Entanglement Bounds on Cardy Growth in Higher Dimensions

We propose bounds relating thermal entropy to entanglement in unitary conformal field theories: $f\le4π^2a$ in four dimensions and $f\le2.99830949\ldots\,F$ in three dimensions. Here $f$ is the dimensionless coefficient of the cosmological constant term in the thermal effective action, $a$ is the Euler anomaly coefficient, and $F$ is the sphere free energy. Spherical Rényi entropy relates the four-dimensional bound to the conformal collider bounds. We formulate a stronger conjectural inequality involving the stress-tensor three-point function, saturated by every free theory, and establish it in controlled regimes of weakly coupled gauge theories, large-$N$ vector models, and holographic CFTs. Finite-$N$ numerical results provide additional tests. The bounds constrain the Cardy growth of the density of states at high energy.