feat(ui): chair approval center — one page for every pending human-gate (#63 follow-up)

Dafna asked for a single page under the prod site listing everything she needs
to approve, so nothing is forgotten — the visible embodiment of INV-G10 (human
gates) and INV-QA1 (halacha backlog must be visible).

Backend — GET /api/chair/pending aggregates every pending chair gate, each as a
direct source query (count + sample + action link):
- halachot review backlog (review_status='pending_review') + oldest
- open missing precedents
- unresolved chair_feedback
- qa_failed cases
- gold-set review (FU-5, file-based, best-effort: total vs source='chair')

Frontend — /approvals page ("מרכז אישורים"):
- src/lib/api/chair.ts — usePendingApprovals() (hand-typed until next api:types)
- src/app/approvals/page.tsx — card per category, severity-coloured count, sample
  rows, oldest-pending date, link to where each is handled; live (60s refetch)
- app-shell nav: "מרכז אישורים" in the work group + total-pending badge (quiet at 0)

Live counts at build time surfaced the value immediately: 226 open missing
precedents, 178 pending halachot, 20 unapplied feedback notes, 1 qa_failed.

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

web-ui/src/components/app-shell.tsx

@@ -16,6 +16,7 @@ import {
 import { GlobalSearch } from "@/components/global-search";
 import { headerSubtitle } from "@/components/header-context";
 import { useMissingPrecedentsOpenCount } from "@/lib/api/missing-precedents";
+import { usePendingApprovals } from "@/lib/api/chair";
 
 /**
  * Ezer Mishpati navigation shell — two-row header.
@@ -39,8 +40,9 @@ const NAV_GROUPS: NavGroup[] = [
   {
     id: "work",
     items: [
-      { href: "/",        label: "בית"    },
-      { href: "/archive", label: "ארכיון" },
+      { href: "/",          label: "בית"          },
+      { href: "/approvals", label: "מרכז אישורים" },
+      { href: "/archive",   label: "ארכיון"       },
     ],
   },
   {
@@ -244,6 +246,7 @@ function NavLink({ item, active }: { item: NavItem; active: boolean }) {
     >
       <span>{item.label}</span>
       {item.href === "/missing-precedents" ? <MissingPrecedentsBadge /> : null}
+      {item.href === "/approvals" ? <ApprovalsBadge /> : null}
       {active && (
         <span
           className="absolute -bottom-[19px] inset-x-2 h-[2px] bg-gold"
@@ -254,6 +257,23 @@ function NavLink({ item, active }: { item: NavItem; active: boolean }) {
   );
 }
 
+/* Total pending-approvals badge next to "מרכז אישורים" — Dafna's outstanding
+ * human-gate items (halachot + missing precedents + feedback + qa_failed).
+ * Renders only when >0 so the nav stays quiet when everything is cleared. */
+function ApprovalsBadge() {
+  const { data } = usePendingApprovals();
+  const total = data?.total_pending ?? 0;
+  if (!total) return null;
+  return (
+    <span
+      className="ms-1 inline-flex items-center justify-center min-w-[1.25rem] h-4 px-1 rounded-full bg-gold text-navy text-[0.65rem] font-semibold"
+      aria-label={`${total} פריטים ממתינים לאישורך`}
+    >
+      {total}
+    </span>
+  );
+}
+
 /* Small open-count badge next to "פסיקה חסרה" — only renders when >0
  * so the nav stays quiet in normal operation. */
 function MissingPrecedentsBadge() {