The Dimension-Shift Category and Its Mellin-Gamma Representation

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