In a bland, sprawling warehouse in Manhattan, election workers carefully inspected piles of absentee ballots on Friday, an exercise that might be described as tedious if it were not so important.

The ballots will most likely decide the winner of New York City’s Democratic mayoral primary, a race that is currently led by Eric Adams, who is seeking to become the city’s second Black mayor. His lead over his two closest rivals, Kathryn Garcia and Maya Wiley, was small enough that either could theoretically pass him once more than 125,000 absentee ballots are factored in.

But Mr. Adams’s campaign suggested that its informal, unofficial tally of the absentee ballots counted so far indicated that he might have slightly widened his lead in first-place votes — placing an even heavier burden on Ms. Garcia and Ms. Wiley to close the gap through the city’s new ranked-choice voting system.

Under that system, voters could rank up to five candidates on their ballots in order of preference. Because Mr. Adams did not collect more than 50 percent of first-choice votes, the process moves to an elimination-round method: The lowest-polling candidates are eliminated a round at a time, with their votes reallocated to whichever remaining candidate those voters ranked next. The process continues until there is a winner.

In the first round, among people who cast their ballots in person during the early-voting period or on Primary Day, Mr. Adams led Ms. Wiley by 9.6 percentage points, and Ms. Garcia by 12.5 points. When a preliminary ranked-choice tabulation was conducted on Wednesday, Ms. Garcia edged slightly ahead of Ms. Wiley, and trailed Mr. Adams by only two points.

The city’s Board of Elections began counting absentee ballots on Monday and plans to release a new ranked-choice tally that includes most of them on Tuesday. A board spokeswoman declined on Friday to discuss the results of the counting until then.

For either Ms. Garcia or Ms. Wiley to overtake Mr. Adams, they must outperform him in first-place absentee votes, or hope that enough absentee voters ranked him low on their ballots or left him off entirely.

Ms. Garcia, who beat Mr. Adams in Manhattan in the in-person vote tally, also showed strength in absentee ballots from the borough, according to a preliminary count of unofficial results obtained by The New York Times. She was the first choice on 9,043 ballots of 23,739 ballots counted as of Thursday night, or about 38 percent. She was the second or third choice on another 7,187 ballots.

Mr. Adams was the first choice on 2,999 absentee ballots from Manhattan, or about 13 percent, and the second or third choice on another 5,304. Manhattan was the only borough he did not win in the in-person tally.

The absentee ballots’ importance was underscored by how many campaign workers and volunteers have been observing the vetting and counting process this week.

On Friday, Ms. Wiley had several volunteers at a site in Manhattan where absentee ballots were being counted. Mr. Adams’s campaign had many more — a sign that his campaign, with more money and institutional support, has often been more muscular and organized than those of his rivals.

Mr. Adams had a team of volunteers seated at every table, diligently tallying his votes and Ms. Garcia’s with pens and notebooks. Occasionally, a volunteer challenged a ballot’s legitimacy over a signature or date.

At one point, a small commotion could be heard.

A volunteer for Mr. Adams was challenging a ballot backing Ms. Garcia because it had a stray pen mark. Election workers and campaign volunteers gathered around the table to scrutinize the ballot, and then set it aside for further examination.

“Our team has been here all week making sure every single vote is counted,” said Ydanis Rodriguez, a City Council member from Upper Manhattan who is an ally of Mr. Adams’s. Mr. Rodriguez was leading the Adams campaign’s presence at the elections board site, leaving only briefly on Wednesday to vote on the city budget.

Ms. Garcia’s campaign did not have volunteers at the Manhattan site, but her representatives said that members of the campaign’s legal team — including the prominent election lawyers Stanley Schlein and Sarah Steiner — were monitoring the proceedings.

Ms. Wiley’s campaign sent an email to supporters on Thursday seeking volunteers to visit absentee sites on Friday.

“All ballots have been cast, and while we cannot persuade any more New Yorkers to vote for Team Maya, we can make sure that every single vote counts, and is counted accurately!” the email said.

The campaigns are allowed to monitor the absentee count and to challenge ballots that may be ineligible. Election workers wore face masks on Friday and held ballots up against a plastic partition so that campaign volunteers could read them. One ballot, for instance, had several first choices marked for mayor. An election worker told volunteers that it would be considered void when it went through a ballot-scanning machine.

Ms. Wiley’s campaign filed a lawsuit on Thursday preserving its right to challenge the election results, following similar moves by the campaigns of Mr. Adams and Ms. Garcia. Ms. Wiley argued that a full hand recount should be required if a decision about who gets eliminated under ranked-choice voting comes down to a “razor-thin” margin.

Under the preliminary ranked-choice tally conducted Wednesday, Ms. Wiley fell just short of getting to the final round, trailing Ms. Garcia by fewer than 350 votes.

“This is a wide-open race and as is standard procedure, my campaign filed a petition to preserve the right to challenge the results should we believe it is necessary,” Ms. Wiley said in a statement on Friday. “For now, we must allow the democratic process to continue and ensure every vote is counted transparently.”

The Board of Elections is under close scrutiny after an embarrassing fiasco on Tuesday that forced it to retract preliminary ranked-choice vote totals it had released just hours before.

The board had mistakenly included more than 130,000 sample ballots, used to test the ranked-choice software, in the preliminary count. The board ran the ranked-choice program again on Wednesday, with the result again showing Mr. Adams ahead of Ms. Garcia and Ms. Wiley.

Despite the vote-count debacle, Dawn Sandow, the board’s deputy executive director, sent a congratulatory email to staff members on Thursday. In the email, Ms. Sandow acknowledged that there had been “negative articles bashing this agency,” but she insisted that board employees had risen to the occasion.

“The amount of changes thrown at us to implement in a short period of time during a worldwide pandemic was unsurmountable,” she wrote, “and WE DID IT ALL SUCCESSFULLY!”

Dana Rubinstein and Katie Glueck contributed reporting.

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