Stable $q$-Hermitian coordinate calculus: Capelli--PBW transport, mixed polarization algebra, and localization obstructions

Abstract

We extend a finite-support $q$-Hermitian radial calculus to Clifford-valued coordinate polynomials. For $N$ bosonic Hermitian vector labels and finite fermionic support, the stable range $m\ge 2N$ admits a canonical Gram--harmonic normal form by Howe separation. A label--Capelli inverse normalizes the Clifford contractions, while a finite Wick--Chevalley conjugation lifts the divided-power scalar gauge to the full Clifford PBW module. Coupling the normalized contractions to contractible fermionic seed complexes gives two conjugate polynomial-preserving $q$-Hermitian coordinate families. They are square-zero, anticommute within each polarization, restrict exactly to the radial PBW calculus, have scalar boundary trace $[2m-2n]_q$, and recover the classical Hermitian super Dirac pair as $q\to1$. For the mixed polarizations we introduce a relative triangular transport. The Grassmann relations for the normalized Capelli contractions hold on the full coordinate module and yield a labelwise factorization. Hence all higher-filtration mixed terms are finite subset products of explicit local defects, and the mixed polarization algebra closes for arbitrary finite label sets and finite fermionic support. The classical complex structure is common to both polarizations, whereas the two full transported Clifford--Weyl products are distinct for $0<q<1$. We also show that the fermionic Cartan denominator is forced in the natural polynomial first-order Berezin--Weyl class and determine the minimal factorwise Ore localization needed for the conjugate all-charge homotopies. The $q$-dependent Cartan orbit is nonresonant for $m\ge\max\{2N,n\}$. Finally, fixed finite-order differential and finite nonzero-shift realizations on the undeformed coordinate algebra are ruled out.