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

biome.json

@@ -0,0 +1,26 @@
+{
+  "$schema": "https://biomejs.dev/schemas/2.4.15/schema.json",
+  "vcs": {
+    "enabled": true,
+    "clientKind": "git",
+    "useIgnoreFile": true
+  },
+  "files": {
+    "ignoreUnknown": true
+  },
+  "linter": {
+    "enabled": true,
+    "rules": {
+      "recommended": true,
+      "suspicious": {
+        "noExplicitAny": "off"
+      },
+      "style": {
+        "useImportType": "off"
+      }
+    }
+  },
+  "formatter": {
+    "enabled": false
+  }
+}