arXiv:2605.02205v2 Announce Type: replace-cross Abstract: Stability Selection resamples the observations but leaves the design untouched, so it cannot reject an irrelevant feature that is correlated with the relevant ones in a way that violates the irrepresentable condition. Such a feature (a ``confounder'') is selected in a substantial fraction of resamples, and the exchangeability behind the false-positive bound of Stability Selection fails. We propose 2D Stability Selection (2D-SS), which aggregates selection frequencies over half-samples and over a grid of injected design noise, and reduces to Stability Selection at zero noise. Because input noise acts, in expectation, as Tikhonov regularisation, a jittered Lasso fit is a noisy elastic net whose support recovery is governed by a noise-indexed irrepresentable condition. Jittering can therefore repair designs in which confounders violate the Lasso condition, although it can also hurt. We give the noise level at which a violated feature's statistic vanishes (for uncorrelated relevant features), exact-recovery guarantees for a single fit, conditions under which aggregation over noise levels recovers the support, and a complementary-pairs bound on the expected number of false positives, carried over from Stability Selection to the noise-aggregated procedure, for a fixed threshold or a size-matched selector. The procedure takes no measurement-error covariance as input, although the guarantees assume i.i.d.\ Gaussian measurement error. In simulations 2D-SS excludes confounders that Stability Selection includes, and its size-matched version stays below the nominal closed-form false-positive bound in all 240 simulated cells, whereas Stability Selection exceeds it in 21; it does not generally improve on Stability Selection in $F_1$ and is lower at low noise. It is useful when stable but wrong selections are the concern.