It has become an article of faith in the New York State Capitol that when Gov. Kathy Hochul enters the Red Room on the building’s second floor to announce a budget agreement, the deal is actually far from sealed.

This year was no different.

Despite declaring that “today is the day” to announce an agreement on a $268 billion state budget, Ms. Hochul on Thursday acknowledged that several key initiatives — including a new tax surcharge on multimillion-dollar second homes in New York City — had been agreed on in principle, but that the details still needed work.

Even the top-line figure had not been finalized.

Lawmakers are fond of saying that the devil is in the details. But in the absence of the lengthy budget bills that include those details, which have yet to be printed and voted on, a host of unanswered questions remain.

Here are five of them:

Why did Hochul announce a deal when one hadn’t really been made yet?

New York’s opaque budget process, which starts in January with the State of the State address and is supposed to be completed by April 1, has become far more than a negotiation over a fiscal document.

Governors have tended to use the budget to wedge in legislative priorities, wielding their leverage over billions of dollars to get their way.

Ms. Hochul has embraced this practice. And, in a re-election year, she wanted to convey to voters that she intended to stand up to President Trump’s immigration crackdown, help out New York City and lower costs for everyday New Yorkers.

She made that case on Thursday at a news conference flanked by several of her top aides. Notably missing were the leaders of the State Assembly and Senate.

When will the budget actually be passed?

Not this week. The Assembly speaker, Carl E. Heastie, said on Thursday that it was “very premature” of the governor to say a deal had been reached. He would not even say that the Legislature had agreed to the $268 billion figure.

He complained about Ms. Hochul’s penchant for jamming nonfiscal policies into the budget and said he would not discuss such matters with his members until he had a better sense of the total amount the state would be spending.

As he spoke, members of the Senate and Assembly, who are currently not being paid, were wrapping up their legislative business for the week in a rush to return to their districts. They will be back in Albany on Monday; it is unclear what bill language, if any, will have been printed and distributed by then.

Did Zohran Mamdani get what he wanted?

Mr. Mamdani, the mayor of New York City, campaigned on wresting more than $10 billion in tax increases from the state to pay for his ambitious agenda. That will not happen this year.

Ms. Hochul did accede to a new tax on second homes that targets the city’s richest property owners whose primary residences are outside New York City. The goal is to raise $500 million each year, which will go toward closing the city’s estimated $5.4 billion budget deficit.

But she spurned the mayor’s request to make changes to a tax credit called the Pass Through Entity Tax that is used by some business owners. Mr. Mamdani had said that the measure, which was also backed by the City Council speaker, Julie Menin, could raise up to $1 billion a year in tax revenue.

Aside from tax increases, Mr. Mamdani’s overarching priority has been expanding child care in the city. Ms. Hochul’s budget does just that, with $4.5 billion allotted for child care and prekindergarten programs across the state.

It’s not the whole loaf, or even half. But Mr. Mamdani can point to that funding and say that he is advancing toward his goal of providing free child care for every New York City child under 5. And while the governor rejected his efforts to fund a program to make buses free, she directed more than $1 billion in additional aid to the city that, combined with revenue from the second-home tax and other proposed measures like delays in pension payments, could help Mr. Mamdani work to close its budget gap.

How will the tax on pieds-à-terre work?

State lawmakers — and just about everyone else — are scratching their heads about the details of this tax surcharge, which Ms. Hochul proposed with great fanfare last month. The New York Times previously reported that one proposal being discussed would apply one tax rate to pieds-à-terre with values between $5 million and $15 million; a higher rate for ones valued between $15 million and $25 million; and an even higher rate for properties valued at $25 million or more, according to three people familiar with the matter.

How much the property owners would pay is still up in the air. Ms. Hochul said on Thursday that more details would be coming in the near future and that the tax would apply to units worth $5 million or more.

Also being sorted out is how, exactly, the value of each co-op or apartment would be determined.

“It’s going to take some time to get to the right number to assess that,” the governor said, noting the city’s complex system for calculating a property’s assessed value.

“We’re looking at the difference between what is currently assessed but what is market value,” she added. “We’re working it out with the city. We have had some really good conversations.”

How will pensions change for state workers?

Facing pressure from the state’s largest public unions, Ms. Hochul has been trying to determine how to restore certain pension benefits that had been cut for public employees hired after 2012.

Any changes could end up costing the state hundreds of millions of dollars, while also saddling local municipalities and school districts with increased spending burdens. Several of the labor groups have prioritized lowering the minimum retirement age to 55 from 63.

Ms. Hochul said on Thursday that the particulars were still being negotiated, but stressed that the cost to the state and local governments would be less than the $1.5 billion that has been requested by the unions.

“We are willing to look at this and make changes, but a much more scaled-back monetary proposal,” she said.

“We will release these numbers as soon as it’s absolutely done,” she added.

Cornell University’s trustees announced on Thursday that they would investigate an April 30 incident in which the president, Michael Kotlikoff, bumped into students with his car after a debate over the conflict between Israelis and Palestinians.

A small group of students confronted Dr. Kotlikoff after he had spoken at the university event about free speech and the Middle East conflict. They posed questions about the suspension of pro-Palestinian student demonstrators in previous years, walking with him to his vehicle, a black Cadillac SUV.

After the students surrounded his vehicle, Dr. Kotlikoff got in, reversed and bumped one student. Another student said that his foot was run over. Emergency medical technicians arrived and checked the foot of that student, who was not seriously injured.

In an email to the campus on Thursday, the university said that an “ad hoc special committee” had been established to oversee the investigation of the events of that night. It said that Dr. Kotlikoff had recused himself from the investigation and “any related university decisions connected to the matter.”

The statement said that the trustees are “committed to ensuring a fair and thorough review guided by adherence to university policies and the best interests of the Cornell community.”

The event highlighted the lingering tensions on college campuses over the Israel-Hamas war, even as the large-scale protests from the spring of 2024 have dissipated.

Some student groups at Cornell have said that the university had unfairly disciplined pro-Palestinian students involved in the demonstrations.

One of the disciplined students was Momodou Taal, a Ph.D. student in Africana studies and a leader of the movement to establish protest encampments on the Ithaca, N.Y., campus. The Trump administration had sought to deport him and students at other universities whom it accused of spreading antisemitism.

Last week’s event, billed as a debate between supporters of the Palestinian cause and Israel, also featured Norman Finkelstein, a political scientist and author who spoke about Israel’s response to the Oct. 7, 2023, Hamas-led attacks on Israel.

The day after last week’s incident, Dr. Kotlikoff framed himself as the victim, asserting that students had harassed him and banged on his car. The students denied doing that, and video of the incident that the students supplied did not show them hitting the vehicle.

With few signs of progress on Mayor Zohran Mamdani’s promise to make New York City’s buses free for all riders, the City Council and some transit advocates are pushing for a more targeted alternative: focusing on the riders who need the help the most, now.

City officials are seeking to revamp an existing and underutilized city program called Fair Fares, which provides half-price subway and bus fares to low-income New Yorkers. Transit riders, advocates and elected officials have argued that in its current form, it excludes many of the roughly one million residents who qualify and need support.

Nodding to those concerns, Julie Menin, the Council speaker, said at a news conference on Wednesday that she supported plans to automatically enroll eligible lower-income New Yorkers into the program, and to make public transit completely free for them.

Fair Fares currently charges New Yorkers who make less than 150 percent of the federal poverty level half price — or $1.50 — to take bus, subway and paratransit rides. Individuals must make less than $23,500 a year to qualify; a family of four needs a household income of about $48,000 or less. Citizenship status is not considered.

Though nearly 380,000 New Yorkers receive the discount, about 575,000 more qualify for the program and do not take part. Supporters of the plans note the application process requires complicated paperwork that they say has stunted enrollment.

The City Council’s push comes as a number of transit advocacy groups have raised concerns about Mr. Mamdani’s free bus plan, which has been projected to cost nearly $1 billion a year, at a time when the city is facing a $5.4 billion budget deficit.

The Council proposals, in contrast, would be more targeted and cost less. The Fair Fares program, which began in 2019, currently costs the city less than $100 million a year. The proposals could more than double that price tag.

Mr. Mamdani’s office did not respond to requests for comment about the plans.

The mayor angered some advocates this year when he did not include funds in his preliminary budget to expand the Fair Fares program.

He has pitched free bus service as a public good, similar to libraries, that would benefit all New Yorkers, regardless of financial need.

Making buses free is key to Mr. Mamdani’s goal of making the fleet, one of the slowest in the nation, faster and more reliable. As an assemblyman, Mr. Mamdani championed a 2023 pilot program that made a select number of bus lines free. (Ridership increased, but the speed of service did not.)

Still, Mr. Mamdani’s plan faces challenges from Gov. Kathy Hochul, who could be vital to its funding. She has been unwilling to raise taxes on the rich at the scale proposed by Mr. Mamdani. And the Metropolitan Transportation Authority, the state agency that controls the city’s buses, has distanced itself from the plan, at a time when it is trying to crack down on fare evasion.

The lack of movement on the free bus plan has opened the door for more specific proposals from the City Council.

Ms. Menin, speaking at the news conference on Wednesday while surrounded by transit advocates on the steps of City Hall, said there was an urgent need to expand the program, amid a growing affordability crisis.

“People are literally forgoing meals because of this cost,” she said. “It is shameful.”

About one in five New Yorkers struggle to pay for public transit, a burden disproportionately borne by Black and Latino commuters and working mothers, according to a 2024 report from the Community Service Society of New York, an economic justice group.

Several transit advocacy groups are seeking more changes to the program.

The Citizens Budget Commission, a fiscal watchdog group that is skeptical of Mr. Mamdani’s free bus plan, said it instead supported raising the income limit for Fair Fares to 250 percent of the federal poverty level. That would include individuals making up to $39,900 a year, or a four-person household making $82,500.

At that threshold, two million New Yorkers would be eligible for the half-fare discount, and it would cover one in four working adults in the city, the group said.

Others want to make the program even more inclusive.

The Community Service Society has called for raising the income threshold to 300 percent of the poverty line, or about $99,000 for a family of four.

“It’s not capturing the real need,” said Rachel Swaner, a vice president at the organization, noting that there are many families who make marginally more than the current income limit. “They make too much to be eligible for public benefits, but they are really struggling to make ends meet.”

At the news conference on Wednesday, supporters of the Council’s proposals said the need for immediate economic relief was urgent.