Dark Energy from Topological Sector Imbalance: A Rigid Falsiable Prediction from the Temporal Direction Topology Principle

We derive the dark energy equation of state from the Temporal Direction Topology Principle (TDTP), which identifies the time direction as a topological invariant of the Pin+ bundle. The Pin- bordism group vanishes in 4D (Ω_4Pin- = 0), making Pin+ the unique spacetime structure capable of producing topological dark energy. The Z_2 exchange symmetry constrains the dark energy density to ρ_DE = ρ_0(1 + c_2 Δ(t)^2), where ρ_0 is the sector background vacuum energy and Δ(t) = e-2Γ_Σ t is the sector imbalance. Combined with Lindblad dynamics (supported by the de Sitter thermalization result of Yu & Wu), this yields two rigid falsifiable constraints: (i) w > -1 for all z (phantom crossing never occurs), and (ii) a specific exponential-decay functional form w(z) + 1 ∝ Δ(z)^2/(1 + c_2 Δ(z)^2) with a non-monotonic peak at z ≈ 0.8. With Γ_Σ = H_0 and c_2 ≈ 10.3, we predict w_0 = -0.762, agreeing with DESI DR2+DESY5 (w_0 = -0.752 ± 0.057) within 0.2σ. DESI DR3 and Euclid will provide decisive tests.