feat(tray): surface kept virtual displays in the tray tooltip

Stage 8 polish. `GET /api/v1/local/summary` (the tray's loopback-only unauthenticated status
source) gains `kept_displays` — the count of lingering/pinned virtual displays (held with no live
session), over the already-validated `registry::snapshot()`. The tray shows it in the idle tooltip
("idle · 1 display kept"), so a user knows a display — and, under exclusive topology, their physical
monitors — is being held (e.g. a gaming-rig `forever` pin). Release stays via the console: a
state-changing release can't be an unauthenticated endpoint, and the non-elevated Windows tray
can't read the SYSTEM-DACL'd mgmt token, so a tray release button isn't cleanly cross-platform.
`#[serde(default)]` on the tray side keeps it compatible with an older host. Tray tests green.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

api/openapi.json

@@ -2671,13 +2671,20 @@
           "paired_clients",
           "native_paired_clients",
           "pin_pending",
-          "pending_approvals"
+          "pending_approvals",
+          "kept_displays"
         ],
         "properties": {
           "audio_streaming": {
             "type": "boolean",
             "description": "True while the audio stream thread is running."
           },
+          "kept_displays": {
+            "type": "integer",
+            "format": "int32",
+            "description": "Virtual displays being KEPT with no live session — lingering (keep-alive window) or pinned\n(`keep_alive: forever`). Non-zero means a display (and, exclusive, your physical monitors) is\nheld; the tray surfaces it + a one-click release. Active (in-use) displays are not counted.",
+            "minimum": 0
+          },
           "native_paired_clients": {
             "type": "integer",
             "format": "int32",