Private Selection from Scalar Gaussian Queries with Logarithmic Expected Loss

For an arbitrary family of N >= 2 sensitivity-one real-valued losses, we construct a selector using only sensitivity-one scalar Gaussian queries with expected excess loss O(log N / sqrt(rho)) and pathwise total Gaussian precision at most rho. The algorithm has a public finite query cap and its selected-label output is rho-zCDP. It admits an exact common-scale implementation, thereby attaining the rate requested in Steinke's Gaussian selection problem and removing the iterated-logarithm factor from the bound of Leeman and Manurangsi (FORC 2026). The construction combines an exponential-race near-minimum count with an independent random placement in a binary tree. The resulting growing gaps support a summable Gaussian allocation. A finite, charged validation procedure converts constant success into a bound on original expected loss without a bounded-loss assumption. The variable- and common-scale statements are formalized in Lean, including their probability laws, resource bounds and selected-output privacy. The constants are explicit from N = 2; the implementation is an ideal-real oracle construction, not a claim of practical numerical efficiency. Preprint; independent human mathematical review has not been established. This research, writing and formalization used substantial AI assistance, disclosed in the manuscript and accompanying provenance file. Formal scope and reproduction limits are documented in FORMAL_SCOPE.md. The manuscript is licensed under CC BY 4.0. Bundled third-party source code retains its stated licenses.