When Curvature Leaves the Density: Regional Geometry at the ReLU Limit

Gaussian curvature is usually treated as an area density. This description is complete on a smooth decoder surface, but not on a piecewise-smooth one, where curvature may also lie on seams and junctions. Pointwise automatic differentiation evaluates only the cell-interior density. We derive the corresponding regional stratified Gauss--Bonnet identity and compare density integrals with angle-defect totals on forty VAE mean decoders with two-dimensional latent spaces. For ReLU decoders on data-adaptive regions scaled by local ten-neighbour metric distances, the signed density accounts for about $5$% of the typical regional curvature on both datasets. Across same-scale audits of anchors, frames, quadrature, and mesh resolution, ReLU recovery remains below $7.2$%; by contrast, the three smooth activations recover the regional total almost completely, remaining within $0.3$ percentage points of full recovery. In a frozen-weight softplus continuation, the density follows the regional total at every numerically resolved finite sharpness. At the exact ReLU endpoint, the total persists but density recovery falls to about $5$% on both datasets. An analytic surface with nonzero area, two-sided seam, and junction terms verifies the complete three-stratum identity. Thus a small pointwise curvature field can coexist with appreciable regional curvature carried by lower-dimensional strata.