A Heisenberg Subdivision Scheme with Central Smoothness Loss
We introduce an interpolatory subdivision scheme for control polygons that take values in the three-dimensional Heisenberg group, the simplest noncommutative model geometry. The scheme keeps existing points at every refinement step and inserts new ones by a coordinate rule whose central correction comes from the group law. The two horizontal coordinates are refined by the classical four-point scheme of Dyn, Gregory and Levin, while the central coordinate acquires a closed-form correction built from a signed area of neighbouring horizontal data. Our main finding concerns the regularity of the limit curve. The horizontal part is exactly the classical four-point limit and inherits its smoothness. The central part behaves very differently. We prove that it converges to a continuous limit that belongs to the Zygmund class, with a logarithmic modulus of continuity. Under an explicit and verifiable condition on the central forcing, this logarithmic bound is sharp, because the scaled first differences then grow linearly with the refinement level, and the limit fails to be continuously differentiable. The effect is confirmed numerically. The correction is harmless at any single refinement step, but its repeated injection at every scale is what impacts smoothness. The example serves as a caution for nonlinear and group-valued subdivision, where a geometrically natural correction can impact regularity.