phiLadderSum

formal source

 127    ∑_n φ^(-n) = 1/(1 - 1/φ) = φ/(φ-1) = φ² (geometric series)
 128
 129    But with alternating signs or other structure, cancellation occurs. -/
 130noncomputable def phiLadderSum : ℝ := phi^2  -- = φ/(φ-1)
 131
 132/-! ## The Dark Energy Density -/
 133
 134/-- The dark energy density ρ_Λ = Λ c² / (8π G).
 135
 136    Observed: ρ_Λ ≈ 6 × 10⁻²⁷ kg/m³
 137
 138    This corresponds to ~70% of the critical density. -/
 139noncomputable def darkEnergyDensity (lambda : ℝ) : ℝ :=
 140  lambda * c^2 / (8 * Real.pi * G)
 141
 142/-- Dark energy equation of state: w = p/ρ = -1.
 143
 144    For a cosmological constant, pressure equals negative density.
 145    This drives accelerated expansion. -/
 146noncomputable def equationOfState : ℝ := -1
 147
 148theorem dark_energy_w :
 149    equationOfState = -1 := rfl
 150
 151/-! ## Why Now? (The Coincidence Problem) -/
 152
 153/-- The coincidence problem: Why is ρ_Λ ~ ρ_matter NOW?
 154
 155    In the past, matter dominated (ρ_m >> ρ_Λ).
 156    In the future, dark energy dominates (ρ_Λ >> ρ_m).
 157    RIGHT NOW, they're comparable. Coincidence?
 158
 159    RS answer: This is not a coincidence!
 160    The transition happens at a specific φ-ladder rung. -/