The Dimension-Shift Category and Its Mellin-Gamma Representation

Abstract

We define a thin category ${\mathrm{Dim}}^{+}$ of dimension shifts and a category $\mathrm{RadMeas}$ of positive Radon measures with Radon--Nikodym density morphisms. We classify scaling-covariant functors ${\mathrm{Dim}}^{+}\to \mathrm{RadMeas}$ whose morphisms are given by homogeneous densities. Gaussian normalization selects a unique functor with values $d\hat{I}{¼}_x(u)=\frac{\ddot{I}{€}^{x/2}}{\hat{I}“(x/2)}{u}^{x/2-1}du.$ Its morphism component yields the radial-integration transport $R(x,r)=\frac{\ddot{I}{€}^r\hat{I}“(x/2)}{\hat{I}“(x/2+r)},$ while the unit-interval observable recovers the Euclidean ball-volume formula $V(x)=\frac{\ddot{I}{€}^{x/2}}{\hat{I}“(x/2+1)}.$ The two transports differ by the multiplicative coboundary of ${\hat{I}}^2(x)=x$, identified with the categorical dimension of the standard object in Deligne's interpolation category $\mathrm{Rep}(O_t)$.

10 pages. Substantial revision and conceptual reorganization of a previous version. Companion analytic paper: "Radial Integration and Ball Volume in Continuous Dimension" (arXiv:2605.04351)