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Exploratory technical note, not peer-reviewed; prepared with AI assistance. v1.1: the closure-condition item is corrected (see below); code and numbers unchanged. 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. The geometry is not new. Four conditional connections are proposed. (1) One geometry, two literatures: Bray–Wheeler's construction with Σ = S³, σ = antipodal, is a candidate identified-sheets counterpart of the Boyle–Turok CPT-symmetric universe; the two literatures 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 convention-dependent and 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 crosscap at the crunch requires the total conformal angle of conformally coupled modes to lie in πℤ; 2π is treated as a candidate closure value, not selected uniquely. A thawing quadratic-potential scalar (m = 1.5 H₀) gives 358.88°, 359.35°, 358.67° at Ω_K = −0.002, −0.005, −0.010. Contents: two-page summary, full note (established items, negative results, literature scan, open questions, falsifiability), conformal-time scalar-field code (thaw6.py) with convergence tables and raw output, README, license. Cross-checks labelled "unarchived" in the note (separate cosmic-time implementation, CPL extrapolation, exact Bardeen-equation solution) are not included. No claim has received independent specialist review.