Hidden multi-state structure versus the piecewise-Poisson flare rate on 50 years of GOES/XRS solar flares

Abstract

The statistics of solar flare occurrence have a mature incumbent description: a time-dependent (nonstationary) Poisson process whose rate varies with the solar cycle and with active-region evolution, usually represented by a piecewise-constant rate. No estimable hidden-state alternative — an exact hierarchical regime filter with a likelihood — has been entered against it. We enter one, on 8,393 M1.0+ and 64,996 C1.0+ flares (1975–2025, era-stitched from the NCEI legacy reports and the GOES-R L2 flare summary), and report what it settles and what it does not. (i) Several latent switching layers beat one, decisively and at fixed parameter count: ΔBIC +3,175 (M1.0) and +10,195 (C1.0), and +460 / +1,436 nats out of sample. (ii) Against the piecewise-constant incumbent the cascade wins on parsimony, not on fit: ΔBIC is +3,102 and +6,172, but the Bayesian-blocks partition carries 531 and 1,423 parameters against the cascade's four and fits the record better in raw likelihood, by +1,035 and +3,876 nats — the entire BIC margin is the parameter penalty net of that deficit. (iii) The specific geometric ladder buys no descriptive advantage over a free transition matrix at these sample sizes: the cascade leads the free-transition Poisson HMM at M1.0 by only +119 BIC, of which +117.8 is the parameter penalty and +1.7 is fit, and loses by −1,475 BIC at C1.0; the sealed calibration shows the axis fires +147 BIC on a synthetic no-hierarchy world, so it cannot be read as a hierarchy test. (iv) A causal, filter-only 24-hour M-class now-cast carries real skill from the event record alone (BSS +0.589 against the training-period climatology, +0.338 against the verification period's own).