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/lib/auth-client.ts

@@ -4,6 +4,7 @@ import { passkeyClient } from '@better-auth/passkey/client'
 
 export const authClient = createAuthClient({
   baseURL: import.meta.env.VITE_API_URL ?? 'http://localhost:4000',
+  basePath: '/api/v1/auth',
   plugins: [organizationClient(), passkeyClient()],
 })