Emergent Sparsity: How Many Effective Variables Survive Coarse-Graining in AlphaEarth Embeddings?

AlphaEarth encodes each 10 m land pixel as a 64-dimensional embedding intended to support diverse global mapping tasks. Yet many scientific applications work at coarser spatial supports, for which fewer than 64 effective variables may suffice. We define effective dimension as a joint function of spatial scale and reconstruction tolerance. At each scale, the same nonlinear $\beta$-VAE architecture produces a rank--distortion envelope from which we read the smallest observed number of effective variables meeting each error budget. At 3% validation tolerance, this number falls from 24 to 7 when the scale is increased from 10 m to 78.9 km, whereas the corresponding PCA rank falls only from 46 to 38. Repeating the analysis on unit-normalized targets leaves this trend unchanged. Coarse-graining thus induces emergent sparsity: within the scanned model family, it reduces the nonlinear rank sufficient for reconstruction much more strongly than the corresponding PCA rank. The result is not one intrinsic dimension but a two-control surface: spatial scale determines what is averaged away when computing the target, while tolerated error determines what must be retained from that target.