Three times a week, millions of Americans participate in a ritual as old as hope itself — the purchase of a small slip of paper that briefly makes the impossible feel plausible. The Powerball jackpot has now grown to $567 million after months without a grand-prize winner, a sum large enough to reshape any life yet still dwarfed by the billion-dollar prizes the game has produced in recent years. Wednesday's drawing offers another moment of collective suspension, when the distance between ordinary life and extraordinary fortune narrows to six numbers.