Safe Under Budget? Verification Budgets and Abstention Failures in Web Agents

Web agents are often deployed under operational limits on steps, tool calls, context, and verification opportunities. Existing safety evaluations show that agents can violate policies or pursue underspecified goals, while budget-aware inference work mainly optimizes cost-performance trade-offs. We study the missing interaction: whether insufficient verification budget changes agent safety behavior. We introduce a controlled diagnostic protocol over deterministic local web environments with policy-constrained, ambiguous, and verification-required tasks. Across a local 7B-model study, insufficient verification budgets produce unsafe side-effect completions: agents sometimes reach task end states while missing required evidence. The effect is clearest under partial verification budgets: agents can take side-effect actions, but cannot inspect all safety-critical evidence. We evaluate Budgeted Safety Gate as a diagnostic mitigation that preserves evidence requirements before side effects. The gate reduces unsafe completion in our suite, at the cost of conservative abstention under tight budgets.