The framework of Large Mathematical Logic Model (LMLM): Gibbs sampling-based restricted Boltzmann machines for financial engineering

Abstract

This paper gives a mathematical account of Gibbs sampling, the classical Boltzmann Machine (BM), and the efficiency breakthrough of the Restricted Boltzmann Machine (RBM), tracing the Markov Chain Monte Carlo lineage from Metropolis–Hastings to coordinate-wise sampling and deriving the Contrastive Divergence (CD k ) algorithm. We extend this lineage into financial engineering: a systemic financial energy function for latent risk contagion is constructed, and a Large Mathematical Logic Model (LMLM) is proposed to suppress hallucination in financial AI agents by embedding Gibbs sampling within an axiomatic Financial Ontology ([Formula: see text]) and compressing the logical space via adversarial negative sampling; the resulting energy-based discriminator is illustrated, rather than empirically benchmarked, through two fully worked numerical examples reported under the standard AUC/F 1 evaluation protocol. Building on this foundation, we position LMLM as the Verifier in a two-agent Proposer/Verifier Consensus Game for AI-driven risk-factor extraction: a semantic-understanding engine (Proposer) generates candidate risk factors, while LMLM (Verifier) checks each candidate against the Financial Ontology’s axiomatic constraints, accepting logically self-consistent candidates and returning a traceable counterexample for rejected ones. We show that whenever LMLM’s verification function satisfies a quasi-concavity condition on its decision space, this iterative game converges within finitely many rounds to a set of risk factors satisfying all rule constraints, and that every rejected candidate is automatically retained as a structured negative sample supporting an auditable decision trail. Finally, we map LMLM’s core technical properties — its energy-based hallucination discriminator, adversarial negative-sampling paradigm, and Proposer/Verifier division of labor — onto specific clauses of China’s National Financial Regulatory Administration Guideline on the Safe Development and Application of Artificial Intelligence in Banking and Insurance (18 June 2026), illustrating how a rigorously derived energy-based verification mechanism can serve as a concrete engineering realization of emerging financial AI-governance requirements for explainability, auditability, and accountability.