Common-mode rejection with a robust LOCC benchmark for four gravitationally coupled oscillators

Removing a common displacement from a gravitational channel test need not preserve its ability to exclude classical communication between quantum laboratories. We give a quantitative construction for four oscillators in a passive Newtonian rotating-wave model, using independent unknown coherent inputs, local heterodyne measurements, and classical subtraction of the common mode. An analytic bound on a bounded score holds for every normal local-operations-and-classical-communication (LOCC) channel, without an output-energy or Gaussian restriction. The ideal quantum mean is 0.125, whereas the LOCC mean is below 0.124068. With residual gain errors of 0.1%, phase errors of 0.01 radians, and an independently bounded common displacement, the respective bounds remain above 0.124581 and below 0.124438. We exhibit a common-noise regime in which the original vacuum-outcome criterion loses its guarantee while the relative score retains one. A conservative fixed-sample test controls false exclusion with memory in the null. The result assumes quantum local matter and trusted measurement structure; it neither excludes all classical phase-space models nor establishes experimental feasibility. This record contains the preprint, editable LaTeX source, and an offline reproducibility archive with exact-arithmetic certificates and 26 computational tests. One project-history test requiring undistributed literature copies is explicitly excluded from the public runner. Substantial use of generative AI is disclosed in the manuscript. This preprint has not undergone journal peer review. Licensing: the manuscript, LaTeX source, documentation and numerical records are licensed under CC BY 4.0; Python software and tests are licensed under MIT. These licences apply to separate file categories, as specified in the archive.