Exact hierarchical polar differential layer

Integrate this as a spatial encoder and differential output head for a neural operator on a disk, annulus-with-center, or spherical patch. First build a polar hierarchy with angular coordinate theta and radial coordinate rho. At level l, store scalar coefficients x0 in P^0_l, vector or 1-form coefficients x1 in P^1_l, and scalar density or 2-form coefficients x2 in P^2_l. Evaluate the corresponding B-spline basis at query points using Cox-de Boor recursion, but tie all basis functions touching rho=0 according to the pole constraints; equivalently, use the reduced dimensions n^0, n^1, and n^2 above. Precompute sparse incidence or derivative matrices D0_l and D1_l from the univariate B-spline derivative relation and tensor-product Kronecker construction, then project them into the pole-compatible bases. Verify numerically that ||D1_l D0_l|| is at machine precision. A neural block can update all three fields using z0 = MLP0([x0, D1^T x1]), z1 = MLP1([x1, D0 x0, D1^T x2]), and z2 = MLP2([x2, D1 x1]); alternatively use D0 x0 and D1 x1 as fixed linear channels. For adaptive refinement, compute a cell score e_K = mean_{q in K} ||r(q)||^2, where r is the PDE residual or supervised prediction error, mark cells with e_K above a percentile threshold, split them radially or angularly, prolongate coefficients with the exact B-spline refinement matrix, and rebuild D0 and D1. Reject a refinement if ||D1D0|| exceeds 1e-8. The mathematical quantities are the pole tying, basis dimensions, prolongation, and exact derivative matrices; network weights and residual thresholds are estimated empirically. First test on a 2D manufactured Poisson problem and a divergence-free vector-field prediction task using a 6-layer CNN or FNO baseline on a 64x64 disk mask. Compare fixed Cartesian, uniform polar, and adaptive polar models at equal parameter count. Success means lower center-region error and fewer active coefficients at the same L2 error, with exact zero divergence for fields produced as D0 phi or exact zero div-curl residual for fields passed through D1.