Topological–Dynamical Origin of the Temporal Arrow from Pin+ Global Anomaly: Anomaly as Physics and Logical Closure

The origin of the arrow of time remains one of the deepest open problems in theoretical physics. In a recent principle-theory approach, the topological holonomy of the time-direction O(1) principal bundle was shown to produce a spin-dependent phase θ(j) = π(2j mod 2) — zero for bosons, π for fermions — leading to the parameter-independent prediction R_FB = ∞ for the fermion–boson temporal decoherence ratio. In this paper, we reveal the deep topological structure underlying these results and establish a logically closed theoretical framework. We prove seven theorems: Proposition 1 identifies the Pin+ structure; Theorem 2 establishes anomaly localization via the unique non-trivial homomorphism Hom(Z₂, Z₁₆) = Z₂; Theorem 3 proves the conditional independence of the spacetime and internal Z₁₆ anomalies under the assumption (w₁ᵀ)² = 0; Theorem 4 shows that ν_R cancels the global anomaly without eliminating the local Berry phase; Theorem 5 upgrades the dynamical assumption Γ_Σ > 0 to a consequence of quantum mechanics; Theorem 6 provides the reverse argument — deriving w₁ᵀ ≠ 0 from macroscopic irreversibility, conditional on the working hypothesis that conventional decoherence mechanisms alone are insufficient; and Theorem 7 establishes the logical closure: five statements are equivalent, w₁ᵀ ≠ 0 ⇔ θ = π ⇔ R_FB = ∞ ⇔ σ_topo > 0 ⇔ 8 ∈ Z₁₆. Our central conceptual contribution is the identification of the anomaly as physics: the uncancelled Z₁₆ global anomaly is not a sign of theoretical inconsistency but the topological–dynamical origin of macroscopic irreversibility. The reverse argument (Theorem 6) is, to our knowledge, the first derivation of spacetime topology from irreversibility, placing the framework on a structural analogy with Einstein's derivation of Lorentz symmetry from the constancy of the speed of light (though the epistemic status of the two postulates differs, as discussed in Section 6.3).