Mellin Kinematics and the Geometry of Scale

The Mellin transform is the Fourier transform of scale. This elementary sentence is the kinematical center of the shadow framework. A positive variable is not naturally additive; it is multiplicative. Its invariant measure is Haar measure d×x = dx/x, its time coordinate is u = log x, its translation group is dilation, and its spectral transform is Mellin analysis. This paper develops that statement as a precise spectral principle. The logarithm identifies L²(R⁺, d×x) with L²(R, du); after the half-density passage to the classical Mellin convention on L²(R⁺, dx), the unitary spectral axis becomes Re(s) = 1/2, and the self-adjoint dilation generator is −i(x∂ₓ + 1/2). In celestial holography, the same positive scale appears as the boost energy ω, and conformal primary wavefunctions are Mellin transforms in ω with dimension Δ. The paper proves a uniqueness theorem for the scale bridge: among homogeneous positive scale laws x = ω^α (proved to be the unique continuous group homomorphisms of (R⁺, ×)), compatibility of the arithmetic half-density axis Re(s) = 1/2 with the four-dimensional celestial principal axis Re(Δ) = 1 forces α = 2. Thus x = ω², and the arithmetic kernel x^{s-1} dx transports to the celestial kernel ω^{Δ-1} dω precisely when Δ = 2s. Consequently the celestial principal series Δ = 1 + iλ is the critical line s = 1/2 + iλ/2, and the celestial shadow Δ → 2 - Δ is the arithmetic reflection s → 1 - s. The result is not a convention but a rigidity statement: once scale, Haar measure, half-densities, and four-dimensional celestial unitarity are fixed, the factor of two is forced. The paper also clarifies the relationship between Mellin spectral parameters (principal-series labels λ ∈ R) and physical conformal dimensions (Δ ∈ R⁺), which are distinct objects arising from the same dilation structure in different contexts.