perf(backend): cut evaluate's DB round-trips; load the game in one query

EvaluatePlay (the hottest gameplay call, fired on every tile placement) now uses
the warm live-game cache directly: an active game stays cached (mutated in place
across moves, evicted only on finish), so the cached engine game and its immutable
seat list answer the membership check and the score with no DB read. The cold path
(eviction / first load) still loads and validates via the store. The seat list is
cached alongside the engine game for the membership fast path.

GetGame also folds its two round-trips (game, then seats) into one LEFT JOIN,
preserving the contract (same Game, a seatless game still returns empty seats, seat
order kept) — one round-trip for every remaining caller.

Measured at 500 players: evaluate p99 halves (200 -> 100 ms) and the per-op query
count drops. It does NOT cut postgres CPU — that is write-bound (per-move CommitMove
plus draft upserts and journal replays), the cheap indexed GetGame reads were never
its bottleneck, and postgres runs with headroom (~1.5 of 2 cores). So this is a
latency / query-volume optimization, not a DB-CPU one.

Regression cover: a non-player evaluate against a warm game asserts the cached-seat
membership path; the integration suite exercises GetGame's join across every game op.

backend/internal/game/store.go

@@ -355,27 +355,33 @@ func (s *Store) ExpiredOpen(ctx context.Context, now time.Time) ([]OpenGame, err
 // GetGame loads the games row joined with its seats (ordered by seat), or
 // ErrNotFound.
 func (s *Store) GetGame(ctx context.Context, id uuid.UUID) (Game, error) {
-	gstmt := postgres.SELECT(table.Games.AllColumns).
-		FROM(table.Games).
+	// One round-trip: the game joined with its seats. A LEFT JOIN keeps a (would-be)
+	// seatless game returning the game with no seats, exactly as the prior two-query
+	// version did; ORDER BY seat preserves seat order. The games columns repeat per seat
+	// row — cheap at 2-4 seats, and one round-trip instead of two, which matters because
+	// GetGame is the universal "load the game" step on every game operation.
+	stmt := postgres.SELECT(table.Games.AllColumns, table.GamePlayers.AllColumns).
+		FROM(table.Games.LEFT_JOIN(table.GamePlayers, table.GamePlayers.GameID.EQ(table.Games.GameID))).
 		WHERE(table.Games.GameID.EQ(postgres.UUID(id))).
-		LIMIT(1)
-	var grow model.Games
-	if err := gstmt.QueryContext(ctx, s.db, &grow); err != nil {
-		if errors.Is(err, qrm.ErrNoRows) {
-			return Game{}, ErrNotFound
-		}
+		ORDER_BY(table.GamePlayers.Seat.ASC())
+	var rows []struct {
+		model.Games
+		model.GamePlayers
+	}
+	if err := stmt.QueryContext(ctx, s.db, &rows); err != nil {
 		return Game{}, fmt.Errorf("game: get %s: %w", id, err)
 	}
-
-	sstmt := postgres.SELECT(table.GamePlayers.AllColumns).
-		FROM(table.GamePlayers).
-		WHERE(table.GamePlayers.GameID.EQ(postgres.UUID(id))).
-		ORDER_BY(table.GamePlayers.Seat.ASC())
-	var srows []model.GamePlayers
-	if err := sstmt.QueryContext(ctx, s.db, &srows); err != nil {
-		return Game{}, fmt.Errorf("game: get seats %s: %w", id, err)
+	if len(rows) == 0 {
+		return Game{}, ErrNotFound
 	}
-	return projectGame(grow, srows)
+	seats := make([]model.GamePlayers, 0, len(rows))
+	for i := range rows {
+		// Skip the phantom all-NULL seat row a LEFT JOIN yields for a seatless game.
+		if rows[i].GamePlayers.GameID == id {
+			seats = append(seats, rows[i].GamePlayers)
+		}
+	}
+	return projectGame(rows[0].Games, seats)
 }
 
 // GetGameVariant reads just a game's variant — a cheap single-column lookup the edge uses