feat(frontend): pass locale to iframe embedded pages via lang parameter

Embedded pages (purchase subscription, custom pages) now receive the
current user locale through a `lang` URL parameter, allowing iframe
content to match the user's language preference.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>

frontend/src/utils/__tests__/embedded-url.spec.ts

@@ -0,0 +1,67 @@
+import { afterEach, beforeEach, describe, expect, it, vi } from 'vitest'
+import { buildEmbeddedUrl, detectTheme } from '../embedded-url'
+
+describe('embedded-url', () => {
+  const originalLocation = window.location
+
+  beforeEach(() => {
+    Object.defineProperty(window, 'location', {
+      value: {
+        origin: 'https://app.example.com',
+        href: 'https://app.example.com/user/purchase',
+      },
+      writable: true,
+      configurable: true,
+    })
+  })
+
+  afterEach(() => {
+    Object.defineProperty(window, 'location', {
+      value: originalLocation,
+      writable: true,
+      configurable: true,
+    })
+    document.documentElement.classList.remove('dark')
+    vi.restoreAllMocks()
+  })
+
+  it('adds embedded query parameters including locale and source context', () => {
+    const result = buildEmbeddedUrl(
+      'https://pay.example.com/checkout?plan=pro',
+      42,
+      'token-123',
+      'dark',
+      'zh-CN',
+    )
+
+    const url = new URL(result)
+    expect(url.searchParams.get('plan')).toBe('pro')
+    expect(url.searchParams.get('user_id')).toBe('42')
+    expect(url.searchParams.get('token')).toBe('token-123')
+    expect(url.searchParams.get('theme')).toBe('dark')
+    expect(url.searchParams.get('lang')).toBe('zh-CN')
+    expect(url.searchParams.get('ui_mode')).toBe('embedded')
+    expect(url.searchParams.get('src_host')).toBe('https://app.example.com')
+    expect(url.searchParams.get('src_url')).toBe('https://app.example.com/user/purchase')
+  })
+
+  it('omits optional params when they are empty', () => {
+    const result = buildEmbeddedUrl('https://pay.example.com/checkout', undefined, '', 'light')
+
+    const url = new URL(result)
+    expect(url.searchParams.get('theme')).toBe('light')
+    expect(url.searchParams.get('ui_mode')).toBe('embedded')
+    expect(url.searchParams.has('user_id')).toBe(false)
+    expect(url.searchParams.has('token')).toBe(false)
+    expect(url.searchParams.has('lang')).toBe(false)
+  })
+
+  it('returns original string for invalid url input', () => {
+    expect(buildEmbeddedUrl('not a url', 1, 'token')).toBe('not a url')
+  })
+
+  it('detects dark mode from document root class', () => {
+    document.documentElement.classList.add('dark')
+    expect(detectTheme()).toBe('dark')
+  })
+})