The Independence Pricing Error in AI Safety: How Correlated Failures Invalidate Defence in Depth

Modern AI safety architectures rely on defence in depth: multiple safety layers—RLHF, constitutional AI, red teaming, output filtering, monitoring—are stacked under the assumption that each catches failures the others miss. If the layers fail independently with probability p, then n layers reduce residual risk to p^n. We show this assumption is structurally false. Safety layers share training distributions, cultural priors about harm, and optimisation targets, producing a dependence parameter d > 1 such that the true residual risk is d^n · p^n—exponentially worse than the independence estimate. We define the independence pricing error R(n) = P_obs(n) / P_ind(n), the ratio of observed co-failure frequency to the frequency predicted under independence, and show that R(n) ≈ d^n for homogeneous layers. We identify seven domains where the independence assumption is load-bearing in current AI governance: multi-model ensembles, safety-layer stacking, evaluation benchmarks, foundation-model concentration, oversight committees, red-team coverage, and deployment-risk assessment. For each, we derive the form of d, estimate its magnitude from public data on transferable jailbreaks, shared hallucination patterns, and benchmark correlation, and propose a measurement protocol. We find that plausible values of d range from 1.5 to 8, implying that four "independent" safety layers provide between 5× and 4,096× less protection than their operators believe. We propose a dependence-adjusted safety framework: a real-time R oracle for AI systems, mandatory dependence disclosure in safety cases, and minimum-independence requirements for regulatory compliance. The framework converts the qualitative intuition that "AI failures are correlated" into a quantitative metric that governance bodies can measure, audit, and enforce.