Fix #6: Add missing DOM elements, fix null references

- Add violet-trial-badge span to index.html header
- Add null guard in updateVioletBadge() for safety
- Remove dead $('tab-ai-violet') reference (Violet tab is dynamic)
- Fix duplicate trialBadge declaration and restore hidden logic

index.html

@@ -87,6 +87,7 @@
       </div>
 
       <div class="header-right">
+        <span id="violet-trial-badge" class="violet-badge hidden"></span>
         <span id="my-username-badge" class="my-badge"></span>
         <button id="logout-btn" class="btn-logout">Exit</button>
       </div>

static/chat.js

@@ -367,6 +367,7 @@ socket.on("ignore_status", (data) => {
 });
 
 function updateVioletBadge() {
+    if (!trialBadge) return;
     if (state.hasAiAccess) {
         trialBadge.classList.add("hidden");
     } else {
@@ -587,7 +588,7 @@ $("sidebar-toggle").onclick = () => {
     sidebar.classList.toggle("open");
 };
 
-$("tab-ai-violet").onclick = () => switchTab("ai-violet");
+// tab-ai-violet is created dynamically when user opens Violet PM
 $("tab-lobby").onclick = () => switchTab("lobby");
 
 $("close-paywall").onclick = () => paywallModal.classList.add("hidden");