Partial Orderings of Curvature Invariants

We establish a new set of pointwise inequalities that order curvature invariants across various Petrov and Segre types of spacetimes. In arbitrary spacetime dimension, we systematically analyze inequalities among contractions of the Ricci tensor. We further explore the conditions under which all Zakhary--McIntosh invariants in $(1+3)$-dimensional spacetimes are bounded above (up to appropriate powers) by the Kretschmann scalar. These results establish a practical hierarchy among curvature scalars and clarify the extent to which higher-order invariants are algebraically controlled by lower-order ones or vice versa.