A box with real, finite walls — a particle can climb out if it has the energy. Two things the infinite box never let you see: the ladder is now finite (a well holds only so many rungs — deepen or widen it and more snap into being), and because the walls are climbable the wave leaks out and decays into the forbidden ground beyond. Drag the rim to deepen the well, drag a wall to widen it — and watch rungs born at the lip or spilling out over it.
Compare it to the box and the bowl next door. The infinite box held an endless ladder (∝ n²) and its wave died dead at the wall; here the walls are only so tall, so only a handful of rungs fit beneath the rim, and each one's wave bleeds through the wall and decays with a length 1/κ set by how far below the rim it sits. Shallow the well toward the edge and the top rungs spill out one by one — a state too shallow to bind simply ceases to exist. There is, however, always at least one bound state, no matter how shallow the well. The self-test proves each rung against an independent from-scratch eigensolve of the actual stepped potential.