We develop a continuous-kernel version of the prime transform on dilation orbits. Instead of the discrete averaging law $$(\mathcal{T} G)(s)=\sum_{k \geq 1} \frac{1}{k} G(k s)$$ we work with a multiplicative kernel on $(0, \infty)$, $$\left(\mathcal{T}_K G\right)(s):=\int_0^{\infty} G(u s) K(u) \frac{d u}{u}$$ whose inversion is governed by the Mellin symbol $$\kappa(z):=\int_0^{\infty} u^{-z} K(u) \frac{d u}{u}$$ When $\kappa$ is invertible on the working strip, the primitive multiplicative content of a nonvanishing holomorphic function $f$ is extracted by the kernel-prime transform $$\left(\mathcal{P}_K f\right)(s):=\left(\mathcal{S}_J \log f\right)(s)$$ where $\mathcal{S}_J$ is the inverse-kernel operator. The corresponding synthesis is $$\left(\mathcal{E}_K G\right)(s):=\exp \left(\left(\mathcal{T}_K G\right)(s)\right)$$ The central new feature is the emergence of a kernel profile $$\Phi_K(x):=\int_0^{\infty} e^{-u x} K(u) \frac{d u}{u}$$ which replaces the classical geometric Euler factor. If $$G(s)=\sum_j c_j e^{-\lambda_j s}$$ then $$\left(\mathcal{E}_K G\right)(s)=\exp \left(\sum_j c_j \Phi_K\left(\lambda_j s\right)\right)$$ so each primitive atom $e^{-\lambda s}$ synthesizes into the factor $\exp \left(\Phi_K(\lambda s)\right)$. The usual Euler factor $\left(1-e^{-\lambda s}\right)^{-1}$ is therefore not universal: it is precisely the profile of the discrete kernel. Under natural operator hypotheses we prove $$\mathcal{P}_K\left(\mathcal{E}_K G\right)=G, \quad \mathcal{E}_K\left(\mathcal{P}_K f\right)=f$$ record stability and truncation remarks, discuss measure-valued spectra, and recover the earlier discrete theory as the special zeta-symbol case. In this way the prime transform is revealed not as an integer-specific device, but as a general multiplicative unmixing calculus whose ambient form is continuous and whose synthesis law is encoded by the kernel profile.