The BID2 Engine: Discretized Binomial Bayesian Induction with Maxent Priors (Part 2/7)

Inference on binomial proportions from limited sample sizes (Small Data) represents a cornerstone challenge in biostatistics, A/B testing, and industrial decision-making. Classical frequentist hypothesis testing relies on asymptotic approximations and p-values that fail on small cohorts, while continuous Bayesian methods depend on subjective conjugate priors or computationally heavy Markov Chain Monte Carlo (MCMC) sampling. This paper presents the mathematical foundations, metrological sensitivity, and implementation architecture of BID2, the foundational binary module of the BID (Discretized Bayesian Induction) framework. Grounded in E.T. Jaynes's Principle of Maximum Entropy (Maxent) via the PEMPI algorithm, BID2 replaces continuous integration with exact arithmetic summations over a discrete, representative virtual urn of configurable resolution (Popu = n). We demonstrate that Laplace's Rule of Succession (s + 1) / (N + 2) is not the theoretical norm from which discretization deviates, but merely the asymptotic limit as Popu approaches infinity of exact summations over physical populations. Through a comprehensive sensitivity study (TC = 200, Successes = 76, raw rate 38.0%), we evaluate performance across Popu = 10, 100, 1,000, and 10,000. We prove that the Bayesian Expected Value (EB) converges instantly to its theoretical Beta-conjugate value (38.119%), while the Closest-Tail quantile calibration method minimizes residual tail errors, achieving 95.089% actual physical coverage at Popu = 1,000 (+0.089% deviation) with an execution time of 1.59 milliseconds. Linear time complexity O(Popu) and a 2 MB precalculated Maxent lookup table establish BID2 as a deterministic, audit-ready, glass-box decision-support engine.