Self-report is the cheapest oversight channel a deployer has, and on capable computer-use agents (CUAs) it fails precisely where oversight matters. On 361 OSWorld tasks our pipeline, a read-only feasibility gate, a planner, and a GUI executor, reaches a mean task score of 82.9, above the 72.4 human reference, yet 64 of its 71 failures (90%) end with a success claim, 61 acknowledging no blocker, and the explicit failure affordance is never used in roughly 9,100 calls. We introduce CURA (Certified Runtime Alarms for Computer-Use Agents), an external monitor that reads only harness-visible telemetry, with no model internals, extra LLM calls, or prompt changes, and turns the running trajectory into a sequential test with certified false-alarm control. At alpha = 0.10 its CUSUM alarm detects 42.3% of failures a median of 31 steps before termination at a realized false-alarm rate of 0.066, and risk is partly resolvable before the first action (gate probe, 0.69 AUROC). Retrospectively the composite reaches 0.828 AUROC (fold-internal floor 0.802), but its margin over a total-token baseline is not significant (Delta = +0.026, p = 0.101); the separation is online, where CURA recalls more at matched certified budgets: 0.41 versus 0.34 at alpha = 0.10, 0.56 versus 0.38 at alpha = 0.20. Alarm-gated mid-execution oversight recovers 23 of 70 failures while spending a frontier overseer on 38, giving a deployable cascade at mean score 86.8 and 84.5% full-solve (305 of 361). The certificate bounds false alarms only. We also report where behavioral monitoring is uninformative.

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