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/src/http/env.ts

@@ -0,0 +1,19 @@
+import type { auth } from '@/modules/auth/auth';
+
+type AuthSession = typeof auth.$Infer.Session;
+
+export type BackendEnv = {
+  Variables: {
+    user: AuthSession['user'] | null;
+    session: AuthSession['session'] | null;
+    requestId: string;
+  };
+};
+
+declare module 'hono' {
+  interface ContextVariableMap {
+    user: AuthSession['user'] | null;
+    session: AuthSession['session'] | null;
+    requestId: string;
+  }
+}