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/sidepanel-terminal.js

@@ -240,6 +240,24 @@
       try {
         ws.send(JSON.stringify({ type: 'resize', cols: term.cols, rows: term.rows }));
       } catch {}
+      // Push a fresh tab snapshot so claude's tabs.json is populated by
+      // the time the lazy spawn finishes booting. Background.js exposes
+      // the snapshot helper via chrome.runtime; we ask for it here and
+      // forward whatever comes back.
+      try {
+        chrome.runtime.sendMessage({ type: 'getTabState' }, (resp) => {
+          if (resp && ws && ws.readyState === WebSocket.OPEN) {
+            try {
+              ws.send(JSON.stringify({
+                type: 'tabState',
+                active: resp.active,
+                tabs: resp.tabs,
+                reason: 'initial',
+              }));
+            } catch {}
+          }
+        });
+      } catch {}
       // Send a single byte to nudge the agent to spawn claude (lazy-spawn trigger).
       try { ws.send(new TextEncoder().encode('\n')); } catch {}
     });
@@ -335,6 +353,22 @@
     els.restartNow?.addEventListener('click', forceRestart);
 
 
+    // Live browser-tab state. background.js → sidepanel.js → us. We
+    // forward over the live PTY WebSocket; terminal-agent.ts writes
+    // <stateDir>/active-tab.json + <stateDir>/tabs.json so claude can
+    // always read the current tab landscape.
+    document.addEventListener('gstack:tab-state', (ev) => {
+      if (!ws || ws.readyState !== WebSocket.OPEN) return;
+      try {
+        ws.send(JSON.stringify({
+          type: 'tabState',
+          active: ev.detail?.active,
+          tabs: ev.detail?.tabs,
+          reason: ev.detail?.reason,
+        }));
+      } catch {}
+    });
+
     // Repaint after a debug-tab → primary-pane transition. The debug
     // tabs (Activity / Refs / Inspector) hide the Terminal pane via
     // .tab-content { display: none }; xterm doesn't auto-redraw when its