Infinitesimal Kernelized Multiplicative Synthesis I: A First Calculus of Local Generators for Kernelized Multiplicative Structure

We propose a first infinitesimal calculus for kernelized multiplicative synthesis. The basic point is simple: if a multiplicative object $f$ is reconstructed from a primitive spectrum $G$ by $$f=\mathcal{E}_K(G)=\exp \left(\int_0^{\infty} G(u \cdot) K(u) \frac{d u}{u}\right),$$ then deformations of $f$ should be studied first on the spectral side, where the geometry is additive and local, and only then transported to the multiplicative side by synthesis. This yields a natural notion of infinitesimal generator: a rule $V$ assigning to each spectrum $G$ a velocity $V[G]$. The corresponding infinitesimal action on the synthesized object is $$\delta_V f:=f \cdot \mathcal{T}_K\left(V\left[\mathcal{P}_K f\right]\right) .$$ We record the basic transport identity, a formal bracket coming from spectral vector fields, and several model flows (constant insertion, scaling, multiplier fields, and motion of atomic kernel-prime scales). The theory is intentionally modest: it is not presented as a full Lie theory, but rather as a local calculus of generators for kernelized multiplicative structure. Still, the analogy with Lie theory is clear in spirit: one studies local generators first and only then the global multiplicative law. The article is intended as the first paper in a projected series. Its role is to isolate the infinitesimal language itself, together with a first normed-class well-posedness layer, before any full global flow theory or sharp admissibility analysis is attempted.