Topological Quantum Foundations Fundamental Mixed States from Spacetime Topology and the Classification of Quantum Mixedness

We propose that spacetime topology fundamentally constrains the structure of quantum state space, leading to a new category of quantum mixed states that cannot be purified by any physical process. On a 4-dimensional manifold with non-trivial first Stiefel-Whitney class w₁ᵀ≠0, the Pin⁺ bordism classification Ω₄^{Pin⁺}=Z₁₆ leads to superselection sector splitting H=H₊⊕H₋. The resulting mixed state ρ = p₊ρ₊ ⊕ p₋ρ₋ is fundamentally mixed—it cannot be purified by including environmental degrees of freedom, because the superselection rule is a topological constraint on the Hilbert space itself. We establish three unconditional results: (1) a topological No-Go theorem proving that fundamental mixed states cannot be purified by any physical operation, (2) a classification of quantum mixedness distinguishing three types—improper (purifiable), strong-to-weak spontaneous symmetry breaking (dynamically non-purifiable), and fundamental (topologically non-purifiable)—and (3) an information protection-shielding duality showing that superselection rules simultaneously protect intra-sector quantum information (via the Knill-Laflamme condition) and shield inter-sector classical information (via topological non-purifiability). These results place fundamental mixed states outside the axiomatic framework of Yang, Shi, and Lee (2025) for topological mixed states, as they violate the local recoverability axiom (P0) for topological rather than dynamical reasons. Conditional on the sector label k being a quantum-mechanical observable, we further show that measurement can be understood as sector determination—a local observer discovering which topological sector the system occupies—and that quantum probabilities arise from irreducible ignorance of global topological quantities (topological epistemology). Our approach requires no new parameters or particles beyond the assumption w₁ᵀ≠0, and makes falsifiable predictions including CMB TB/EB polarization from global topological CPT modification (dual-channel falsifiability architecture) and a fermion-boson decoherence asymmetry. This work opens the direction of topological quantum foundations—the study of how spacetime topology constrains the structure of quantum theory.