Localized Approximation and Short-Block Randomized One-Way NOF Lifting

For every fixed k >= 2 and prime q > 2k, this paper proves that generalized inner product of dimension r = O_k(sqrt(q)/log(q)) lifts the public-coin one-way randomized communication complexity of every partial Boolean function on F_q x F_q to the general one-way number-on-forehead model, up to an additive constant and error parameters 1/6 and 1/3. This improves the sufficient O_k(q/log(q)) dimension in Wang's preceding general randomized lifting theorem. The proof uses a localized polynomial approximation on a cone, controlled coefficient norm, and an error-parameter gap to absorb local dual mass. The manuscript includes complete definitions, the approximation and transfer proofs, exact integer gadget parameters, source comparisons, limitations, and an AI-assistance disclosure. No necessary-dimension lower bound or constant-dimension lifting result is claimed. The record contains the manuscript PDF, standalone LaTeX file, and reproducible source archive.