The paper investigates the dimensions of scalar operators in 4D conformal field theories (CFTs), focusing on the operator $\phi^2$ defined as the lowest dimension scalar appearing in the operator product expansion (OPE) of $\phi$ with itself. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, the authors derive a theory-independent inequality, $|\phi^2| \leq f(|\phi|)$, where $f(d)$ is a continuous function that approaches 2 as $d \to 1$. This bound is evaluated numerically and shown to be satisfied by all weakly coupled 4D conformal fixed points. The bound is violated by Wilson-Fischer fixed points by a constant factor, likely due to extrapolation issues in $4-\varepsilon$ dimensions. The method is also applied to derive an analogous bound in 2D, which is checked against Minimal Models, with the Ising model nearly saturating the bound. The paper discusses the phenomenological motivation for studying these bounds, particularly in the context of constructing models of dynamical Electroweak Symmetry Breaking without flavor problems.The paper investigates the dimensions of scalar operators in 4D conformal field theories (CFTs), focusing on the operator $\phi^2$ defined as the lowest dimension scalar appearing in the operator product expansion (OPE) of $\phi$ with itself. Using general considerations of OPE, conformal block decomposition, and crossing symmetry, the authors derive a theory-independent inequality, $|\phi^2| \leq f(|\phi|)$, where $f(d)$ is a continuous function that approaches 2 as $d \to 1$. This bound is evaluated numerically and shown to be satisfied by all weakly coupled 4D conformal fixed points. The bound is violated by Wilson-Fischer fixed points by a constant factor, likely due to extrapolation issues in $4-\varepsilon$ dimensions. The method is also applied to derive an analogous bound in 2D, which is checked against Minimal Models, with the Ising model nearly saturating the bound. The paper discusses the phenomenological motivation for studying these bounds, particularly in the context of constructing models of dynamical Electroweak Symmetry Breaking without flavor problems.