feat(FN-1254): add inbox-lite task selection for heartbeat agents

- Add InboxTask typing and TaskStore.selectNextTaskForAgent() with priority ordering, dependency checks, paused filtering, and FIFO selection
- Wire heartbeat execution to auto-select and assign inbox work when no task is set, with optional checkout attempts and graceful conflict fallback
- Add POST /api/agents/:id/inbox to expose next-task selection details (task, priority, reason) and return task:null when no work is available
- Expand core, engine, and dashboard tests to cover selection priorities, heartbeat precedence/metadata, checkout-conflict handling, and route behavior with type-safe mocks

packages/dashboard/src/routes.ts

@@ -8689,6 +8689,45 @@ Output ONLY the prompt text (no markdown, no explanations).`;
     }
   });
 
+  /**
+   * POST /api/agents/:id/inbox
+   * Select the next inbox-lite task candidate for an agent.
+   *
+   * Returns `{ task, priority, reason }` when work is available,
+   * or `{ task: null }` when no matching work is found.
+   */
+  router.post("/agents/:id/inbox", async (req, res) => {
+    try {
+      const scopedStore = await getScopedStore(req);
+      const { AgentStore } = await import("@fusion/core");
+      const agentStore = new AgentStore({ rootDir: scopedStore.getFusionDir() });
+      await agentStore.init();
+
+      const agentId = req.params.id;
+      const agent = await agentStore.getAgent(agentId);
+      if (!agent) {
+        throw notFound("Agent not found");
+      }
+
+      const selection = await scopedStore.selectNextTaskForAgent(agentId);
+      if (!selection) {
+        res.json({ task: null });
+        return;
+      }
+
+      res.json({
+        task: selection.task,
+        priority: selection.priority,
+        reason: selection.reason,
+      });
+    } catch (err: any) {
+      if (err instanceof ApiError) {
+        throw err;
+      }
+      rethrowAsApiError(err);
+    }
+  });
+
   /**
    * POST /api/agents/:id/heartbeat
    * Record a heartbeat for an agent.