Uniform Kretschmann Control of Zakhary–McIntosh Invariants in Aligned Petrov Type D Spacetimes

This work studies pointwise control of the full set of 17 Zakhary–McIntosh curvature invariants in aligned Petrov type D spacetimes. The main result gives a necessary-and-sufficient characterization of uniform algebraic control by the reduced Kretschmann scalar. In particular, control of the complete Zakhary–McIntosh system is shown to be equivalent to a three-test criterion involving the Weyl pair I1 and I2 together with the Ricci invariants I7 and I8. The three tests are shown to be irredundant by explicit obstruction families. The analysis also yields an enlarged Ricci coercivity cone, extending beyond the real-spectrum and null-convergence sector, together with exact sharp algebraic constants on a two-parameter family of cones. The relation to Einstein spacetimes satisfying the null energy condition is treated separately. As a geometric application, the Weyl-phase condition is analyzed throughout the exterior of subextremal Kerr–Newman spacetimes, giving an explicit parameter criterion and an optimal exterior phase constant. A Kerr example demonstrates why an unconditional Kretschmann bound cannot hold throughout general vacuum Petrov type D. The manuscript is accompanied by supplementary material and reproducibility files containing independent symbolic checks of the Zakhary–McIntosh contractions and the algebraic results.