feat(21-06): add conversation search input and Cmd+K shortcut

- Add Search/X icons and Input to ChatConversationList with 300ms debounce
- Listen for nexus:focus-chat-search event to focus search input
- Add onSearch handler to useKeyboardShortcuts (fires before input guard)
- Wire Layout Cmd+K to open chat panel and dispatch focus event

ui/src/components/ChatConversationList.tsx

@@ -1,11 +1,12 @@
 import { useEffect, useRef, useState } from "react";
-import { Plus } from "lucide-react";
+import { Plus, Search, X } from "lucide-react";
 import { useChatConversations } from "../hooks/useChatConversations";
 import { useChatPanel } from "../context/ChatPanelContext";
 import { ChatConversationItem } from "./ChatConversationItem";
 import { ScrollArea } from "@/components/ui/scroll-area";
 import { Skeleton } from "@/components/ui/skeleton";
 import { Button } from "@/components/ui/button";
+import { Input } from "@/components/ui/input";
 import {
   Dialog,
   DialogContent,
@@ -22,8 +23,25 @@ interface ChatConversationListProps {
 
 export function ChatConversationList({ companyId }: ChatConversationListProps) {
   const { activeConversationId, setActiveConversationId } = useChatPanel();
+
+  const [searchTerm, setSearchTerm] = useState("");
+  const [debouncedSearch, setDebouncedSearch] = useState("");
+  const searchInputRef = useRef<HTMLInputElement>(null);
+
+  useEffect(() => {
+    const timer = setTimeout(() => setDebouncedSearch(searchTerm), 300);
+    return () => clearTimeout(timer);
+  }, [searchTerm]);
+
+  // Listen for focus-chat-search custom event (dispatched by Cmd+K in Layout)
+  useEffect(() => {
+    const handler = () => searchInputRef.current?.focus();
+    window.addEventListener("nexus:focus-chat-search", handler);
+    return () => window.removeEventListener("nexus:focus-chat-search", handler);
+  }, []);
+
   const { data, isLoading, hasNextPage, fetchNextPage, createMutation, updateMutation, deleteMutation } =
-    useChatConversations(companyId);
+    useChatConversations(companyId, { search: debouncedSearch || undefined });
 
   const [deletingId, setDeletingId] = useState<string | null>(null);
   const sentinelRef = useRef<HTMLDivElement>(null);
@@ -112,6 +130,29 @@ export function ChatConversationList({ companyId }: ChatConversationListProps) {
         </Button>
       </div>
 
+      {/* Search input */}
+      <div className="px-2 pb-1 pt-1">
+        <div className="relative">
+          <Search className="absolute left-2 top-1/2 -translate-y-1/2 h-3.5 w-3.5 text-muted-foreground" />
+          <Input
+            ref={searchInputRef}
+            value={searchTerm}
+            onChange={(e) => setSearchTerm(e.target.value)}
+            placeholder="Search conversations..."
+            className="h-7 pl-7 pr-7 text-xs"
+          />
+          {searchTerm && (
+            <button
+              type="button"
+              onClick={() => setSearchTerm("")}
+              className="absolute right-2 top-1/2 -translate-y-1/2 text-muted-foreground hover:text-foreground"
+            >
+              <X className="h-3 w-3" />
+            </button>
+          )}
+        </div>
+      </div>
+
       <ScrollArea className="flex-1">
         <div className="p-1 space-y-0.5">
           {isLoading ? (

ui/src/components/Layout.tsx

@@ -54,7 +54,7 @@ export function Layout() {
   const { sidebarOpen, setSidebarOpen, toggleSidebar, isMobile } = useSidebar();
   const { openNewIssue, openOnboarding } = useDialog();
   const { togglePanelVisible, setPanelVisible } = usePanel();
-  const { chatOpen, toggleChat } = useChatPanel();
+  const { chatOpen, setChatOpen, toggleChat } = useChatPanel();
   const {
     companies,
     loading: companiesLoading,
@@ -167,6 +167,10 @@ export function Layout() {
     onNewIssue: () => openNewIssue(),
     onToggleSidebar: toggleSidebar,
     onTogglePanel: togglePanel,
+    onSearch: () => {
+      if (!chatOpen) setChatOpen(true);
+      requestAnimationFrame(() => window.dispatchEvent(new Event("nexus:focus-chat-search")));
+    },
   });
 
   useEffect(() => {

ui/src/hooks/useKeyboardShortcuts.ts

@@ -6,6 +6,7 @@ interface ShortcutHandlers {
   onNewIssue?: () => void;
   onToggleSidebar?: () => void;
   onTogglePanel?: () => void;
+  onSearch?: () => void;
 }
 
 export function useKeyboardShortcuts({
@@ -18,6 +19,13 @@ export function useKeyboardShortcuts({
     if (!enabled) return;
 
     function handleKeyDown(e: KeyboardEvent) {
+      // Cmd+K / Ctrl+K → Search (global, works even from inputs)
+      if (e.key === "k" && (e.metaKey || e.ctrlKey)) {
+        e.preventDefault();
+        onSearch?.();
+        return;
+      }
+
       // Don't fire shortcuts when typing in inputs
       if (isKeyboardShortcutTextInputTarget(e.target)) {
         return;