Compile the API instead of running its TypeScript in production.

The unit's ExecStart named src/index.ts, so every start depended on the host
having Node 22.18 or newer for native type stripping. A deploy onto a host with
Node 20 met ERR_UNKNOWN_FILE_EXTENSION, exited in under a second, and was
restarted 464 times over fifteen hours with nothing anywhere going red.

api/tsconfig.json now emits to api/dist. The source keeps its explicit .ts import
specifiers, which is what makes `node --watch src/index.ts` work in development;
rewriteRelativeImportExtensions turns them into .js on the way out, so what runs
in production is ordinary ESM that any Node from 20.18 up will start.

`pnpm build` builds shared, then api, then web. `pnpm dev` is unchanged.

deploy/ is tracked rather than ignored: the unit files are the thing an operator
copies to /etc/systemd/system, and the alert unit added next has to live
somewhere a deploy can find it.

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

api/package.json

@@ -4,8 +4,9 @@
   "private": true,
   "type": "module",
   "scripts": {
+    "build": "tsc -p tsconfig.json",
     "dev": "node --env-file-if-exists=../.env --watch src/index.ts",
-    "start": "node --env-file-if-exists=../.env src/index.ts",
+    "start": "node --env-file-if-exists=../.env dist/index.js",
     "seed": "node --env-file-if-exists=../.env src/seed.ts",
     "migrate": "node --env-file-if-exists=../.env src/migrate.ts",
     "typecheck": "tsc -p tsconfig.json --noEmit",