feat(FN-4029): expose database health in API and store contracts

Adds database health endpoints to the API layer (FN-4029), exposing a new store health accessor through `api-node.ts` and `api/legacy.ts`, with aligned auth middleware integration tests and updated architecture documentation.

Fusion-Task-Id: FN-4029

packages/core/src/__tests__/store-health.test.ts

@@ -0,0 +1,45 @@
+import { beforeEach, describe, expect, it } from "vitest";
+
+import { TaskStore } from "../store.js";
+import { createTaskStoreTestHarness } from "./store-test-helpers.js";
+
+describe("TaskStore.getDatabaseHealth", () => {
+  const harness = createTaskStoreTestHarness();
+  let store: TaskStore;
+
+  beforeEach(async () => {
+    await harness.beforeEach();
+    store = harness.store();
+  });
+
+  it("reports healthy by default before corruption is detected", () => {
+    const health = store.getDatabaseHealth();
+
+    expect(health.healthy).toBe(true);
+    expect(health.isRunning).toBe(false);
+    expect(health.lastCheckedAt).toBeNull();
+  });
+
+  it("reports an in-progress integrity check", () => {
+    const db = store.getDatabase();
+    db.integrityCheckPending = true;
+
+    const health = store.getDatabaseHealth();
+
+    expect(health.healthy).toBe(true);
+    expect(health.isRunning).toBe(true);
+  });
+
+  it("reports unhealthy when corruption has been detected", () => {
+    const db = store.getDatabase();
+    db.corruptionDetected = true;
+    db.integrityCheckPending = false;
+    db.integrityCheckLastRunAt = "2026-05-11T12:34:56.000Z";
+
+    const health = store.getDatabaseHealth();
+
+    expect(health.healthy).toBe(false);
+    expect(health.isRunning).toBe(false);
+    expect(health.lastCheckedAt?.toISOString()).toBe("2026-05-11T12:34:56.000Z");
+  });
+});