Support Selection Beyond Smooth DAG Exactness: Completion Geometry, Score Margins, and Selective Certificates

Continuous DAG learning often relies on a smooth exact constraint to enforce acyclicity. Exactness specifies the zero set, but leaves two questions unanswered. How informative is a small nonzero residual? Which edges should be removed when several repairs restore acyclicity? For the first question, we prove sharp local limits for sufficiently smooth exact representations on signed candidate neighborhoods of a fixed DAG. The bounds reveal that the smallest minimal cycle completion controls the earliest possible response, while the largest controls the best uniform distance-error exponent. They also distinguish signed or vector residuals from nonnegative scalars. For the second question, we show how a score margin breaks a feasibility tie in an isolated-cycle flow. We then use simultaneous parent-set score intervals to account for sampling uncertainty and certify features shared by all population optima of a frozen score, without requiring a unique graph. Controlled experiments recover the predicted orders and verify certification under nonunique optima, while comparisons with standard optimizers show the flow model’s limitations. The results clarify what changing a smooth constraint can improve and what requires evidence from scores and data.