Principal-agent hypothesis testing studies statistical contracts in which a principal rewards an agent according to data the agent generates. Existing work focuses on the single-shot setting, where the agent produces a single dataset. We extend the framework to sequential settings, where evidence accumulates over time, rewards may be deferred to the end or disbursed incrementally, and the agent may stop the contract at a strategically chosen stopping time. We call a contract null-deterring if no ''null agent'' (those whose data-generating distribution lies in the null hypothesis) can profit in expectation under any stopping rule, and our main conceptual result shows that such contracts are characterized by e-processes. We then turn to the agent's best response and identify the optimal stopping rule through the Snell envelope of the reward process. We develop simpler, more interpretable approximations that often require the agent to have only partial knowledge of its distribution.
Statistical procedures often control expected loss at a prescribed level while leaving part of their risk budget unused. Building on the e-value representation of risk-controlled decisions and the e-closure principle for multiple testing, we formulate recertification, a general paradigm for improving procedures by exploiting computable slack in their validity certificates. A valid procedure's losses normalized by their risk budgets form a state-indexed e-value collection; conversely, decisions whose normalized losses are bounded by this collection inherit its risk guarantees. Recertification uses distributional structure to quantify slack, enlarge the certifying e-values while preserving validity, and construct less conservative decisions. We develop two applications. For conformal risk control (CRC) with a known loss family, recertification redistributes unused risk budget among the held-out points of each hypothetical augmented sample. The resulting controller is valid under every finite exchangeable law and no more conservative than CRC on any calibration sample. For convex losses on an interval of actions, repeated recertification converges geometrically to an admissible controller. For false discovery rate control with arbitrarily dependent e-values, recertification exploits unused conditional budgets in randomized e-BH. It yields a closed-form randomized procedure whose rejection set contains that of randomized e-BH for every realization of the shared randomization and whose expected number of rejections is maximal subject to the same conditional budget constraints. Recertifying with the larger e-collection of closed e-BH gives a second closed-form procedure whose rejection set contains that of the first. Experiments on CIFAR-100 hierarchical classification and multiple testing simulations demonstrate improved budget utilization, informativeness and power over the baselines.