Add admin identity module: better-auth, per-app permissions, invitations

Introduces src/models/admin/, a dedicated identity and permissions module on
its own nachklang_admin database, and the shared authenticator that feedback
and tickets will move onto in the cutover step. Nothing swaps over yet:
feedback.auth.ts and tickets.auth.ts still authenticate against the legacy
calendar sessions, so production behaviour is unchanged.

- better-auth 1.7 mounted at /admin/auth/*, sessions as httpOnly cookies
  scoped to .nachklang.art so one sign-in covers every *.nachklang.art app.
- Accounts are invite-only: public sign-up is disabled, and the invitations
  plugin is the only code that creates users. Tokens are stored as SHA-256
  hashes and travel in the request body, never in a URL.
- Per-app permissions in user_app_permissions; requireAppAccess(app) queries
  the database on every request (no cookie cache) so disabling a user or
  revoking a session takes effect immediately.
- ADMIN_BOOTSTRAP_EMAIL guarantees a way in on an empty database, idempotently
  and without crashing the API if the database is unreachable at boot.
- Guards prevent an admin from removing their own admin permission, disabling
  themselves, or stripping the last active admin.

The admin pool uses the callback-style mysql2, not mysql2/promise: Kysely's
MysqlDialect drives the pool with callbacks, and the promise wrapper ignores
them, so every query hangs silently. Only the integration tests caught this.

Schema in sql/admin/001_init.sql, derived from getAuthTables() on the
installed better-auth rather than the published CLI, which lags the library
and omits account.issuer.

app.ts is split into src/app.factory.ts so the integration tests drive the
real middleware order rather than a copy of it.

Tests: 131 unit, plus 41 integration tests against a throwaway MariaDB
started by test/integration/setup.ts (docker or podman).

Co-Authored-By: Claude Opus 5 <noreply@anthropic.com>

.env.example

@@ -25,6 +25,20 @@ TICKETS_DB=
 TICKETS_RATE_LIMIT_MAX=10
 TICKETS_RATE_LIMIT_WINDOW_MIN=10
 
+ADMIN_DB=
+# 32+ random bytes, e.g. `openssl rand -base64 48`. Rotating it signs everyone
+# out and invalidates outstanding password-reset links.
+BETTER_AUTH_SECRET=
+API_BASE_URL=http://localhost:3000
+ADMIN_APP_URL=http://localhost:3002
+# Comma-separated origins of the apps that may call /admin/* with credentials.
+APP_ORIGINS=http://localhost:3001
+# nachklang.art in production; passkeys are bound to this value.
+PASSKEY_RP_ID=localhost
+# On start-up, makes sure this address can get in (invite, or grant admin if the
+# user already exists). Idempotent, safe to leave set.
+ADMIN_BOOTSTRAP_EMAIL=
+
 MEMBER_CREDENTIAL=123
 CHOIR_CREDENTIAL=123
 MANAGEMENT_CREDENTIAL=123