refactor: migrate backend to pnpm monorepo with Hono module architecture

- Replace Bun with pnpm 9 + Turborepo + tsx; apps/api renamed to apps/backend
- Split backend into http/, modules/, lib/ layers mirroring tai-specguard
- Add use-case + Result pattern, Problem Details RFC 7807, basePath /api/v1
- Mount Better Auth at /api/v1/auth, keep session-auth whitelist
- Split Prisma schema into prisma/models/*, generate into generated/
- Rework packages/shared into lib/ + schemas/ with pagination DTOs
- Implement health, groups, students, payments, waitlist modules
- Add vitest suite with prisma mocks (21 tests), biome lint
- Point web client to /api/v1/auth and /api/v1/groups/from-organization

apps/backend/test/health-check.test.ts

@@ -0,0 +1,45 @@
+import { Hono } from 'hono';
+import { beforeEach, describe, expect, it, vi } from 'vitest';
+
+vi.mock('@/lib/prisma', () => ({
+  default: {
+    $queryRaw: vi.fn(),
+  },
+  getPrismaClient: vi.fn(),
+  UnitOfWork: class {},
+}));
+
+import prisma from '@/lib/prisma';
+import { healthCheckRoutes } from '@/modules/health-check';
+
+const app = new Hono();
+app.route('/health', healthCheckRoutes);
+
+describe('health check routes', () => {
+  beforeEach(() => {
+    vi.clearAllMocks();
+  });
+
+  it('reports ok when the database answers', async () => {
+    vi.mocked(prisma.$queryRaw).mockResolvedValue([{ '?column?': 1 }]);
+
+    const res = await app.request('/health');
+
+    expect(res.status).toBe(200);
+    const body = await res.json();
+    expect(body).toMatchObject({ status: 'ok', checks: { database: 'ok' } });
+    expect(typeof body.timestamp).toBe('string');
+  });
+
+  it('reports degraded when the database is unreachable', async () => {
+    vi.mocked(prisma.$queryRaw).mockRejectedValue(new Error('connection refused'));
+
+    const res = await app.request('/health');
+
+    expect(res.status).toBe(503);
+    expect(await res.json()).toMatchObject({
+      code: 'database_unavailable',
+      status: 503,
+    });
+  });
+});