Unbounded Line Pinning by Pairwise Disjoint Convex Bodies

For every integer m >= 3 we construct m+4 pairwise disjoint compact convex bodies with nonempty interiors in R^3 and a common transversal line such that every subfamily pinning that line contains at least m members. Thus isolated line transversals to arbitrary disjoint convex bodies admit no pinning bound depending only on the dimension. Four separated polytopes restrict the nearby transversals to a classical one-parameter family on a hyperbolic paraboloid. The remaining bodies are convex epigraphs of flat oscillatory perturbations: together they exclude every nonzero parameter, while each has an exclusive sequence of excluded parameters tending to zero. All contacts with the pinned line are distinct single points. An explicit extension gives the same conclusion in every dimension at least three. The construction is nonsemialgebraic and does not settle the separate question for pairwise disjoint polytopes. This is an AI-assisted, self-audited, unrefereed preprint; no external review or absolute priority is claimed.