feat(chat): unread count + sidebar badge (Q1)

Track a per-conversation "last seen" watermark (message timestamp) on App:
- chatUnreadCount() sums incoming messages newer than each conversation's
  watermark; surfaced as SidebarStatus.chatUnreadCount → a badge on the Chat nav
  item (mirrors the History/Peers badges).
- Viewing a thread marks it seen (markChatConversationSeen while displayed).
- Baseline on load: existing stored messages are marked seen, so only messages
  that arrive while the app is open badge as unread.
- Wiped in resetChatSession so unread state never leaks across a wallet switch.
In-memory only (resets on app restart); persistence is a later refinement.

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

src/app_network.cpp

@@ -1227,6 +1227,26 @@ void App::processWalletSwitchRevert()
     daemon_restarting_ = false;   // the connect loop now restarts the previous wallet's daemon
 }
 
+int App::chatUnreadCount() const
+{
+    if (!chat::hushChatFeatureEnabledAtBuild() || !chat_service_.hasIdentity()) return 0;
+    int unread = 0;
+    const auto& store = chat_service_.store();
+    for (const auto& cid : store.conversationIds()) {
+        std::int64_t seen = 0;
+        const auto it = chat_seen_watermark_.find(cid);
+        if (it != chat_seen_watermark_.end()) seen = it->second;
+        for (const auto& m : store.conversation(cid))
+            if (m.direction == chat::ChatDirection::Incoming && m.timestamp > seen) ++unread;
+    }
+    return unread;
+}
+
+void App::markChatConversationSeen(const std::string& cid, std::int64_t latestTs)
+{
+    if (latestTs > 0) chat_seen_watermark_[cid] = latestTs;
+}
+
 void App::wipePendingTransactionHistoryCachePassphrase()
 {
     if (!pending_transaction_history_cache_passphrase_.empty()) {
@@ -2658,6 +2678,13 @@ void App::provisionChatIdentityFromSecret(std::string secret)
         chat_service_.setPersistence(&chat_db_);
         if (chat_db_.unlockWithSecret(trimmed)) {
             chat_service_.loadFromDatabase();
+            // Baseline unread: treat everything already in the store at load as read, so only messages
+            // that arrive while the app is open badge as unread (Q1).
+            const auto& store = chat_service_.store();
+            for (const auto& cid : store.conversationIds()) {
+                const auto msgs = store.conversation(cid);
+                if (!msgs.empty()) chat_seen_watermark_[cid] = msgs.back().timestamp;
+            }
         }
     } else {
         chat_identity_unavailable_ = true;
@@ -2677,6 +2704,7 @@ void App::resetChatSession()
     // Drop identity + decrypted plaintext in RAM, lock the seed-encrypted DB, re-arm provisioning.
     chat_service_.clearIdentity();
     chat_service_.store().clear();
+    chat_seen_watermark_.clear();  // unread state is per-wallet — don't leak it across a switch
     ui::ResetChatTab();  // wipe the chat UI's typed plaintext + selection so it can't leak into the next wallet
     chat_db_.lock();
     chat_identity_provisioned_ = false;