fix(game): clear nudges on game completion

Nudge badges lingered in the lobby on games that ended by turn-timeout,
resignation or forfeit: those paths commit the finish directly, bypassing
the move path's per-mover NudgeClearer, so the awaited seat's nudge was
never marked read. A finished game's nudges are stale, so clear them all.

Add a wired NudgeExpirer (social.ExpireNudges) called from the shared
commit finish block — covering every completion path — and from the
voidGame recovery path. It clears every seat's nudge bits for the game
and leaves chat messages unread; unlike ClearNudges it records no
publish-to-read latency, since a completion is an expiry, not a read.

An integration test reproduces the timeout case (nudge cleared, chat
kept). Docs: ARCHITECTURE §9.1, FUNCTIONAL (+_ru), backend/README.

docs/ARCHITECTURE.md

@@ -646,7 +646,11 @@ in either direction (the enqueue excludes the caller's `BlockedWith` set);
   robot instead clears when the robot answers by moving, as for a human. A seat's bit clears when that player **opens the move history or the chat**
   (`POST /games/:id/chat/read`, which the client sends only when it holds unread, so a
   history open is not a constant backend call), and a **nudge additionally clears when its
-  recipient answers by moving** (the move path calls a wired `NudgeClearer`). The mask is
+  recipient answers by moving** (the move path calls a wired `NudgeClearer`); and **every nudge in
+  a game is marked read when that game finishes** — on any completion path (a closing move, a
+  resignation, a forfeit or a turn-timeout, all funnelling through the shared `commit`), since the
+  nudge badge is stale once the game is over (a wired `NudgeExpirer`; chat messages stay unread and
+  this expiry records no read latency). The mask is
   inverted so "anything unread" is a plain `unread_seats <> 0`, which the per-viewer
   `unread_chat` game-view flag (seeding the lobby and in-game unread **dot**), the admin
   unread filter and the unread gauge all use. A second per-viewer flag, **`unread_messages`**
@@ -656,8 +660,9 @@ in either direction (the enqueue excludes the caller's `BlockedWith` set);
   REST-seed-then-event-bump lifecycle (the live-event game-view leaves them false; a nudge
   event raises only `unread_chat`, a message event raises both). The lobby additionally
   **floats games with any unread entry to the top** of the your-turn and opponent-turn
-  sections (the finished section keeps its activity order). On each clear the publish-to-read
-  latency is recorded; the read time itself is not retained.
+  sections (the finished section keeps its activity order). On each player-driven clear the
+  publish-to-read latency is recorded — the completion expiry records none (it is not a read);
+  the read time itself is not retained.
 - **Profile**: `preferred_language` (en/ru; tracks the interface language — §4), display name, email
   (confirm-code binding, see §4), **timezone**, the daily **away window**, the
   **variant preferences** (`variant_preferences`, the matchable-variant set that gates New

docs/FUNCTIONAL.md

@@ -260,7 +260,8 @@ entry — a message **or a nudge** from an opponent — raises a small **dot** n
 in the **lobby** and on the game's **score bar**, **red when an unread message is waiting and a
 softer amber when only nudges are**. 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**.
+also clears the moment its recipient **takes their move**, and **all of a game's nudges clear once
+the game finishes** (the badge is stale once the game is over) — but unread chat messages stay unread.
 
 ### Profile & settings
 Edit the display name (letters joined by a single space / "." / "_" separator, with an

docs/FUNCTIONAL_ru.md

@@ -269,7 +269,8 @@ _Вход сейчас только через провайдера, поэто
 **лобби** и на **строке счёта** партии, **красную при непрочитанном сообщении и мягкую жёлтую,
 когда непрочитаны только nudge'и**. Открытие **истории ходов** считается прочтением чата, даже
 без захода в него: точка гаснет, а значок 💬 **дважды мигает**. Nudge также гаснет в момент, когда
-его получатель **делает ход**.
+его получатель **делает ход**, и **все nudge'и партии гаснут по её завершении** (после конца
+партии бейдж неактуален) — но непрочитанные сообщения чата остаются непрочитанными.
 
 ### Профиль и настройки
 Редактирование отображаемого имени (буквы, разделённые одиночным пробелом / «.» /