Anti-successors and anti-replicators, Manin products and Koszul duality

We develop an operadic theory of anti-splitting and anti-replication for binary operads. For a binary operad $\text{opd}$, we define its anti-bisuccessor $\text{ABSu}(\text{opd})$ and anti-trisuccessor $\text{ATSu}(\text{opd})$, with the property that an operation of $\text{opd}$ splits into a negative sum of two or three operations, yielding canonical operad morphisms $\text{opd}\to \text{ABSu}(\text{opd})$ and $\text{opd}\to \text{ATSu}(\text{opd})$. For binary quadratic operads, the process gives Manin black products, $$\text{ABSu}(\text{opd})\cong \text{antiprelie}\bullet \text{opd},\text{ATSu}(\text{opd})\cong \text{antipostlie}\bullet \text{opd}\mathrm{.}$$ Dually, we introduce the anti-duplicator $\text{ADup}(\mathcal{P})$ and anti-triplicator $\text{ATrip}(\mathcal{P})$ and prove $$\text{ADup}({\text{opd}}^{!})\cong \text{ABSu}(\text{opd}{)}^{!},\text{ATrip}({\text{opd}}^{!})\cong \text{ATSu}(\text{opd}{)}^{!}\mathrm{.}$$ Consequently, the anti-duplicator gives a new way to calculate Manin white products. %, in particular, $\text{ADup}(\text{opd})\cong \text{antiperm}\circ \text{opd}$. The framework recovers known anti-splittings, including anti-dendriform, anti-pre-Lie, and anti-pre-Poisson structures, in a uniform operadic form and produces further examples and dual pairs.