The Nassau County Police Department did not receive a formal criminal complaint into whether someone hacked Nassau County Edward Mangano’s cellphone to make it appear that he exchanged sexually charged text messages with a public relations executive — contrary to public statements made by Mangano and the department.

Acting Nassau Police Commissioner Thomas Krumpter and four separate other sources said Thursday no written criminal complaint had been filed in the case.

Police issued a news release Feb. 14 titled “unusual incident,” which said it “received a criminal complaint” from Mangano on Feb. 4 about allegations that Mangano’s cellphone and that of public relations executive Karin Murphy Caro were hacked.

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Mangano himself issued a statement on Feb. 14 saying he filed a “criminal complaint” with the police department.

Criminal complaints are accusatory documents filed by the state that can launch a criminal prosecution and a court case. They can also be used by prosecutors to seek subpoenas.

Asked about the assertions from both Mangano and the department that a criminal complaint was filed, Krumpter said only: “A criminal complaint? There’s different meanings of a criminal complaint.” He declined to elaborate on what he called an active investigation.

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Krumpter later said the release used the phrase “criminal complaint” to indicate Mangano had verbally reported the alleged hacking to the department.

A spokesman for Nassau District Attorney Madeline Singas said: “We have offered our assistance, but have not been asked to assist with the police department’s investigation.”

Krumpter said in an interview Thursday that the department had not generated any documents in its investigation into the texting. But he said that did not indicate the lack of a serious inquiry.

Krumpter, who is awaiting Mangano’s nomination as the permanent commissioner, said the department has also not generated a case report, although the probe has a case number. He said the department “routinely” assigns an investigative number to sensitive cases — without generating reports or other documents — in order to “ensure that the investigation will not be compromised and only those participating in the investigation will have access to the investigation.”

“This is a common practice,” said Krumpter. “We do not need a case report to get a subpoena.”

Mangano, who has vehemently denied exchanging any messages with Caro, called the allegations “disgusting,” told reporters at a heroin task force summit in Baldwin Thursday: “I have not sexted.”

He added: “I don’t know what else to tell you, other than it’s untrue. You will see there’s no communication.”

Responding to questions about whether a criminal complaint was filed, Mangano spokesman Brian Nevin said: “The matter was reported to the police department. The Intelligence Unit interviewed the county executive and advised him that an active investigation is ongoing.”

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Mangano had said that he would not have been able to send any illicit texts because he has a police security detail with him.

But Krumpter said Sunday Mangano has a security detail with him only “whenever he’s in a public space, whenever he’s working during the day,” and the officer in the detail would not likely interfere with the county executive’s texting.

“I don’t know that it would prevent him from having texts,” said Krumpter.

Joseph Giacalone, a retired NYPD sergeant and adjunct professor at John Jay College of Criminal Justice, said the police department should recuse itself from the investigation and hand it over the district attorney’s office.

“They remove themselves from any type of allegations or speculation down the road that they didn’t find him guilty of anything because their jobs depend on it,” said Giacalone. “Especially for a guy like Krumpter, who’s the acting PC. He’s between a rock and hard place. If they want to have transparency, I think they have that by giving it over the district attorney squad to see if there was any wrongdoing.”

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Krumpter, who has called Mangano and Caro “credible,” and referred to them as victims, challenged the notion that he or his investigators could be compromised by the power Mangano wields over the department.

“If at any time Ed Mangano becomes the subject of the investigation, the investigation will be referred to the appropriate agency,” Krumpter said.

Asked why police waited 10 days to put out a news release announcing that the incident had occurred, Krumpter said: “We don’t protect anybody. I’m gonna make this very clear. The Nassau County Police Department is independent, and we conduct all investigations with sensitivity. And we don’t protect anybody — not the county executive, not anybody else. We conduct investigations. We’re impartial fact-finders. That’s what we do.”

Caro, 40, of Smithtown, president of Hauppauge-based BluChip Marketing, had been awarded two no-bid county contracts and has more than 35,000 Twitter followers.

On Instagram and her personal website, she has posted scores of photos posing at events with public officials. She added that she frequently takes photos with people at public events.

“Again, I’m in public relations,” she said. “I love good relationships with like you and the news. I’m always looking to network.”

Mangano’s wife, Linda Mangano, who emceed the heroin summit, said she would not respond to questions on the purported texts between her husband and Caro when approached by WCBS-TV reporter Marcia Kramer, who first reported the texts.

“This is not the time or the place for that,” Linda Mangano said.

With Kevin Deutsch and Laura Figueroa