Sharp variance ordering for rotating Gaussian samplers

Unrefereed research candidate. A Markov chain sampler estimates averages from one long trajectory; its accuracy is the long-run (asymptotic) variance of the time average. Adding a target-preserving rotation (a nonreversible perturbation) to overdamped Langevin samplers is known to help every observable. For kinetic (underdamped) Langevin samplers with friction γ, Duncan, Nüsken and Pavliotis (J. Stat. Phys. 2017) proved a local improvement for linear and quadratic observables of a Gaussian target when γ > √2 and conjectured it for every square-integrable observable. Results (exact stationary Gaussian dynamics, matched position and momentum rotations, position observables). (1) Every square-integrable observable's long-run variance never increases with rotation strength if and only if γ ≥ √2, endpoint included: a global form of the conjecture. The mechanism: the oscillator's power spectrum S(ξ) = 2γ/(ξ4 + (γ2 − 2)ξ2 + 1) is single-peaked exactly when there is no resonance, rotation shifts the spectrum, and convolution keeps single peaks single. (2) The literal second-derivative form holds exactly on an explicit weighted Hermite class containing all polynomials; rough observables have sharper cusps. (3) Below the threshold, for every friction, even observables still improve at every rotation strength, only finitely many odd Hermite degrees can be hurt, and one finite rotation strength improves every observable. (4) Sharp large-rotation rates (uniform constant √(2π/e); slower for incommensurate rotation speeds). (5) For noncommuting stiffness K and friction Γ: ordering iff Γ3 − ΓK − KΓ ≽ 0; the least safe friction (2K)1/2 is a safety boundary, not an optimal friction; two coupled replicas make the criterion apply to any Gaussian target. (6) Fixed horizons and regular sampling can reverse the ordering; independent Poisson sampling preserves it. Scope. Gaussian targets only; no claim about numerical cost or discretised integrators (Duncan, Nüsken and Pavliotis found small friction best in practice). Classical ingredients (Breuer–Major Hermite identities, closure of symmetric unimodality under convolution, the convolution-power variance formula of Maini 2026) are credited, not claimed. The search covered the originating paper and thesis, the overdamped variance-reduction literature, work citing Duncan, Nüsken and Pavliotis, related nonreversible-sampling work, and formula-level aliases across Gaussian spectral analysis and convolution theory; Maini (2026) and the published text of Franke et al. (2021) were read in full. No earlier statement of the main results was found. Archive: manuscript (Markdown source, generated LaTeX and PDF), three deterministic verification scripts with their JSON reports, figure, extension notes, novelty reports, frozen targets, rating-free internal review record, bibliography with registry records, CI and SHA-256 manifests. Assurance: internal AI-assisted review only (not human peer review, independent reproduction or formal verification). Creator: Anonymous. Text CC0-1.0; code MIT. Version 0.3.0-candidate corrects the citations of Maini (2026) and Franke et al. (2021) after both were read in full, and restates what was searched. No theorem, proof, computation or recorded output changed. Version 0.1.0-candidate remains archived at doi:10.5281/zenodo.23941497. Version 0.3.0-candidate corrects the citations of Franke et al. (2021) to the journal text and its numbering after that text was read in full. Version 0.2.0-candidate removed the report's statement that the paper discusses an irrational rotation; that removal was mistaken, and version 0.1.0-candidate was correct. The statement is restored. No theorem, proof, computation or recorded output changed. Versions 0.1.0-candidate (doi:10.5281/zenodo.23941497) and 0.2.0-candidate (doi:10.5281/zenodo.23946160) remain archived.