test(web): add Vitest tests for search, auth, public, and admin layouts

- SearchLayout: verifies children pass-through (3 tests)
- AuthLayout: verifies role=main, #main-content, max-w-md centering (5 tests)
- PublicLayout: verifies navbar, ticker strip, footer, compare bar, #main-content (8 tests)
- AdminLayout: verifies sidebar nav, auth guard, loading state, logout, mobile toggle (10 tests)

All 156 web test files pass (1157 total web tests). Pre-existing API test
failures in unrelated modules (auth OTP handler, projects, search indexer,
admin settings encryption) are outside scope of this task.

Co-Authored-By: Paperclip <noreply@paperclip.ing>

apps/web/app/[locale]/(public)/search/__tests__/layout.spec.tsx

@@ -0,0 +1,42 @@
+/* eslint-disable import-x/order */
+/**
+ * Kiểm thử SearchLayout: layout đơn giản chỉ render children.
+ */
+import { render, screen } from '@testing-library/react';
+import * as React from 'react';
+import { describe, expect, it } from 'vitest';
+
+import SearchLayout from '../layout';
+
+describe('SearchLayout', () => {
+  it('renders children', () => {
+    render(
+      <SearchLayout>
+        <div data-testid="child">Nội dung tìm kiếm</div>
+      </SearchLayout>,
+    );
+    expect(screen.getByTestId('child')).toBeInTheDocument();
+    expect(screen.getByText('Nội dung tìm kiếm')).toBeInTheDocument();
+  });
+
+  it('renders multiple children', () => {
+    render(
+      <SearchLayout>
+        <p data-testid="a">A</p>
+        <p data-testid="b">B</p>
+      </SearchLayout>,
+    );
+    expect(screen.getByTestId('a')).toBeInTheDocument();
+    expect(screen.getByTestId('b')).toBeInTheDocument();
+  });
+
+  it('does not add extra wrapper markup', () => {
+    const { container } = render(
+      <SearchLayout>
+        <span id="only-child">Span</span>
+      </SearchLayout>,
+    );
+    // The layout returns children directly, so the span should be the root child
+    expect(container.querySelector('#only-child')).toBeInTheDocument();
+  });
+});