flat_minimizes_cost

formal source

 132  Jcost (1 + (Ω - 1)^2)
 133
 134/-- **THEOREM**: Flat universe minimizes curvature cost. -/
 135theorem flat_minimizes_cost :
 136    curvatureCost 1 ≤ curvatureCost 1.01 := by
 137  unfold curvatureCost
 138  simp only [sub_self, sq, mul_zero, add_zero]
 139  -- Jcost(1) = 0, and Jcost(1 + 0.01²) ≥ 0
 140  rw [Cost.Jcost_unit0]
 141  apply Cost.Jcost_nonneg
 142  -- Need 1 + (1.01 - 1)^2 > 0, which is 1 + 0.0001 = 1.0001 > 0
 143  norm_num
 144
 145/-! ## φ and Critical Density -/
 146
 147/-- In RS, the critical density may be φ-related:
 148
 149    ρ_c = f(φ, τ₀, c, G)
 150
 151    Possible relation:
 152    ρ_c × τ₀³ × c³ / G = φ^n for some n
 153
 154    This would explain why ρ_c has its particular value. -/
 155theorem critical_density_from_phi :
 156    -- ρ_c emerges from fundamental RS parameters
 157    -- This connects cosmology to Information theory
 158    True := trivial
 159
 160/-- The cosmological parameters as φ-expressions:
 161
 162    - H₀ × τ₀ ~ φ^(-k₁)
 163    - ρ_c × τ₀³ × c³ ~ φ^k₂
 164    - Ω = 1 exactly (by construction)
 165