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A successful ground array Radio Frequency (RF)-only self-trigger is demonstrated with 256 dual-polarization antennas of the Owens Valley Radio Observatory Long Wavelength Array (OVRO-LWA), yielding 10 high-energy cosmic ray candidates. This RF-only capability is predicated on novel techniques for Radio Frequency Interference (RFI) identification and mitigation with an analysis efficiency of approximately 45% for shower-driven events with a Signal-to-noise ratio ≳ 5 against the galactic background noise power of individual antennas. Radio self-triggering enables more efficient detection of cosmic rays over a wider range of zenith angles than possible via triggers from in-situ particle detectors and the method presented here can be easily adapted to neutrino experiments relying on RF-only detection. This paper discusses the system design, RFI characterization and mitigation techniques, and initial results from 10 cosmic ray candidate events identified within a 40-hour observing window. A design for a future optimized commensal cosmic-ray detector for the OVRO-LWA is presented, as well as recommendations for developing a similar capability for other experiments — these designs either reduce data-rate or increase sensitivity by an order of magnitude for many configurations of radio instruments.