Exceptional supersphere integration and logarithmic Pizzetti kernels

Abstract

We study orthosymplectically invariant supersphere integration at the exceptional superdimensions $M=-2u$, where the harmonic Fischer structure becomes nonsemisimple and the Pizzetti pairing degenerates. For the meromorphically continued homogeneous inverse kernels we obtain the generating function \[ \mathscr G_μ(ρ;x,y) =\frac{Γ(μ/2)}{2π^{μ/2}} \bigl(1+ρ\{x,y\}+ρ^2x^2y^2\bigr)^{-μ/2}. \] At $\hat{I}¼=-2u$, its Laurent expansion has a polynomial residue and a logarithmic finite part. We prove that these coefficients recover the complete degreewise duality structure on a fixed superspace with nonzero bosonic dimension. In degrees $k\le u$, the residue inverts a canonical renormalized pairing on ${\mathcal{P}}_k$. In the collision range $u<k\le 2u$, the ordinary pairing has radical $(x^2{)}^{k-u}{\mathcal{P}}_{2u-k}$; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For $k\ge 2u+1$, the finite part is the ordinary inverse kernel. We also establish the nondegenerate head--socle pairing on the generalized harmonic modules. As an application, we derive covariant right--left radial $q$-monogenic zonal symbols and identify precise degree-one obstructions to transferring scalar Pizzetti reproduction through a one-sided $q$-Fischer projection.