Infinitesimal Kernelized Multiplicative Synthesis II: Global Flow Theory for Infinitesimal Kernelized Multiplicative Synthesist

Paper I introduced the local infinitesimal calculus of kernelized multiplicative synthesis: primitive spectra $G$ evolve additively, and the synthesized objects $$f=\mathcal{E}_K(G)=\exp \left(\mathcal{T}_K G\right)$$ inherit their motion through transport. The present paper develops the next layer: a global flow theory on selected Banach spectral classes. We work on right half-planes $$H_\sigma:=\{s \in \mathbb{C}: \operatorname{Re} s>\sigma\}, \quad \sigma>0$$ and introduce exponentially weighted spectral spaces $$\mathcal{S}_{a, \sigma}\left(H_\sigma\right)=\left\{G \in \mathcal{O}\left(H_\sigma\right):\|G\|_{a, \sigma}:=\sup _{\operatorname{Res}>\sigma} e^{a \operatorname{Res}}|G(s)|0$$ For kernels $K$ satisfying a natural boundedness profile, these spaces are stable under the kernelized averaging operator $$\left(\mathcal{T}_K G\right)(s):=\int_0^{\infty} G(u s) K(u) \frac{d u}{u}$$ hence support the multiplicative synthesis map $$\mathcal{E}_K(G):=\exp \left(\mathcal{T}_K G\right) .$$ For locally Lipschitz generators on $\mathcal{S}_{a, \sigma}$ we prove local well-posedness, maximal interval theory, blow-up alternatives, continuous dependence on initial data, and semiflow structure. We then give several global existence criteria: affine-linear growth, dissipative estimates, positively invariant bounded sets, and bounded exhaustion principles. Finally, we transport the resulting spectral dynamics to the multiplicative side and record detailed examples: affine generators, multiplier flows, finite-dimensional atomic sectors, and measure-valued continuous spectra. The purpose of this paper is not sharp admissibility or optimal global theory in complete generality. Rather, it is to provide the first honest global flow package for infinitesimal kernelized multiplicative synthesis, in a form robust enough for repeated later use.