fix: close SSE stream for non-running threads, fix Live badge

- routes-live: check .running marker before keeping SSE open;
  if thread is not running, emit existing records and close
- thread-detail: only show Live badge when connected AND not completed

小橘 <xiaoju@shazhou.work>

packages/cli-workflow/src/commands/serve/routes-live.ts

@@ -1,4 +1,4 @@
-import { statSync, watch } from "node:fs";
+import { existsSync, statSync, watch } from "node:fs";
 import { join } from "node:path";
 import { createCasStore, getContentMerklePayload } from "@uncaged/workflow-cas";
 import {
@@ -307,6 +307,12 @@ export function createLiveRoutes(storageRoot: string): Hono {
         return;
       }
 
+      // If thread is not actively running, emit all records and close — don't keep SSE open
+      const runningPath = join(storageRoot, "logs", threadTarget.bundleHash, `${threadId}.running`);
+      if (!existsSync(runningPath)) {
+        return;
+      }
+
       const controller = new AbortController();
       let completed = false;
 

packages/workflow-dashboard/src/components/thread-detail.tsx

@@ -80,7 +80,7 @@ export function ThreadDetail({ agent, threadId, onBack }: Props) {
 
       <h2 className="text-xl font-semibold mb-2 font-mono flex items-center gap-2 flex-wrap">
         <span>{threadId}</span>
-        {sse.connected && (
+        {sse.connected && !sse.completed && (
           <span
             className="text-xs font-medium px-2 py-0.5 rounded"
             style={{ background: "var(--color-success)", color: "var(--color-bg)" }}