2026/07/23 by Juan Bory-Reyes, Baruch Schneider, Diana Schneiderova +1
#math.CV
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 Gμ(ρ;x,y) =\fracΓ(μ/2)2πμ/2 (1+ρ\x,y\+ρ2x2y2)-μ/2. At μ=-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≤ u, the residue inverts a canonical renormalized pairing on \mathcal Pk. In the collision range u<k≤2u, the ordinary pairing has radical (x2)k-u\mathcal P2u-k; the finite part reproduces the quotient, while the residue reproduces the radical after transport from the reflected degree. For k≥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.