Generalization of Discretized Bayesian Induction: From Binary BID2 to Multinomial BIDn and Contextual BIDnC Architectures (Part 3/7)

Evaluating multinomial outcomes and contextual spaces presents severe computational challenges: direct multidimensional joint probability distribution (JPD) construction encounters an exponential combinatorial wall of O(Popu^(n-1)). This paper presents the formal generalization of the Discretized Bayesian Induction (BID) framework from binary variables (BID2) to multinomial outcomes (BIDn, BID3) and high-dimensional contextual spaces (BIDnC). BID overcomes the combinatorial wall through four modular paradigms: (1) BIDn for linear-time O(n * Popu) 1-vs-Rest marginal candidate tracking (< 15 ms for n = 10 candidates); (2) BID3 for exact joint trinomial induction over the 2D barycentric simplex (501,501 discrete hypotheses at Popu = 1,000 resolution in 90–140 ms), extracting exact lead gap distributions dp(SA - SB) and crossover probabilities P(SB > SA); (3) BIDnC for contextual inference across multi-variable demographic and clinical profile hypercubes using Tableau des Cumuls d'Observations Brutes (TCOB) matrices across dimensions 2^m x n; and (4) EvalCout for projecting multinomial gain/utility vectors onto a 1D equivalent axis in linear O(n * Popu) time. Inheriting the SICP structure from industrial knowledge compilation (Renault SICG/C2G), BIDnC executes geometric Focalisation (column masking) and Fusion (row reduction) operations before or after induction, projecting complex hypercubes down to 1 x 2 atomic TCOB matrices in 1.59 milliseconds. Post-induction consultation via the C2GP interactive configurator enables a closed-loop knowledge cycle for bedside clinical decision support and political polling analysis.