The Topological Invariance Principle: Exact Spatial Flatness and the Spin–Torsion Link: an Open Problem in Einstein–Cartan Cosmology

Abstract

Standard cosmology treats the observed spatial flatness of the Universe (Ω_K ≃ 0) as an observational coincidence, to be explained dynamically by inflation. We adopt a different stance and state it as a postulate: spatial flatness is exact, Ω_K ≡ 0, as a topological property of spacetime rather than a dynamical outcome. Unlike inflationary flattening, which predicts only Ω_K ≈ 0, this is a sharp and falsifiable claim: a measurement of Ω_K ≠ 0 at any precision would exclude it. We are equally explicit about what this postulate does not do. Because Σ Ω_i = 1 is an identity once k = 0 — the critical density ρ_crit = 3H²/8πG being recomputed with H — the closure condition cannot enforce any compensating dark-energy component, and constrains neither ρ_DE nor its equation of state. Earlier versions of this Letter presented such a compensation as a mechanism; that claim is withdrawn. 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, which we state rather than solve. 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 Ω_spin and Ω_τ is therefore expected on physical grounds — but it must be derived from the action, not imposed by Ω-bookkeeping. We formulate this problem precisely, show why the route through the closure condition fails, and identify the propagating torsion mode as the direction in which a derivation could be sought. --- Key changes in PREPRINT v5 (internal version 8, August 2026) This version withdraws the compensation mechanism presented in v4–v7 and reformulates the Letter around the open problem it leaves. The postulate is retained, with its scope restricted to exact spatial flatness. (i) The closure condition is shown to be an identity, incapable of enforcing a compensating debt. (ii) The static dark-energy term is attributed to the FLRW reduction of the action (Foundation II, Appendix J), independently of the postulate. (iii) The Ω_spin–Ω_τ relation is stated as the central open question rather than assumed. (iv) The analogy with charge conservation is withdrawn. --- Key changes in PREPRINT v4 (internal version 7, July 2026), superseding preprint v3(June 2026). Conceptual clarification and cross-reference maintenance;no numerical value is altered. (1) TIP restated as a self-contained geometric gauge postulate. TheTopological Invariance Principle is presented as a postulate aboutspacetime geometry, not as a theorem derived from the minimal ECKSLagrangian. This is a clarification of epistemic status, not a changeof content. (2) Omega_spin^(peak) = 0.093 explicitly identified as an observationalinput inherited from Foundation I, not as a logical prerequisite of theprinciple. A robustness range Omega_spin in [0.05, 0.15] is added: theTIP mechanism and its qualitative predictions (geometric dark energy,non-singular bounce, cosmic birefringence) hold across that range. (3) DESI comparison qualified. The comparison with DESI DR1 is nowlabelled a qualitative consistency check rather than a parameter-freeprediction. This aligns the Letter with the more careful treatmentadopted in Foundation I v10.2, where the same comparison was rebuiltwith the full published covariance matrix. (4) H = 0 bounce derivation made self-contained, so that the Letter canbe read without reference to the Foundation papers. (5) Epoch labels clarified. The QCD transition (T ~ 200 MeV) is statedas the *physical origin* of the primordial spin fluid, and keptexplicitly distinct from the much later *observational calibrationpoint* at which the peak amplitude is fitted against H0/S8/BAO data.No numerical value is affected; only the epoch label attached to it.The same class of error was corrected the same day in Foundation I andFoundation II. (6) Cross-paper DOIs migrated to concept DOIs. All references tocompanion papers now use concept DOIs, which always resolve to the mostrecent version, instead of frozen version DOIs: PIT 10.5281/zenodo.19798923 Foundation I 10.5281/zenodo.19577447 Foundation II 10.5281/zenodo.20629237Previous deposits cited frozen version DOIs, which would have directedreaders to superseded versions. H0 tension context: the H0 Distance Network Collaboration reportsH0 = 73.50 +/- 0.81 km/s/Mpc (arXiv:2510.23823), at 7.1 sigma tensionwith flat LambdaCDM and consistent with the ECF calibration at 73.04(0.6 sigma difference). Numerical values unchanged: Omega_spin^(peak) = 0.093, (w0, wa) =(-0.904, -0.153), F_ion = 1.2765. --- Code: github.com/pfichant/spin-torsion-cosmology (CC-BY 4.0)