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