We study geometric covering problems: stabbing axis-parallel squares and cubes by coordinate hyperplanes, stabbing horizontal segments by horizontal and vertical lines, and transversal of separated $d$-intervals. We prove matching NP-hardness bounds for their natural covering-LP thresholds. For every $\varepsilon>0$, square stabbing is NP-hard to approximate within $2-\varepsilon$, horizontal-segment stabbing within $e/(e-1)-\varepsilon$, and, for every fixed $d\ge2$, cube stabbing and separated $d$-interval transversal within $d-\varepsilon$. The results hold with finite candidate sets and when arbitrary admissible lines, hyperplanes, or track points are allowed. We construct harmonic integrality-gap families for squares and cubes and use the Kovaleva--Spieksma gap family for horizontal segments. In each family, every object meets a candidate of every orientation. We lift these gaps through the strict-CSP framework of Kumar, Manokaran, Tulsiani, and Vishnoi, using the Unique Games theorem to obtain NP-hardness. An ordered sampler preserves the LP marginals, coupled shifts make every local sample feasible, and a small perturbation through an absorbing label connects every local support. Cartesian cloning removes rational weights, after which disjoint blocks of candidate coordinates realize each output constraint by one square, cube, or horizontal segment. Applying the same transfer theorem to rectangle-hitting gaps of Pach and Tardos, we prove that stabbing axis-parallel boxes by supplied candidate lines in $\mathbb{R}^3$ is NP-hard to approximate within any fixed constant factor. Incidence-preserving algebraic maps yield the same hardness for hitting boxes in $\mathbb{R}^3$ and halfspaces in $\mathbb{R}^6$ by points from supplied finite candidate sets.

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