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

.gitignore

@@ -136,19 +136,15 @@ dist
 .yarn/install-state.gz
 .pnp.*
 
-# Bun
-.bun
-bun.lockb
-
 # Turborepo
 .turbo
 
 # Prisma
-apps/api/prisma/dev.db
-apps/api/prisma/migrations/dev/
-apps/api/prisma/generated/
+apps/backend/prisma/dev.db
+apps/backend/prisma/migrations/dev/
+apps/backend/generated/
 *.db
 
 # env
-apps/api/.env
+apps/backend/.env