We derive a topological dark/bright classification from the bordism group Ω₄Pin⁺ = Z₁₆ of 4-dimensional non-orientable spacetime with Pin⁺ structure. The forgetful map f: Ω₄Pin⁺ → Ω₄ᴼ ≅ (Z/2) ⁴, with Im (f) = (0, 0, 0, 0), (1, 0, 0, 1) as the diagonal subgroup, partitions the 16 topological phases into 8 topologically dark phases (ker (f) ≅ Z₈, invisible to orientable probes) and 8 topologically bright phases, yielding an algebraic dark-to-bright ratio of 1: 1. The element k = 8 ∈ ker (f) is distinguished as the unique element of order 2—the “deepest dark” phase that is self-inverse under Z₈ addition and a candidate for purely gravitational dark matter. Excluding the vacuum (k = 0), the physical dark-to-bright ratio is 7: 8. We develop a three-tier hierarchy of topological protection, discuss the physical mapping from topological darkness to dark matter (inferred, not rigorously derived), and explore cosmological implications including dark matter phase transitions as gravitational wave sources.
Fangyuan Hao (Fri,) studied this question.