feat(lanes): badge codex lanes and document the lane tool

client/src/components/lanes/__tests__/LaneCard.test.tsx

@@ -58,6 +58,7 @@ function makeLane(overrides: Partial<Lane> = {}): Lane {
     cwd: "/work/demo",
     branch: "lane/demo",
     kind: "adopted",
+    tool: "claude",
     source_repo: null,
     pipeline: "default",
     session_id: null,
@@ -85,6 +86,17 @@ function makeLane(overrides: Partial<Lane> = {}): Lane {
   } as Lane;
 }
 
+describe("LaneCard lane tool badge", () => {
+  it("badges a codex lane and leaves a claude lane unbadged", () => {
+    const { rerender } = render(
+      <LaneCard {...{ lane: makeLane({ tool: "codex" }), onAction: vi.fn() }} />
+    );
+    expect(screen.getByText("Codex")).toBeInTheDocument();
+    rerender(<LaneCard {...{ lane: makeLane({ tool: "claude" }), onAction: vi.fn() }} />);
+    expect(screen.queryByText("Codex")).not.toBeInTheDocument();
+  });
+});
+
 describe("LaneCard child worktrees", () => {
   it("lists worktrees provisioned from this lane and jumps to one on click", async () => {
     const onSelectLane = vi.fn();