feat(sagi): spawn a real claude session for Sagi's Ask handoff

Unmatched Ask questions now adopt a dedicated "Sagi" lane (POST
/api/lanes/ensure, cwd defaults to $HOME) and spawn a tmux-backed claude
run there instead of just prefilling the Run page composer.

client/src/components/Sagi/Sagi.tsx

@@ -78,6 +78,7 @@ import { useSagiBrain } from "./useSagiBrain";
 import { useSagiPosition, SAGI_SIZE } from "./useSagiPosition";
 import { matchIntent } from "./intents";
 import { sagiPrefs } from "./prefs";
+import { api } from "../../lib/api";
 import "./sagi.css";
 
 const FLYOUT_GAP = 10; // px between avatar and flyout
@@ -196,11 +197,31 @@ export function Sagi() {
     (query: string): string | null => {
       const result = matchIntent(query, brain.status);
       if (result.kind === "answer") return result.text;
-      // Handoff: spawn a real claude via the existing Run page. `autostart=1`
-      // tells Run to fire the prompt automatically once it's prefilled, so the
-      // question is actually sent instead of just dropped into the composer.
-      navigate(`/run?prompt=${encodeURIComponent(result.prompt)}&autostart=1`);
-      setOpen(false);
+      // Handoff: ensure the "Sagi" lane exists for this dashboard, spawn a
+      // fresh claude session in it carrying the user's prompt as its first
+      // message, then jump to the Run page so the live pane is observable.
+      // Fire-and-forget — the panel already clears its composer on null and
+      // closes; errors surface as an answer instead of silently swallowing.
+      const prompt = result.prompt;
+      brain.setThinking(true);
+      (async () => {
+        try {
+          const { lane } = await api.lanes.ensure({ title: "Sagi" });
+          await api.run.start({
+            laneId: lane.id,
+            cwd: lane.cwd,
+            initialPrompt: prompt,
+            permissionMode: "acceptEdits",
+          });
+          navigate(`/run?laneId=${lane.id}&autostart=1`);
+          setOpen(false);
+        } catch (err) {
+          const msg = err instanceof Error ? err.message : String(err);
+          brain.setBubble(`couldn't start that session: ${msg}`);
+        } finally {
+          brain.setThinking(false);
+        }
+      })();
       return null;
     },
     [brain, navigate]