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 v6 = internal v11, September 2026) ────────── Version 6 (September 2026) carries five corrections; all headlinenumerical results are unchanged (Δχ² = −39.5, H₀ = 73.04,r_s = 135.8 Mpc, S₈ = 0.766). (1) SPIN-DENSITY SIGN STATED AS CALIBRATED, NOT DERIVED. The sign of thepost-bounce spin-density term in the Friedmann equation is now statedas calibrated rather than derived, consistently with the epistemicconvention of this corpus. The gap is tracked as open problemPO-F1-SIGMA0. (2) NORMALISATION ANCHOR REALIGNED TO z_ref = 5600. Several passagesquoted z ≈ 7500 as the acoustic-era reference redshift. That valueappears in no script: script_01_solve_sound_horizon.py normalises atz_ref = 5600, and 5600 is what reproduces r_s = 135.8 Mpc. The twoanchors give sound horizons compatible at 1.25σ, so this is atraceability correction, not the repair of a wrong result — but thetext and every affected figure have been realigned on 5600. Thissupersedes the z ≈ 7500 figure quoted in the v4 version notes (item 3below), where the QCD/acoustic-epoch distinction remains valid. (3) APPENDIX CROSS-REFERENCES REPAIRED. References to ten appendicesresolved to the wrong letter, broken by unnumbered \section*declarations — a rendering defect invisible at compilation, since thereference was defined and merely pointed at the wrong number. (4) H_b CLARIFIED. The Hubble rate at bounce exit is now explicitlydistinguished from the bounce value itself. (5) KOPCZYŃSKI ATTRIBUTION CORRECTED. A bibliography defect was repaired in passing: a \textit command hadbeen mangled into "extit" by a non-raw generation string, causing thefirst reference to render without italics. The same defect was presentin six documents of this corpus and has been corrected throughout. All figures were regenerated from the deposited scripts (40/40 scripts,0 failures) and verified to be embedded in the compiled PDFs. ── 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