[P6.7.2] Deliver background notifications when batch monitor is unmounted (or document)

[frankbria]

Problem

Async notifications only fire while BatchExecutionMonitor is mounted (i.e. while the user is on /execution?batch=…). BatchExecutionMonitor is the sole dispatcher of batch.completed/blocker.created, and gate.run.failed has the same constraint on /proof. Navigate away and a batch that finishes in the background produces no notification — in-app or browser. Since the entire value of browser notifications is being alerted while not watching, this materially undercuts the feature.

This is already acknowledged as a known limitation in CLAUDE.md.

Evidence

• web-ui/src/components/execution/BatchExecutionMonitor.tsx:108-155 — sole dispatcher of batch.completed/blocker.created, only active on the execution page

Fix (pick one)

• Preferred: a cross-page background poller (mounted in NotificationProvider / root layout) that watches active batches and emits notifications regardless of current route.
• Minimum: document the limitation clearly in release notes / UI copy so users don't expect background alerts.

Acceptance criteria

• Either background notifications fire when the monitor is unmounted, or the limitation is explicitly documented for beta.

Source: release-readiness audit 2026-06-13 (frontend agent).

[coderabbitai]

🔗 Related PRs

#593 - feat(web-ui): async notifications — browser + in-app center (#559) [merged]

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