Completed SDSS-IV extended Baryon Oscillation Spectroscopic Survey: Cosmological implications from two decades of spectroscopic surveys at the Apache Point Observatory

We present the cosmological implications from final measurements of clustering using galaxies, quasars, and Ly$\alpha$ forests completed Sloan Digital Sky Survey (SDSS) lineage experiments in large-scale structure. These experiments, composed data SDSS, SDSS-II, BOSS, eBOSS, offer independent baryon acoustic oscillation (BAO) angular-diameter distances Hubble relative to sound horizon, $r_d$, eight different samples six growth rate parameter, $f\sigma_8$, redshift-space distortions (RSD). This composite sample is most constraining its kind allows us perform a comprehensive assessment model after two decades dedicated spectroscopic observation. show that BAO alone are able rule out dark-energy-free models at more than standard deviations an extension flat, $\Lambda$CDM for curvature. When combined with Planck Cosmic Microwave Background (CMB) temperature polarization provide nearly order magnitude improvement on curvature constraints. The RSD indicate consistent predictions primary General Relativity. combining results SDSS external data, all multiple-parameter extensions remain model. Regardless model, precision $\Omega_\Lambda$, $H_0$, $\sigma_8$, remains roughly 1\%, showing changes less 0.6\% central values between models. inverse distance ladder measurement under o$w_0w_a$CDM yields $H_0= 68.20 \pm 0.81 \, \rm km\, s^{-1} Mpc^{-1}$, remaining tension several direct determination methods. (abridged)