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/src/server.ts

@@ -48,6 +48,7 @@ import { type ApiRuntimeConfig, createApiRuntimeConfig } from "./config.js";
 import { type ApiDatabaseHandle, createApiDatabase } from "./database.js";
 import { withDevicePrincipals } from "./device-principal.js";
 import { loadDisplayHostAssets } from "./display-host-assets.js";
+import { READINESS_DATABASE_TIMEOUT_MS } from "./health-routes.js";
 import { prepareStartupModuleInventory } from "./platform-permissions.js";
 
 const connectionTimeoutMs = 10_000;
@@ -576,6 +577,10 @@ export async function startApi(
     throw error;
   }
 
+  // Set by the first `shutdown()` call, before anything is drained or stopped, so readiness turns
+  // `503` while the instance still finishes in-flight requests.
+  let shuttingDown = false;
+
   let app: Hono;
   try {
     app = createApiApp({
@@ -620,6 +625,12 @@ export async function startApi(
       },
       // A typed view over the same single pool as identity, authorization, and the module facade.
       dashboard: createDashboardStore(databaseHandle.dashboard),
+      // The app is only composed after `startDiscoveredModuleRuntime` resolved, so module startup
+      // has completed whenever it can answer; it stops counting as complete once shutdown begins.
+      health: {
+        checkDatabase: () => databaseHandle.ping(READINESS_DATABASE_TIMEOUT_MS),
+        isModuleStartupComplete: () => !shuttingDown,
+      },
       devices: {
         // `Secure` follows the deployment's own scheme, read from the already-validated identity
         // base URL rather than from any request. Production is https by configuration policy, so a
@@ -677,6 +688,7 @@ export async function startApi(
   let shutdownPromise: Promise<void> | undefined;
   const shutdown = (): Promise<void> => {
     if (!shutdownPromise) {
+      shuttingDown = true;
       shutdownPromise = (async () => {
         // Close long-lived streams so HTTP can drain, but leave publishers active until the drain
         // completes: an in-flight module command may still commit and publish its resulting state.