feat(acks): surface 999 rejections via sidebar badge + table sort

The 1,156 accepted acks were burying the 5 rejections — the operator
had to scroll the whole table to find them. Two changes:

1. Sidebar '999 ACKs' nav item gets a warning-toned badge showing
   the total rejected segment count from aggregates.rejected_count
   (mirrors the Reconciliation unmatched badge pattern). One number,
   visible from any page.

2. Acks page table now sorts rejected rows to the top, then
   newest-id-first. All 5 rejections fit on page 1 since they're a
   tiny fraction of the total — no pagination needed.

New files:
- src/hooks/useAckStats.ts (lightweight aggregates-only fetch)
- src/hooks/useAckStats.test.tsx (2 tests)
- src/components/Sidebar.test.tsx (3 tests)

Pre-existing baseline: 10 frontend failures (api.test.ts exportBatch837,
tail-stream.test.ts acks/ta1_acks targeting, Inbox/InboxHeader copy).
Unchanged.

src/hooks/useAckStats.test.tsx

@@ -0,0 +1,51 @@
+// @vitest-environment happy-dom
+(globalThis as { IS_REACT_ACT_ENVIRONMENT?: boolean }).IS_REACT_ACT_ENVIRONMENT = true;
+
+import { describe, expect, it, vi, beforeEach } from "vitest";
+import { QueryClient, QueryClientProvider } from "@tanstack/react-query";
+import { renderHook, waitFor } from "@testing-library/react";
+import type { ReactNode } from "react";
+
+// Mock the API module so the hook doesn't touch the real /api/acks endpoint.
+// The mock mirrors the wire shape that listAcks returns.
+vi.mock("@/lib/api", () => ({
+  api: {
+    isConfigured: true,
+    listAcks: vi.fn().mockResolvedValue({
+      items: [],
+      total: 1161,
+      returned: 0,
+      has_more: false,
+      aggregates: { accepted: 1156, rejected: 5, received: 1161 },
+    }),
+  },
+}));
+
+import { useAckStats } from "./useAckStats";
+
+function withClient(): (props: { children: ReactNode }) => ReactNode {
+  const qc = new QueryClient({ defaultOptions: { queries: { retry: false } } });
+  return ({ children }) => (
+    <QueryClientProvider client={qc}>{children}</QueryClientProvider>
+  );
+}
+
+describe("useAckStats", () => {
+  beforeEach(() => {
+    vi.clearAllMocks();
+  });
+
+  it("returns server-computed aggregates for sidebar badge use", async () => {
+    const { result } = renderHook(() => useAckStats(), { wrapper: withClient() });
+    await waitFor(() => expect(result.current.isSuccess).toBe(true));
+    expect(result.current.data?.aggregates.rejected).toBe(5);
+    expect(result.current.data?.aggregates.accepted).toBe(1156);
+  });
+
+  it("calls listAcks with limit=1 so the lightweight payload is smallest", async () => {
+    const { api } = await import("@/lib/api");
+    renderHook(() => useAckStats(), { wrapper: withClient() });
+    await waitFor(() => expect(api.listAcks).toHaveBeenCalled());
+    expect(api.listAcks).toHaveBeenCalledWith({ limit: 1 });
+  });
+});

src/hooks/useAckStats.ts

@@ -0,0 +1,29 @@
+import { useQuery } from "@tanstack/react-query";
+import { api, type AckAggregates } from "@/lib/api";
+
+/**
+ * Lightweight 999 ACK aggregates for sidebar / nav badges.
+ *
+ * Hits `/api/acks?limit=1` so the server-side aggregates (which are
+ * computed by iterating every persisted row) come back without
+ * pulling 100 rows for a sidebar badge. The 1-row payload is the
+ * smallest the endpoint accepts (limit ge=1).
+ *
+ * Used by the Sidebar's "999 ACKs" nav badge to surface the
+ * total rejected segment count without forcing a full page fetch.
+ * 60s stale time matches the page-level fetch cadence — the badge
+ * refreshes when the user navigates to the page anyway.
+ *
+ * Distinct from useAcks (which fetches paginated rows + aggregates
+ * for the page body). The query key intentionally differs so React
+ * Query doesn't dedupe the two into a single entry.
+ */
+export function useAckStats() {
+  return useQuery<{ aggregates: AckAggregates }>({
+    queryKey: ["acks", "stats"],
+    queryFn: () => api.listAcks({ limit: 1 }) as Promise<{ aggregates: AckAggregates }>,
+    enabled: api.isConfigured,
+    staleTime: 60_000,
+    refetchOnWindowFocus: false,
+  });
+}