Top-$k$ selection determines which components of a sparse model remain active. Hard selection blocks gradients, while continuous relaxations often couple mask hardness to the selected mass. We introduce LaPrune, a mathematically exact-budget differentiable layer that controls the normalized second moment while preserving the selected mass. A LapSum barrier preserves the selection mass, and a normalized second-moment constraint moves the mask from a dense equal-mass allocation toward hard top-$k$ at each budget. We derive a population prediction of the saturated fraction, a near-binary limiting law, and a tight worst-case guarantee on the near-zero fraction. The normalized hardness parameter is invariant to score scale, while a fixed LapSum temperature is not.

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