The Independent-Constraint Framework (ICF): Systematic analysis of multi-Observable Systems - application to Black Holes

Title: The Independent-Constraint Framework (ICF): Systematic Analysis of Multi-Observable Systems - Application to Black Holes Description: This work presents the Nadia Independent-Constraint Framework (ICF), a systematic methodology for analyzing physical systems where multiple independent observations constrain parameters. The framework provides a structured approach to: (1) identify conceptually independent constraints from distinct physical processes, (2) derive dependent quantities algebraically, and (3) perform consistency tests to validate theoretical models. This book demonstrates ICF application to black hole parameter determination using real observational data. For stellar-mass black holes, gravitational-wave observations from LIGO-Virgo-KAGRA provide independent constraints through inspiral dynamics (chirp mass, effective spin) and ringdown oscillations (remnant mass, remnant spin). These enable algebraic derivation of component masses, radiated energy, and systematic consistency tests of general relativity predictions. Example analysis of GW150914 shows excellent agreement (Δ < 0.2σ) between inspiral-predicted and ringdown-measured remnant properties. For supermassive black holes, Event Horizon Telescope observations provide independent constraints through shadow diameter (mass determination) and brightness asymmetry/polarization (spin determination). These enable derivation of horizon radius, ISCO location, and Bekenstein-Hawking entropy. Analysis of M87* and Sgr A* demonstrates consistency (Δ < 0.2σ) between EHT imaging and stellar dynamics mass measurements. ICF builds upon previous work organizing ΛCDM cosmological parameters (DOI: 10.5281/zenodo.18139885), extending the methodology to astrophysical systems. The framework emphasizes pedagogical clarity by explicitly separating measured versus derived quantities, making the logical structure of parameter inference transparent. ICF does not replace standard Bayesian inference methods but provides an organizational layer that clarifies which observations constrain which parameters. The methodology is applicable to any multi-observable system with clear physical distinctions between measurement processes and algebraic relationships between parameters. Potential applications include neutron star equation of state constraints, exoplanet characterization, supernova parameter determination, and gravitational lensing analysis. This work contributes to scientific pedagogy and observational strategy by making implicit relationships in comprehensive statistical analyses explicit, facilitating understanding of how independent measurements combine to constrain complex systems. Using ICF, we can define a black hole purely in terms of observable, independently constrained quantities: "A black hole is a compact astrophysical object whose mass M and spin aaa can be independently constrained by distinct physical processes within a specified gravitational theory, such as spacetime geometry (shadow size, photon orbits) and matter dynamics (accretion disk asymmetry, gravitational-wave inspiral and ringdown)". Keywords: black holes, gravitational waves, event horizon telescope, parameter estimation, cosmological parameters, multi-messenger astronomy, Bayesian inference, general relativity, LIGO, Virgo, KAGRA, M87, Sagittarius A*, methodology, pedagogy