Cmdr. Aaron Rugh, second from right, is officially sworn in as an appeals court judge at the Washington Navy Yard alongside Col. Brian Palmer, Cmdr. Anne Marks, and Lt. Col. Troy Campbell. (Natalie Morehouse/U.S. Navy)

The Navy has concluded that one of its top judges did not lie when he provided erroneous information about witnesses while testifying under oath at an administrative hearing.

Last year, The Washington Post found that Cmdr. Aaron Rugh made false statements in 2014 to a board of officers deciding whether a Marine he had ­prosecuted for sexual misconduct should be expelled from the ­service.

That discovery led to an ethics inquiry — just part of the fallout from the flawed investigation into Marine Maj. Mark Thompson, a former history instructor at the U.S. Naval Academy convicted of having sex with two female midshipmen. He is accused of lying to the same board and faces a second court-martial because of The Post’s revelations about his case.

The Navy launched the probe into Rugh after Thompson’s attorney, Kevin B. McDermott, filed an ethics complaint.

“Ironically, Major Thompson is being criminally prosecuted for making false statements before the same board,” McDermott wrote last year, arguing that Rugh had either lied or failed to correct a “grossly negligent misrepresentation.”

Vice Adm. James W. Crawford III — the judge advocate general of the Navy — notified McDermott earlier this week that the case had been closed, as first reported by Stars and Stripes.

“After careful consideration,” Crawford wrote, “I found the available evidence failed to support a violation” of the Navy’s rules of professional responsibility.

But the controversy surrounding Rugh, who sits on the appeals court, has once again exposed the military’s lack of transparency when judges are accused of misconduct. For weeks last year, the Navy would not acknowledge that the inquiry ­delayed Rugh’s promotion to captain or that the 20-year veteran was under ­review.

Now, the Navy has declined to explain how or why the Judge Advocate General’s Corps (JAG) decided that one of its most popular members should be cleared of wrongdoing.

“We cannot discuss the details of the investigation, or any review deliberations,” the Navy’s deputy chief of information, Capt. Amy Derrick, wrote in an email.

In 2013, Rugh — once the Navy’s top prosecutor — led the legal case against Thompson, who was accused of raping one student and having a consensual relationship with another. Thompson was found guilty of five charges but was acquitted of sexual assault.

After serving two months in a Virginia brig, Thompson faced what is known as a board of inquiry, which would decide his fate in the military.

The government called Rugh, then still a prosecutor, to testify by phone about one narrow aspect of the case. But the three board members wanted to know more.

Sarah Stadler, one of Thompson’s accusers, had alleged at trial that they had sex a final time on the night of her graduation — but he had a compelling alibi.

So one board member asked Rugh how the government had proved that Thompson and Stadler had seen each other that night. Investigators, Rugh said, had contacted family members who had rented a house in Annapolis for the ­graduation ceremony.

“Now, we went and interviewed her family, her brother, etcetera,” he said, according to a transcript of the proceeding. “They remember her, you know, they verified that they had the rental house, that she was there for a little while and then left the house and came back later that evening.”

Stadler’s mother, Laurel, later said that Rugh’s statement was “a complete and total lie.” She and her son denied that they had spoken to investigators about the case, and both said they did not clearly recall what had happened that night.

On Wednesday, Rugh’s attorney, retired Rear Adm. Christian L. Reismeier, offered the most detailed explanation yet for his client’s testimony.

“He recalled and still recalls today that this information was relayed to him by a member of the prosecution,” said Reismeier, former chief judge of the Navy. “There’s just no corroboration that that actually occurred. It appears that his memory is just inaccurate.”

But Reismeier, who expects Rugh’s promotion in rank will now proceed, would not say who told his client about witness interviews that apparently never happened.

Before taking on Rugh’s case, Reismeier had sharply criticized the military’s lack of transparency on judicial misconduct, saying that such opaqueness could undermine confidence in the system.

Eugene R. Fidell, a longtime attorney who teaches at Yale Law School, agrees, suggesting that the secrecy may lead members of the public to conclude JAG is simply protecting one of its own.

“There’s a problem of incest. There’s no doubt that’s because these are micro-jurisdictions where everybody knows everybody else,” Fidell said. “If somebody has done a responsible, thorough investigation and concluded Rugh has done nothing wrong, why wouldn’t they give chapter and verse about the investigation?”