Metabolic Saliency as a KL-Divergence Estimator: Information-Geometric Attribution of Systemic Stress in JSE Equity Networks

The attribution of systemic financial stress to specific market sectors requires metrics that are simultaneously faithful to the model’s internal computations, statistically consistent as the sample size grows, and connected to a physically meaningful measure of directed information flow. This paper addresses all three requirements through the lens of information geometry. We present and empirically verify the Entropy-Saliency Equivalence Theorem: the Metabolic Saliency Sms(i, t) introduced in the companion paper (Paper 1 of this series) is an asymptotically unbiased estimator of the local Kullback-Leibler divergence KL(q(i) t ∥q(i) 0 ) between the stressed and resting sector-level return distributions, where the convergence is governed by the Fisher information matrix of the Power Mapping Network (PMNet) output distribution. We also derive the finite-sample bias-variance decomposition of the Kraskov-Stögbauer-Grassberger (KSG) transfer entropy estimator used to construct the saliency weights, establishing a minimax-optimal convergence rate of O(T−2/(d+2)) for a d-dimensional density support. A novel evaluation metric, the Spatio-Temporal Information Flux (STIF), is proposed to quantify the directed flow of stress-relevant information between JSE sectors in bits per trading day, providing a sector-level causal audit trail that satisfies the interpretability requirements of the South African Financial Sector Regulation Act (FSRA, 2017) and MiFID II. Empirical validation on the JSE canonical panel (N = 87 securities, T = 2,731 trading days, January 2015 to December 2025) with Eskom load-shedding stages as exogenous stress injectors confirms that Sms tracks KL(qt||q0) with a Pearson correlation of ( hat{rho} ) = 0.81 (p < 0.001) and that the STIF metric identifies the energy sector as the primary information source during Stage 4+ events, with information flux to the financial sector peaking at 0.43 bits/day—a 3.1× increase above the resting baseline of 0.14 bits/day. These results complete the information-theoretic glass-box characterisation of the GWS-STNet architecture and bridge topological stability theory with a fully information-theoretic characterisation of financial stress attribution.