Real-Time Standard Siren H₀ Inference: Numerical Stability and Low-Latency Web Implementation

We present the development and validation of a real-time web application for performing galaxy catalog based H₀ inference from gravitational wave event skymaps. The system continuously monitors public LVK alerts, performs hierarchical Bayesian inference using the ICAROGW package, and publishes H₀ posterior results to a live Streamlit interface. To ensure robustness in low-latency operation, we conducted a detailed study of the numerical stability of the H₀ likelihood evaluation. We find that random fluctuations and artificial zeros in event posteriors can be mitigated by appropriate parameter choices and sampling procedures. We also characterize how sky-localization area and event distance influence the reliability of combined posteriors, providing guidance for real-time standard siren analyses. Although our preliminary application to O4 events yields no strong constraints on H₀, these stability studies demonstrate that low-latency cosmological inference is feasible on modest computational resources and scalable to future observing runs.