Galois Coordinates for Three-Dimensional Shape Representation and Spectral Description of Convolutional Networks

We construct Galois coordinates that connect general three-dimensional shape representations with a layerwise spectral description of convolutional networks. Two spirals on an auxiliary ellipsoid form a closed contour; for q ≡ 3 (mod 4), their nodes are indexed by the quadratic-residue subgroup and its negative coset in the Galois field GF(q). A shape is represented by a radial profile or volumetric cell occupancies, allowing for concavities, cavities, and disconnected components. The contour produces cyclic arrays of length q − 1, while each branch has length (q − 1)/2; roots of unity of both orders exist in the original field. We derive branch-symmetry conditions, explicit spectral mixing for polynomial activations, and unique reconstruction of convolution kernels from prescribed transfer matrices. We also prove covariance of the occupancy representation and convergence under decreasing cell diameters, with a conditional error bound for a spherical carrier. Numerical checks cover 72 shape–resolution combinations and quadrature sensitivity of occupancies and quantized codes. A fixed spectral projector distinguishes centered and displaced cavities in a prescribed family of 84 shapes at two quadrature densities, with exact agreement between direct and spectral calculations. Generalization to real data and hardware efficiency remain untested.