The grim campaign ads portended carnage to come: New York City, in the hands of a Democratic mayor, thrust back into the days of squeegee men, street prostitutes and 2,000 murders a year.

Yet four years after those predictions of the lawlessness that would take hold should Bill de Blasio win, he comes before voters on Tuesday with a record of keeping crime at bay. Murders stand at 242, on pace to be the lowest on record.

There is little doubt that Mr. de Blasio will win: No major Democrat rose to challenge him in the primary; his Republican opponent, Nicole Malliotakis, trails by more than 30 points in polls; and while many have criticized him, most voters in this heavily Democratic city appear ready to give him their support. Unlike 2013, it has been an election without municipal crises prompting calls for new leadership.

What it does have that 2013 did not is Mr. de Blasio’s four-year mayoral record. And perhaps no issue presented bigger risks for Mr. de Blasio than policing. He came to office critical of past practices and promising reforms. In his first year, he found himself at odds with the Police Department’s rank-and-file after the killing of two officers in 2014, which some blamed on his remarks.

As The Times reviewed Mr. de Blasio’s record, on issues like early childhood education and housing , public safety emerged as the most delicate ? and the one where his record, on crime reduction at least, has provided his clearest success.

Even opponents grant him a kind of grudging respect. The Patrolmen’s Benevolent Association, the city’s largest police union, declined to endorse the mayor for re-election, but, tellingly, it did not endorse any of his opponents, either.

Now, with violent crime on the rise in some of the country’s largest cities and President Trump apparently encouraging police aggression , Mr. de Blasio is hoping to make his mark on public safety in New York City and beyond with a policing experiment on a massive scale ? to make officers nicer.

Among the ways to see how the effort is working is to spend time with the officers who are trying to execute it, and the people in the city’s neighborhoods with whom they are trying to connect. So far, the record has been mixed and mostly anecdotal.

Mr. de Blasio has promoted two prongs of this approach to local and national audiences: officers engaging in deeper outreach with residents and, at the same time, looking inward to the internal biases they maintain. The mayor is convinced that the crime numbers and a declining number of official citizen complaints about the police prove that the changes have already taken hold.

“We’re bringing into this city neighborhood policing, real neighborhood policing, cops walking the beat again,” he said in a 2016 television interview in Philadelphia, with the stage of the Democratic National Convention behind him. “One thing that’s crucially important is implicit bias training. Helping our police understand, like every human being, they have biases that we can overcome.”

On the ground, however, things look different than how they have been portrayed by the mayor. Officers do not walk a beat and are not expected to. Neighborhood meetings organized by newly trained “neighborhood coordination officers” are sparsely attended in many cases, and sometimes not attended at all. Firm metrics do not exist for measuring the new approach, which city officials call neighborhood policing and not community policing, which for some officers carries a stigma of laxity and past failure.

And despite Mr. de Blasio’s promise nearly two years ago in his State of the City address to begin implicit bias training for officers that spring, and repeated invocations of the idea since then, the Police Department has yet to train a single officer. A $4.5 million contract for the training is still being completed.

“We have implicit bias training now as part of our curriculum,” the mayor said in July 2016 . Last month in a conversation with students at Columbia University, Mr. de Blasio conceded that it had still not yet begun . “We’re starting implicit bias training soon,” he said.

A spokesman for Mr. de Blasio, Austin Finan, said the department tried to start the training internally before turning to outside help.

A New ‘Philosophy’

The slow beginnings contrasted with swift and sweeping changes that the Police Department has undertaken to carry out the new patrolling plan.

Fifty-one of the city’s 77 precincts have been carved into new subsections where patrol officers spend all their time. Specialized units tasked with addressing crime, known as conditions teams, are being scuttled in favor of these groups of neighborhood-based officers. Arrests are not only declining, they are being played down by supervisors and chiefs. At crime-trend meetings in Police Headquarters, numbers are out; explanations of how community concerns are being addressed are in.

“For the commanders, they were worried,” said Chief of Patrol Terence A. Monahan, who oversees the effort, which he calls an overarching new “philosophy,” more than a single program. “It took a little pushing to get this going. But they saw the results.”

The Police Department added hundreds of counterterrorism officers under Mr. de Blasio, a priority at a time when Paris, London and last week, New York, have been targets of attacks involving trucks .

How much the drop this year can be attributed to the new approach is unknown. Citywide, the seven major felony crimes ? murder, rape, robbery, assault, burglary, grand larceny and auto theft ? tracked by the Federal Bureau of Investigation declined by nearly 6 percent.

An analysis of precinct-by-precinct crime data through Oct. 22 by The New York Times showed that those with the program ? anywhere from a few months to more than two years ? saw declines of nearly 7 percent; those without declined more slowly, a little over 4 percent. Yet at the same time, crime has increased in some of the first neighborhoods to get the program, including in parts of the northern Bronx and Washington Heights.

Even less certain is whether the new approach, meant as a lasting balm for minority neighborhoods rattled by decades of harsh policing tactics, will heal the rifts that reopened in 2014 with the police killing of Eric Garner on Staten Island and the national protests over similar killings of unarmed black men by officers in Baltimore, Ferguson, Mo., and elsewhere.

“That’s sort of like the secret ingredient in McDonald’s or something,” said Chuck Wexler, the executive director of the Police Executive Research Forum, a national group. “The magic sauce, you know, ‘build trust,’ like, if we could figure out how to do that it would be like the Manhattan Project. How do you build trust?”

Complaints by citizens, often proportional to police enforcement activity, are down from four years ago, as are stop-and-frisk encounters, summonses and arrests.

Chief Monahan said overall arrests were falling faster in precincts with the new program.

Meeting With Residents

That would not surprise Detective Edwin A. Rodriguez, a neighborhood-based officer. Part of his strategy for gaining trust in areas of Washington Heights is to let some crimes go unpunished. In one instance, he let a marijuana dealer go, and later that dealer helped with information on gangs.

“He’s up there selling weed and stuff, a bunch of small stuff,” he said before heading out on patrol on a Friday last month. “And we’re worried about violent stuff.”

Over the summer, a new phase of the plan began around the city: small neighborhood meetings, often with coffee and cupcakes, aimed at helping officers learn the crime concerns in their small patrol areas, known as sectors.

Local commanders are not present, as they would be at the traditional precinct community council meetings. The meetings are run by the neighborhood-based officers, not supervisors.

In the basement of St. Luke’s Episcopal Church in the northern Bronx on a recent Thursday night, an officer brought pizzas. His colleagues wrote the concerns of residents on a white sketch pad.

A man stood to complain about drug use in a playground. Others spoke of unlicensed cabs and double-parked cars. But the officers perked up as a woman described a problem at a business.

“They put their drugs right there in the Chinese place, I can tell you that,” the woman said, one of 17 mostly older residents who came to the church. “I’m not trying to get my name involved,” she added.

“Let’s talk after,” one officer, Detective Alejandro Colon, told her, before turning to the group. “This is what this program is all about.”

But not all meetings are as active. In more than a dozen visited by The Times over the last few months, several were sparsely attended, one in Harlem included no one from the area and another in Washington Heights, listed on the website for a June evening, did not take place.

“I was going to speak my mind,” said Orlando Ramos, 69, a retired doorman who was outside the designated location.

By contrast, more than 40 people at a meeting in Bedford-Stuyvesant told officers about problem spots, and officers explained, in an informal way, the challenges they face: the legal rules of stop-question-and-frisk; the limitations of stopping people from smoking marijuana in privately owned apartment buildings.

“It eased my mind because I realized that they’re aware” of the problems in the area, Odessa Watson, 39, said of the officers.

Officer Patrick Malone, 32, who helped to lead the meeting, said he gave out the number for his city-issued cellphone and received calls from residents at all hours. “It’s good information,” he said.

His sergeant, Shaun Brown, said that while some people wanted to see more officers on foot, “we’re in the car 100 percent of the time.” Despite adding 1,300 officers to the Police Department , for counterterrorism and to facilitate the new neighborhood-based program, they would need far more to cover their area on foot, he said.

Few people under 40-years-old were at the meetings. At several, including one in the Bronx on Thursday, officers implored those who came to bring their younger relatives. Assistant Chief Rodney Harrison said that the department was looking to help neighborhood-based officers reach problematic young people by going “to them” in schools and community centers. They would entice the young people to show up, he said, by offering “some type of gift,” like a sporting goods gift card or movie tickets.

“It’s a fluid philosophy,” said Chief Monahan. “This is all about trusting our cops.” He added that the department did not want officers to make arrests for the sake it.

Pragmatism on Patrols

That message has reached the street.

As Detective Rodriguez walked through a desolate corner of his Upper Manhattan sector this month, near an otherwise empty skateboard park, he and his partner, Detective Thomas W. Troppmann, came upon three men huddled under a footbridge. One had his shirt sleeve rolled up. Another appeared to be cooking heroin.

As the officers approached, the men quickly skirted away, leaving belongings behind.

In another era, the encounter might have ended with an arrest, or at least a formal police stop. But Detectives Rodriguez and Troppmann did not see a case to be made, or much point in locking up clearly drug-addicted men who later admitted they were struggling with heroin addiction.

“Hey, come get your jacket!” Detective Rodriguez called out to the men.

Two turned back, and the officers spoke to them about addiction services. The third, whom the officers suspected of being a dealer, kept walking.