The Ichimura-Austern-Vincent (IAV) sum-rule formalism for inclusive breakup reactions $a+A\to b+\mathrm{anything}$ treats the detected fragment $b$ as a spectator by replacing its interaction with the target by an optical potential. This assumption becomes questionable when $b$ is a loosely bound composite particle such as a deuteron. I derive a generalization that removes the spectator approximation and retains $b\text{"s}$ internal degrees of freedom, providing state-resolved inclusive cross sections. Within the distorted-wave Born approximation, all nonspectator effects enter through the source function via the operator $V_{bA}-U_{bA}$. The exact sum rule involves the full $x+A$ resolvent ${(E_{x,0}^{+}-H_{xA})}^{-1}$, while a single-channel IAV-like expression is recovered only when the explicit target dependence of $V_{bA}$ is neglected; post-prior equivalence is preserved in both cases, since $V_{bA}$ cancels in the difference between the post- and prior-form residual interactions, and is verified explicitly in the single-channel limit. A key conceptual finding is that the standard IAV result for structureless $b$ corresponds, under closure and a reduction of $b\text{"s}$ internal space, to the total inclusive cross section summed over all of $b\text{"s}$ internal states, rather than the cross section for $b$ in a specific state. An operator-level estimate for $b=d$ on ${}^{208}\mathrm{Pb}$ shows that the nonspectator correction is not a small perturbation at the nuclear surface. The present work is purely formal: it establishes the theoretical framework and identifies the relevant operators, while quantitative assessment of the cross-section impact awaits a full numerical evaluation of the source integrals.

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• Breakup reactions
• Direct reactions
• Elastic scattering reactions
• Nuclear reactions