Exact Spatial Flatness and the Spin-Torsion Link: an Open Problem in Einstein-Cartan Cosmology

Standard cosmology treats the observed spatial flatness of the Universe (ΩK≃0) as an observational coincidence, to be explained dynamically by inflation. This Letter adopts a different stance and states it as a postulate: spatial flatness is exact, ΩK≡0, as a topological property of spacetime rather than a dynamical outcome. Unlike inflationary flattening, which predicts only ΩK≈0, this is a sharp and falsifiable claim: a measurement of ΩK≠0 at any precision would exclude it. What this postulate does not do. Because ΣΩi=1 is an identity once k=0, the closure condition cannot enforce any compensating dark-energy component, and constrains neither ρDE nor its equation of state. Earlier versions of this Letter (v4–v7) presented such a compensation as a mechanism; that claim is withdrawn in v8. What Einstein–Cartan cosmology does supply, independently of this postulate, is a static dark-energy term derived from the action: the FLRW reduction of the axial torsion sector yields a background condensate with w=−1, its fluctuations behaving as radiation. The observed evolution of the equation of state does not follow, and remains calibrated. This leaves a well-posed open problem, stated rather than solved. The Weyssenhoff spin fluid that reduces the sound horizon before recombination, and the frozen condensate that sources dark energy, are two regimes of the same torsion field. A relation between Ωspin and Ωτ is expected on physical grounds, but must be derived from the action, not imposed by Ω-bookkeeping. The Letter formulates this problem precisely, shows why the route through the closure condition fails, and identifies the propagating torsion mode as the direction in which a derivation could be sought. Part of the Einstein–Cartan Framework corpus: Foundation I (10.5281/zenodo.19577447), Foundation II (10.5281/zenodo.20629237), Foundation III — Cosmological Physics (10.5281/zenodo.22359982). Reproducibility code: github.com/pfichant/spin-torsion-cosmology (CC-BY-4.0).