Only the Outermost Bound System Carries the Phantom: Hierarchical Ownership in a Lambda-Tied MOND Law, Its Scorecard, and an Exactly Newtonian Wide-Binary Prediction for Gaia DR4

Abstract

The programme's central relation ties the MOND acceleration scale to the vacuum, a0 = kappa c sqrt(G rho_Lambda), with kappa = 1/2 fitted, not derived. This note reports a principle that decides which systems the law acts on, the scorecard it earns, and the prediction it makes for Gaia DR4 before the data exist. Hierarchical ownership. The law acts on a system's own baryons only if that system is the outermost bound system. An accreted satellite keeps the dark component it owned at infall; a system that formed embedded in a host (a tidal dwarf, a globular cluster, collision debris, a wide binary, the Solar System) is Newtonian. The rule adds no constant and removes one, the Solar-System screening length. Scorecard. On the record's population gates it passes 9 of 14, against 6 for the same law with the host's external field. With the infall baryons its own logic calls for, the M31 satellites pass (1.2-1.6 sigma), and the combined target law passes 42 of 48 harness gates. It fails two: the Milky Way's ultra-faint satellites (3.5-3.8 sigma once refereed, first quoted at 7-8 sigma on statistical errors alone; the external-field reading fails them at 13 sigma) and the external-field signal of Chae et al. (4.1-4.3 sigma in their fits; 1.7-2.2 sigma refit under the framework's own law at the canonical scale). Shape. A cold component in local virial equilibrium around a point mass reproduces the kernel nu = sqrt(1 + 1/y) exactly, with a0 as its charge per unit baryonic mass; SPARC disfavours that kernel's transition shape at about 2 sigma. Dark sector. The dark sector's edge meets a 5.4 sigma tension between KiDS-1000 lensing and the cosmic cold budget. The same lensing machinery misfits standard LCDM halos by at least Delta chi^2 = 50, so the tension is not decidable there. The Local Group's zero-velocity radius conflicts with standard halos at least as strongly as with the framework. Prediction. Wide binaries are exactly Newtonian: gamma_v = 1.000 on the pre-registered Gaia DR4 estimator, falsified at gamma_hat >= 1.084. The same law with the external field predicts 1.16-1.23 and is falsified by a Newtonian result. Gaia DR4 is scheduled for 2 December 2026. Every quoted number is re-derived from committed outputs by PAPER35_audit.py (98 of 98). AI-assisted research programme; not peer reviewed; nothing is claimed closed.