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
95% to under
5%. Whether this trade-off has a theoretical floor was open. We prove that it does. For any VLA policy, capability
I(\Astar;\Api) and robustness
I(\Api;\Atildepi)−I(\Api;δ) sum to at most
H(\Astar)+I(X;\Xtilde), the task entropy plus adversarial channel capacity. The proof reduces to two applications of the Data Processing Inequality. The pixel-level bound is loose by
∼103 nats and serves as a ceiling guarantee; an encoder-specific corollary tightens it by over an order of magnitude, into a regime where realized capability already consumes
5--
9% of the budget. We validate Theorem~\ref{thm:main} with zero violations across
308 cells:
252 closed-form Gaussian-VLA,
48 OpenVLA-7B
+LIBERO
+PGD (
4 suites
× 4 \eps × 3 seeds),
4 Square-Attack, and
4 multi-step (
T=10). A complementary measurability inequality
\Robdisc≤⋒disc further holds across
144 cross-architecture cells spanning OpenVLA, OpenVLA-OFT (continuous-
L1), and SmolVLA (flow-matching). The same construction yields three label-free diagnostics: a pre-flight encoder ceiling, a defense-forensics probe that localizes input-side vs.\ language-model intervention, and a head-agnostic robustness ratio comparable across discrete-token,
L1-regression, and flow-matching policies. Together these provide the cross-setting axis defense and architecture comparisons currently lack.