feat(plex): Test-connection media mount probe

Extend admin POST /api/plex/test to resolve one real media file through the
local mount (a mirrored sample, else a single leaf fetched from Plex) and report
whether it is readable. A missing media bind-mount or a wrong plex_path_map is
now caught at Test time rather than only surfacing later as a dead player.

backend/app/routes/plex.py

@@ -37,11 +37,14 @@ _TS_CONFIG = "public.unaccent_simple"
 @router.post("/test")
 def test_connection(_: User = Depends(admin_user), db: Session = Depends(get_db)) -> dict:
     """Admin: verify the configured Plex server URL + token, returning the server name and the
-    movie/show sections (for the config library-picker)."""
+    movie/show sections (for the config library-picker). Also probes ONE real media file through
+    the local mount, so a missing bind-mount / wrong path-map is caught here rather than only
+    surfacing as a dead player later."""
     try:
         with PlexClient(db) as plex:
             info = plex.server_info()
             sections = plex.sections()
+            media_check = _media_check(db, plex, sections)
     except PlexNotConfigured as e:
         raise HTTPException(status_code=400, detail=str(e))
     except PlexError as e:
@@ -55,9 +58,44 @@ def test_connection(_: User = Depends(admin_user), db: Session = Depends(get_db)
             for s in sections
             if s.get("type") in ("movie", "show")
         ],
+        "media_check": media_check,
     }
 
 
+def _sample_plex_path(db: Session, plex: PlexClient, sections: list[dict]) -> str | None:
+    """A real Plex-side media file path to validate the mount + path-map against. Prefers the
+    already-mirrored catalog (cheap); before the first sync, pulls one movie leaf from an enabled
+    section straight from Plex."""
+    row = db.query(PlexItem.file_path).filter(PlexItem.file_path.isnot(None)).first()
+    if row and row[0]:
+        return row[0]
+    wanted = plex_sync._enabled_section_keys(db)
+    for s in sections:
+        if s.get("type") != "movie":
+            continue
+        key = str(s.get("key"))
+        if wanted is not None and key not in wanted:
+            continue
+        items, _ = plex.section_items(key, 1, 0, 1)
+        if items:
+            fp = plex_sync._media_facts(items[0]).get("file_path")
+            if fp:
+                return fp
+    return None
+
+
+def _media_check(db: Session, plex: PlexClient, sections: list[dict]) -> dict:
+    """Resolve a sample Plex file path to its local mount path and confirm it's readable, so the
+    admin sees at Test time whether the physical media is actually reachable (bind-mount +
+    path-map correct). ``checked=False`` when no sample is available (empty library)."""
+    plex_path = _sample_plex_path(db, plex, sections)
+    if not plex_path:
+        return {"checked": False}
+    local_path = plex_paths.map_path(db, plex_path)
+    ok = plex_paths.local_media_path(db, plex_path) is not None
+    return {"checked": True, "ok": ok, "plex_path": plex_path, "local_path": local_path}
+
+
 @router.post("/sync")
 def trigger_sync(_: User = Depends(admin_user), db: Session = Depends(get_db)) -> dict:
     """Admin: mirror the enabled Plex sections into the local catalog. Synchronous (can take a