1Distributed maximum likelihood estimation trades statistical efficiency for communication. We study a two-phase protocol in which each of $K$ workers corrects the bias of its local maximum likelihood estimator by an $r$-th order split-sample extrapolation over nested sub-blocks, the coordinator averages the corrected estimates, and $q$ globally synchronized Newton corrections follow. For fixed-dimensional regular likelihoods with $N$ observations we prove a joint order--round frontier: first-order equivalence to the centralized estimator holds whenever $K = o(N^{1-1/(2^{q+1} r)})$, so that one extra local extrapolation level and one extra synchronization round are exchangeable in the exponent, and the half-block jackknife followed by one Newton step tolerates $K = o(N^{7/8})$ partitions, against the $K = O(N^{1/2})$ regime stated for the plain-average one-step estimator. In the exponential model we derive exact bias, variance and Newton identities, show that the exponent is sharp for the construction with limiting shift $-\theta_0(\beta_r c^r)^{2^q}$ when $K \sim cN^{\alpha(r,q)}$, and characterize the Newton basin of attraction. A perturbation theorem for inexact curvature recovers the known single-machine-Hessian frontiers as special cases and gives the number of bits to which transmitted derivatives may be quantized at each local order. Monte Carlo experiments with paired replications (exponential, logistic and Poisson likelihoods), an exact-model boundary study, and timed multi-node benchmarks on an Apple-silicon cluster quantify the finite-sample variance cost of higher-order corrections and the wall-clock exchange rate between local fits and synchronization rounds. The components we combine are known; the joint frontier, its sharpness and the bit-precision result are new. No universal communication lower bound is claimed.2