A No-Go Theorem for Background-Only Interpretations of Low-Energy Nuclear Recoil Signals Under Joint Experimental Constraints

Background-only explanations are commonly invoked to interpret low-energy nuclear recoil signals reported across multiple experimental platforms. Such interpretations implicitly assume the simultaneous validity of independently published constraints describing nuclear recoil energetics, atomic excitation probabilities, detector response statistics, and background noise correlations. In this work, we perform a global consistency test of this assumption by formulating the background-only hypothesis as an existence problem: does any configuration exist that satisfies all published constraints jointly, without parameter fitting, access to raw event data, or introduction of new physical models? Using a constraint-interaction framework operating exclusively on publicly reported summary statistics, we conduct a large-ensemble joint-consistency analysis. We find that no admissible background-only configuration exists. Across all tested initializations and noise realizations, the constraint space collapses to an empty set. This result constitutes a computational no-go theorem. It demonstrates that the background-only interpretive framework, as currently formulated in the literature, is internally inconsistent under joint enforcement of its own constraints. At least one assumption commonly treated as valid must therefore be revised, relaxed, or replaced.