Tension in the solar energetic proton paradigm for historical cosmogenic nuclide events

'Miyake events' are exceptional increases in atmospheric 14C recorded in tree rings that, if confirmed, are accompanied by simultaneous 10Be and 36Cl enhancements in ice cores. During the last 15 000 years, six such events have been confirmed. They are generally thought to originate in solar flares that produce strong high-energy solar proton events. The 774 CE Miyake event is currently estimated to have been produced by a flare with a soft X-ray (SXR) flare classification of X430. A recently constructed size distribution of SXR events from 1975 to 2022 permits an estimate of the 'return time' for any given SXR event. We extrapolate the observed distribution to obtain a flare return time of ⋟106 years for all six of the Miyake events. We examine various possibilities (e.g. overestimates of greater than 200 MeV proton fluences, episodes of multiple smaller flares) to reduce the tension between the estimated long return (waiting) times for these extreme flares and the 15 000-year observation interval. We conclude that the most likely explanation for the inferred outsized proton events is a yet unknown Dragon-King type mechanism or scenario that can increase high-energy solar proton fluences in certain events by one to two orders of magnitude. This article is part of the Theo Murphy meeting issue 'Radiocarbon and cosmic radiation events'.