test: Add comprehensive test coverage for untested modules

Add tests for previously untested components and services:
- apiKeyService.ts: localStorage operations (14 tests)
- ApiKeyModal.tsx: form validation, submission (22 tests)
- App.tsx: state transitions, error handling (23 tests)
- latexService.ts: API calls, template detection (35 tests)
- latex_service.py: LaTeX escaping, compilation (59 tests)
- server.py: Flask routes, field mapping (38 tests)

Also fix geminiService tests by adding proper apiKeyService mock.

Total new test coverage: 173 TypeScript tests, 97 Python tests

https://claude.ai/code/session_01D4k6b4nUjwfcHMvectsri2

tests/services/apiKeyService.test.ts

@@ -0,0 +1,125 @@
+import { describe, it, expect, beforeEach, afterEach, vi } from 'vitest';
+import { getApiKey, setApiKey, clearApiKey, hasApiKey } from '../../services/apiKeyService';
+
+describe('apiKeyService', () => {
+  const STORAGE_KEY = 'gemini_api_key';
+
+  beforeEach(() => {
+    // Clear localStorage before each test
+    localStorage.clear();
+  });
+
+  afterEach(() => {
+    localStorage.clear();
+  });
+
+  describe('getApiKey', () => {
+    it('should return null when no key is stored', () => {
+      const result = getApiKey();
+      expect(result).toBeNull();
+    });
+
+    it('should return the stored API key', () => {
+      localStorage.setItem(STORAGE_KEY, 'AItest123');
+
+      const result = getApiKey();
+
+      expect(result).toBe('AItest123');
+    });
+
+    it('should return empty string if empty string was stored', () => {
+      localStorage.setItem(STORAGE_KEY, '');
+
+      const result = getApiKey();
+
+      expect(result).toBe('');
+    });
+  });
+
+  describe('setApiKey', () => {
+    it('should store the API key in localStorage', () => {
+      setApiKey('AItest456');
+
+      expect(localStorage.getItem(STORAGE_KEY)).toBe('AItest456');
+    });
+
+    it('should overwrite existing key', () => {
+      localStorage.setItem(STORAGE_KEY, 'AIold');
+
+      setApiKey('AInew');
+
+      expect(localStorage.getItem(STORAGE_KEY)).toBe('AInew');
+    });
+
+    it('should allow storing empty string', () => {
+      setApiKey('');
+
+      expect(localStorage.getItem(STORAGE_KEY)).toBe('');
+    });
+  });
+
+  describe('clearApiKey', () => {
+    it('should remove the API key from localStorage', () => {
+      localStorage.setItem(STORAGE_KEY, 'AItest');
+
+      clearApiKey();
+
+      expect(localStorage.getItem(STORAGE_KEY)).toBeNull();
+    });
+
+    it('should not throw when no key exists', () => {
+      expect(() => clearApiKey()).not.toThrow();
+    });
+  });
+
+  describe('hasApiKey', () => {
+    it('should return false when no key is stored', () => {
+      const result = hasApiKey();
+
+      expect(result).toBe(false);
+    });
+
+    it('should return false when empty string is stored', () => {
+      localStorage.setItem(STORAGE_KEY, '');
+
+      const result = hasApiKey();
+
+      expect(result).toBe(false);
+    });
+
+    it('should return true when a non-empty key is stored', () => {
+      localStorage.setItem(STORAGE_KEY, 'AItest789');
+
+      const result = hasApiKey();
+
+      expect(result).toBe(true);
+    });
+
+    it('should return true for single character key', () => {
+      localStorage.setItem(STORAGE_KEY, 'A');
+
+      const result = hasApiKey();
+
+      expect(result).toBe(true);
+    });
+  });
+
+  describe('integration', () => {
+    it('should work correctly with set and get', () => {
+      setApiKey('AIintegration');
+
+      expect(getApiKey()).toBe('AIintegration');
+      expect(hasApiKey()).toBe(true);
+    });
+
+    it('should work correctly with set, clear, and get', () => {
+      setApiKey('AItest');
+      expect(hasApiKey()).toBe(true);
+
+      clearApiKey();
+
+      expect(getApiKey()).toBeNull();
+      expect(hasApiKey()).toBe(false);
+    });
+  });
+});