A categorical kinematical framework for quantum fluctuations in Shape Dynamics, and its explicit realization in Bianchi IX minisuperspace

This paper proposes a kinematical framework for linearized quantum fluctuations in Shape Dynamics. For each shape in the regular stratum of conformal superspace, a Hilbert space of quantum fluctuations is constructed by quantizing the tangent space using the Jacobi metric and white‑noise analysis. Classical parallel transport along Jacobi geodesics is lifted to unitary operators between these Hilbert spaces, under certain analytic assumptions. These structures form a dagger category with a symmetric monoidal structure. The framework is illustrated concretely in the Bianchi IX minisuperspace, where all constructions become explicit and the metaplectic double cover is examined. The work establishes a consistent geometric foundation for coupling quantum fluctuations to classical shape degrees of freedom, while leaving a full dynamical theory for future research.