feat(offline): offline lobby lists + creates local vs_ai games

In offline mode the lobby now shows only the device-local games and its
New-vs-AI entry creates one through the in-browser engine — the visible
payoff of the offline mode.

- LocalSource.list() reconstructs a lobby GameView per stored local game by
  replay, exposed through the lazy gamesource proxy; unit-tested via an
  in-memory store.
- Lobby.load() branches on offlineMode: lists local games and skips every
  gateway call (no online games/invitations/incoming); the Stats tab is
  disabled offline.
- NewGame offline: find() creates a device-local vs_ai game via
  LocalSource.create using the profile's advertised dict version + a local
  seed; the friends flow and the random-opponent option are hidden, and the
  variant picker / Start are enabled offline (were gated on connection).
- id.ts: newLocalGameId + randomSeed (tested).

Docs: ARCHITECTURE + FUNCTIONAL(+ru) offline-mode section.

Deferred to fast-follow: the Settings Friends/Profile/Feedback affordance
gating, the flip-to-offline readiness wait, the offline mock e2e (needs
mock-dawg support), and the local-hint UI. The offline flow is verified on
the test contour — the mock e2e cannot enter offline mode (the toggle is
gated to an installed PWA).

ui/src/lib/localgame/source.list.test.ts

@@ -0,0 +1,61 @@
+import { describe, it, expect, vi, beforeEach } from 'vitest';
+import type { LocalGameRecord } from './serialize';
+
+// An in-memory stand-in for the IndexedDB store, so list() — which reads the *persisted* records,
+// not the source's in-memory cache — can be exercised in the node env (the real store is a
+// best-effort no-op without IndexedDB). Declared before the vi.mock factory that closes over it.
+const store = new Map<string, LocalGameRecord>();
+vi.mock('./store', () => ({
+  saveLocalGame: async (r: LocalGameRecord) => void store.set(r.id, r),
+  getLocalGame: async (id: string) => store.get(id) ?? null,
+  listLocalGames: async () => [...store.values()],
+  deleteLocalGame: async (id: string) => void store.delete(id),
+}));
+vi.mock('../dict', () => ({
+  getDawg: async () => {
+    const { Dawg } = await import('../dict/dawg');
+    const { readFileSync } = await import('node:fs');
+    return new Dawg(new Uint8Array(readFileSync(new URL('../dict/testdata/sample_en.dawg', import.meta.url))));
+  },
+}));
+
+import { LocalSource } from './source';
+import type { Seat } from './serialize';
+
+const seats: Seat[] = [
+  { kind: 'human', name: 'You' },
+  { kind: 'robot', name: 'Robot' },
+];
+const opts = (id: string, seed: bigint) => ({
+  id,
+  variant: 'scrabble_en' as const,
+  dictVersion: 'sample',
+  seed,
+  multipleWords: true,
+  seats,
+});
+
+beforeEach(() => store.clear());
+
+describe('LocalSource.list', () => {
+  it('is empty when no games are stored', async () => {
+    expect(await new LocalSource().list()).toEqual([]);
+  });
+
+  it('returns a lobby GameView for every stored local game', async () => {
+    const src = new LocalSource();
+    await src.create(opts('local:a', 1n));
+    await src.create(opts('local:b', 2n));
+    const list = await src.list();
+    expect(new Set(list.map((g) => g.id))).toEqual(new Set(['local:a', 'local:b']));
+    expect(list.every((g) => g.vsAi && g.status === 'active')).toBe(true);
+  });
+
+  it('reconstructs a game persisted by a previous session (not in the source cache)', async () => {
+    // Seed the store via one source, then list from a fresh source whose in-memory cache is empty.
+    await new LocalSource().create(opts('local:old', 7n));
+    const list = await new LocalSource().list();
+    expect(list.map((g) => g.id)).toEqual(['local:old']);
+    expect(list[0].variant).toBe('scrabble_en');
+  });
+});