Revisiting the S8 Gap: Non-Metricity in Light of DESI DR2 and Gravitational Wave Standard Sirens

The $\Lambda \mathrm{CDM}$ model faces persistent discrepancies between early- and late-Universe measurements of the Hubble constant ($H_0$) and matter clustering ($S_8$). We investigate whether the metric-affine framework of $f\left(Q\right)$ symmetric teleparallel gravity can accommodate these tensions. Constraining the reciprocal $f\left(Q\right)$ model with DESI DR2 BAO, Pantheon+ Supernovae, and RSD data, our joint MCMC analysis reveals a mild preference for a negative geometric density parameter (${\Omega }_Q=-0.164\pm 0.070$). Within the reciprocal $f\left(Q\right)$ framework, such values modify the late-time growth history through a reduction of the effective gravitational coupling, yielding an intermediate clustering amplitude ($S_8=0.815\pm 0.029$) that accommodates the $S_8$ tension. Conversely, the model does not intrinsically shift $H_0$ using uncalibrated low-redshift geometric data ($H_0=67.88\pm 3.27$ km $s^{-1}$ ${\mathrm{Mpc}}^{-1}$), though high-$H_0$ solutions remain compatible when local SH0ES calibrations are included. Finally, we forecast the constraints from future Einstein telescope and LISA standard sirens. By probing the non-metricity sector via modified wave propagation, future sirens could reduce $H_0$ uncertainties to sub-percent levels ($\sigma \left(H_0\right)\sim 0.3$ km $s^{-1}$ ${\mathrm{Mpc}}^{-1}$) conditional on specific astrophysical assumptions, while placing significantly tighter constraints on deviations from general relativity.