Multicriticality and Scaling: Mellin Spectral Theory, and the Decoupling of Geometric and Spectral Exponents

We develop a spectral theory of scale-invariant operators on the multiplicative half-line $({\mathbb{R}}_{+},dx\mathrm{/}x)$. A symmetric kernel $M(x,y)$ satisfying $M(kx,ky)=k^{-a}M(x,y)$ necessarily factorizes as $(xy{)}^{-a\mathrm{/}2}F(x\mathrm{/}y)$, where the shape function $F$ depends only on the ratio of its arguments. The Mellin transform diagonalizes such operators: the generalized eigenfunctions are ${\psi }_{\omega }(x)=x^{-a\mathrm{/}2+i\omega }$, and the eigenvalues are the Mellin multiplier $\tilde{F}(\omega )$. This structure reveals a fundamental decoupling of two exponents. The geometric exponent $a$, carried by the power-law envelope $(xy{)}^{-a\mathrm{/}2}$, governs the matrix scaling under dilation. The spectral exponent $b$, measured from the eigenvalue decay of the finite-dimensional truncation, is an effective quantity determined by the shape of $\tilde{F}(\omega )$. For the explicit kernel $F(t)=c{\rho }^{\mathrm{\mid }\ln t\mathrm{\mid }}$, the Mellin multiplier is a Lorentzian of width $\sigma =-\ln \rho$, not a power law – so $b$ is generically distinct from $a$. This decoupling provides a precise mathematical characterization of multicriticality: the equality $a=b$ corresponds to a simple critical fixed point of the Renormalization Group, while $a\mathrm{\ne }b$ signals the presence of multiple independent scaling dimensions. We prove that the discrete self-similarity condition forces eigenvector collapse on the lattice, motivating the continuum formulation. Finite-size corrections from lattice sampling are quantified numerically.