fix: two-player lobby deadlock

test/web/test_lobby_controls.js

@@ -0,0 +1,18 @@
+const assert = require('node:assert/strict');
+const fs = require('node:fs');
+const path = require('node:path');
+const test = require('node:test');
+
+const app = fs.readFileSync(path.join(__dirname, '../../data/app.js'), 'utf8');
+
+test('Play requests the server-authoritative next player slot', () => {
+  assert.match(app, /join\('player'\)/);
+  assert.doesNotMatch(app, /info\?\.player1Available \? 'player1'/);
+});
+
+test('lobby controls are visible only to player1 and enable when player2 is present', () => {
+  assert.match(app, /const playerOne = role === 'player1'/);
+  assert.match(app, /ui\.modeControls\.hidden = !playerOne/);
+  assert.match(app, /state\.players\[1\]\.trim\(\) !== '' \|\| info\?\.player2Available === false/);
+  assert.match(app, /const canStart = playerOne && \(state\.mode === 'bot' \|\| playerTwoReady\)/);
+});