Fermions on a reflective topological big bang: Pin+, Z16, the CPT-symmetric universe, and a closure condition for dark energy

Exploratory technical note, not peer-reviewed; prepared with AI assistance. The object is a closed FRW universe with the free identification (η, n) ~ (−η, −n): time reflection through the bang combined with the antipodal map on the spatial S³. The bang becomes a one-sided RP³ crosscap; the quotient is non-orientable and non-time-orientable, and in the pure-Λ limit it is Schrödinger's elliptic de Sitter space. This is the S³/antipodal case of Bray–Wheeler's "reflective topological big bang" (CQG 2025) and a candidate identified-sheets counterpart of the Boyle–Finn–Turok CPT-symmetric universe. Four conditional connections are proposed. (1) One geometry, two literatures: the Bray–Wheeler construction and the Boyle–Turok CPT programme describe the same object and have not been connected. (2) Pin⁺: in the standard Riemannian obstruction convention the quotient admits Pin⁺ but not Pin⁻, so Bray–Wheeler's spin obstruction is not a Pin obstruction; the matching to the Lorentzian CRT/T action is left open. (3) Z₁₆: if the physically relevant lift is this Pin⁺ structure, RP⁴ and its Z₁₆ bordism invariant become relevant, within the gauged-CRT framework of Harlow–Numasawa; whether this yields a mod-16 condition on Standard Model fermion content is open. (4) A closure condition for dark energy: in a recollapsing closed universe with DESI-like dynamical dark energy, a second reflection at the crunch would require total conformal time 2π; a thawing quadratic-potential scalar falls 0.65°–1.33° short at Ω_K = −0.002, −0.005, −0.010. Contents: two-page summary, full note (established items, negative results, literature scan, open questions), scalar-field code with convergence tables, raw output, README, license.