What does the statement "the truffle he had routed out of earth" was "Rotten at the center - - rotten to the core!" mean?
Ah, an interesting question. Truffles are considered quite precious. (They are very expensive food!) For one to be rotten is a terrible disappointment.
Pigs are often used to sniff out truffles, and if pigs aren't, other lower animals like dogs are. This part is implied, not stated, but it refers to how much lower the jeweler sees himself as being than the duchess or Diana. Taken together, the lines therefore mean he sees himself as a lower creature who found something very valuable, only to have it be ruined by being rotten.
Since this line comes after the question about whether the pearls are real or fake, this line refers to the pearls being fake. In that case, he then gave away money for nothing, disappointing his mother's memory and costing himself a lot of money, all to get a weekend with Diana. It would also refer to his sense of self. He had been thinking of himself as impressive, but he now knows he didn't check the pearls, and made a stupid mistake.
that the truffle he got- diana and invitation to the ball thing- was rotten in the way of how he got it