“Through constantly shifting theories, McBride has dragged this languishing case along long after discovery — including the deposition testimony of the U.S. Army’s own Federal Rule of Civil Procedure 30(b)(6) witnesses — made clear that KBR’s recreation-center headcounts were immaterial,” the brief said.

The companies urged the court to reject McBride’s appeal without oral argument, citing years of litigation and high cost of its defense. In addition, the companies pointed out that only KBR Services is the proper defendant in the case, and the litigation has put a substantial burden on the other companies involved.

“The time has come to finally close the book on this costly litigation, which, as the district court noted, has forced KBR to spend ‘a king’s ransom’ defending against baseless claims,” the brief said.

McBride’s appeal argued that the judge did not apply a two-prong test, laid out in U.S. v. Science Applications International Corp., in which a court weighs whether adherence to contractual requirements was either a precondition for payment or material to payment.

Had U.S. District Judge Frederick J. Scullin Jr. used this test, he would have found that contract clauses made clear that contract adherence was a condition for KBR’s payment, she said. In addition, a government contracting officer has said officials “may not have paid” had they known KBR violated its contractual obligations, McBride said.

KBR did not meet a contractual obligation related to keeping sufficient records of its staffing levels at a morale, welfare and recreation center for Marines in Iraq, and therefore filed false claims by not disclosing this when it billed the government for payment, according to McBride.

An attorney for KBR has said that the contractor did not inflate its staffing numbers and did not use these figures as a basis for its billing. Thursday’s brief built on that argument, saying that contracting officials based their staffing for billing on the overall population of the base, not any measure of the use of the facility.

In a statement from counsel John Elwood, KBR defended its record in Iraq, saying it was proud of the work that it had done on behalf of the military and that it hoped the DC Circuit would affirm Judge Scullin’s decision.

“The district court correctly held that the alleged inflation of headcounts at KBR-run recreation centers in Iraq was immaterial to government payment decisions and thus cannot give rise to False Claims Act liability. Relator Julie McBride’s appeal seeks to set aside that well-reasoned decision,” the statement said.

McBride, along with two former KBR employees, had filed a qui tam suit against KBR, then-parent Halliburton Co. and several units of the companies in April 2005, alleging violations of the FCA, retaliation, and breach of employment contract.

In 2001, KBR had been awarded a contract to carry out the third round of the U.S. Department of Defense’s Logistics Civil Augmentation Program in support of the global war on terror. According to the suit, the defendants overbilled the government on one of the task orders under the multibillion-dollar contract, which allowed KBR to bill the government for a base fee, plus a potential performance fee and costs.

In October 2015, Judge Scullin declined to reconsider his summary judgment grant to KBR, saying he already wrestled with the case law and was unconvinced by McBride’s arguments there was sufficient evidence to support claims against the contractor.

On Monday, columns of the Iraqi army, Iraqi Peshmerga and counter-terrorism forces launched “the biggest military operation” yet to rid Iraq of the Islamic State group, Pentagon Press Secretary Peter Cook told reporters.

The advisers — who up until now had been assisting Iraqi military divisions, which have about 10,000 troops — will extend that assistance to units of about 2,000 soldiers who are more directly involved in day-to-day combat.