Split out from #1409, which sat under #1402 (Core/App split Epic). The Epic's stated goal is "no user-visible regression on the dashboard, history view, archive, or settings"; a tray icon — even a minimal one — is a user-visible UX surface, not a refactor step. Carving the tray out into its own feature issue keeps #1402 a pure refactor Epic and lets the tray be designed as part of the background-mode UX (#1275) it actually serves.

Goal

A minimal adapters::tray::TrayAdapter (src-tauri/src/adapters/tray.rs) that gives users a way back to the main window in background mode. No live metric rendering — that lands in #1401.

Scope

• Tray icon registered on app startup.
• Tray menu with Open (restore main window) and Quit (calls lifecycle::request_quit from Phase 5, which drives core::monitoring::MonitoringState to Stopped).
• Left-click on the tray icon also restores the window.
• Windows / macOS / Linux. Document any AppIndicator caveats discovered during implementation.

Out of Scope

• Live metric values in the tray (feat: live hardware metrics in system tray / menu bar (Stats-like widget) #1401).
• Pause / Resume monitoring entry — owned by feat: add background monitoring mode with system tray support #1275.
• Custom icons / theming.
• Visibility policy: whether the tray is always-on, gated by background mode, or driven by a persisted user setting. That decision is a UX call owned by feat: add background monitoring mode with system tray support #1275.

Definition of Done

• Closing the main window (in background mode) leaves a tray icon visible.
• Clicking Open restores the window with state intact.
• Quit exits the app cleanly through the lifecycle path from Phase 5.
• Graceful fallback when the user's environment does not support a tray (some Linux desktops): the app still quits cleanly via Quit from another path.

Relationships

• Part of feat: add background monitoring mode with system tray support #1275 — background-mode + tray umbrella. This is child [A]; siblings are feat: close-to-tray setting + first-run dialog #1423 ([B], close-to-tray setting + first-run dialog) and feat: pause / resume monitoring (tray menu + runtime wiring) #1424 ([C], pause / resume monitoring).
• Extended by feat: live hardware metrics in system tray / menu bar (Stats-like widget) #1401 — live tray/menu-bar widget reuses this adapter; depends only on this issue, not on the rest of feat: add background monitoring mode with system tray support #1275.
• Depends on Phase 5 of Epic: split backend into Tauri-independent Core and thin App adapters #1402 (already merged in refactor(core): Phase 5 — MonitoringState machine and lifecycle decoupling (#1408) #1420) — uses lifecycle::request_quit.

Notes

Replaces #1409, which is closed as rescoped to this issue. A skeleton implementation was written on refactor/1409-phase-6-tray-adapter (commit 02ae710, closed PR #1421) and is preserved for reuse — see the follow-up comment for what carries over and what should be reworked when this issue is picked up.