feat(chat): unread read-receipts with lobby/game dot and history-open ack

Persist per-message read state as a chat_messages.unread_seats bitmask
(migration 00008): a text message seeds every recipient seat's bit, a nudge
only the awaited seat's. A seat's bit clears when the player opens the move
history or chat (POST /games/:id/chat/read, sent only when something is
unread), and a nudge additionally clears when its recipient answers by moving
(a wired game NudgeClearer, dependency-inverted so game keeps off social).

UI shows a per-viewer unread dot in the lobby (next to the opponent) and the
game score bar — the unread_chat game-view flag seeds it from authoritative
REST views, live chat/nudge events raise it. Opening the move history counts
as reading (even without entering chat): the 💬 fade-blinks twice and the
client acks. Admin Messages gains an unread-only filter, a read/unread column,
and a per-message card with the per-seat read breakdown. Observability:
chat_read_duration histogram + chat_unread_messages gauge + social tracing.

docs/FUNCTIONAL.md

@@ -191,8 +191,11 @@ is awaited at most once per hour (the
 nudge is part of the game chat); both the in-app toast and the out-of-app push (delivered
 via the platform) **name the nudger** ("<opponent>: waiting for your move").
 Chat and the word-check tool share one **comms screen** with **💬 chat** / **🔎 dictionary**
-tabs, reached from the 💬 in the move-history header (with a back to the game); a new chat
-message raises an **unread badge** on the game's score bar and the 💬 until the chat is opened.
+tabs, reached from the 💬 in the move-history header (with a back to the game). An unread chat
+entry — a message **or a nudge** from an opponent — raises a small **red dot** next to the game
+in the **lobby** and on the game's **score bar**. Opening the **move history** counts as reading
+the chat, even without entering it: the dot clears and the 💬 icon **fade-blinks twice**. A nudge
+also clears the moment its recipient **takes their move**.
 
 ### Profile & settings
 Edit the display name (letters joined by a single space / "." / "_" separator, with an