feat(backend): per-tier, per-kind active-game limits with a guest funnel

Cap a player's simultaneous unfinished games per kind (vs_ai, random,
friends) with independent guest and durable-account tiers, held in a new
single-row backend.config table (-1 = unlimited) behind an in-memory cache
and editable live in the admin console (/_gm/limits). Each game is tagged
with games.game_kind on creation.

This replaces the earlier flat MaxActiveQuickGames=10 combined cap: the
per-tier/kind config is the single mechanism, enforced at the same handler
gate (ensureUnderGameLimit by kind on lobby/enqueue) plus the durable
friends cap in CreateInvitation. game.Service.AtGameLimit only resolves the
tier and counts; the limit policy stays at the request edge.

Guests are now refused friend requests, friend-code redemption,
befriend-in-game and invitation creation outright (403 guest_forbidden) --
previously only the UI hid these.

Admin: a kind column in both game lists and the config editor.

Defaults: guest 1 vs_ai / 1 random / 0 friends; durable 10 / 10 / 10.

backend/internal/game/store.go

@@ -15,6 +15,7 @@ import (
 
 	"scrabble/backend/internal/account"
 	"scrabble/backend/internal/engine"
+	"scrabble/backend/internal/gamelimits"
 	"scrabble/backend/internal/postgres/jet/backend/model"
 	"scrabble/backend/internal/postgres/jet/backend/table"
 )
@@ -45,6 +46,8 @@ type gameInsert struct {
 	multipleWordsPerTurn bool
 	// vsAI marks an honest-AI game (games.vs_ai).
 	vsAI bool
+	// kind tags the game's origin (games.game_kind) for the active-game limits.
+	kind gamelimits.Kind
 	// 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.
@@ -138,6 +141,20 @@ func (s *Store) CreateGame(ctx context.Context, ins gameInsert, seats []seatInse
 	})
 }
 
+// CountActiveByKind counts the account's active (open or in-progress) games of the given kind — the
+// per-tier active-game limit is checked against it before a new game of that kind is created.
+func (s *Store) CountActiveByKind(ctx context.Context, accountID uuid.UUID, kind gamelimits.Kind) (int, error) {
+	var n int
+	if err := s.db.QueryRowContext(ctx,
+		`SELECT count(*) FROM backend.games g
+		   JOIN backend.game_players p ON p.game_id = g.game_id
+		  WHERE p.account_id = $1 AND g.game_kind = $2 AND g.status IN ('open', 'active')`,
+		accountID, int16(kind)).Scan(&n); err != nil {
+		return 0, fmt.Errorf("game: count active by kind %s: %w", accountID, err)
+	}
+	return n, nil
+}
+
 // insertGameTx inserts the games row and one game_players row per seat (seat 0
 // first) on tx, stamping each seat's display-name snapshot. A seat whose account id is
 // uuid.Nil is written with a NULL account_id (and an empty snapshot) — the still-empty
@@ -155,9 +172,9 @@ func insertGameTx(ctx context.Context, tx *sql.Tx, ins gameInsert, seats []seatI
 		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.VsAi,
+		table.Games.DropoutTiles, table.Games.MultipleWordsPerTurn, table.Games.VsAi, table.Games.GameKind,
 	).VALUES(ins.id, ins.variant, ins.dictVersion, ins.seed, status, ins.players,
-		ins.turnTimeoutSecs, ins.hintsAllowed, ins.hintsPerPlayer, deadline, ins.dropoutTiles, ins.multipleWordsPerTurn, ins.vsAI)
+		ins.turnTimeoutSecs, ins.hintsAllowed, ins.hintsPerPlayer, deadline, ins.dropoutTiles, ins.multipleWordsPerTurn, ins.vsAI, int16(ins.kind))
 	if _, err := gi.ExecContext(ctx, tx); err != nil {
 		return fmt.Errorf("insert game: %w", err)
 	}
@@ -501,28 +518,6 @@ func (s *Store) ListGamesForAccount(ctx context.Context, accountID uuid.UUID) ([
 	return out, nil
 }
 
-// CountActiveQuickGames counts the account's in-progress quick games — the ones the
-// simultaneous-game limit (MaxActiveQuickGames) is checked against. It includes both
-// active and still-open (awaiting-opponent) games, the honest-AI ones among them, and
-// excludes friend games (those linked to a game_invitations row) and finished games.
-// A hidden game still occupies a slot, so this is a dedicated count rather than a
-// filter over ListGamesForAccount (which drops hidden games). Joining on the account's
-// own seat counts each game once (an open game's empty opponent seat has no account).
-func (s *Store) CountActiveQuickGames(ctx context.Context, accountID uuid.UUID) (int, error) {
-	// The status literals are game.StatusActive / game.StatusOpen, matching the
-	// games.status CHECK in the baseline migration.
-	const q = `
-SELECT COUNT(*) FROM backend.games g
-JOIN backend.game_players gp ON gp.game_id = g.game_id
-LEFT JOIN backend.game_invitations gi ON gi.game_id = g.game_id
-WHERE gp.account_id = $1 AND g.status IN ('active', 'open') AND gi.game_id IS NULL`
-	var n int
-	if err := s.db.QueryRowContext(ctx, q, accountID).Scan(&n); err != nil {
-		return 0, fmt.Errorf("game: count active quick games: %w", err)
-	}
-	return n, nil
-}
-
 // HideGame hides a game from the account's own lobby list (idempotent). The caller validates the
 // game is finished and the account is a player.
 func (s *Store) HideGame(ctx context.Context, accountID, gameID uuid.UUID) error {
@@ -1361,6 +1356,7 @@ func projectGame(g model.Games, seats []model.GamePlayers) (Game, error) {
 	}
 	out.MultipleWordsPerTurn = g.MultipleWordsPerTurn
 	out.VsAI = g.VsAi
+	out.Kind = gamelimits.Kind(g.GameKind)
 	if g.EndReason != nil {
 		out.EndReason = *g.EndReason
 	}