feat: honest AI opponent in quick game

New Game's quick game gains an explicit opponent selector — 🤖 AI (default)
or 👤 Random player. AI starts a game seated with a pooled robot that joins
and moves at once: no per-move timeout (a 7-day inactivity loss reusing the
turn-timeout sweeper), chat/nudge disabled, no statistics, the opponent shown
as 🤖 everywhere. The random path (disguised robot) is unchanged.

Driven by one game flag (games.vs_ai), set only on AI-started games so the
disguised path is never revealed; Matchmaker.StartVsAI seats the robot
directly (no open pool); the robot replies event-driven via the game service's
after-commit/after-create hook. Wire: vs_ai on EnqueueRequest + GameView.

backend/internal/game/store.go

@@ -41,6 +41,8 @@ type gameInsert struct {
 	dropoutTiles    string
 	// multipleWordsPerTurn false selects the single-word rule for the game.
 	multipleWordsPerTurn bool
+	// vsAI marks an honest-AI game (games.vs_ai).
+	vsAI bool
 	// status is the lifecycle state to create the game in: StatusActive for a normal
 	// seated game, StatusOpen for an auto-match game still awaiting an opponent. An
 	// empty string defaults to StatusActive.
@@ -130,9 +132,9 @@ func insertGameTx(ctx context.Context, tx *sql.Tx, ins gameInsert, seats []uuid.
 		table.Games.GameID, table.Games.Variant, table.Games.DictVersion, table.Games.Seed,
 		table.Games.Status, table.Games.Players, table.Games.TurnTimeoutSecs,
 		table.Games.HintsAllowed, table.Games.HintsPerPlayer, table.Games.OpenDeadlineAt,
-		table.Games.DropoutTiles, table.Games.MultipleWordsPerTurn,
+		table.Games.DropoutTiles, table.Games.MultipleWordsPerTurn, table.Games.VsAi,
 	).VALUES(ins.id, ins.variant, ins.dictVersion, ins.seed, status, ins.players,
-		ins.turnTimeoutSecs, ins.hintsAllowed, ins.hintsPerPlayer, deadline, ins.dropoutTiles, ins.multipleWordsPerTurn)
+		ins.turnTimeoutSecs, ins.hintsAllowed, ins.hintsPerPlayer, deadline, ins.dropoutTiles, ins.multipleWordsPerTurn, ins.vsAI)
 	if _, err := gi.ExecContext(ctx, tx); err != nil {
 		return fmt.Errorf("insert game: %w", err)
 	}
@@ -916,7 +918,7 @@ func (s *Store) RobotTurns(ctx context.Context, ids []uuid.UUID) ([]RobotTurn, e
 	}
 	stmt := postgres.SELECT(
 		table.Games.GameID, table.Games.ToMove, table.Games.TurnStartedAt,
-		table.Games.MoveCount, table.Games.Seed,
+		table.Games.MoveCount, table.Games.Seed, table.Games.VsAi,
 		table.GamePlayers.Seat, table.GamePlayers.AccountID,
 	).FROM(
 		table.Games.INNER_JOIN(table.GamePlayers, table.GamePlayers.GameID.EQ(table.Games.GameID)),
@@ -934,24 +936,85 @@ func (s *Store) RobotTurns(ctx context.Context, ids []uuid.UUID) ([]RobotTurn, e
 	}
 	out := make([]RobotTurn, 0, len(rows))
 	for _, r := range rows {
-		// The filter matches only the robot's (non-null) seat, so AccountID is set.
-		robotID := uuid.Nil
-		if r.GamePlayers.AccountID != nil {
-			robotID = *r.GamePlayers.AccountID
-		}
-		out = append(out, RobotTurn{
-			GameID:        r.Games.GameID,
-			RobotID:       robotID,
-			RobotSeat:     int(r.GamePlayers.Seat),
-			ToMove:        int(r.Games.ToMove),
-			TurnStartedAt: r.Games.TurnStartedAt,
-			MoveCount:     int(r.Games.MoveCount),
-			Seed:          r.Games.Seed,
-		})
+		out = append(out, robotTurnFrom(r.Games, r.GamePlayers))
 	}
 	return out, nil
 }
 
+// RobotTurnByGame returns the robot turn for a single active game — the seat held by
+// one of ids (the robot pool) — and true, or false when the game is not active, holds
+// no pooled robot, or is gone. It backs the honest-AI after-commit trigger, which
+// drives one game at once rather than scanning the whole pool (RobotTurns).
+func (s *Store) RobotTurnByGame(ctx context.Context, gameID uuid.UUID, ids []uuid.UUID) (RobotTurn, bool, error) {
+	if len(ids) == 0 {
+		return RobotTurn{}, false, nil
+	}
+	exprs := make([]postgres.Expression, len(ids))
+	for i, id := range ids {
+		exprs[i] = postgres.UUID(id)
+	}
+	stmt := postgres.SELECT(
+		table.Games.GameID, table.Games.ToMove, table.Games.TurnStartedAt,
+		table.Games.MoveCount, table.Games.Seed, table.Games.VsAi,
+		table.GamePlayers.Seat, table.GamePlayers.AccountID,
+	).FROM(
+		table.Games.INNER_JOIN(table.GamePlayers, table.GamePlayers.GameID.EQ(table.Games.GameID)),
+	).WHERE(
+		table.Games.GameID.EQ(postgres.UUID(gameID)).
+			AND(table.Games.Status.EQ(postgres.String(StatusActive))).
+			AND(table.GamePlayers.AccountID.IN(exprs...)),
+	).LIMIT(1)
+
+	var rows []struct {
+		model.Games
+		model.GamePlayers
+	}
+	if err := stmt.QueryContext(ctx, s.db, &rows); err != nil {
+		return RobotTurn{}, false, fmt.Errorf("game: robot turn by game: %w", err)
+	}
+	if len(rows) == 0 {
+		return RobotTurn{}, false, nil
+	}
+	return robotTurnFrom(rows[0].Games, rows[0].GamePlayers), true, nil
+}
+
+// robotTurnFrom projects a games row joined with the robot's seat into a RobotTurn.
+// The query matches only the robot's (non-null) seat, so AccountID is set.
+func robotTurnFrom(g model.Games, p model.GamePlayers) RobotTurn {
+	robotID := uuid.Nil
+	if p.AccountID != nil {
+		robotID = *p.AccountID
+	}
+	return RobotTurn{
+		GameID:        g.GameID,
+		RobotID:       robotID,
+		RobotSeat:     int(p.Seat),
+		ToMove:        int(g.ToMove),
+		TurnStartedAt: g.TurnStartedAt,
+		MoveCount:     int(g.MoveCount),
+		Seed:          g.Seed,
+		VsAI:          g.VsAi,
+	}
+}
+
+// GameVsAI reports whether a game is an honest-AI game (games.vs_ai) — a cheap
+// single-column read for the social chat/nudge gate, which must reject both in an
+// AI game even though it reports status 'active'. ErrNotFound when the game is gone.
+func (s *Store) GameVsAI(ctx context.Context, id uuid.UUID) (bool, error) {
+	stmt := postgres.SELECT(table.Games.VsAi).
+		FROM(table.Games).
+		WHERE(table.Games.GameID.EQ(postgres.UUID(id))).
+		LIMIT(1)
+	var row model.Games
+	if err := stmt.QueryContext(ctx, s.db, &row); err != nil {
+		if errors.Is(err, qrm.ErrNoRows) {
+			return false, ErrNotFound
+		}
+		return false, fmt.Errorf("game: get vs_ai %s: %w", id, err)
+	}
+	return row.VsAi, nil
+}
+
 // GameSeed returns the bag seed a game was dealt from, used to replay it. The
 // seed is server-only state and never travels in the public Game view.
 func (s *Store) GameSeed(ctx context.Context, id uuid.UUID) (int64, error) {
@@ -1046,6 +1109,7 @@ func projectGame(g model.Games, seats []model.GamePlayers) (Game, error) {
 		UpdatedAt:      g.UpdatedAt,
 	}
 	out.MultipleWordsPerTurn = g.MultipleWordsPerTurn
+	out.VsAI = g.VsAi
 	if g.EndReason != nil {
 		out.EndReason = *g.EndReason
 	}