An Analytical Distance Formula for Ultra-High-Energy Cosmic Ray Sources: Connecting Black Hole Parameters to Observables

This collection contains three preprints on ultra-high-energy cosmic ray (UHECR) source identification using black hole parameters: Paper I: "An Analytical Distance Formula for Ultra-High-Energy Cosmic Ray Sources: Connecting Black Hole Parameters to Observables" We derive a closed-form analytical formula connecting observed UHECR energy to source black hole parameters (mass M, spin a_*, magnetic field B₀) through GZK energy loss. Validated against M87, Centaurus A, NGC 1068, Markarian 501, and the TA Hotspot. Predicts heavy-nucleus composition for Cen A UHECRs and introduces a critical BH mass separating neutrino-only from UHECR-capable sources. Paper II: "Dark Black Holes: Ultra-High-Energy Cosmic Ray Backtracking as a New Observational Channel" A novel method for discovering electromagnetically invisible supermassive black holes using UHECR backtracking. Four candidate locations identified at 21-82 million light-years. Highest-confidence candidate in Ursa Major (TA Hotspot, 5. 1 sigma). Paper III: "The Amaterasu Particle and Precessing Black Holes at the Edge of the Local Void" A precessing, quiescent SMBH model for the 244 EeV Amaterasu particle. Source constrained to d = 10-17 Mpc with M >= 1. 1e9 Mₛun. Predicted source constellation: Hercules (protons), Ophiuchus (CNO), or Aquila (iron). Keywords ultra-high-energy cosmic rays, black holes, cosmic ray propagation, multi-messenger astrophysics, GZK cutoff, Amaterasu particle, dark black holes