arXiv:2610.30034v1 Announce Type: cross Abstract: This paper revisits the relation between dual effect and certainty equivalence (CE) through a scalar linear-Gaussian model. In dual control, the current input affects not only the immediate cost but also the information available for future decisions, and hence the future cost. Dynamic programming (DP) provides a natural framework for formulating dual control problems. At each step of the Bellman recursion, an optimal input minimizes the sum of the stage cost and the expected cost-to-go. We restrict attention to stage costs minimized by the CE input. However, CE may not minimize the Bellman objective because of the input-dependent expected cost-to-go. Classical no-dualeffect (NDE) conditions guarantee CE optimality in certain problem classes by requiring the conditional distribution of the estimation error to be independent of past controls. In the present model, we observe that CE can remain optimal at the level of a single Bellman recursion even when the classical NDE conditions fail, provided that the expected cost-to-go is independent of the current input. Motivated by this observation, we introduce a notion of NDE formulated directly at the level of the Bellman objective, which we call Bellman-side NDE. Under some mild regularity conditions, we show that Bellman-side NDE can be characterized by a heat equation. This characterization identifies the class of Bellman-side NDE functions, and further yields a decomposition of a generic cost-to-go function into a Bellman-side NDE component and a dual-effect component.