feat: live tab awareness for the Terminal pane

claude in the PTY now has continuous tab-aware context. Three pieces:

1. Live state files. background.js listens to chrome.tabs.onActivated /
   onCreated / onRemoved / onUpdated (throttled to URL/title/status==
   complete so loading spinners don't spam) and pushes a snapshot. The
   sidepanel relays it as a custom event; sidepanel-terminal.js sends
   {type:"tabState"} text frames over the live PTY WebSocket.
   terminal-agent.ts writes:
     <stateDir>/tabs.json          all open tabs (id, url, title, active,
                                   pinned, audible, windowId)
     <stateDir>/active-tab.json    current active tab (skips chrome:// and
                                   chrome-extension:// internal pages)
   Atomic write via tmp + rename so claude never reads a half-written
   document. A fresh snapshot is pushed on WS open so the files exist by
   the time claude finishes booting.

2. New $B tab-each <command> [args...] meta-command. Fans out a single
   command across every open tab, returns
   {command, args, total, results: [{tabId, url, title, status, output}]}.
   Skips chrome:// pages; restores the originally active tab in a finally
   block (so a mid-batch error doesn't leave the user looking at a
   different tab); uses bringToFront: false so the OS window doesn't
   jump on every fanout. Scope-checks the inner command BEFORE the loop.

3. --append-system-prompt hint at spawn time. Claude is told about both
   the state files and the $B tab-each command up front, so it doesn't
   have to discover the surface by trial. Passed via the --append-system-
   prompt CLI flag, NOT as a leading PTY write — the hint stays out of
   the visible transcript.

Tests:
- browse/test/tab-each.test.ts (new) — registration + source-level
  invariants (scope check before loop, finally-restore, bringToFront:false,
  chrome:// skip) + behavior tests with a mock BrowserManager that verify
  iteration order, JSON shape, error handling, and active-tab restore.
- browse/test/terminal-agent.test.ts — three new assertions for
  tabState handler shape, atomic-write pattern, and the
  --append-system-prompt wiring at spawn.

Verified live: opened 5 tabs, ran $B tab-each url against the live
server, got per-tab JSON results back, original active tab restored
without OS focus stealing.

extension/background.js

@@ -287,6 +287,7 @@ chrome.runtime.onMessage.addListener((msg, sender, sendResponse) => {
   const ALLOWED_TYPES = new Set([
     'getPort', 'setPort', 'getServerUrl', 'getToken', 'fetchRefs',
     'openSidePanel', 'sidebarOpened', 'command', 'sidebar-command',
+    'getTabState',
     // Inspector message types
     'startInspector', 'stopInspector', 'elementPicked', 'pickerCancelled',
     'applyStyle', 'toggleClass', 'injectCSS', 'resetAll',
@@ -302,6 +303,11 @@ chrome.runtime.onMessage.addListener((msg, sender, sendResponse) => {
     return true;
   }
 
+  if (msg.type === 'getTabState') {
+    snapshotTabs().then(snap => sendResponse(snap || { active: null, tabs: [] }));
+    return true; // async sendResponse
+  }
+
   if (msg.type === 'setPort') {
     savePort(msg.port).then(() => {
       checkHealth();
@@ -506,11 +512,48 @@ chrome.runtime.onInstalled.addListener(() => {
 // Fire on every service worker startup (covers persistent context reuse)
 autoOpenSidePanel();
 
-// ─── Tab Switch Detection ────────────────────────────────────────
-// Notify sidepanel instantly when the user switches tabs in the browser.
-// This is faster than polling — the sidebar swaps chat context immediately.
+// ─── Tab Awareness ───────────────────────────────────────────────
+// Push live tab state to the sidepanel so claude in the Terminal pane
+// always has up-to-date tabs.json + active-tab.json on disk. The
+// sidepanel relays these to terminal-agent.ts over the live WebSocket;
+// terminal-agent writes the files for claude to read.
+
+async function snapshotTabs() {
+  try {
+    const [active] = await chrome.tabs.query({ active: true, currentWindow: true });
+    const all = await chrome.tabs.query({});
+    const slim = all.map(t => ({
+      tabId: t.id,
+      url: t.url || '',
+      title: t.title || '',
+      active: !!t.active,
+      windowId: t.windowId,
+      pinned: !!t.pinned,
+      audible: !!t.audible,
+    }));
+    return {
+      active: active ? { tabId: active.id, url: active.url || '', title: active.title || '' } : null,
+      tabs: slim,
+    };
+  } catch {
+    return null;
+  }
+}
+
+async function pushTabState(reason) {
+  const snapshot = await snapshotTabs();
+  if (!snapshot) return;
+  chrome.runtime.sendMessage({
+    type: 'browserTabState',
+    reason,
+    ...snapshot,
+  }).catch(() => {}); // expected: sidepanel may not be open
+}
 
 chrome.tabs.onActivated.addListener((activeInfo) => {
+  // Keep the legacy event for any consumer still listening to it (the chat
+  // path is gone but the message type is harmless), and also fire the new
+  // unified state push so claude's tabs.json reflects the new active tab.
   chrome.tabs.get(activeInfo.tabId, (tab) => {
     if (chrome.runtime.lastError || !tab) return;
     chrome.runtime.sendMessage({
@@ -518,8 +561,20 @@ chrome.tabs.onActivated.addListener((activeInfo) => {
       tabId: activeInfo.tabId,
       url: tab.url || '',
       title: tab.title || '',
-    }).catch(() => {}); // expected: sidepanel may not be open
+    }).catch(() => {});
   });
+  pushTabState('activated');
+});
+
+chrome.tabs.onCreated.addListener(() => pushTabState('created'));
+chrome.tabs.onRemoved.addListener(() => pushTabState('removed'));
+chrome.tabs.onUpdated.addListener((_id, changeInfo) => {
+  // Throttle: only re-push on URL or title changes, not on every loading
+  // tick. We don't want to spam claude with a state push every 50ms while
+  // a page loads.
+  if (changeInfo.url || changeInfo.title || changeInfo.status === 'complete') {
+    pushTabState('updated');
+  }
 });
 
 // ─── Startup ────────────────────────────────────────────────────