English

Right q-vector calculus at integral superdimension: localized decompositions and resonance

Complex Variables 2026-07-16 v1

Abstract

We specialize the intrinsic right qq-vector derivative on radial algebras to integral superdimension. The formal dimension is encoded by an independent coefficient QQ, and formal radial superspace of superdimension M=m2nM=m-2n is obtained by the coefficient specialization QqMQ\mapsto q^M. This gives a rigorous universal calculus and, on finite blocks containing at most mm abstract vectors, a faithful coordinate realization on Rm2n\mathbb{R}^{m|2n}. The localized exterior result is formulated as a Green decomposition by complementary projector images. Whenever the specialized finite determinant is nonzero, full left multiplication yields a determinant-localized right-monogenic Fischer decomposition. Beyond this base-change theory, we determine the exceptional one-vector calculus completely: for M=2M=-2\ell there is one additional singular monomial and one missing image monomial, whereas all other integral superdimensions give a surjective derivative with constants as its kernel. We then prove that the degree-zero Fischer operator is exactly diagonal on exterior blades, obtain its determinant explicitly, classify all support-resonance values in 0<q<10<q<1, and give an exact kernel-rank formula as a sum of support multiplicities. On a block with NN auxiliary vectors, an even support rank pp has pure multiplicity (Np)\binom Np on the support truncation. At an odd-support root, every lower odd factor is nonzero and any simultaneous lower resonance is unique, even, and characterized by one strictly monotone scalar equation. These results distinguish persistent nonpositive-even-superdimension defects from isolated support-dependent qq-resonances. An appendix records that constant scalar projection of two independent orthogonal right qq-vector derivatives does not descend to the Hermitian quotient.

Keywords

Cite

@article{arxiv.2607.14449,
  title  = {Right q-vector calculus at integral superdimension: localized decompositions and resonance},
  author = {Juan Bory-Reyes and Baruch Schneider and Diana Barseghyan Schneiderová and Yifan Zhang},
  journal= {arXiv preprint arXiv:2607.14449},
  year   = {2026}
}