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/lifecycle.test.ts

@@ -36,6 +36,11 @@ import { PermissionRegistryError } from "@ekklesia/core/authorization";
 import { createApiApp } from "../src/app.js";
 import { createApiRuntimeConfig } from "../src/config.js";
 import { DisplayHostAssetError } from "../src/display-host-assets.js";
+import {
+  HEALTH_LIVE_PATH,
+  HEALTH_READY_PATH,
+  READINESS_DATABASE_TIMEOUT_MS,
+} from "../src/health-routes.js";
 import { installShutdownSignalHandlers, startApi } from "../src/server.js";
 
 function validConfig() {
@@ -447,6 +452,36 @@ describe("Ekklesia API shutdown lifecycle", () => {
     expect(poolEnd).toHaveBeenCalledTimes(1);
   });
 
+  it("answers readiness with one bounded ping on the shared pool, and turns it 503 the moment shutdown begins", async () => {
+    const query = emptyQuery();
+    poolConstructor.mockImplementation(function fakePool() {
+      return { end: vi.fn().mockResolvedValue(undefined), on: vi.fn(), query };
+    });
+    betterAuthMock.mockReturnValue({ api: { getSession: vi.fn() }, handler: vi.fn() });
+    mockSuccessfulServe();
+
+    const running = await startApi(validConfig());
+    query.mockClear();
+
+    const ready = await running.app.request(HEALTH_READY_PATH);
+    expect(ready.status).toBe(200);
+    expect(query).toHaveBeenCalledTimes(1);
+    // The statement timeout is server-side and scoped to this one transaction, so PostgreSQL itself
+    // cancels a stalled ping rather than the route merely abandoning it.
+    expect(query).toHaveBeenCalledWith(
+      `BEGIN READ ONLY; SET LOCAL statement_timeout = ${READINESS_DATABASE_TIMEOUT_MS}; SELECT 1; COMMIT`,
+    );
+
+    const shutdown = running.shutdown();
+    const draining = await running.app.request(HEALTH_READY_PATH);
+    expect(draining.status).toBe(503);
+    expect(await draining.json()).toEqual({ code: "NOT_READY" });
+    // Draining is not dead: liveness still answers while in-flight requests finish.
+    expect((await running.app.request(HEALTH_LIVE_PATH)).status).toBe(200);
+    expect(query).toHaveBeenCalledTimes(1);
+    await shutdown;
+  });
+
   it("keeps shutdown idempotent for calls made after the first has already resolved", async () => {
     const poolEnd = vi.fn().mockResolvedValue(undefined);
     poolConstructor.mockImplementation(function fakePool() {