1\section{Conclusion}\label{sec:conclusion}23For regular fixed-dimensional likelihoods, local split-sample extrapolation of order $r$ followed by $q$ aggregated Newton corrections is first-order equivalent to the centralized MLE whenever $K = o(N^{1 - 1/(2^{q+1}r)})$, and this exponent is sharp for the construction in an exactly solvable model, where the limiting shift at the boundary is $-\theta_0(\betar c^r)^{2^q}$. One synchronization round and a doubling of the local order are exchangeable in the exponent, which turns the choice between them into a question about hardware that a calibrated cost model and measured benchmarks can answer for a given installation. Inexact curvature changes the exponent in a computable way, recovering the known single-machine-Hessian frontiers and showing that higher local order tolerates coarser quantization of the transmitted derivatives. The finite-sample evidence tempers the exponents: higher-order corrections inflate variance and can fail to exist at small block sizes, and the boundary regime is reached only at very large $N$. The components combined here are known; the joint frontier, its sharpness and the bit-precision result are the contribution.4