A top federal official branded it terrorism. Dauphin County, Pa., authorities cautioned against "wild speculation." And lawmakers from Harrisburg to Washington say they'll wait to see what develops.

Four days after Ahmed El-Mofty, a naturalized U.S. citizen from Egypt, opened fire just steps from the Pennsylvania Capitol in what officials have described as a deliberate attack on police, investigators in Harrisburg continue to piece together a profile of the man – and, in the process, have run headlong into a politically charged debate over how to describe his act of public violence.

Tyler Q. Houlton, acting spokesman for the U.S. Department of Homeland Security, took to Twitter over the weekend to declare the shots El-Mofty fired Friday night at three separate groups of officers a "terror attack."

Local authorities, meanwhile, cautioned that they had not completed their investigation and didn't know what motivated the man's violent turn.

The distinction may seem trivial. But determining what types of violence to label "terrorism," what conclusions to draw from such a distinction, and when in an investigation it is appropriate to use that word has become particularly fraught when mass shootings and public attacks occur with alarming frequency.

In El-Mofty's case, for instance, authorities have said that the 51-year-old security guard had traveled to the Middle East in October, although it remains unclear for what reason, or when he returned to the United States.

They have made no public pronouncements linking his shooting spree to a wider cause such as Islamic extremism or suggesting that he was inspired by or acting in concert with international terrorist groups.

Dauphin County District Attorney Ed Marsico said that determining a motive for the shooting spree remains the top priority.

"Terrorism link is still under investigation," Marsico said in a text message Tuesday afternoon. "Authorities [are] conducting more interviews today and hope to have more information to talk about in the next 24 hours."

One thing is clear, Marsico told PennLive.com on Sunday: There is "no doubt" that El-Mofty meant to target police.

"He fired several shots at a Capitol Police officer and at a Pennsylvania State Police trooper in marked vehicles," the district attorney said.

El-Mofty's shooting spree began at the start of Friday's rush hour as he fired several shots at a Capitol Police officer, hitting his cruiser multiple times but missing him.

About a half-hour later, El-Mofty surfaced again, shooting at and hitting a State Police trooper before fleeing into a residential area in Harrisburg's Allison Hill neighborhood.

There, he opened fire with two handguns on pursuing officers, investigators have said. The officers fired back, striking and killing him.

The trooper was the only officer injured in the attack. She was released from a hospital Saturday.

That same day, Houlton, the Homeland Security spokesman, cited El-Mofty's shooting spree to make a political point and condemn a policy that he said was allowing terrorists to infiltrate the country.

In a tweet, Houlton criticized the United States' family-based immigration policy known as "chain migration," through which immigrants already admitted to the country can sponsor relatives for visas.

Although it is unclear when El-Mofty came to the United States, he gained entry and later earned citizenship thanks to an extended family member from Egypt who had been granted a visa "years ago," Houlton said.

El-Mofty's shooting spree Friday, Houlton tweeted, "highlights the Trump administration's concerns with extended family chain migration," which has been "exploited by terrorists to attack our country."

But those remarks elicited a quick response from U.S. Rep. Scott Perry (R., Pa.), a member of the House Homeland Security Committee.

"We must let federal authorities conclude their investigations before making any judgments or policy recommendations," the congressman, who represents parts of York and Dauphin Counties, said in a statement.

The back-and-forth recalled a similar dispute that emerged after Edward Archer, a man pledging support for ISIS, shot Philadelphia Police Officer Jesse Hartnett in 2015.

U.S. Sen. Marco Rubio, then running as a Republican candidate for president, quickly pointed to that shooting as an ISIS-launched terror attack on the United States. Mayor Kenney, meanwhile, sought to squelch such speculative talk before the FBI completed its investigation, drawing attacks from critics across the country for that stance.

The FBI ultimately concluded that Archer had acted alone and, despite travelling to Saudi Arabia and Egypt in previous years, had no known terrorist contacts.

Speaking on Archer's case, then-FBI Director James Comey described the threat that he and others posed as one of self-radicalization, not infiltration.

"Wherever there is the internet and an unmoored soul who might conclude they have to engage in violence to find meaning, there is the nature of the threat," he said at the time.

Dauphin County investigators along with the FBI, the Pennsylvania State Police, and the U.S. Bureau of Alcohol, Tobacco, Firearms and Explosives are still working to develop a profile on El-Mofty that will answer key questions such as where he had been living, whether he attended local mosques, and whether he had been employed since his return to the U.S. Marsico said Tuesday that agents know where El-Mofty purchased his guns.

But El-Mofty appears to have kept a relatively low profile, authorities said. Public records show that he had lived at various addresses in Camp Hill, just across the Susquehanna River from Harrisburg, starting in 2010.

Family members there struggled to reconcile his actions Friday with the timid family man they said they knew.

Ahmed Soweilam, co-owner of a halal grocery in Camp Hill, said that El-Mofty had been married to his sister, Ola Soweilam, and that they had had two children before becoming estranged six years ago.

"He is a chicken," Soweilam told ABC News. "He's not a terrorist."

El-Mofty, he said, had worked in the private security industry before returning to Egypt and had no history of violence or mental illness.

"He's not a perfect guy," Soweilam said. "But he's not an aggressive person."

Staff writer Angela Couloumbis contributed to this article, which also contains information from the Associated Press.