arXiv:2611.05533v1 Announce Type: new Abstract: This paper presents a probabilistic framework for the direct, data-driven synthesis of the stochastic linear- quadratic output-feedback control for linear systems sub- ject to process and measurement noise. Using delayed input-output measurements, we derive a lifted parameter- ization together with an equivalent lifted linear quadratic cost, enabling controller synthesis without explicitly identi- fying the lifted dynamics. Two complementary closed-loop formulations, based on trajectory and covariance represen- tations, are then developed, whose uncertainties depend only on the realized but unknown lifted noise. We char- acterize the relative performance of the two formulations in finite-data regimes. High-probability uncertainty bounds are developed for realized noises, leading to a convex optimization solution to the problem, whose feasibility cer- tifies probabilistic mean-square stability (p-MSS) and per- formance guarantees. To reduce conservatism, Gaussian zonotopes are used to refine uncertainty sets by elimi- nating noise realizations inconsistent with measured data, and a novel intersection rule enables the consistent fusion of data-driven and physics-informed information. These components are integrated into an adaptive framework that unifies indirect (identification-based) and direct (control- oriented) data-driven linear-quadratic output-feedback con- trol. Unlike existing direct data-driven LQR methods that verify stability a posteriori, our by-construction approach achieves robust stability alongside reduced conservatism and superior performance. Simulation results show that the proposed controller approaches the ideal optimal solution while outperforming existing data-driven methods under noisy measurements.