feat: floating chat bubble with pulse indicator

- Floating chat icon in bottom-right corner
- Opens board-level chat (not tied to any task)
- Pulsing green dot when new response arrives while closed
- Hides bubble when chat panel is open
- Clears unread indicator on open

web/src/App.tsx

@@ -12,6 +12,7 @@ import { AuthGuard } from './components/auth';
 import { ErrorBoundary } from './components/shared/ErrorBoundary';
 import { SkipToContent } from './components/shared/SkipToContent';
 import { LiveAnnouncerProvider } from './components/shared/LiveAnnouncer';
+import { FloatingChat } from './components/chat/FloatingChat';
 
 // Main app content (only rendered when authenticated)
 function AppContent() {
@@ -38,6 +39,7 @@ function AppContent() {
                 </main>
                 <Toaster />
                 <KeyboardShortcutsDialog />
+                <FloatingChat />
               </div>
             </TaskConfigProvider>
           </BulkActionsProvider>

web/src/components/chat/FloatingChat.tsx

@@ -0,0 +1,69 @@
+import { useState, useEffect } from 'react';
+import { Button } from '@/components/ui/button';
+import { MessageSquare } from 'lucide-react';
+import { ChatPanel } from './ChatPanel';
+import { chatEventTarget } from '@/hooks/useTaskSync';
+import { cn } from '@/lib/utils';
+
+/**
+ * Floating chat bubble — bottom-right corner.
+ * Opens a board-level ChatPanel (no taskId).
+ * Pulses when a new response arrives while closed.
+ */
+export function FloatingChat() {
+  const [open, setOpen] = useState(false);
+  const [hasUnread, setHasUnread] = useState(false);
+
+  // Listen for incoming chat messages when panel is closed
+  useEffect(() => {
+    const handler = (e: Event) => {
+      const msg = (e as CustomEvent).detail;
+      // Only pulse for board-level chat messages (no taskId in sessionId)
+      if (
+        !open &&
+        (msg.type === 'chat:message' || msg.type === 'chat:delta') &&
+        msg.sessionId &&
+        !msg.sessionId.includes('task_')
+      ) {
+        setHasUnread(true);
+      }
+    };
+
+    chatEventTarget.addEventListener('chat', handler);
+    return () => chatEventTarget.removeEventListener('chat', handler);
+  }, [open]);
+
+  // Clear unread when opening
+  const handleOpen = () => {
+    setOpen(true);
+    setHasUnread(false);
+  };
+
+  return (
+    <>
+      {/* Floating button */}
+      <Button
+        onClick={handleOpen}
+        size="icon"
+        className={cn(
+          'fixed bottom-6 right-6 z-40 h-14 w-14 rounded-full shadow-lg',
+          'bg-primary hover:bg-primary/90 text-primary-foreground',
+          'transition-all duration-200 hover:scale-105',
+          open && 'hidden'
+        )}
+        aria-label="Open chat"
+      >
+        <MessageSquare className="h-6 w-6" />
+        {hasUnread && (
+          <span className="absolute -top-1 -right-1 flex h-4 w-4">
+            <span className="animate-ping absolute inline-flex h-full w-full rounded-full bg-emerald-400 opacity-75" />
+            <span className="relative inline-flex rounded-full h-4 w-4 bg-emerald-500" />
+          </span>
+        )}
+      </Button>
+
+      {/* Chat panel — board-level (no taskId) */}
+      <ChatPanel open={open} onOpenChange={setOpen} />
+    </>
+  );
+}