For 17 years, the schools on the Upper West Side of Manhattan have coexisted in harmony. But when the families of their students gathered on a recent night in their shared auditorium, there was nothing neighborly about it.

On one side was Public School 9, a coveted but overcrowded elementary school where parents raise $2 million annually to pay for extra teachers in every classroom. On the other was Center School, a beloved middle school with ingrained traditions like allowing its students to eat lunch off campus. The topic that night was their proposed breakup.

P.S. 9 wanted to take over the entire building, kicking out Center School, so that it could expand, reduce class sizes and, perhaps, attract more families from the neighborhood. Center School would move to a building about 20 blocks south that it would occupy alongside a chronically low-performing school, Riverside School for Makers and Artists, whose middle school is losing students and would be eliminated. Center School families agreed that it should move — but not to Riverside, which they say lacks everything it needs.

Across the country in recent years, a similar landscape of schools — some hollowing out, others teeming with students — has emerged as public school systems confront a yearslong, sustained decline in enrollment. The exodus accelerated during the Covid-19 pandemic, as parents considered other options for their children, and has started to strain budgets and force tough decisions.

Since the pandemic, more than 123,000 students have departed New York City public schools, while nearly 1.3 million have left public schools nationwide. From California to Texas to Maine, school leaders facing half-empty schools and stark forecasts of continued enrollment declines have few options, including closing or merging schools.

But those are deeply unpopular, politically radioactive and tear apart neighborhoods. That was clear that night on the Upper West Side.

“If you don’t want families to go to charter schools, if you don’t want families to go to private schools, then stop closing schools,” said one of the first speakers, Dawn Goddard, a mother of a sixth grader at Center School.

A fifth grader at Center School said she was being punished to help a “larger, wealthier school.” If P.S. 9 really needed space, a sixth-grade boy said, it should start by eliminating its science lab. And a Center School mother wondered why P.S. 9 parents hadn’t expressed concern for the Riverside students, many of whom are recent asylum seekers, including one who, the mother noted, had witnessed the decapitation of his parents.

Seated in the first rows, parents and teachers from P.S. 9 shook their heads in disgust. A mother of a girl with special needs took the microphone and spoke about the shame her daughter feels because her therapy sessions have to be held in a room with other students because of space constraints.

Gale Brewer, a City Council member who represents the area, had tried to broker a deal to appease all sides. “They’re very, very nasty to each other,” she said. “I don’t know what to do.”

This has been the atmosphere during the past four months after New York City’s Education Department announced a plan to break up those schools and close or downsize others before the next school year. Opponents said that the proposals had been rushed.

Versions of this feud on the Upper West Side have been playing out across the country. Parents have protested proposed closures, shouted at public meetings and, on rare occasion, been escorted away by the police, including recently in Houston just before its school board approved the shuttering of 12 schools.

Facing overwhelming opposition, school leaders in some places, such as Philadelphia, have scaled back their closure plans, revealing the difficulty in trying to address declining enrollment.

“Closing a school is an incredibly sharp pain point for parents and communities,” said Thomas S. Dee, a professor of education at Stanford University, who has been tracking school closures nationwide since the pandemic. “Local schools are often a focal point for neighborhood identity.”

On the Upper West Side, where families carefully study school attendance zones before buying or renting and sometimes pay more to live near higher-achieving schools, Education Department leaders insisted in town hall-style meetings that they would not back down from the proposal. It could not be negotiated, there was no Plan B.

But two days before a panel of education advisers was expected to approve it, New York’s new schools chancellor called it off, an inauspicious start to what could be a wave of closures and mergers in the coming years as enrollment declines.

The chancellor, Kamar Samuels, said the proposal was too much change too quickly into a new administration, even though Mr. Samuels had crafted it himself in his previous job overseeing Upper West Side schools.

But the deal was not dead. He said it would be revised by local school leaders in consultation with parents.

While the clash pitted families against one another and strained friendships, it also cast a harsh light on the differences and inequities among schools, even those just blocks apart, and also brought up fraught questions about race and class. A battle a decade ago on the Upper West Side over school attendance boundaries centered on the same issues.

Across the country, school closures have disproportionately affected Black and Hispanic students, and it would have been no different for the schools marked for shuttering or downsizing on the Upper West Side.

At one of them, Community Action School, nearly every student is Black or Hispanic. During one of the first meetings about the closures, an eighth-grade girl from the school pleaded for it to be saved, describing how it had been a refuge after a tumultuous experience earlier in middle school.

While she spoke, a mother, who was watching remotely and speaking on a hot mic, said, “They’re too dumb to know they’re in a bad school.” (Afterward, the woman said that her comment had been taken out of context.) Mr. Samuels later announced that Community Action School would stay open.

Of all the changes Mr. Samuels had pursued, one school would have come out ahead of the rest: P.S. 9, one of the most-sought after in the city.

Most of its students are white, and it has resources — a science lab, a computer room, a library and two art rooms — that are a rarity among New York elementary schools. Among the city’s nearly 1,600 schools, only five raised more money than P.S. 9’s parent organization last year.

The Education Department believes that P.S. 9 could lure families back into the public schools, which remain popular in neighborhoods that are home to many middle-class and upper-middle-class families.

This school year, more than 800 students applied for the school’s kindergarten class, which had just 100 seats. With an expansion, P.S. 9 could accept more of those students, school leaders said, and also lower its class sizes to comply with a new state cap.

More than 40 years ago, education leaders in New York saw the same potential in Center School during another period when schools were losing students. A group of educators aimed to create a school unlike any other in the city — and it remains that way today.

It has four grades, unlike the three in most middle schools. The roughly 250 students are grouped in classes that span every grade. Collaboration and fun are prioritized. Students play together during recess, eat lunch together — often off-campus at pizza shops and empanada spots — and direct, perform and produce the school’s highly anticipated variety shows.

“Families who might otherwise opt for charter, private, suburban middle schools see Center as a rare gem,” said Michael Fram, a high school principal whose child attends the school and who attended a meeting in January.

Relocating Center School to Riverside would kill Center, parents and students said. Riverside does not have a dedicated auditorium, its play area is on a rooftop and the neighborhood has fewer restaurants.

That night in the auditorium, a mother, Tiffany Rodriguez-Noel, who has children at Riverside, said it had been overlooked in the clash among the other schools. It needed more resources, which were promised years ago, she said.

In the school boundary fight a decade ago, much of it centered on the creation of Riverside. Back then, it was known as Public School 191, and almost every student who attended it lived in the Amsterdam Houses, a public housing complex near Lincoln Center.

It was renamed and relocated, placed on the ground floor of a luxury high-rise in a wealthier neighborhood, as part of an effort by the Education Department to give it a new start. It would have a diverse student population, school leaders said, and just 20 percent of its students would come from low-income families. Today, that number is 86 percent.

“It feels a lot like our land is being stolen to cover up being neglected, ignored and robbed of an adequate education,” Ms. Rodriguez-Noel said.

Kitty Bennett and Georgia Gee contributed research.

Trying to devise a one-size-fits-all description for ultraprocessed foods is flummoxing federal regulators.

For months, the health secretary, Robert F. Kennedy Jr., has promised to create a definition of ultraprocessed, a crucial part of the Make America Healthy Again agenda. In mid-April, he testified before Congress that the Food and Drug Administration had forwarded a definition to other agencies, including the Department of Agriculture.

But behind the scenes, officials said, the process of defining ultraprocessed foods is still very much up in the air. Agencies are struggling to agree, and it is unclear when a definition will be released.

“It’s not final until it’s final,” said Calley Means, a senior adviser to Mr. Kennedy, adding that the definition would ultimately be the result of hundreds of conversations with scientists, agency staff and other stakeholders.

For the food industry, which is already seeing demand for many of its products soften as some consumers cut back on spending while others use weight-loss drugs, the debate about what an ultraprocessed food is or isn’t has potentially far-reaching consequences.

Under one classification widely used among the scientific community, essentially any foods or drinks made with ingredients you wouldn’t find in a home kitchen are defined as ultraprocessed. If regulators adopt that sort of definition, nearly three-quarters of foods sold in the United States could be deemed ultraprocessed.

The food industry is arguing against a strict definition that would label chicken nuggets, strawberry yogurt and whole-grain tortillas as ultraprocessed.

Based on that definition, deli turkey could be categorized the same as a snack cake, the National Turkey Federation wrote in a comment letter to regulators last fall. It said that certain food additives and processing steps were critical to keep turkey fresh and that those “benefits are especially important for lower-income households, where access to nutrient-dense, high-quality protein can otherwise be limited.”

One fear that emerged in interviews with food companies, lobbyists and regulators, most of whom declined to be quoted, is that foods tagged as ultraprocessed could be restricted or eliminated entirely from the nation’s school meal programs. They are multibillion-dollar revenue streams for the companies that make sandwich breads, cereals, salsas and other foods. Others are worried that regulators could create new rules for warning labels on the packaging of foods in grocery stores.

But Mr. Kennedy, who has frequently referred to ultraprocessed foods as “poison” and has suggested restricting them from the diets of Americans, particularly children, is facing intensifying pressure from the MAHA movement, which is credited with helping elect President Trump to his second term. A Politico poll published in April showed that removing ultraprocessed foods from the American diet was a core principle for people identifying as MAHA followers.

“If we can have a federal definition that is strong and science-based, it opens the door for meaningful policy,” said Vani Hari, a health advocate who is known as the “Food Babe” online and is a prominent voice in the MAHA movement. And, she added, “anything positive that is coming out of the administration on food reform is popular with voters.”

Many scientists support a strong definition, and noted that evidence linking ultraprocessed foods to a host of chronic diseases, including obesity, Type 2 diabetes, heart disease and some types of cancer, has grown in the last decade.

Defining ultraprocessed foods is “one of the most important policy actions around food that the U.S. government has done for probably 25 years or more,” said Dr. Dariush Mozaffarian, a cardiologist and the director of the Food Is Medicine Institute at Tufts University. “Industry is going to fight tooth and nail because this is a fundamental threat to their entire model,” he added.

Dr. Mark Hyman, a physician and friend of Mr. Kennedy’s, called ultraprocessed foods the “single biggest threat to public health ever.” He characterized the food industry’s lobbying on the definition as “their usual shenanigans.” It is “obfuscating, confusing, undermining credibility of scientists,” he said.

While the food industry is trying to make its case to federal regulators, it is also facing a dizzying array of new regulations by states. Emboldened by the MAHA movement, many states are not waiting for the federal government. Instead, they are writing their own rules for food. Last year, Texas and Louisiana passed laws that required warning labels on foods that contained any of 44 additives, while California banned certain ultraprocessed foods from school lunch programs. Other states are weighing similar moves.

Dr. Mozaffarian at Tufts and some other health experts argue that the federal definition should adhere closely to a food classification system called Nova, which is used in the vast majority of the research linking ultraprocessed foods to poor health.

The Nova definition deems a wide swath of products ultraprocessed, including candy, soda and hot dogs. It also includes some foods traditionally considered to be healthy, like many whole wheat breads, peanut butters and yogurts.

The definition is focused primarily on ingredients. Peanut butter, if it contains nothing more than peanuts or salt, would not be considered ultraprocessed. But most peanut butters sold in grocery stores contain hydrogenated vegetable oils, kicking them into the ultraprocessed category, according to Nova. Likewise, plain yogurt would not be considered ultraprocessed. But when ingredients like emulsifiers and flavorings are added, the food becomes ultraprocessed.

In his testimony before Congress in mid-April, Mr. Kennedy said that once a definition was in place, his agency would move forward with a plan that would mandate placing color-coded labels on the front of food and beverage packages.

“If it’s a red light, don’t eat it,” he said during his testimony. Green, he said, would signal to consumers the food is healthy.

Mr. Kennedy also suggested that following a federal definition of ultraprocessed foods, states could restrict them from their food stamp programs — jeopardizing billions more in revenue to food companies.

In conversations with regulators and members of Congress, the food industry broadly asserts that an overly strict definition of ultraprocessed foods like the one from the Nova system could target nutritious foods, that more research is needed, and that increased regulation could result in higher prices for consumers.

Using a broad brush to categorize foods “fails to capture the important reality that all processed foods are not equal,” the cereal manufacturer WK Kellogg wrote in a comment letter last year when the F.D.A. asked about how ultraprocessed foods should be defined. The National Chicken Council wrote in another comment letter that “gummy bears and chicken nuggets in this comparison are technically ‘ultraprocessed’ by common definitions. Yet, it is clear which one has potential to contribute meaningful nutrients to a higher quality diet.”

Some of those arguments are gaining traction inside the Department of Agriculture, which oversees the nation’s school meal programs. Officials there say it is unclear if the definition will be tied to policy and, if it is, expressed concern that food manufacturers could quit the programs, leaving school officials with fewer choices for children’s lunches.

Lindsey Smith Taillie, a professor of nutrition at the University of North Carolina’s Gillings School of Global Public Health, who met with F.D.A. officials in mid-April, said there was a middle ground. She argues that regulators can still create a strong federal definition similar to Nova’s while constructing useful policies — such as for warning labels or school meals — by exempting products that are “healthy,” according to F.D.A. standards.

The F.D.A. defines a healthy food as one containing a certain amount of actual food — like fruits, vegetables or milk, as opposed to processed ingredients like cornstarch — and not too much saturated fat, sodium or added sugars.

Dr. Taillie said most ultraprocessed foods don’t meet the “healthy” definition. But a policy like this could motivate food companies to reformulate products to fit that criterion, she added.

The president of Bard College, who has run the unorthodox liberal-arts school for more than a half century, announced his retirement on Friday, after the release of documents that showed he had a closer relationship with Jeffrey Epstein than previously known.

The president, Leon Botstein, was known for his fund-raising prowess and outsize personality, but came under scrutiny after the release of a trove of documents collected by the Justice Department related to Mr. Epstein. The files showed Dr. Botstein had exchanged messages and visits with Mr. Epstein for years, including after Mr. Epstein’s conviction on solicitation of a minor for prostitution.

In one 2013 note, Dr. Botstein signed off with “Miss you.” He spoke of his cherished “new friendship” with the financier, and wished him well after the publication of news article that detailed his abuse.

The college commissioned an independent review, conducted by the law firm WilmerHale, and the findings were released on Friday.

The review found that Dr. Botstein had done nothing illegal but that his relationship with Mr. Epstein raised concerns about his leadership. The review said that Dr. Botstein had ignored the concerns of a senior faculty member who advised him that Bard should avoid Mr. Epstein.

“President Botstein forcefully argues that Bard’s need for funds was paramount,” the review concluded. “His view was, ‘I would take money from Satan if it permitted me to do God’s work.’”

The review noted that Mr. Botstein had visited Mr. Epstein’s island, invited Mr. Epstein to stay at Bard and to visit a high school affiliated with Bard, and had taken payments from Mr. Epstein. Mr. Botstein said he had in turn funneled those payments to Bard under his own name.

Dr. Botstein has long maintained that his relationship with Mr. Epstein was entirely about coaxing him to give money to the school, which is about 100 miles north of New York City.

Dr. Botstein became president of Bard in 1975, when he was only 28 years old and the college was in dire financial shape. He earned a reputation as a talented fund-raiser, and is credited by his supporters with keeping Bard afloat at a time when many colleges are facing difficulties and some have closed.

In 2021, the billionaire George Soros pledged to donate $500 million to Bard’s endowment, which now tops $1 billion.

The new documents did not show any criminal wrongdoing on Dr. Botstein’s part, but Dr. Botstein is the latest powerful person to leave a top position after their communications with Mr. Epstein were revealed.

Dr. Botstein said in a statement Friday that he believed it was in the “best interest of Bard” to wait until the review was complete before he announced his retirement.

In the statement, he said that he would continue working as a professor and participating in music programs connected to Bard. Since 1992, Dr. Botstein has been the principal conductor and music director of the American Symphony Orchestra. In his statement, he said he would also live at Finberg House, an on-campus residence hall.

Billing itself as “a private college in the public interest,” Bard has long prided itself on bucking the conventions of higher education. The college doubled down on its bohemian sensibility under Dr. Botstein’s leadership.

The out-of-the-box thinking extended to college admissions. Bard applicants, for example, can skip the traditional process and instead submit three lengthy essays.

And Dr. Botstein has lampooned the U.S. News and World Report rankings, which many college leaders swear by.

He has also been a strong advocate for early college, creating some of the first programs that allow teenagers, often from underrepresented backgrounds, to earn college credit tuition-free while still in high school.

But when it came to another convention of modern higher education — the need to raise private money — Bard embraced the practice. Dr. Botstein said he hated raising money from the wealthy, describing it as a humiliating experience.

Still, in his statement Friday, he said that the college under his watch had secured nearly $3 billion in philanthropy. He said he would stay as president until the end of this academic year, June 30.

It was fund-raising that brought Dr. Botstein into contact with Mr. Epstein. Dr. Botstein has said that the relationship with Mr. Epstein began with a small, unsolicited donation by the sex offender in 2011. “A guy sent us money and we followed up,” Dr. Botstein told The New York Times in 2023. “It’s a simple story.”

But after the latest release of documents, Dr. Botstein’s explanations for various interactions with Mr. Epstein often left community members with even more questions.

For instance, after the documents showed that Dr. Botstein’s office had planned a trip to Mr. Epstein’s island in 2012, Dr. Botstein said he had become sick during the trip and wasn’t sure whether he actually stayed on the island. When The Times reported an email from Dr. Botstein from the day after that 2012 trip, in which the president thanked Mr. Epstein and wrote “the place is great,” Dr. Botstein, through a spokesman, said he was referring to “the overall environment of St. Thomas.”

The WilmerHale report said Dr. Botstein was not “fully accurate” in describing his relationship with Mr. Epstein in public statements.

The documents also showed that the two had worked together to buy an expensive watch. Dr. Botstein, a watch collector, explained that he was helping Mr. Epstein, who had expressed interest in a watch, buy one.

Dr. Botstein kept the timepiece for about a year before Mr. Epstein demanded Dr. Botstein return it or begin making payments to cover the $56,000 cost.

In one email, Mr. Epstein even excoriated Dr. Botstein, describing his purchase of the watch as “careless.”

The initial response to the news was subdued on the Bard campus, and it appeared many, including board members, were willing to stand by a leader viewed by some as central to the college’s success over the decades. But the pressure mounted after a slow drip of news coverage.

This spring, the board of trustees, headed by the billionaire James Cox Chambers, announced it had hired WilmerHale to investigate Mr. Epstein’s relationship with the president and Bard.

The faculty senate eventually weighed in on the matter, urging trustees to “plan for a transition in leadership.” The faculty statement also called for envisioning a Bard after the man who had led it for more than half a century.

The board on Friday thanked Dr. Botstein for “his countless accomplishments and the lasting impact of his leadership.” It said it will soon announce an interim leader and the details of a national search for the next president.