Foundation I: The Metric Universe - Extended Version

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 comoving sound horizon to rs = 135.8 Mpc, enabling H0 = 73.0 km s−1 Mpc−1 consistent with local measurements. The same effective amplitude Fion = 1.2765 — calibrated on the data rather than derived — governs both the sound-horizon reduction and the 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 versus ΛCDM (AIC = −31.5, BIC = −8.2). Resolving H0 natively exposes an irreducible H0–Age–BAO tension — 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 the H0 Distance Network (Casertano et al. 2026): H0 = 73.50 ± 0.81 km s−1 Mpc−1, in 7.1σ tension with flat ΛCDM (Planck+SPT+ACT) and 5.0σ with BBN+BAO. Additional falsifiable signatures: cosmic birefringence βECF ≈ 0.35° (LiteBIRD); JWST high-redshift seeds; a blue-tilted primordial gravitational-wave background. What changed in this version (PREPRINT v7). One factual error and two unreproducible values are corrected: the 4.9σ DESI DR2 figure quoted in earlier versions does not appear in the cited paper, which reports 3.1σ (BAO+CMB) and 2.8–4.2σ once supernovae are included; the recombination epoch, previously given as 380,000 yr (ΛCDM) and 365,000 yr (ECF), is recomputed at z* = 1089.9 as 371,000 and 343,000 yr; four attributions are narrowed, and one open problem (PO-F1-SSC, on the spin supplementary condition) is opened. ── Version notes (PREPRINT v7 = internal v11, corrected 13 September 2026) ── This version corrects one factual error and narrows four claims. No resultis withdrawn; two bounds are weakened. CORRECTION OF RECORD — DESI DR2 significance.Earlier versions stated that DESI DR2 prefers dynamical dark energy overLCDM at 4.9 sigma. That value does not appear in the cited paper(arXiv:2503.14738), which reports 3.1 sigma for DESI BAO + CMB and2.8-4.2 sigma once supernovae are included, depending on the sample. The4.9 sigma figure was the distance of the LCDM point from the DESI centralvalues in the marginalised plane -- a quantity computed here, not asignificance published by DESI. Corrected throughout. NARROWED ATTRIBUTIONS.- Poplawski (2010, 2012): earlier versions stated that the robustness of the ECKS bounce against spin inhomogeneities "has been established" by these works. They are carried out in a homogeneous, isotropic FLRW setting, where homogeneity is an assumption rather than a result. The locality argument is the framework's own, and stability in an inhomogeneous regime remains to be demonstrated.- Kibble (1961) and Sciama (1964) are now cited separately, with their correct years; earlier versions presented them as co-authors of a 1961 work. NEW OPEN PROBLEM — PO-F1-SSC.Closing the Weyssenhoff description requires a spin supplementary condition.Bohmer & Bronowski (arXiv:gr-qc/0601089) show that the Frenkel,Papapetrou-Corinaldesi and Tulczyjew conditions are each incompatible withthe cosmological principle. The macroscopic a^-6 scaling used here followsthe standard spin-fluid averaging and is not claimed to resolve that tension;whether the objection bears on a homogeneously averaged cosmological fluid,or only on the kinematics of individual spinning elements, is tracked asPO-F1-SSC. CORRECTION OF RECORD -- recombination age.The "Comparison of timescales" table quoted 380,000 yr (LCDM) and 365,000 yr(ECF) at z = 1100. Neither reproduces: the 380,000 figure is the textbookround number, not a value computed for these parameters, and the ECF entryfollowed from it by subtracting an estimated 15,000 yr shift. Both columnsare now evaluated at the Planck decoupling redshift z* = 1089.9, giving371,000 yr and 343,000 yr; the shift is 28,000 yr. The calculation isreproducible with Scripts_F1/A_Core_Calculations/script_09_cosmic_ages.py,which returns the published Planck age to within 1 kyr. F_ion does not enterthe age integral separately: it is already absorbed into the calibrated H0. FIGURE REGENERATED.Figure_H0_S8_Contours.png displayed Delta chi2 = -39.4 in its internallegend, while the text (three occurrences), the value registry and its ownproducing script all state -39.5. The figure was the only piece carrying thestale value. Regenerated; no other figure differs from its script output. CLARIFICATIONS.- The trilemma scan is now described as numerical evidence within a three-parameter tanh-crossing family, not as a proof of irreducibility.- gamma_spin is quoted with its uncertainty (0.3116 +/- 0.04).- The S8 constraint notes that the KiDS-1000 3x2pt value is asymmetric (0.766 +0.020 -0.014); the symmetric +/-0.019 used in the chi-squared budget is a simplification.- r_s = 147.1 Mpc is identified as r_drag (end of the baryon drag epoch, z_d ~ 1060), distinct from the horizon at last scattering (~144.4 Mpc).- n_target = 2.99e-4 Mpc^-3 is the midpoint of the (2-4)e-4 range inferred from Labbe et al. (2023); its three significant figures reflect the arithmetic of the midpoint, not the precision of the measurement. ── 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 th