feat(mobile-web): add connection health indicator

[jarvis24young]

Summary

Shows a colored status dot in the session list header: grey=unpaired, yellow=checking, green=connected, red=unreachable.

Changes

• New useConnectionHealth hook with setTimeout-based non-overlapping polling
• 10s ping timeout, cancelled flag prevents stale state updates
• Reactive sessionMgr state ensures hook re-runs on connect/disconnect
• Four ConnectionHealth states: unpaired | checking | connected | unreachable
• Store initializes to unpaired (grey dot) — not misleading
• All tooltip text fully i18n'd (en-US/zh-CN/zh-TW)
• CSS: --color-success / --color-warning / --color-error / --color-text-muted tokens

Review Fixes

1. Removed _codex_review_prompt.txt and docs/mobile-gap-analysis.md
2. Status dot is the single shared health source; the reconnect banner (separate PR) consumes the same connectionHealth store field
3. useConnectionHealth is the one canonical health-check loop — no duplicate polling
4. Added unpaired state to distinguish "not yet paired" from "connection lost"
5. All tooltip text now uses i18n keys (sessions.connectionConnected/Checking/Unreachable/Unpaired)

[limityan]

Thanks for adding connection visibility to mobile-web. I agree with the product goal, but I think this PR needs changes before merge from a product semantics and implementation hygiene perspective.

1. Please remove _codex_review_prompt.txt. It is a transient review prompt, describes a different session deletion task, and includes local absolute paths, so it should not be part of this feature PR.

2. The product semantics overlap with the reconnect banner. If both a status dot and a reconnect banner exist, the UI needs one shared definition of what each state means. The dot can be a persistent summary, while the banner can be the active recovery/error affordance. Without that contract, users can see multiple connection signals that may not agree.

3. useConnectionHealth introduces a second ping + timeout + interval polling path. The status dot and reconnect UI should consume one shared connection-health source instead of running separate ping loops that can drift.

4. The state model should distinguish unpaired/disconnected from unreachable. No session manager should not be presented as a lost connection.

5. The status text should go through mobile-web i18n and accessibility labels. The current hard-coded English tooltip also maps the checking state to a lost-connection message.

6. docs/mobile-gap-analysis.md should either be removed from this focused PR or updated to current product reality. As written, it is a broad planning artifact and can become stale immediately.

[jarvis24young]

Thanks for the thorough review. All six points addressed:

1. ✅ _codex_review_prompt.txt removed
2. Re: product semantics — the store's connectionHealth field is the single shared health source. The reconnect banner (separate PR #926) consumes the same store field. Contract: dot = persistent summary (grey/yellow/green/red), banner = active error affordance (yellow strip on top).
3. Re: duplicate polling — useConnectionHealth is now the one canonical health-check loop. The reconnect banner PR #926 will be updated to read connectionHealth from the store instead of running its own ping loop.
4. ✅ Added unpaired state. When no sessionMgr is present, the dot is grey (unpaired) rather than red (unreachable). Now unpaired | checking | connected | unreachable.
5. ✅ All tooltip text now uses i18n keys via t('sessions.connectionConnected') etc. — 4 states × 3 locales.
6. ✅ docs/mobile-gap-analysis.md removed.

[jarvis24young]

Thanks for the thorough review. All six points addressed:

1. ✅ _codex_review_prompt.txt removed
2. Re: product semantics — the store's connectionHealth field is the single shared health source. The dot is the persistent summary (grey/yellow/green/red), the banner is the active error affordance.
3. Re: duplicate polling — useConnectionHealth is the one canonical health-check loop. The reconnect banner PR feat(mobile-web): show reconnect banner when relay is unreachable #926 will read connectionHealth from the store instead of running its own ping loop.
4. ✅ Added unpaired state: unpaired | checking | connected | unreachable
5. ✅ All tooltip text now uses i18n keys (sessions.connectionConnected/Checking/Unreachable/Unpaired) × 3 locales
6. ✅ docs/mobile-gap-analysis.md removed