Goodness-of-fit for multi-distribution neutrino cross-section measurements with shared events
When multiple differential cross-section measurements are extracted from a common event sample, the same events contribute simultaneously to multiple distributions. This event-sharing structure imposes exact linear constraints among the bin counts, reducing the effective dimensionality of the measurement below the total number of bins. Since the covariance matrix obeys these inter-distribution constraints, it contains a rank deficiency. While limited numerical precision may inadvertently restore invertibility, the $χ^{2}$ contributions along constrained dimensions will be arbitrary, yielding unphysically inflated $χ^{2}$ values in global goodness-of-fit tests. We present the range-projected $χ^{2}$, a test statistic that restricts the goodness-of-fit test to the subspace carrying independent statistical information, yielding a $χ^{2}$ with $N_\text{bins} - N_\text{null}$ degrees of freedom, where $N_\text{null}$ is the number of independent constraints. We show that this rank deficiency is a predictable consequence of the event-sharing structure, and that $N_\text{null}$ decomposes into a structural contribution determined a priori from the binning geometry and a kinematic contribution that depends on the phase-space occupancy. The method is validated with an analytical toy model and a simulated neutrino--argon cross-section measurement including unfolding and multi-source systematic uncertainties.