The fund was opened by the federal government in 2011 to compensate for deaths and illnesses linked to toxic exposure at the World Trade Center, the Pentagon, and in Shanksville, Pa., after terrorists crashed four hijacked airliners in 2001. To date, the $7.3 billion fund has paid about $5 billion to roughly 21,000 claimants. About 700 were for deaths that occurred long after the attacks.

Now, faced with more than 19,000 additional unpaid claims, the math has become painful.

“We recognize that this is horribly unfair, particularly because we have spent the balance of this program paying claims at full value, and claimants who are coming in now are going to receive less,” said Rupa Bhattacharyya, who administers the fund. “Unfortunately, the law really leaves us no choice. This is the fairest way we could come up with to do it.”