A Kolmogorov fourth-moment bound on Poisson chaos via a martingale core

For any finite family of Poisson multiple integrals and any finite $p\geq2$, we construct a common increasing filtration generated by finitely many exact Poisson counts such that the associated conditional expectations converge in $L^p$, remain in their original chaoses, and have bounded step kernels with finite-measure support. This finite-count martingale core allows regular fixed-chaos identities and estimates to be extended under the sole assumption of a finite fourth moment. In particular, if $F$ lives in a Poisson chaos with unit variance and finite fourth moment, we prove that the Kolmogorov distance between $F$ and a standard normal is bounded by $15.6(\mathbb{E}[F^4]-3)^{1/2}$. This removes Assumptions $\mathbf A$ and $\mathbf A^{\textbf{loc}}$ from the Kolmogorov bound of D\"obler and Peccati (Ann. Probab., 2018). We also obtain quantitative $L^4$ estimates for all iterated Malliavin derivatives and, for $F$ in a Poisson chaos, the fourth moment assumption of $F$ forces the $L^4$-integrability of its kernel.