fix(bridge): forward Pi agent_end so the spinner stops at workflow completion

The FE derived 'working' purely as activeSession && !pendingWidget, so the
final workflow turn — the only one that ends without a follow-up gate —
left the spinner on forever (observed live: 21592s after F8 approve).

- SessionBridge maps Pi's agent_end -> SSE system_event {event: agent_end}
- PiProcessManager notifies the client (info error + synthetic agent_end)
  when the child dies unexpectedly; expected teardowns stay silent
- sessionStore tracks agentActive (on: user entry/text_delta/ui_request,
  off: agent_end); AppShell working now requires it; resume sets it
  optimistically

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

frontend/src/store/sessionStore.test.ts

@@ -39,6 +39,35 @@ test("ui_request without a phase keeps the existing currentPhase", () => {
   expect(useSessionStore.getState().currentPhase).toBe("F2");
 });
 
+test("agentActive lifecycle: off by default, on with activity, off on agent_end", () => {
+  const st = useSessionStore.getState();
+  expect(useSessionStore.getState().agentActive).toBe(false);
+  // Any streamed text means the turn is alive (also covers reattaching mid-turn).
+  st.applyEvent({ type: "text_delta", text: "Analisi" });
+  expect(useSessionStore.getState().agentActive).toBe(true);
+  // Pi's end-of-turn signal (mapped by the backend) releases the working state.
+  st.applyEvent({ type: "system_event", event: "agent_end" });
+  expect(useSessionStore.getState().agentActive).toBe(false);
+});
+
+test("setLastUserEntry hands the ball back to the harness (agentActive on)", () => {
+  useSessionStore.getState().setLastUserEntry({ kind: "choice", text: "approve" });
+  expect(useSessionStore.getState().agentActive).toBe(true);
+});
+
+test("setAgentActive drives the flag directly (resume optimistic spin)", () => {
+  useSessionStore.getState().setAgentActive(true);
+  expect(useSessionStore.getState().agentActive).toBe(true);
+  useSessionStore.getState().resetSession();
+  expect(useSessionStore.getState().agentActive).toBe(false);
+});
+
+test("a ui_request keeps agentActive on (turn blocked on the gate, not ended)", () => {
+  const st = useSessionStore.getState();
+  st.applyEvent({ type: "ui_request", ui_request: { id: "u1", widget: "select" } });
+  expect(useSessionStore.getState().agentActive).toBe(true);
+});
+
 test("pushToast appends an error toast", () => {
   useSessionStore.getState().pushToast({ level: "error", text: "boom" });
   expect(useSessionStore.getState().toasts.at(-1)).toEqual({ level: "error", text: "boom" });