The U.S. Supreme Court has declined to take up the high-stakes legal fight between Mayor Mitch Landrieu’s administration and the city’s firefighters, an unsurprising outcome that leaves the city with few options but to keep negotiating over how much money it owes the firefighters’ pension fund.

The mayor’s final appeal was obviously a long shot. It is mostly state rather than federal law at issue in the firefighters’ case against Landrieu, and the Supreme Court accepts only a tiny fraction of the appeals it receives.

The Louisiana 4th Circuit Court of Appeal has already upheld the trial ruling from Civil District Court Judge Robin Giarrusso, who ordered Landrieu’s administration to pay the fund $17.5 million in unpaid obligations just for 2012. The state Supreme Court declined to review the case.

In the meantime, the two sides in the lawsuit have made some progress in courtroom negotiations over the past couple of months. Essentially, the independent board of trustees that governs the pension fund has agreed to alter its actuarial accounting methods in order to shrink what Landrieu’s administration owes.

First, the board is now assuming that it will pay off the fund’s long-term obligations over 30 years instead of about 15, the equivalent of extending a home mortgage over a longer period with smaller monthly payments.

Second, the board’s actuary will apply a technique called “asset smoothing” to the fund’s books, an accounting method that spreads volatile ups or downs in the value of a pension fund’s assets over a number of years, rather than taking a big gain or loss all at once.

Those steps will have a considerable impact on what the city owes. As the value of the pension fund’s assets drops — as it has sharply over the past few years — the city is legally obligated to contribute more in order to keep it healthy. So any accounting method that allows losses to be spread over time will mean the city can get away with smaller monthly contributions.

What’s more, the new accounting methods will be applied retroactively; the courts have ruled that Landrieu has been shortchanging the fund since 2010.

Louis Robein, an attorney for the firefighters who are suing the city, said the new financial reckoning will reduce the city’s overall debt for 2010 through 2014 from more than $70 million to perhaps less than $32 million, although the ultimate figure remains in dispute.

For the coming year, the city’s projected contributions to the fund would drop from about $35 million to $24 million.

What the two sides cannot agree on is whether all of this will be cumulative — in other words, whether the city will owe $24 million next year, or $24 million plus what the courts have decided it should have been paying since 2010. That would add up to about $56 million.

The parties are due back in court on Oct. 21.