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

packages/shared/src/schemas/problem-details.ts

@@ -0,0 +1,12 @@
+import { z } from 'zod';
+import type { ProblemDetails } from '../lib/problem-details.js';
+
+export const ProblemDetailsSchema: z.ZodType<ProblemDetails> = z.object({
+  type: z.string().optional(),
+  title: z.string(),
+  status: z.number().int().min(100).max(599),
+  detail: z.string().optional(),
+  instance: z.string().optional(),
+  code: z.string().optional(),
+  errors: z.record(z.string(), z.array(z.string())).optional(),
+});