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/module-ui-host-e2e.test.ts

@@ -36,6 +36,7 @@ import {
 } from "@ekklesia/shell";
 import { afterEach, describe, expect, it, vi } from "vitest";
 import { createInMemoryDashboardStore } from "./support/in-memory-dashboard-store.js";
+import { readyHealth } from "./support/ready-health.js";
 import { createApiApp } from "../src/app.js";
 import { createDisplayHostAssets } from "../src/display-host-assets.js";
 import {
@@ -288,6 +289,7 @@ async function startHarness(options: HarnessOptions = {}): Promise<Harness> {
     moduleConfiguration: moduleConfigurationOptionFor(modules),
     moduleStorage: moduleStorageOptionFor(modules),
     dashboard: createInMemoryDashboardStore(),
+    health: readyHealth(),
     runtime,
     trustedOrigins: ["https://dashboard.example.test"],
   });