chore: scaffold Nuxt 4 app with Postgres, tooling and health route

Nuxt 4 (app/ structure) + TypeScript, ESLint, Vitest + @nuxt/test-utils,
Drizzle + pg, a /api/health route with a DB ping, and a dev docker-compose
(app + Postgres). Verified: pnpm test, lint, and build pass; the built
server answers /api/health.

server/db/client.ts

@@ -0,0 +1,21 @@
+import pg from 'pg'
+
+let pool: pg.Pool | undefined
+
+/** Lazily-created shared Postgres connection pool. */
+export function getPool(): pg.Pool {
+  if (!pool) {
+    pool = new pg.Pool({ connectionString: process.env.DATABASE_URL })
+  }
+  return pool
+}
+
+/** Liveness probe for the database, used by /api/health. */
+export async function checkDb(): Promise<'up' | 'down'> {
+  try {
+    await getPool().query('select 1')
+    return 'up'
+  } catch {
+    return 'down'
+  }
+}

server/db/schema.ts

@@ -0,0 +1,2 @@
+// Drizzle (Postgres) schema — tables are defined in Task 3.
+export {}