fix(chat): no chat field or nudge on a finished game

A finished game is read-only: gate the nudge behind canNudge (active game,
opponent's turn) so it no longer shows on a finished (or otherwise
non-active) game — where the backend rejects it anyway. The message field
is already hidden off your own turn. Extend the finished-game e2e to assert
neither Send nor Nudge is offered.

ui/src/game/Chat.svelte

@@ -9,6 +9,7 @@
     busy,
     myTurn = false,
     canSend = false,
+    canNudge = false,
     waiting = false,
     nudgeOnCooldown = false,
     onsend,
@@ -17,16 +18,18 @@
     messages: ChatMessage[];
     myId: string;
     busy: boolean;
-    // The input area has three turn-driven states: on your own turn the message field +
-    // send show while you may still post (canSend); once the one allowed message is sent a
-    // short caption replaces them until the next turn; on the opponent's turn only the
-    // nudge button shows (nudging your own turn makes no sense — there is no one to hurry).
-    // While the hourly nudge cooldown is active the nudge is disabled with an "awaiting
-    // reply" caption.
+    // The input area is turn-driven, in priority order: while you may still post (canSend)
+    // the message field + send show; once your one allowed message is sent a short caption
+    // replaces them until the next turn (myTurn); on the opponent's turn of an active game
+    // the nudge button shows (canNudge) — disabled with an "awaiting reply" caption while the
+    // hourly cooldown is active; a finished game offers nothing (it is read-only).
     myTurn?: boolean;
     // canSend is true while the player may still post this turn (own turn and the per-turn
     // limit not yet reached); it gates the message field.
     canSend?: boolean;
+    // canNudge is true on the opponent's turn of an active game; it gates the nudge button
+    // (nudging your own turn or a finished game makes no sense).
+    canNudge?: boolean;
     // waiting is true while an auto-match game still has no opponent: both send and nudge
     // are disabled (there is no one to message or hurry yet).
     waiting?: boolean;
@@ -71,7 +74,7 @@
       <!-- Own turn, the one allowed message already sent: a caption holds the row until the
            next turn (no nudge — there is no one to hurry on your own turn). -->
       <span class="cooldown">{t('chat.sentThisTurn')}</span>
-    {:else}
+    {:else if canNudge}
       <!-- A flex:1 caption keeps the nudge pinned right whether or not the cooldown text shows. -->
       <span class="cooldown">{nudgeOnCooldown ? t('chat.awaitingReply') : ''}</span>
       <button class="iconbtn" onclick={onnudge} disabled={busy || waiting || nudgeOnCooldown || !connection.online} aria-label={t('chat.nudgeAction')}>🛎️</button>

ui/src/game/ChatScreen.svelte

@@ -26,6 +26,9 @@
   // the turn advances. canSend hides the field once the limit is reached (and always on
   // the opponent's turn).
   const canSend = $derived(canSendChat(isMyTurn, view ? sentThisTurn(messages, myId, view.game.lastActivityUnix) : 0));
+  // The nudge is offered only on the opponent's turn of an active game (you hurry the player
+  // who is to move). A finished game is read-only: neither chat nor nudge is offered.
+  const canNudge = $derived(!!view && view.game.status === 'active' && view.game.toMove !== view.seat);
   // While the auto-match game still has no opponent, chat and nudge are both disabled.
   const waiting = $derived(!!view && view.game.status === 'open');
   const nudgeCooldownSecs = 3600;
@@ -110,4 +113,4 @@
   }
 </script>
 
-<Chat {messages} {myId} {busy} myTurn={isMyTurn} {canSend} {waiting} {nudgeOnCooldown} onsend={sendChat} onnudge={nudge} />
+<Chat {messages} {myId} {busy} myTurn={isMyTurn} {canSend} {canNudge} {waiting} {nudgeOnCooldown} onsend={sendChat} onnudge={nudge} />