Effective Information Gain in Multi-Organ Systems: A Lateral Inhibition Model of Vagus-Mediated Gut-Immune Signaling with Clinical Validation in Gastrointestinal Adenocarcinoma

Inter-organ communication is essential for systemic homeostasis, yet no existing framework quantifies the information content of organ-to-organ signaling channels. We introduce the effective information gain G_eff, a function grounded in the Kullback-Leibler divergence of the von Mises phase distribution, that measures how much structured signal a neuro-immune channel carries relative to noise. A single equation, G_eff(kappa,alpha) = G_info(kappa) * exp(-alpha S_suppress), with G_info(kappa) = kappa I_1(kappa)/I_0(kappa) - ln I_0(kappa), naturally produces three dynamical regimes without fitted parameters: subthreshold (G_eff < 1), near-threshold (1 <= G_eff <= 1.5), and saturated (G_eff > 1.5). The subthreshold boundary G_eff = 1 corresponds to kappa_c = 3.8163, the unique root of G_info(kappa) = 1 -- one nat of phase information -- derived from first principles. When embedded in a five-organ ODE network with lateral inhibition, the equation produces cascade failure: saturating one organ drives its neighbors into the subthreshold regime. In two independent TCGA transcriptomic platforms for gastrointestinal adenocarcinoma (n = 767 on cBioPortal, n = 670 on UCSC Xena), channel activity predicts overall survival (cBioPortal: adjusted HR = 0.480, p = 0.001; Xena: continuous HR = 1.129, p = 0.029), with direction concordant across platforms. Permutation-based negative control (B = 1000) confirms signal specificity (p < 0.0001). Immune profiling shows that the channel-collapse group harbors a hotter tumor microenvironment (COAD: 10/11 cell types, FDR < 0.05).