Bayesian Analysis of the Nexus Paradigm Predictions for Supermassive Black Hole Observations by the Event Horizon Telescope

We present a Bayesian statistical analysis to evaluate the Nexus Paradigm (NP) of quantum gravity, using horizon-scale observations of supermassive black holes (SMBHs) Sagittarius A* (Sgr A*) and M87* from the Event Horizon Telescope (EHT). The NP predicts angular diameters for the dark depression, emission ring, and base diameter, which we compare to EHT measurements. Employing Gaussian likelihoods and priors informed by mass-to-distance ratio uncertainties, we compute the posterior distribution for the angular scale parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mi>θ</mi></mrow><mrow><mi>g</mi></mrow></msub></mrow></semantics></math></inline-formula>, achieving a combined <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≈</mo><mn>0.0062</mn></mrow></semantics></math></inline-formula> (four degrees of freedom) corresponding to a 4.37<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo> </mo><mi>σ</mi></mrow></semantics></math></inline-formula> (99.9972%) confidence level. Individual features show deviations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo><</mo><mn>0.1</mn><mo> </mo><mi>σ</mi></mrow></semantics></math></inline-formula> supporting the NP’s claim of 99th percentile agreement. Compared to General Relativity (GR), which predicts a shadow diameter inconsistent with the observed dark depression (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mrow><mi>χ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>≈</mo><mn>168</mn><mo>,</mo><mo> </mo><mo>~</mo><mn>12.97</mn><mo> </mo><mi>σ</mi></mrow></semantics></math></inline-formula>) the NP is favored with a Bayes factor of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>~</mo><msup><mrow><mn>10</mn></mrow><mrow><mn>36</mn></mrow></msup></mrow></semantics></math></inline-formula>. These results validate the NP’s predictions and highlight its potential as a quantum gravity framework, though refined uncertainties and broader model comparisons are recommended.