Utility-Invariant Support Selection and Eventwise Decoupling for Simultaneous Independent Multi-Outcome Bets

For simultaneous independent events with finitely many outcomes, consider the expected-utility problem with nonnegative wagers and an endogenous cash position. We prove a short support theorem for a broad class of strictly increasing strictly concave utilities. On any fixed support family and at any optimal portfolio with positive cash, summing the active first-order conditions and comparing that sum with cash stationarity yields the exact identity $$\frac{\lambda }{K_{\mathrm{\ell }}^{(U)}}=\frac{1-P_{\mathrm{\ell },A}}{1-Q_{\mathrm{\ell },A}},$$ where $P_{\mathrm{\ell },A}$ and $Q_{\mathrm{\ell },A}$ are the active probability and price masses of event $\mathrm{\ell }$, $\lambda$ is the budget multiplier, and $K_{\mathrm{\ell }}^{(U)}$ is the continuation factor seen by inactive outcomes of that event. Consequently, after sorting each event by the edge ratio $p_{\mathrm{\ell }i}\mathrm{/}{\pi }_{\mathrm{\ell }i}$, the exact active support is the eventwise union of the single-event supports, and this support is independent of the utility function. The single-event utility-invariant support theorem is already explicit in the free-exposure pari-mutuel setting in Smoczynski and Miles; the point of the present note is that the simultaneous independent-events analogue follows from the same state-price geometry once the right continuation factor is identified.