Paper TS-G: The Origin of Dark Energy — Sector III as the Cosmological Constant: A Derivation from the Four-Sector Cosmology Framework

Abstract We derive the cosmological constant Λ > 0 from the Four-Sector Cosmology (FSC) framework by identifying Sector III — the time-reversed sector with coordinate −t — as the physical origin of dark energy. Three results are established. (G1) Sign of Λ: The time reversal t → −t in Sector III reverses the sign of the pressure in the stress-energy tensor as seen from Sector I, yielding the equation of state w = −1 and a repulsive contribution to Einstein's equations. This is the first derivation of the sign of Λ from a sector-based symmetry principle. (G2) Form of Λ: The stress-energy tensor of Sector III, coupled to Sector I via the Israel junction conditions, takes the form T^III_μν = −ρ_III g_μν, which is identical to the cosmological constant term. The identification Λ = 8πG ρ_III/c⁴ follows immediately. (G3) Magnitude Ω_Λ = 0.683: Beginning from the CPT-symmetric initial state Ω_I = Ω_II = Ω_III = Ω_IV = 25% (Paper D), the decay of Sector IV at rate λ = 1.25×10⁻¹⁷ s⁻¹ and the subsequent redistribution of density governed by α(t) = 1 − e^(−t/τ) (τ ≈ 4.6 Gyr) produce the present-day composition Ω_I : Ω_II : Ω_III = 5 : 27 : 68 at t = 13.8 Gyr. These results resolve both the old cosmological constant problem (why Λ ≪ M_Pl⁴) and the coincidence problem (why Ω_Λ ≈ Ω_m), and provide the first self-consistent derivation of all three properties of dark energy — sign, form, and magnitude — from a single underlying framework. Five observational predictions are made, including w_a ≈ +0.003 (opposite in sign to quintessence), detectable by the Euclid satellite.