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The Different Varieties of the Suyama-Yamaguchi Consistency Relation

We present the different consistency relations that can be seen as variations of the well known Suyama-Yamaguchi (SY) consistency relation τ_{NL} \geqslant (\frac{6}{5} f_{NL})^2, the latter involving the levels of non-gaussianity f_{NL} and τ_{NL} in the primordial curvature perturbation ζ, they being scale-invariant. We explicitly state under which conditions the SY consistency relation has been claimed to hold in its different varieties (implicitly) presented in the literature since its inception back in 2008; as a result, we show for the first time that the variety τ_{NL} ({\bf k}_1, {\bf k}_1) \geqslant (\frac{6}{5} f_{NL} ({\bf k}_1))^2, which we call "the fifth variety", is always satisfied even when there is strong scale-dependence and high levels of statistical anisotropy as long as statistical homogeneity holds: thus, an observed violation of this specific variety would prevent the comparison between theory and observation, shaking this way the foundations of cosmology as a science.