Infinitesimal Kernelized Multiplicative Synthesis V: Ambient Mittag–Leffler Mechanisms for Continuous Kernelized Multiplicative Synthesis

The purpose of this paper is not to reprove the Helson-specific Mittag-Leffler theorem of Andersson, nor to reprove the blur/Fourier route developed in the companion work. Its purpose is ambient and structural. We isolate a general continuous-kernel setting in which moving singular data, moving primitive spectra, and moving synthesis laws may be treated as parts of a single transported mechanism. The basic point is that a Mittag-Leffler phenomenon has three logically distinct channels:(i) an additive channel, where one prescribes or models a meromorphic logarithmic field;(ii) a multiplicative synthesis channel, where primitive spectra are turned into global objects through a kernelized exponential law;(iii) a singularity channel, where zeros and poles move through a family of principal parts. In the discrete Helson setting these three channels are tied to a very specific kernel, namely the zeta-symbol kernel. In the present paper we show that, at the structural level, the same interaction is not peculiar to that case. We define coherent moving Mittag-Leffler data on domains containing $\{\operatorname{Res}>1\}$, formulate kernelized realization modulo entire error, and prove an ambient transport theorem showing that if the additive model, the kernel law, and the singular pattern move coherently with controlled defect, then the resulting multiplicative family inherits the prescribed moving zero/pole pattern. We also isolate a guarded transfer principle from a model family to a target family. The gain is conceptual: it makes clear that a fluid passage between additive data, Euler-like synthesis, and moving singularities is a feature of the continuous-kernel environment itself, rather than a Helson-specific miracle. In particular, the discrete Helson world appears as one distinguished kernel choice inside a much larger ambient deformation theory. Thus, if one later verifies the hypotheses in a concrete Helson/blur setting, the Mittag-Leffler behavior should be read as a special selection from a continuous family of multiplicative synthesis laws, not as a phenomenon coming from nowhere.