Convolutional Neural Networks (CNNs) are deployed in more and more classification systems, but adversarial samples can be maliciously crafted to trick them, and are becoming a real threat. There have been various proposals to improve CNNs' adversarial robustness but these all suffer performance penalties or other limitations. In this paper, we provide a new approach in the form of a certifiable adversarial detection scheme, the Certifiable Taboo Trap (CTT). The system can provide certifiable guarantees of detection of adversarial inputs for certain $l_{\infty}$ sizes on a reasonable assumption, namely that the training data have the same distribution as the test data. We develop and evaluate several versions of CTT with a range of defense capabilities, training overheads and certifiability on adversarial samples. Against adversaries with various $l_p$ norms, CTT outperforms existing defense methods that focus purely on improving network robustness. We show that CTT has small false positive rates on clean test data, minimal compute overheads when deployed, and can support complex security policies.