feat(api): add unauthenticated health endpoints

Add GET /api/v1/health/live and GET /api/v1/health/ready per ADR 0044.

- live returns a fixed 200 {"status":"live"} whenever the process serves.
- ready returns 200 {"status":"ready"} only when module startup has
  completed (and shutdown has not begun) and a database ping succeeds
  within 2s; otherwise a fixed 503 {"code":"NOT_READY"}.
- The ping runs on the shared pool as one simple-protocol message with
  a transaction-local statement_timeout, so PostgreSQL cancels a stalled
  statement; the pool discards the connection on failure.
- At most one ping is ever unsettled: probes share its outcome and, once
  its deadline passes, answer 503 immediately until it settles.
- Both routes set Cache-Control: no-store, accept GET only (405 with
  Allow: GET otherwise, HEAD included), never resolve a principal, and
  are mounted in app.ts beside, never inside, /api/v1/modules/.
- createApiApp gains a required `health` option; startApi supplies it
  and clears module readiness when shutdown begins.

Co-Authored-By: Claude Opus 5.5 <[email protected]>

apps/api/tests/api-http.integration.ts

@@ -27,6 +27,12 @@ import { Client, type QueryResultRow } from "pg";
 import { afterAll, describe, expect, it, vi } from "vitest";
 import type { ClientAddressConfigInput } from "../src/client-address.js";
 import { createApiRuntimeConfig } from "../src/config.js";
+import { createApiDatabase } from "../src/database.js";
+import {
+  HEALTH_LIVE_PATH,
+  HEALTH_READY_PATH,
+  READINESS_DATABASE_TIMEOUT_MS,
+} from "../src/health-routes.js";
 import {
   MODULE_UI_CACHE_CONTROL,
   MODULE_UI_CONTENT_TYPE,
@@ -2977,6 +2983,56 @@ describe.sequential("live API — published module UI artifact", () => {
   });
 });
 
+describe.sequential("live API — health", () => {
+  it("answers liveness and readiness with 200 against a migrated disposable PostgreSQL", async () => {
+    const database = await createDisposableDatabase("health");
+    await applyCoreMigrations(database.url);
+    const running = await startMigratedApi(database.url);
+    try {
+      const origin = apiOrigin(running);
+
+      const live = await fetch(`${origin}${HEALTH_LIVE_PATH}`);
+      expect(live.status).toBe(200);
+      expect(live.headers.get("cache-control")).toBe("no-store");
+      expect(await live.json()).toEqual({ status: "live" });
+
+      const ready = await fetch(`${origin}${HEALTH_READY_PATH}`);
+      expect(ready.status).toBe(200);
+      expect(ready.headers.get("cache-control")).toBe("no-store");
+      expect(await ready.json()).toEqual({ status: "ready" });
+    } finally {
+      await running.shutdown();
+    }
+  });
+
+  it("pings through the shared pool, and PostgreSQL itself cancels a statement that outlives the ping's transaction-local timeout", async () => {
+    const database = await createDisposableDatabase("health_ping");
+    const handle = createApiDatabase(database.url);
+    try {
+      await expect(handle.ping(READINESS_DATABASE_TIMEOUT_MS)).resolves.toBeUndefined();
+    } finally {
+      await handle.close();
+    }
+
+    // The ping's exact statement shape, with a statement that cannot finish inside its timeout: the
+    // server cancels it (query_canceled) long before the sleep would end, on a client that has no
+    // client-side timeout of its own.
+    const client = new Client({ connectionString: database.url });
+    await client.connect();
+    try {
+      const startedAt = Date.now();
+      await expect(
+        client.query(
+          "BEGIN READ ONLY; SET LOCAL statement_timeout = 50; SELECT pg_sleep(5); COMMIT",
+        ),
+      ).rejects.toMatchObject({ code: "57014" });
+      expect(Date.now() - startedAt).toBeLessThan(2_000);
+    } finally {
+      await client.end();
+    }
+  });
+});
+
 describe.sequential("live API — public signup is disabled by default", () => {
   it("rejects sign-up with a stable sanitized 400 when publicSignupEnabled is not set, creating no user row", async () => {
     const database = await createDisposableDatabase("signup_disabled");