Resolving the S₈ Clustering Tension via the Ψ-Field Cosmology: Matter Power Spectrum Modulation and Cosmic Web Smoothing

We investigate the evolution of linear and non-linear matter perturbations within the modified scalar-tensor framework of the Ψ-field cosmology. Using fixed late-time structural coupling parameters (A ≈ 0.25, m ≈ 3.0, qΨ ≈ 0.15), we derive the modified growth equation for matter density perturbations (δ) where the effective gravitational coupling Geff(z) dynamically modulates the gravitational attraction across cosmic epochs. Numerical evaluation reveals that while the system converges identically to standard General Relativity today (Geff/G → 1.0000 at $z=0$), the historical presence of the cosmic damper suppresses the aggressive growth of dark matter overdensities during the structure formation era. Consequently, our simulation yields a systematic $10.12%$ reduction in the modern matter clustering amplitude, shifting the standard ΛCDM prediction down to S₈ = 0.7460. This value is in remarkable agreement with local weak lensing surveys (e.g., DES, KiDS), effectively resolving the long-standing S₈ tension that plagues the standard model without fine-tuning. We present the resulting Matter Power Spectrum $P(k)$ as a definitive observational footprint of the Ψ-field on large-scale cosmic web structures.