From soft-collinear effective theory one can derive a factorization formula for the e+e- thrust distribution dsigma/dtau with tau = 1-T that is applicable for all tau. The formula accommodates available O(alpha_s^3) fixed-order QCD results, resummation of logarithms at NNNLL order, a universal nonperturbative soft function for hadronization effects, factorization of nonperturbative effects in subleading power contributions, bottom mass effects and QED corrections. We emphasize that the use of Monte Carlos to estimate hadronization effects is not compatible with high-precision, high-order analyses. We present a global analysis of all available e+e- thrust data measured at Q = 35 to 207 GeV in the tail region, where a two-parameter fit can be carried out for alpha_s(m_Z) and Omega_1, the first moment of the soft function. To obtain small theoretical errors it is essential to define Omega_1 in a short-distance scheme, free of an O(Lambda_QCD) renormalon ambiguity. We find alpha_s(m_Z) = 0.1135 +- (0.0002)_expt +- (0.0005)_Omega_1 +- (0.0009)_pert with chi^2/dof = 0.9.
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