Capability and Robustness Cannot Both Be Free: An Information-Theoretic Bound for Vision-Language-Action Models

Vision-Language-Action (VLA) models reach high success rates on clean inputs but collapse under small adversarial perturbations: a $16/255$ PGD attack drops OpenVLA-7B's LIBERO success from above $95\%$ to under $5\%$. Empirical defenses recover part of the loss at a cost in clean accuracy, but the literature does not say whether the trade-off has a theoretical floor. We prove that it does, giving the first information-theoretic bound for action-generating policies. For any VLA policy, capability (mutual information between policy action and oracle action) and robustness (mutual information preserved under attack, minus the action-channel leakage that policies can passively transmit through their output) sum to at most a policy-independent budget: task entropy plus adversarial channel capacity. The leakage term has no analogue in classifier formulations, and is what keeps the inequality tight on action spaces, which can carry attack signal directly. The proof reduces to two applications of the Data Processing Inequality, and an encoder-specific corollary tightens the pixel-level bound by over an order of magnitude on a per-experiment basis. We validate the bound with zero violations across $320$ cells spanning closed-form Gaussian-VLAs, OpenVLA-7B under PGD and Square attacks across all four LIBERO suites, multi-step horizons up to $T{=}10$, and two structurally different action heads (continuous-$L_1$ regression and flow-matching). The bound also yields three diagnostics that practitioners can compute from $\le 200$ samples without ground-truth labels: a pre-flight encoder ceiling for deployment audits, a defense-forensics probe that identifies which channel stage a defense intervenes in, and a head-agnostic robustness ratio that compares discrete-token, $L_1$-regression, and flow-matching policies on equal footing where success-rate-under-attack cannot.