feat(rooms): Fix NaN online stats + Add DAARWIZZ CTA on homepage

- Fixed NaN in online stats by using nullish coalescing (?? 0)
- Added members_online, zone, room_type to /api/v1/city/rooms response
- Added DAARWIZZ chat CTA section on homepage with link to city-lobby
- Created task files for next phases:
  - TASK_PHASE_CITY_ROOMS_FINISH_v2.md
  - TASK_PHASE_AGENT_MANAGEMENT_v1.md
  - TASK_PHASE_CITIZENS_DIRECTORY_v1.md

apps/web/src/app/city/page.tsx

@@ -31,10 +31,10 @@ export default function CityPage() {
   // Use SSE presence data if available, otherwise fallback to API data
   const totalOnline = cityOnline > 0 
     ? cityOnline 
-    : rooms.reduce((sum, r) => sum + r.members_online, 0)
+    : rooms.reduce((sum, r) => sum + (r.members_online ?? 0), 0)
   
   const activeRooms = Object.values(roomsPresence).filter(p => p.online > 0).length ||
-    rooms.filter(r => r.members_online > 0).length
+    rooms.filter(r => (r.members_online ?? 0) > 0).length
 
   if (loading) {
     return (

apps/web/src/app/page.tsx

@@ -1,5 +1,5 @@
 import Link from 'next/link'
-import { ArrowRight, Building2, Bot, Users, Zap, Shield, Globe } from 'lucide-react'
+import { ArrowRight, Building2, Bot, Users, Zap, Shield, Globe, MessageCircle } from 'lucide-react'
 import { StatusIndicator } from '@/components/StatusIndicator'
 
 const features = [
@@ -95,6 +95,35 @@ export default function HomePage() {
         </div>
       </section>
 
+      {/* DAARWIZZ Chat CTA */}
+      <section className="px-4 py-12 sm:py-16">
+        <div className="max-w-4xl mx-auto">
+          <div className="glass-panel p-8 sm:p-10 text-center bg-gradient-to-br from-fuchsia-950/50 to-violet-950/50 border-fuchsia-500/20">
+            <div className="inline-flex p-4 rounded-2xl bg-gradient-to-br from-fuchsia-500/20 to-violet-600/20 mb-6">
+              <MessageCircle className="w-10 h-10 text-fuchsia-400" />
+            </div>
+            
+            <h2 className="text-2xl sm:text-3xl font-bold text-white mb-4">
+              Головний публічний чат DAARION
+            </h2>
+            
+            <p className="text-slate-300 max-w-xl mx-auto mb-8 leading-relaxed">
+              Поспілкуйтеся з <span className="text-fuchsia-400 font-semibold">DAARWIZZ</span> — 
+              головним AI-помічником міста, та іншими громадянами в City Lobby.
+            </p>
+            
+            <Link
+              href="/city?room=city-lobby"
+              className="inline-flex items-center justify-center gap-2 px-8 py-4 text-lg font-semibold bg-gradient-to-r from-fuchsia-500 to-violet-600 rounded-2xl hover:from-fuchsia-400 hover:to-violet-500 transition-all duration-300 shadow-lg shadow-fuchsia-500/25 hover:shadow-fuchsia-500/40 hover:scale-105"
+            >
+              <Bot className="w-5 h-5" />
+              Поспілкуватися з DAARWIZZ
+              <ArrowRight className="w-5 h-5" />
+            </Link>
+          </div>
+        </div>
+      </section>
+
       {/* Vision Section */}
       <section className="px-4 py-16 sm:py-24">
         <div className="max-w-4xl mx-auto">