Phase 2: API client, typed hooks, live probe

- Add api:types script (openapi-typescript against live FastAPI)
- Generate src/lib/api/types.ts (2972 lines, 55 paths, 16 schemas)
- lib/api/client.ts: typed apiRequest + ApiError + makeQueryClient
  (staleTime 5s, no refetchOnWindowFocus to preserve editor state)
- lib/providers.tsx: QueryClientProvider client component, useState
  singleton so App Router re-renders don't dump the cache
- lib/api/cases.ts: Case type + casesKeys + useCases hook (pragmatic
  hand-typed Case pending backend response-model annotations)
- layout.tsx: wrap children with <Providers>
- Smoke test: CasesLiveProbe component on home page hitting live FastAPI
  via /api/cases rewrite proxy

Phase 2 deliverable check: useCases() returns typed Case[] from the
production FastAPI through the Next.js proxy. End-to-end wiring proven.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

web-ui/src/app/page.tsx

@@ -1,4 +1,5 @@
 import { AppShell } from "@/components/app-shell";
+import { CasesLiveProbe } from "@/components/cases/cases-live-probe";
 
 export default function HomePage() {
   return (
@@ -15,11 +16,8 @@ export default function HomePage() {
         <hr className="border-0 h-[2px] bg-gradient-to-l from-transparent via-gold to-transparent" />
 
         <div className="rounded-lg bg-surface shadow-sm p-6 border border-rule">
-          <p className="text-ink">
-            שלב 1 של תוכנית השדרוג הושלם: scaffold של Next.js 16, פורט של design
-            tokens ל-Tailwind v4, RTL עברית עם Heebo. הפאזות הבאות יביאו את 10
-            המסכים ל-parity עם ה-UI הקיים.
-          </p>
+          <h2 className="text-navy text-xl mb-3">פאזה 2 — חיבור חי ל-API</h2>
+          <CasesLiveProbe />
         </div>
       </section>
     </AppShell>