Topological Dark Energy: Three Roles, Three Components, and a Two-Layer Architecture for Time-Direction Selection

Dark Energy Series-Paper 2

We propose a unified framework for topological dark energy that identifies three distinct roles of topology—classification, constraint, and implementation—and derives a three-component dark energy structure from the interplay of the Pin⁺ bordism classification and the Nieh-Yan topological invariant. The classification role: Ω₄Pin⁺ = Z₁₆ provides sector labels k ∈ Z₁₆ that classify spacetime topological phases. The constraint role: the Nieh-Yan term, despite being a total derivative, constrains the contorsion current to degenerate onto the torsion 3-form (J₂ → |θ|²) via Hodge decomposition and the null-gauge fixing condition. The implementation role: Nieh-Yan torsion provides a geometric mechanism for time-reversal symmetry breaking, which is necessary but not sufficient for time-direction selection. The resulting dark energy density decomposes into three mathematically independent components: ρ_DE = |dϕ|²_quint + |θ|²_g + E(k), where |dϕ|² is the quintessence component, |θ|²_g is the topological component, and E(k) is the sector energy with k ∈ Z₁₆ the independent sector label. The mathematical independence is established through three independent proofs: (i) Hodge decomposition orthogonality, (ii) Pin⁺ torsion invisibility, (iii) independence of Z₁₆ sector and Nieh-Yan T-breaking strength. This framework yields a complete testable chain: strict null-gauge fixing implies w = −1 (pure topological dark energy), while its violation implies w ≠ −1 (mixed dark energy). DESI DR2 data (w₀ = −0.752 ± 0.057, DESI+CMB+DESY5) supports a small quintessence component. We establish a two-layer architecture for time-direction selection: the classification layer (Z₁₆ sector) and the implementation layer (Nieh-Yan T-breaking). The framework is falsifiable through: (i) w > −1 always, (ii) dark energy–black hole correlation, (iii) CMB parity-odd correlations.