Pure-DP Statistical Query Release at the Conjectured Square-Root Rate

Nikolov and Ullman asked whether k statistical queries on a universe of size T can be released under pure differential privacy with expected worst-coordinate error at the square-root rate suggested by known lower bounds. We prove their conjectured upper bound. For every database size n and privacy parameter epsilon>0, there is an epsilon-differentially private mechanism with expected error O(1, sqrt (2T) (2k)/(epsilon n)). This matches the lower-bound dependence in the standard high-dimensional regimes where those bounds apply; the shifted logarithms and outer minimum make the upper bound valid without additional parameter assumptions. The construction starts from a selection-only private multiplicative weights transcript, then replaces its probability mass function by a distance-penalized likelihood envelope. To prove that the modification preserves accuracy, a likelihood-level Maurey argument upper-bounds each Hamming-ball maximum by a small family of auxiliary PMW laws. Renyi moment bounds control nearby balls, a direct mixture bound controls distant balls, and grouping radii at the privacy scale prevents an additional 1/epsilon factor in the error. The mechanism is information-theoretic. A companion Lean 4 development machine-checks the finite construction, pure privacy after deterministic decoding, and the displayed all-regimes upper bound.