Subgrid features on a 2D grid

The continuous-field analog of the interface page, and the same payload question: the grid is fixed, and each node stores either the field value alone or its 1-jet — the value with the partials ∂x, ∂y, ∂xy. The ground truth is a compact feature — a blob, an elongated lens, or a two-blob pair — traveling diagonally through the square as the shift slider moves, at an angle the grid can never line up with.

The nearest panel is the naive floor — every point copies its closest node, tiling the square into flat cells. Bilinear looks far better between the nodes, but a bilinear patch takes its maximum at a cell corner, so its peak is exactly as quantized as the nearest panel's: slide the feature and watch both yellow peak markers snap from node to node while their height sags whenever the true peak crosses a cell interior — with the blob about a cell wide, the sampled values barely see it at mid-cell and the feature all but vanishes, then reinflates at the next node. The level contour tells the same story: blocky tiles, then lumpy cell-diamonds. The peak-height and peak-offset tiles are the continuous analogs of the interface page's misfit and offset.

The jet payload feeds bicubic Hermite reconstruction: the peak marker glides, the height tile stays far closer to 100%, and the contour stays round at every shift and angle. Same storage sites, same grid — the derivative data is what lets the reconstruction place structure between the nodes, exactly as the distance values did for interfaces. The honest limit is here too: shrink the grid slider and once the blob drops well below a cell width, even the jets lose it — carry more of the Taylor tower (the 1D page goes to third order) or refine the grid.