Cosmological Origin of the Time Direction Bundle Inflation Era Symmetry Breaking and Topological Fossils

The principle-theory approach of Ref. [1] established that a non-trivial time-direction bundle (w_1^T ≠ 0) produces a spin-dependent topological phase θ(j) = π(2j mod 2) and the parameter-independent prediction R_FB = ∞ for the fermion--boson temporal decoherence ratio. However, the origin of w_1^T ≠ 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 w_1^T ≠ 0 through inflation-era Z_2 symmetry breaking. We establish three main results. (1) Topological constraints (Theorems 1--3): Given w_1^T ≠ 0, the existence of the flip surface Σ = PD(w_1^T), 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 Z_2 symmetry of the time-direction field is spontaneously broken during inflation, domain walls are pushed beyond the causal horizon, and w_1^T ≠ 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 w_1^T ≠ 0. (3) Physical consequences: The holonomy rate Γ_Σ ~ H_0 follows from gauge-invariant and dimensional arguments, since w_1^T ≠ 0 has no local dynamics and the only cosmological timescale is H_0^{-1}; the prediction R_FB = ∞ is robust under this scenario. Our framework resolves the domain-wall overclosure problem, provides a falsifiability basis for w_1^T ≠ 0 comparable to that of cosmic inflation, and predicts that the time-direction topological structure is a fossil record of inflation-era physics.