The principle-theory approach of Ref. 1 established that a non-trivial time-direction bundle (w₁T ≠ 0) produces a spin-dependent topological phase θ (j) = π (2j mod 2) and the parameter-independent prediction RFB = ∞ for the fermion--boson temporal decoherence ratio. However, the origin of w₁T ≠ 0---why the time-direction structure of spacetime should be topologically non-trivial---was left as an open question. In this paper, we provide a cosmological realization of w₁T ≠ 0 through inflation-era Z₂ symmetry breaking. We establish three main results. (1) Topological constraints (Theorems 1--3): Given w₁T ≠ 0, the existence of the flip surface Σ = PD (w₁T), its orientability, and the Berry holonomy phase θ = π are mathematical theorems requiring no physical assumptions beyond the topology itself. (2) Cosmological realization (Scenario 1): If a Z₂ symmetry of the time-direction field is spontaneously broken during inflation, domain walls are pushed beyond the causal horizon, and w₁T ≠ 0 is preserved as a topological fossil---a topological invariant guaranteed to persist through any smooth deformation including inflation. The flip surface Σ becomes a mathematical submanifold analogous to a Dirac string: gauge-dependent but encoding the physical invariant w₁T ≠ 0. (3) Physical consequences: The holonomy rate Γ_Σ ~ H₀ follows from gauge-invariant and dimensional arguments, since w₁T ≠ 0 has no local dynamics and the only cosmological timescale is H₀^-1; the prediction RFB = ∞ is robust under this scenario. Our framework resolves the domain-wall overclosure problem, provides a falsifiability basis for w₁T ≠ 0 comparable to that of cosmic inflation, and predicts that the time-direction topological structure is a fossil record of inflation-era physics.
Fangyuan Hao (Sun,) studied this question.