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/web/src/routes/organizations.tsx

@@ -35,7 +35,7 @@ function OrganizationList({ onRefresh }: { onRefresh: () => void }) {
   const handleSyncGroup = async (org: OrganizationRow) => {
     setSyncing(org.id)
     try {
-      const res = await fetch(`${API_URL}/api/groups/from-organization`, {
+      const res = await fetch(`${API_URL}/api/v1/groups/from-organization`, {
         method: 'POST',
         headers: { 'Content-Type': 'application/json' },
         credentials: 'include',