Fractal-Informational Seismic Regime Detection (QO3/FIO) v2.2: Declustering, KL-SID, Block Bootstrap, Calibration Composition Operator Ψ: Definition and Axiomatic Foundation

This record presents QO3/FIO Framework v2.2, an open-source scientific software framework for seismic regime detection based on fractal–informational observables and operator-spectral invariants. QO3/FIO is designed to identify transitions between distinct seismic regimes (background vs elevated activity) using robust statistical and informational features. The methodology emphasizes regime detection rather than deterministic earthquake prediction and includes anti-leakage safeguards, rigorous validation, and reproducible evaluation. Key methodological components in this release include: • Gardner–Knopoff declustering with causality constraints to avoid aftershock leakage; • Bias-corrected Gutenberg–Richter b-value estimation (Tinti–Mulargia); • Robust coefficient of variation (MAD-based) estimators; • Seismic Information Deficit using discrete Kullback–Leibler divergence (KL-SID); • Moving block bootstrap confidence intervals for temporally correlated data; • Rolling/expanding window cross-validation for time-series evaluation; • Probability calibration metrics including Brier Skill Score (BSS) and Expected Calibration Error (ECE). All formal statements (e.g., operator structures, scaling behavior) are categorized as Theorem, Proposition, or Conjecture to maintain mathematical rigor. The archive includes source code, default configurations, and manuscript artifacts to ensure complete reproducibility. Keywords: seismology; earthquake statistics; regime detection; fractal analysis; KL divergence; block bootstrap; calibration; b-value; declustering; time series. This software is released under the Creative Commons Attribution 4.0 International (CC BY 4.0) license; permitted reuse includes modification and redistribution provided proper citation is given to the author.