feat: add single player snake mode

Add solo mode where players survive as long as possible while tracking
score (food eaten). Single player games start with a shorter 3-second
countdown vs 10 seconds for multiplayer, maintain exactly 1 food item
for classic snake feel, and end when the player dies rather than when
one player remains.

- Add GameMode type (ModeMultiplayer/ModeSinglePlayer) and Score field
- Filter single player games from "Join a Game" lobby list
- Show "Ready?" and "Score: X" UI for single player mode
- Hide invite link for single player games
- Preserve game mode on rematch

db/persister.go

@@ -171,6 +171,7 @@ func (p *SnakePersister) SaveSnakeGame(sg *snake.SnakeGame) error {
 			Status:     int64(sg.Status),
 			GridWidth:  gridWidth,
 			GridHeight: gridHeight,
+			GameMode:   int64(sg.Mode),
 		})
 		return err
 	}
@@ -193,6 +194,7 @@ func (p *SnakePersister) SaveSnakeGame(sg *snake.SnakeGame) error {
 		Status:        int64(sg.Status),
 		WinnerUserID:  winnerUserID,
 		RematchGameID: rematchGameID,
+		Score:         int64(sg.Score),
 		ID:            sg.ID,
 	})
 }
@@ -220,6 +222,8 @@ func (p *SnakePersister) LoadSnakeGame(id string) (*snake.SnakeGame, error) {
 		State:   state,
 		Players: make([]*snake.Player, 8),
 		Status:  snake.Status(row.Status),
+		Mode:    snake.GameMode(row.GameMode),
+		Score:   int(row.Score),
 	}
 
 	if row.RematchGameID.Valid {