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/problem-builders.ts

@@ -0,0 +1,89 @@
+import type { ProblemDetails } from '@gruperly/shared';
+import { notFoundProblem } from './problem-details';
+import { PROBLEM_DOMAIN } from './problem-domain';
+
+export function validationProblem(params: {
+  detail?: string;
+  instance?: string;
+  errors?: Record<string, string[]>;
+}): ProblemDetails {
+  return {
+    type: `${PROBLEM_DOMAIN}/problems/validation`,
+    title: 'Bad Request',
+    status: 400,
+    detail: params.detail ?? 'Invalid request data',
+    instance: params.instance,
+    errors: params.errors,
+  };
+}
+
+export function conflictProblem(params: {
+  detail: string;
+  code?: string;
+  instance?: string;
+}): ProblemDetails {
+  return {
+    type: `${PROBLEM_DOMAIN}/problems/conflict`,
+    title: 'Conflict',
+    status: 409,
+    detail: params.detail,
+    instance: params.instance,
+    code: params.code,
+  };
+}
+
+export function forbiddenProblem(params: {
+  detail: string;
+  code?: string;
+  instance?: string;
+}): ProblemDetails {
+  return {
+    type: `${PROBLEM_DOMAIN}/problems/forbidden`,
+    title: 'Forbidden',
+    status: 403,
+    detail: params.detail,
+    instance: params.instance,
+    code: params.code,
+  };
+}
+
+export function organizationNotFoundProblem(id: string): ProblemDetails {
+  return {
+    type: `${PROBLEM_DOMAIN}/problems/organization-not-found`,
+    title: 'Not Found',
+    status: 404,
+    detail: `Organization ${id} was not found.`,
+    code: 'organization_not_found',
+  };
+}
+
+export function groupOwnerRequiredProblem(): ProblemDetails {
+  return forbiddenProblem({
+    detail: 'Only the organization owner can create the group.',
+    code: 'group_owner_required',
+  });
+}
+
+export function databaseUnavailableProblem(params?: {
+  instance?: string;
+}): ProblemDetails {
+  return {
+    type: `${PROBLEM_DOMAIN}/problems/database-unavailable`,
+    title: 'Service Unavailable',
+    status: 503,
+    detail: 'Database health check failed',
+    instance: params?.instance,
+    code: 'database_unavailable',
+  };
+}
+
+export function noGroupAccessProblem(): ProblemDetails {
+  return forbiddenProblem({
+    detail: 'You do not have access to this group.',
+    code: 'group_access_denied',
+  });
+}
+
+export function notFoundResourceProblem(resourceName: string, id: string): ProblemDetails {
+  return notFoundProblem(`${resourceName} ${id}`);
+}