Place n unit squares on a grid in some configuration. You want to rearrange them into a different configuration by sliding squares one at a time to adjacent empty cells. The question: how many moves does each square need?
The naive bound is O(n) — in the worst case, a square might need to traverse the entire grid. Standard reconfiguration results give polynomial total moves, but the per-square cost could be high if some squares are shuttled back and forth as obstacles for others.
Akitaya et al. (arXiv:2603.05203) show that O(1) moves per square suffice. Not O(1) amortized over all squares — O(1) worst case for every individual square. Each square moves a bounded number of times regardless of n, the grid size, or the complexity of the source and target configurations.
The construction uses a universal intermediate configuration — a canonical arrangement that any valid configuration can reach in O(1) moves per square. The reconfiguration from source to target routes through this intermediate: source → canonical → target. Each leg of the trip costs O(1) per square because the canonical form is structured to be reachable from any valid placement with purely local operations.
The constant is not small — the paper doesn't optimize it. But its existence is the result. The per-square cost of reconfiguration is independent of scale. A square in a 10×10 grid and a square in a 10,000×10,000 grid both move the same bounded number of times.
This is remarkable because most reconfiguration problems have per-element costs that grow with system size. Sorting n items requires Ω(log n) swaps per element. Reconfiguring robots on a grid typically requires Ω(diameter) moves per robot. The grid-of-squares setting has enough geometric structure — specifically, enough room for local rearrangements — that global reconfiguration reduces entirely to local operations.