γ(s+1) = s·Γ(s) with s = 1/2.
56 have : Real.Gamma ((1 / 2 : ℝ) + 1) = (1 / 2 : ℝ) * Real.Gamma (1 / 2 : ℝ) := by
57 simpa using (Real.Gamma_add_one (s := (1 / 2 : ℝ)) (by norm_num))
58 -- Rewrite (1/2)+1 = 3/2 and Γ(1/2) = √π.
59 simpa [Real.Gamma_one_half_eq, add_comm, add_left_comm, add_assoc, div_eq_mul_inv] using this
60
61 -- Simplify π^(3/2) = π^(1 + 1/2) = π * π^(1/2) = π * √π.
62 have hPiPow : Real.pi ^ ((3 : ℝ) / 2) = Real.pi * Real.sqrt Real.pi := by
63 have hpi_pos : (0 : ℝ) < Real.pi := Real.pi_pos
64 -- 3/2 = 1 + 1/2
65 have : (Real.pi : ℝ) ^ ((3 : ℝ) / 2) = Real.pi ^ ((1 : ℝ) + (1 / 2 : ℝ)) := by
66 norm_num
67 -- Expand with rpow_add.
68 calc
69 Real.pi ^ ((3 : ℝ) / 2)
70 = Real.pi ^ ((1 : ℝ) + (1 / 2 : ℝ)) := this
71 _ = Real.pi ^ (1 : ℝ) * Real.pi ^ (1 / 2 : ℝ) := by
72 simpa [add_comm, add_left_comm, add_assoc] using (Real.rpow_add hpi_pos (1 : ℝ) (1 / 2 : ℝ))
73 _ = Real.pi * Real.sqrt Real.pi := by
74 -- π^1 = π, and √π = π^(1/2) (via `sqrt_eq_rpow`).
75 simp [Real.rpow_one, Real.sqrt_eq_rpow]
76
77 -- Now compute the surface area.
78 -- 2 * π^(3/2) / (√π/2) = 4π
79 -- First normalize the coercion: (↑3 / 2) = (3 / 2 : ℝ)
80 have hcoerce : ((3 : ℕ) : ℝ) / 2 = (3 / 2 : ℝ) := by norm_num
81 simp only [hcoerce] at *
82 rw [hPiPow, hGamma]
83 -- Clear denominators carefully.
84 have hsqrt_pos : (0 : ℝ) < Real.sqrt Real.pi := by
85 have : (0 : ℝ) < Real.pi := Real.pi_pos
86 exact Real.sqrt_pos.mpr this
87 field_simp [hsqrt_pos.ne']
88 ring
89
90/-- For D = 2, the "unit sphere" is a circle with circumference 2π. -/