Infinitesimal Kernelized Multiplicative Synthesis III: Sharp Admissibility Criteria for Infinitesimal Kernelized Multiplicative Synthesis

Paper I introduced infinitesimal kernelized multiplicative synthesis as a local calculus on primitive spectra. Paper II developed the first global flow theory on selected Banach spectral classes. The present paper isolates the remaining analytic bottleneck: admissibility. Fix $\sigma, a>0$ and the weighted spectral class $$\mathcal{S}_{a, \sigma}=\left\{G \in \mathcal{O}\left(H_\sigma\right):\|G\|_{a, \sigma}:=\sup _{\operatorname{Res}>\sigma} e^{a \operatorname{Res}}|G(s)|\sigma\} .$$ We determine exact or near-exact criteria for natural operations to preserve $\mathcal{S}_{a, \sigma}$. These include the kernelized dilation-average operator, multiplier actions, scalar composition operators, polynomial nonlinearities, atomic sectors, and positive measure spectra. In particular, we prove: the sharp boundedness criterion for the kernel-average operator; the exact criterion $m \in H^{\infty}\left(H_\sigma\right)$ for multiplier admissibility; the sharp criterion $F(0)=0$ for scalar composition $G \mapsto F \circ G$; the precise weight threshold for products; the exact atomic threshold $\lambda \geq a$ for exponential atoms; and, for positive measures, the support barrier $\operatorname{supp} \rho \subset[a, \infty)$ as the exact admissibility condition. We also record counterexamples showing failure at the border: kernels with divergent admissibility constant, nondecaying forcing, unbounded multipliers, nonzero scalar constant term, weight overshoot, and atoms or mass crossing below the spectral barrier $a$. The purpose of the paper is not merely to list sufficient conditions, but to turn admissibility into a reusable threshold theory for kernelized multiplicative synthesis.