The Gaussian jet of an image is a graded vector space: a dilation by $λ$ multiplies derivatives of order $k$ by $λ^{-k}$. Local graded convolutions are exactly homogeneous differential operators, and a local feature is dilation-invariant if and only if it factors through the scale-normalized jet or, without the scale, through the retraction onto the weighted sphere of the grading. Under unknown Gaussian blur the retraction changes only by a change of scale, whereas the scale-normalized jet acquires a grading factor that no radial normalization removes. At the zero jet the retraction separates image families that radial normalizations and continuous classifiers cannot. Networks built on the retraction are universal for invariant features. On MNIST such a network keeps 88.4% accuracy under a two-pixel blur, against 66.2% for the same network on scale-normalized derivatives; in Lindeberg's Gaussian derivative network the retraction raises accuracy at that blur from 91.5% to 95.1%.
Every real spherical harmonic of even degree is congruent to its mirror image. We prove that a real spherical harmonic of odd degree $n\ge3$ with distinct Maxwell axes is congruent to its mirror image if and only if the hyperelliptic curve of genus $n-1$ defined by its binary form has a reduced automorphism of even order whose lift acts on the form by $-1$. We then determine the locus of these curves: it is the locus of curves with an extra involution if and only if $n-1$ is a power of two, and otherwise it also contains, for each even divisor $k$ of $n-1$ with odd cofactor $d\ge3$, an irreducible family $y^2=x K(x^{2k})$, $°K=d$, with generic reduced automorphism group $\Z/2k$. For $n=3$ the condition is the vanishing of the skew invariant of the binary sextic. As an application, the invariants of the truncated spherical harmonic expansion of a genus-zero surface, in a conformal parameterization normalized by its conformal barycenter, give a descriptor that is exactly invariant under parameterization, position and scale, records handedness, and is stable in the orbit distance.