fix(booth): backfill the live-feed plate + make plate search work

Two booth feed fixes:

- Plate not showing until refresh. Plate recognition is async/advisory
  (snapshot.ts recognizePlate → a kind:"read" device_event keyed by the session
  identity), so it lands AFTER the entry/exit event already shipped over the WS
  without a plate; a refresh re-fetched via the bulk enrich path and showed it.
  Added a `plate-recognized` bus event (device-events.ts) emitted when the read
  is written; ws.ts forwards it; the client patchPlate(identity, plate)
  (live-store) backfills the already-rendered feed row in place and invalidates
  the Query-owned active-sessions list. No refresh.

- Plate search didn't filter. Both the live-feed (BoothScreen) and active-sessions
  (ActiveSessions) search haystacks matched the wrong field — the displayed plate
  is the ENRICHED top-level e.plate/s.plate (set by enrichEvent), not payload.plate
  (the plate is unsigned, never in the signed payload). Switched the haystacks to
  the displayed field.

Claude-Session: https://claude.ai/code/session_01Xcm6ikLgGoCxxHrxtjkk5V

apps/web/src/lib/live-store.ts

@@ -52,6 +52,9 @@ interface LiveState {
   setLanes: (l: LaneStatus) => void;
   /** Set lane radar presence (WS hello + each lane-presence push). */
   setRadar: (r: LanePresence) => void;
+  /** Backfill the enriched plate on every feed event matching `identity` (a late async
+   *  recognition that landed after the event's own push). No-op if no row matches. */
+  patchPlate: (identity: string, plate: string) => void;
   reset: () => void;
 }
 
@@ -80,5 +83,10 @@ export const useLiveStore = create<LiveState>((set) => ({
   upsertDevice: (d) => set((s) => ({ devices: { ...s.devices, [d.deviceId]: d } })),
   setLanes: (lanes) => set({ lanes }),
   setRadar: (radar) => set({ radar }),
+  patchPlate: (identity, plate) =>
+    set((s) => {
+      if (!s.feed.some((e) => e.identity === identity && !e.plate)) return s; // nothing to fill
+      return { feed: s.feed.map((e) => (e.identity === identity && !e.plate ? { ...e, plate } : e)) };
+    }),
   reset: () => set({ status: "connecting", occupancy: null, feed: [], devices: {}, lanes: null, radar: null }),
 }));

apps/web/src/lib/use-live-feed.ts

@@ -19,12 +19,14 @@ type WsMessage =
   | { kind: "printer-status"; event: unknown }
   | { kind: "device-status"; event: DeviceStatus }
   | { kind: "lane-status"; lanes: LaneStatus }
-  | { kind: "lane-presence"; radar: LanePresence };
+  | { kind: "lane-presence"; radar: LanePresence }
+  | { kind: "plate-recognized"; plate: { identity: string; plate: string; direction: "entry" | "exit" } };
 
 
 export function useLiveFeed(): void {
   const qc = useQueryClient();
-  const { setStatus, setOccupancy, pushEvent, setDevices, upsertDevice, setLanes, setRadar } = useLiveStore();
+  const { setStatus, setOccupancy, pushEvent, setDevices, upsertDevice, setLanes, setRadar, patchPlate } =
+    useLiveStore();
   // Hold the socket + reconnect timer across renders; guard against StrictMode
   // double-invoke and unmount.
   const sockRef = useRef<WebSocket | null>(null);
@@ -64,6 +66,11 @@ export function useLiveFeed(): void {
           setLanes(msg.lanes);
         } else if (msg.kind === "lane-presence") {
           setRadar(msg.radar);
+        } else if (msg.kind === "plate-recognized") {
+          // Backfill the badge on the already-rendered feed row, and refetch the
+          // Query-owned active-sessions list (re-runs enrichEvents → the now-written plate).
+          patchPlate(msg.plate.identity, msg.plate.plate);
+          void qc.invalidateQueries({ queryKey: qk.activeSessions });
         } else if (msg.kind === "ledger") {
           setOccupancy(msg.occupancy);
           pushEvent(msg.event);