Localized Big Bang Stability of Spacetime Dimensions $n\geq4$

We prove the past nonlinear stability of the sub-critical Kasner-scalar field solutions to the Einstein-scalar field equations on a truncated cone domain in spacetime dimensions $n\geq4$. Our analysis demonstrates that the perturbed solutions are asymptotically pointwise Kasner, geodesically incomplete in the contracting direction and terminate at quiescent and crushing singularities characterized by the blow-up of curvature invariants. This work generalizes the result of Beyer-Oliynyk-Zheng in [arXiv:2502.09210v2] to all higher dimensional spacetimes.