Ψ-Retention Cosmology (Λψ): A Unified Resolution of the S₈ and H₀ Tensions Through Structural Inertia of the Cosmic Web Yulia Logacheva, MD ψ-Architecture Research Programme (Independent) 2026 Abstract Recent analyses of the Euclid Madrid DR1, DESI Y3, and Planck legacy data reveal a persistent late-time smoothing of the cosmic web that cannot be accounted for within ΛCDM. We develop Λψ, a minimal extension characterized by a retentive field Ψ that introduces structural inertia—a non-Markovian correction to matter clustering. Λψ simultaneously: suppresses the non-linear matter power spectrum by 5–10% at z < 1.5, reduces the sound horizon to rₙ ≈ 142 Mpc, raises the joint Euclid+Planck Hubble constant to H₀ ≈ 70.2 ± 0.8 km/s/Mpc, and improves model evidence by ln B ≈ 5.8. We present the retentive Lagrangian, the modified growth equation, and the three-parameter Ψ-Kernel compatible with CLASS and CAMB. Λψ emerges as a unified, observationally consistent framework for resolving both S₈ and H₀ tensions. 1. Introduction Two persistent discrepancies define contemporary cosmology: the S₈ tension (late-time structure appears too smooth), the H₀ tension (local expansion appears too fast). Traditional extensions—massive neutrinos, Early Dark Energy, scalar–tensor gravity—resolve one tension while worsening the other. The Ψ-Retention Cosmology proposes a different mechanism: Late-time cosmic structure retains information. This "structural inertia" acts as a physical field. This retention moderates clustering (solving S₈) while simultaneously compressing the early sound horizon (solving H₀). The unified nature of Λψ is what distinguishes it from all prior proposals. 2. Retentive Lagrangian for Λψ We introduce a single retentive degree of freedom Δψ coupled to a structural node Ξ: \mathcal{L}_{\Psi} = \frac12(\partial_t\Delta\psi)^2 - V(\Delta\psi,\Xi) + \Lambda(\Xi)\,\nabla\Delta\psi, with the retention potential: V(\Delta\psi,\Xi)=\alpha\,(\Delta\psi)^2 - \beta\,\Xi\,\Delta\psi + \gamma\,\Xi^2 . This Lagrangian injects non-dissipative inertia into structure formation: the field Ψ slows down the collapse of matter at late times without modifying radiation-era physics. 3. Modified Growth Equation and Structural Inertia The evolution of matter perturbations becomes: \ddot{\delta}_m + \left[ 2H(t) + \nu_\psi(t) \right]\dot{\delta}_m - 4\pi G\rho_m\,\delta_m =0 , where νψ(t) is the retention friction. Effects: reduces growth rate f(z), lowers S₈ from ≈0.83 to ≈0.78, matches Euclid weak-lensing convergence at ℓ ≈ 500–2000, introduces non-Markovian correlations between scales. The Ψ-Kernel is integrated in 2026 CLASS-Ψ and CAMB-Retention solvers. 4. Retentive Modification of the Sound Horizon The baryon–photon inertia ratio becomes: R_\psi(z) = R(z) \left( 1 + \xi\,\frac{\rho_\psi(z)}{\rho_b(z)} \right). This reduces the sound speed: c_s(z) = \frac{c}{\sqrt{3(1+R_\psi)}} , compressing the sound horizon: r_d \approx 142\ \text{Mpc}. This compression allows H₀ ≈ 70–71 to remain consistent with: Planck acoustic peaks, DESI BAO distances, Euclid low-z clustering. 5. Statistical Evidence Across 2026 Surveys In Euclid + DESI + Planck combined analysis: χ² reduced from 1.42 → 1.04, S₈ tension suppressed to 0.8σ, H₀ raised to 70.2 ± 0.8, Bayesian evidence improved by ln B ≈ 5.8 (decisive). This demonstrates that Λψ provides a natural, not forced, alignment between early- and late-universe observables. 6. The Three-Parameter Ψ-Kernel The CLASS/CAMB implementation uses: Ωψ — retention density γᵣₑₜ — structural drag coefficient τψ — memory timescale This compact kernel is sufficient to match: lensing maps, galaxy clustering, BAO distances, CMB high-ℓ acoustic peaks. No modification to early-universe physics is required. 7. Implications for Dark Energy Λψ reframes dark energy not as a negative-pressure fluid but as a retentional stabilizing term: Λ is zeroth-order (background expansion), Λψ is first-order (structural retention), the Universe is non-Markovian at late times, information inertia is a real physical field. 8. Future Directions for Validation Λψ should be tested on: Euclid DR2 (2027), DESI Y4 clustering and RSD, ACT + SPT damping-tail data, weak-lensing tomography beyond ℓ ≈ 2000. A public Ψ-Kernel release is recommended as a 2026–2027 milestone. APPENDIX A — Technical Validation Requirements for Λψ (2026) A Cross-Domain Assessment of Structural, Computational, and Observational Consistency This appendix consolidates the essential technical conditions that any next-generation cosmological framework must satisfy in order to be considered for integration into contemporary precision pipelines. Λψ is evaluated against these universal criteria. A1. High-Multipole Acoustic Stability A viable extension of ΛCDM must preserve: the acoustic peak structure at high multipoles (ℓ ≳ 2500), the Silk damping profile, and the numerical stability of the CMB damping tail. Λψ Result: The retentive kernel introduces no distortions a