Resolving the $S_8$ Clustering Tension via the $\Psi$-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 $\Psi$-field cosmology. Using fixed late-time structural coupling parameters ($A \approx 0.25$, $m \approx 3.0$, $q_{\Psi} \approx 0.15$), we derive the modified growth equation for matter density perturbations ($\delta$) where the effective gravitational coupling $G_{\text{eff}}(z)$ dynamically modulates the gravitational attraction across cosmic epochs. Numerical evaluation reveals that while the system converges identically to standard General Relativity today ($G_{\text{eff}}/G \to 1.0000$ 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 $\Lambda$CDM prediction down to $S_8 = 0.7460$. This value is in remarkable agreement with local weak lensing surveys (e.g., DES, KiDS), effectively resolving the long-standing $S_8$ 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 $\Psi$-field on large-scale cosmic web structures.