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 (Ω₄^Pin- = 0), making Pin+ the unique spacetime structure capable of producing topological dark energy. The Z₂ exchange symmetry constrains the dark energy density to ρDE = ρ₀ (1 + c₂ Δ (t) ²), where ρ₀ 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 + c₂ Δ (z) ²) with a non-monotonic peak at z ≈ 0. 8. With Γ_Σ = H₀ and c₂ ≈ 10. 3, we predict w₀ = -0. 762, agreeing with DESI DR2+DESY5 (w₀ = -0. 752 ± 0. 057) within 0. 2σ. DESI DR3 and Euclid will provide decisive tests.