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A stringent test of CP symmetry in hyperon decays is performed using 1.0 x 10(10) J/psi events and 2.7 x 10(9) psi(3686) events collected by the BESIII experiment. The CP asymmetry in the two-body nonleptonic weak decays Sigma(+) -> p pi(0) and (Sigma) over bar (-) -> (p) over bar pi(0) is determined to be A(CP) = -0.0118 +/- 0.0083(stat) +/- 0.0028(syst), consistent with CP conservation. The average decay parameter = -0.9869 +/- 0.0011(stat) +/- 0.0016(syst) represents the most precise measurement in the baryon sector to date. A model-independent analysis of the hyperon polarization is performed for the first time at J/psi and psi(3686) resonances, revealing a clear sign reversal and offering new insight into SU(3) symmetry breaking and baryon internal structure. These results provide the most stringent constraints on CP violation in Sigma decays and establish a benchmark for future precision studies.