feat: Chrome extension Side Panel + Conductor API proposal

Chrome extension (Manifest V3, sideload):
- Side Panel with live activity feed, @ref overlays, dark terminal aesthetic
- Background worker: health polling, SSE relay, ref fetching
- Popup: port config, connection status, side panel launcher
- Content script: floating ref panel with @ref badges

Conductor API proposal (docs/designs/CONDUCTOR_SESSION_API.md):
- SSE endpoint for full Claude Code session mirroring in Side Panel
- Discovery via HTTP endpoint (not filesystem — extensions can't read files)

TODOS.md: add $B watch, multi-agent tabs, cross-platform CDP, Web Store publishing.
Mark CDP mode as shipped.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

extension/background.js

@@ -0,0 +1,130 @@
+/**
+ * gstack browse — background service worker
+ *
+ * Polls /health every 10s to detect browse server.
+ * Fetches /refs on snapshot completion, relays to content script.
+ * Updates badge: green (connected), gray (disconnected).
+ */
+
+let serverPort = null;
+let isConnected = false;
+let healthInterval = null;
+
+// ─── Port Discovery ────────────────────────────────────────────
+
+async function loadPort() {
+  const data = await chrome.storage.local.get('port');
+  serverPort = data.port || null;
+  return serverPort;
+}
+
+async function savePort(port) {
+  serverPort = port;
+  await chrome.storage.local.set({ port });
+}
+
+function getBaseUrl() {
+  return serverPort ? `http://127.0.0.1:${serverPort}` : null;
+}
+
+// ─── Health Polling ────────────────────────────────────────────
+
+async function checkHealth() {
+  const base = getBaseUrl();
+  if (!base) {
+    setDisconnected();
+    return;
+  }
+
+  try {
+    const resp = await fetch(`${base}/health`, { signal: AbortSignal.timeout(3000) });
+    if (!resp.ok) { setDisconnected(); return; }
+    const data = await resp.json();
+    if (data.status === 'healthy') {
+      setConnected(data);
+    } else {
+      setDisconnected();
+    }
+  } catch {
+    setDisconnected();
+  }
+}
+
+function setConnected(healthData) {
+  if (!isConnected) {
+    isConnected = true;
+    chrome.action.setBadgeText({ text: '' });
+    chrome.action.setBadgeBackgroundColor({ color: '#4ade80' });
+    // Small green dot via badge
+    chrome.action.setBadgeText({ text: ' ' });
+  }
+  // Broadcast health to popup and side panel
+  chrome.runtime.sendMessage({ type: 'health', data: healthData }).catch(() => {});
+}
+
+function setDisconnected() {
+  if (isConnected) {
+    isConnected = false;
+    chrome.action.setBadgeText({ text: '' });
+  }
+  chrome.runtime.sendMessage({ type: 'health', data: null }).catch(() => {});
+}
+
+// ─── Refs Relay ─────────────────────────────────────────────────
+
+async function fetchAndRelayRefs() {
+  const base = getBaseUrl();
+  if (!base || !isConnected) return;
+
+  try {
+    const resp = await fetch(`${base}/refs`, { signal: AbortSignal.timeout(3000) });
+    if (!resp.ok) return;
+    const data = await resp.json();
+
+    // Send to all tabs' content scripts
+    const tabs = await chrome.tabs.query({});
+    for (const tab of tabs) {
+      if (tab.id) {
+        chrome.tabs.sendMessage(tab.id, { type: 'refs', data }).catch(() => {});
+      }
+    }
+  } catch {}
+}
+
+// ─── Message Handling ──────────────────────────────────────────
+
+chrome.runtime.onMessage.addListener((msg, sender, sendResponse) => {
+  if (msg.type === 'getPort') {
+    sendResponse({ port: serverPort, connected: isConnected });
+    return true;
+  }
+
+  if (msg.type === 'setPort') {
+    savePort(msg.port).then(() => {
+      checkHealth();
+      sendResponse({ ok: true });
+    });
+    return true;
+  }
+
+  if (msg.type === 'getServerUrl') {
+    sendResponse({ url: getBaseUrl() });
+    return true;
+  }
+
+  if (msg.type === 'fetchRefs') {
+    fetchAndRelayRefs().then(() => sendResponse({ ok: true }));
+    return true;
+  }
+});
+
+// ─── Side Panel ─────────────────────────────────────────────────
+
+chrome.sidePanel.setPanelBehavior({ openPanelOnActionClick: false }).catch(() => {});
+
+// ─── Startup ────────────────────────────────────────────────────
+
+loadPort().then(() => {
+  checkHealth();
+  healthInterval = setInterval(checkHealth, 10000);
+});