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

client/src/components/lanes/LaneCard.tsx

@@ -381,11 +381,18 @@ export default function LaneCard({
           </div>
         )}
 
-        {lane.ci_status && (
+        {(lane.ci_status || lane.tool === "codex") && (
           <div className="mb-2 flex flex-wrap items-center gap-1.5 text-[11px]">
-            <span className="rounded bg-surface-2 px-1.5 py-0.5 text-fg-secondary">
-              CI {lane.ci_status}
-            </span>
+            {lane.ci_status && (
+              <span className="rounded bg-surface-2 px-1.5 py-0.5 text-fg-secondary">
+                CI {lane.ci_status}
+              </span>
+            )}
+            {lane.tool === "codex" && (
+              <span className="rounded border border-border bg-surface-2 px-1.5 py-0.5 text-[10px] text-fg-muted">
+                Codex
+              </span>
+            )}
           </div>
         )}
 

client/src/components/lanes/LaneStripCard.tsx

@@ -69,6 +69,11 @@ export default function LaneStripCard({
         <span className="truncate rounded bg-surface-4 px-1.5 py-0.5 text-fg-secondary">
           {lane.stage}
         </span>
+        {lane.tool === "codex" && (
+          <span className="shrink-0 rounded border border-border bg-surface-2 px-1.5 py-0.5 text-[10px] text-fg-muted">
+            Codex
+          </span>
+        )}
         {detectionLeads(lane) && (
           <span
             data-testid={`lane-tile-auto-${lane.id}`}

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();