Foundation I: The Metric Universe - Extended Version

Abstract

We investigate a minimal cosmological extension of General Relativity based on the Einstein-Cartan-Kibble-Sciama (ECKS) framework. A primordial spin component — with stiff equation of state w=1, geometrically distinct from scalar kination — induces a transient phase reducing the comovingsound horizon to rₛ = 135.8 Mpc, enabling H₀ = 73.0 km/s/Mpc consistent with local measurements. The same stiffness parameter F_ion = 1.2765 governs both sound-horizon reduction and scale-dependent suppression of structure growth (S₈ = 0.766), unifying H₀ and S₈ under a single degree of freedom. A global fit over the stiff-phase parameters yields Δχ² = −39.5 vs ΛCDM (AIC = −31.5, BIC = −8.2). Resolving H₀ natively exposes an irreducible H₀–Age–BAOtension — shown in v10.2 to be part of a four-way (H₀–Age–BAO–θ*) quadrilemma — with the w=−1 ansatz predicting t₀ = 12.74 Gyr; the H₀ value is corroborated by H0DN 2026 (H₀ = 73.50 ± 0.81 km/s/Mpc, >6σ tension). Additional falsifiable signatures: cosmic birefringence β_ECF ≈ 0.35° (LiteBIRD), JWST high-redshift seeds, and a blue-tilted primordial GW background. ── Version notes (PREPRINT v5= internal v10.2, August 2026) ────────── Version 5 (August 2026) corrects an erroneous derivation in Appendix B, detected by internal audit. The dark-energy parameters (w₀, wₐ) = (−0.904, −0.153) were previously presented as derived from a single-parameter torsion ansatz; this is mathematically incorrect (no value of the coupling reproduces the pair). They are now stated as calibration outputs of the global χ² fit, not an ab initio prediction (open problem PO-F1-B). A partial reconstruction is added: w₀ is geometrically derived, w₀ = −1 + Ω_spin,peak = −0.907 (matching the calibrated −0.904 to 0.3%), while wₐ retains a phenomenological factor requiring the torsion-condensate dynamics ⟨δS²⟩(z) (cf. Foundation II, PO-F2-4). New script: calibrate_w0wa.py. All headline results are unchanged. ── Version notes (PREPRINT v4 = internal v10.2, July 2026) ──────────Editorial and correctness revision. Headline numerical resultsUNCHANGED: Δχ² = −39.5, H₀ = 73.04, r_s = 135.8 Mpc,(w₀,wₐ) = (−0.904,−0.153). What changes is the epistemic statusattached to three claims, not any measured or derived quantity. (1) Cartan density harmonised to ρ_c = c⁵/ℏG² = 5.15×10⁹⁶ kg m⁻³throughout (three occurrences previously rounded to 5.2), aligningthis paper with Foundation III and the chiral-bounce companion. (2) Nine inherited LaTeX defects repaired; the document now compileswith 0 errors (previously 16). One overstatement softened("definitively rules out" → "strongly disfavours"). (3) QCD-epoch mislabel corrected. A passage paired the acoustic-eracalibration redshift z≈7500 with the QCD confinement temperature(T~200 MeV) as if they were one epoch — they differ by ~8 orders ofmagnitude in redshift. The spin fluid's physical origin (QCDtransition) is now kept distinct from its observational calibrationpoint. No numerical value affected. (4) TRILEMMA OVER-CLAIM WITHDRAWN — the most substantive fix. A footnotestated that the calibrated (w₀,wₐ) "removes this BAO penaltyentirely" and resolves the H₀–Age–BAO trilemma. This is contradictedby this corpus' own evidence: a systematic 2500-point scan of thetanh-crossing scalar w(z) family (plot_trilemma_irreducibility.py,Appendix L) finds ZERO configurations satisfying both the BAOdistance and the t₀ ≥ 13.32 Gyr globular-cluster bound at H₀=73.04.Reworded as an open problem, tracked as PO-F2-5 in Foundation II. (5) QUADRILEMMA REFINEMENT. The scan above fixed Ω_m = 0.315 (Planck-h)without recomputing it for H₀ = 73.04; the consistent value isΩ_m ≈ 0.268. Re-scanning (Ω_m,w₀,wₐ) jointly reaches t₀ ≈ 13.0–13.2Gyr at Ω_m ≈ 0.28–0.29. Adding the CMB acoustic scale θ* shows thetrilemma is part of a QUADRILEMMA: age, BAO and θ* pull Ω_m inmutually incompatible directions. The required Ω_m shift (roughly19–46% below the CMB-measured value, depending on the weight givento θ*) is robust in sign and order of magnitude but not in its exactvalue; definitive quantification awaits a CLASS-EC treatment. (6) DESI MCMC SECTION REBUILT WITH THE FULL COVARIANCE MATRIX. Theindependent consistency check previously used diagonal errors only,reporting d = 0.78σ (SH0ES) and 0.77σ (H₀DN), with a caveat thatfull covariance "may shift these values by 10–30%". That estimate issuperseded: the full published covariance matrix (Adame et al. 2024,intra-bin D_M–D_H correlations) gives Mahalanobis distances of2.22σ (Planck prior) and 7.32σ (SH0ES prior) — an order of magnitudebeyond the earlier estimate. H₀DN has not been re-run with fullcovariance and is no longer reported. The full covariance alsoreveals a near-perfect degeneracy ρ(w₀,wₐ) ≈ −0.99, along which theECF point sits perpendicular, explaining the strong priorsensitivity. The SH0ES prior is itself in ~5σ tension with DESI+CMB,so that combination is not internally consistent to begin with; the7.32σ figure should be read in that light rather than as an isolatedfailure of the ECF prediction. Script: plot_ecf_desi_mcmc_v3_0.py(supersedes plot_ecf_desi_mcmc.py). (7) DISTINCTION FROM CLASSICAL BOUNCING COSMOLOGIES (new). The ECKSspin–torsion bounce is now explicitly distinguished from the classof classical GR bouncing cosmologies critically reviewed byBattefeld & Peter (Phys. Rept. 571, 1, 2015). The ECKS mechanismescapes the null-energy-condition-violation branch of that critique,since the spin–spin interaction is local and algebraic and torsionis non-propagating. The shear/BKL branch is NOT thereby resolved andis stated as such: shear scales as a⁻⁶, the same power as theeffective torsion term, so the two compete on equal footing.Popławski's anisotropic (Kantowski–Sachs) treatment averts thesingularity only if particle production dominates over shear(arXiv:2007.11556), isotropisation arising from the post-bounce erarather than from torsion itself; a recent dynamical-systems analysisof an Einstein–Cartan ekpyrotic model damps homogeneous shear viathe ekpyrotic scalar, not via torsion, and is confined tohomogeneous backgrounds (arXiv:2512.11885). No claim is made thatthe ECKS mechanism resolves the BKL problem. ── Companion short paper (PRD Short v6, same deposit) ───────────────Corrected in parallel, and brought into line with the Extended version:- Macro-Knot seed formation mislabelled "QCD transition scale",relabelled "Great Annihilation freeze-out" (T~1 MeV).- THREE stale DESI figures corrected: the short paper still reportedd = 0.78σ and stated the prediction "lies within the 1σ posterior",values superseded by the full-covariance analysis above. Body text,cross-reference and figure caption now carry 2.22σ / 7.32σ with theprior-sensitivity context.- Broken figure reference repaired: the paper calledfig_ecf_desi_mcmc_contours_shoes.png, which no script produces;now uses Fig_ECF_DESI_Contours.png (two-panel, both priors), thefigure actually generated and used in the Extended version.- Version alignment: two contradictory mentions of the Extendedversion it summarises ("Extended v10" and "Extended Version 3")both corrected to v10.2.- Condensed form of the classical-bounce distinction added. ── Cross-paper references ───────────────────────────────────────────All companion-paper citations now use CONCEPT DOIs, which alwaysresolve to the latest version (previous deposits cited frozen versionDOIs, directing readers to superseded versions):PIT Letter 10.5281/zenodo.19798923Foundation I 10.5281/zenodo.19577447Foundation II 10.5281/zenodo.20629237 GitHub: https://github.com/pfichant/spin-torsion-cosmology