Notifications: report standalone widget errors to Sentry

[StevenDufresne]

Proposed Changes

• Add a small Sentry wrapper (apps/notifications/src/lib/sentry.ts) — reuses the shared Calypso DSN and tags events as the wp-admin notifications surface. captureException is a no-op until initSentry runs, so the shared REST client stays silent on surfaces that don't opt in (the old panel, the in-Calypso popover).
• Wire init into the standalone-app boot; catch the previously-unhandled createClient() rejection and the swallowed tracks/locale catches.
• Add a React error boundary with a fallback (message + Reload) so a render crash is reported and recoverable instead of unmounting to a dead spinner.
• Report the first top-window getNotes() failure of a streak — the path that retries forever without ever clearing the loading state.

Why are these changes being made?

The wp-admin notifications widget shipped with no error reporting, so the recent redesign's failure modes (a hung boot, a render crash, a stuck fetch) reached users only as an endless spinner with zero signal on our side. This makes the next occurrence observable and gives users a way to recover. It is instrumentation, not the root-cause fix — it lets us confirm which trigger is actually firing in production.

Testing Instructions

• yarn test-apps apps/notifications — new suites for the Sentry wrapper and the error boundary pass (44 tests total).
• apps/notifications typechecks and lints clean.
• Manual: load the widget with the network offline or third-party cookies blocked → expect a Sentry event plus the fallback message instead of an endless spinner.

Notes / follow-ups

• Source-map upload (sentry-webpack-plugin) for symbolicated stack traces is a sensible follow-up.

Pre-merge Checklist

• Has the general commit checklist been followed? (PCYsg-hS-p2)
• Have you written new tests for your changes?
• Have you tested the feature in Simple (P9HQHe-k8-p2), Atomic (P9HQHe-jW-p2), and self-hosted Jetpack sites (PCYsg-g6b-p2)?
• Have you checked for TypeScript, React or other console errors?
• For UI changes, have you tested the affected components in dark mode?
• Have you tested accessibility for your changes?
• Have you used memoizing on expensive computations?
• Have we added the "[Status] String Freeze" label as soon as any new strings were ready for translation (p4TIVU-5Jq-p2)?
• For changes affecting Jetpack: Have we added the "[Status] Needs Privacy Updates" label if this pull request changes what data or activity we track or use (p4TIVU-aUh-p2)?

🤖 Generated with Claude Code

[github-actions]

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[matticbot]

This PR modifies the release build for the following Calypso Apps:

For info about this notification, see here: PCYsg-OT6-p2

• help-center
• notifications
• wpcom-block-editor

To test WordPress.com changes, run install-plugin.sh $pluginSlug add/notifications-sentry-error-logging on your sandbox.