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Abstract: We develop a novel conformal bootstrap formulated directly in terms of conformal symmetry, reflection positivity, and crossing symmetry, without explicitly relying on the operator product expansion or conformal blocks. By constructing Gram matrices from states created by pairs of local operators and imposing spectral constraints through conformal generators, we obtain a hierarchy of semidefinite programs that directly constrain correlation functions. We demonstrate the method in one to three dimensions, obtaining correlator and spectral bounds that capture generalized free field behavior and reproduce the characteristic Ising kinks in two and three dimensions. We further show that joint positivity of states at different insertion configurations naturally constrains correlators away from self-dual kinematics.