Dark Energy from Topological Sector Imbalance: A Falsiable Prediction from the Pin+ Framework

[Dark Energy Series-Paper 1, An important update] We derive the dark energy equation of state from the Pin⁺ framework, which identifies the time direction as a topological invariant of the Pin⁺ bundle. The Pin⁻ bordism group vanishes in 4D (Ω₄Pin⁻ = 0), making Pin⁺ the unique spacetime structure capable of producing topological dark energy; this bordism exclusion is reinforced by representation-theoretic exclusion and observational constraints that remain valid even under the hypothesis that the bordism result could be bypassed, establishing progressive redundancy in the Pin⁺ selection. The bordism convention T̂² = +1 on the base manifold generates a compatibility operator T′ satisfying (T′)² = +1 on sector labels, producing the Z₂ exchange symmetry that constrains the dark energy density to ρ_DE = ρ₀(1 + c₂Δ(t)²), where ρ₀ is the sector background vacuum energy and Δ(t) = e^(−2Γ_Σ t) is the sector imbalance. The bordism classification Ω₄Pin⁺ = Z₁₆ further implies CPT transparency: the η-invariant is a mod-Z bordism invariant and admits no local density, so the SME coefficient (a_L) for CPT-odd effects vanishes identically. Combined with Lindblad dynamics (supported by the de Sitter thermalization result of Yu [29]), this yields two falsifiable constraints: (i) w > −1 for all finite redshifts when c₂ > 0 (phantom crossing never occurs within the consistent one-parameter truncation, as a joint consequence of topological, dynamical, and gravitational constraints; the mathematical domain is all finite z ≥ 0, with z < 1 being the observationally relevant regime), and (ii) a specific exponential-decay functional form w(z) + 1 = 4Γ_Σ c₂Δ(z)²/[3H(z)(1 + c₂Δ(z)²)] with a non-monotonic peak at z ≈ 0.77 (from the competition between the decreasing Δ(z)² and the increasing H(z)), converging to w → −1 at high redshift as H(z) dominates the correction term. With Γ_Σ = H₀ and c₂ ≈ 10.3 (fitted from the observed w₀, analogous to the Ginzburg–Landau parameter in superconductivity), we obtain w₀ = −0.762 and w_a ≈ +0.68 (standard CPL convention), consistent with DESI DR2+DESY5 (w₀ = −0.752 ± 0.057) within 0.2σ. The core predictive content is the topologically constrained functional form and the bound w > −1 (for c₂ > 0), not the specific w₀ value. The framework is falsifiable through two independent channels: CMB parity-odd correlations (positive detection) and neutrino CPT tests (negative exclusion). DESI DR3 and Euclid will provide decisive tests.