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.

browse/test/terminal-agent.test.ts

@@ -169,6 +169,38 @@ describe('Source-level guard: terminal-agent', () => {
     expect(dispose).toContain("'SIGKILL'");
     expect(dispose).toContain('3000');
   });
+
+  test('tabState frames write tabs.json + active-tab.json', () => {
+    expect(AGENT_SRC).toContain("msg?.type === 'tabState'");
+    expect(AGENT_SRC).toContain('function handleTabState');
+    const fn = AGENT_SRC.slice(AGENT_SRC.indexOf('function handleTabState'));
+    // Atomic write via tmp + rename for both files (so claude never reads
+    // a half-written JSON document).
+    expect(fn).toContain("'tabs.json'");
+    expect(fn).toContain("'active-tab.json'");
+    expect(fn).toContain('renameSync');
+    // Skip chrome:// and chrome-extension:// pages — they're not useful
+    // targets for browse commands.
+    expect(fn).toContain("startsWith('chrome://')");
+    expect(fn).toContain("startsWith('chrome-extension://')");
+  });
+
+  test('claude is spawned with --append-system-prompt tab-awareness hint', () => {
+    expect(AGENT_SRC).toContain('function buildTabAwarenessHint');
+    const hint = AGENT_SRC.slice(AGENT_SRC.indexOf('function buildTabAwarenessHint'));
+    // The hint must mention the live state files and the fanout command —
+    // those are the two affordances that distinguish a gstack-PTY claude
+    // from a plain `claude` session.
+    expect(hint).toContain('tabs.json');
+    expect(hint).toContain('active-tab.json');
+    expect(hint).toContain('tab-each');
+    // And it must be passed via --append-system-prompt at spawn time
+    // (NOT written into the PTY as user input — that would pollute the
+    // visible transcript).
+    const spawn = AGENT_SRC.slice(AGENT_SRC.indexOf('function spawnClaude'));
+    expect(spawn).toContain("'--append-system-prompt'");
+    expect(spawn).toContain('tabHint');
+  });
 });
 
 describe('Source-level guard: server.ts /pty-session route', () => {