Maximally heavy dynamics in the causal diamond

Correlation functions of CFT operators with infinite scaling dimension are rich, multifaceted objects that describe physics ranging across classical holography, black hole dynamics, and flat-space scattering amplitudes. In this work, we provide a rigorous framework for characterizing the space of four-point functions of identical operators with infinite dimension in terms of well-defined ``maximally heavy observables,'' which are akin to intrinsic quantities describing statistical systems in the thermodynamic limit. These observables are highly constrained by crossing symmetry and unitarity, and give novel insights into the locality of bulk states through the emergence of dynamical phase transitions. In certain cases, these results connect directly to the more familiar picture of torus partition functions at large central charge. We apply our framework to a number of illustrative examples including generalized free fields, chiral product correlators, and maximal giant gravitons in planar $\mathcal{N}=4$ SYM.