Brocco's Threshold: A Universal Phase Transition in Accreting Systems at s = 1/(4π)
We present evidence for a universal phase transition governing the activation of accretion systems across 11 orders of magnitude in mass. Analyzing 22 AGN from the BASS DR2 survey (Koss et al. 2022, Table 9) — an independent dataset not used in any fitting — we identify a critical threshold at s = 0.0787 ± 0.0005 in the entropy coordinate s = λ/(λ + κ), where λ = L/L_Edd is the Eddington ratio and κ = 0.176 ± 0.002. This threshold separates quiescent from active states with 90.9% accuracy, statistically indistinguishable from the geometric value 1/(4π) = 0.079577 (p > 0.99). Only 2 of 22 sources (9.1%) lie below threshold — both low-Eddington AGN near the detection limit.
The same threshold — with the same κ and zero refitting — correctly classifies white dwarfs (10/10), neutron stars (29/29), X-ray binaries (15/15), JWST cosmic dawn AGN (10/10), GRB 170817A, and the super-Eddington pulsar M82 X-2. The coupling constant satisfies κ = (4π − 1)λ_crit, where λ_crit ≈ 0.0153 is the independently established ADAF threshold (Ho 2008). The cancellation of λ_crit in the threshold condition yields s = 1/(4π) exactly, independent of accretion scale — a direct consequence of spherical geometry.
Brocco's Constant s = 1/(4π) represents the critical information density on a spherical boundary: the point at which any accreting system transitions from quiescent disorder to active coherent emission.
Keywords: accretion, AGN, black holes, neutron stars, white dwarfs, phase transition, 1/(4π), ADAF, BASS DR2, universal threshold