Since being hospitalized with acute pneumonia for two weeks, Pope Francis has signed off on the appointment of 20 bishops, accepted the resignations of at least four and approved moving five people along the road to sainthood, according to the daily bulletin on the Vatican website.

Some of the events noted were more esoteric, like his appointing a cardinal to represent him in April at the 1,000th anniversary celebration of the coronation of Boleslaw I the Brave, the first king of Poland.

Overall, the public tenor of the Holy See has changed absent its most public face and voice — lacking the weekly appearances of the pope and his pronouncements on important global issues, like rebuking the Trump administration over immigrant deportations. Yet the machinery of the Vatican state churns on even while the 88-year-old pontiff lies in a hospital with no discharge on the calendar.

“Francis is still governing, he is in contact with his aides,” Matteo Bruni, a Vatican spokesman, said in response to a question about the pope’s work. “What he is missing is the contact with the faithful, the general audiences, the pastoral activity.”

In a statement on Thursday evening, the Vatican said that the pope’s clinical condition continued to improve. He was alternating between using a high flow of oxygen and a ventilation mask, it said, and had spent the morning doing respiratory physiotherapy and resting, before an afternoon round of physiotherapy.

“In view of the complexity of the clinical picture, further days of clinical stability are required to determine the prognosis,” the Vatican statement said. The Vatican press office added that the pope was no longer considered in critical condition, but was still not out of danger.

Yet the whole ordeal of his extended stay at Gemelli hospital, the longest of his papacy, with no public appearances, has awakened old concerns among church experts and observers about who would govern the church if this or any pope were to fall into a coma or otherwise be incapacitated.

To a certain extent, the Holy See operates much like a political system with a president and a prime minister. The pope, the president in this scenario, deals with major policy pronouncements and appointments, while the secretary of state, Cardinal Pietro Parolin, oversees day-to-day matters involved in running the Vatican itself and the global church.

The powers limited to the pope alone include appointing bishops, accepting retirements and issuing new decrees to change Vatican law. In addition, the bureaucracy would be unlikely to make policy pronouncements on issues of global import, like cease-fire negotiations in Gaza, without his imprimatur.

Inevitably, Francis’ illness postpones some decisions, which critics of his agenda do not mind. Before he fell ill, for example, Francis dissolved the Sodality of Christian Life, a controversial Catholic organization founded in Peru. Some of the group’s supporters had hoped his illness might slow the implementation of the decision, said John Allen, the editor of Crux, an independent online news site covering the Catholic Church. But Vatican officials and experts alike said there has been no particular slowdown during this hospitalization because the pope, while ailing, remains alert and able to engage with those around him.

Business rolls along as usual “as long as the pope can understand what people are saying to him and can sign his name,” said the Rev. Thomas J. Reese, a longtime Vatican analyst and the author of the book “Inside the Vatican: The Politics and Organization of the Catholic Church.”

But there are no established legal guidelines for what to do when that is not the case, especially if any incapacity endured for months rather than weeks. In the case of a brief hospitalization, a lot of those actions already in the pipeline could continue without a hitch.

But major decisions, like whom to appoint as archbishop in a high-profile city, would be problematic. It would also be extremely difficult to remove an errant cardinal or other prominent figure, experts said.

Pope Francis, like Paul VI in the 1960s, announced that soon after assuming the papacy in 2013, he signed a resignation letter in case he was ever incapacitated. But its contents have remained secret, and experts point to a number of legally murky issues.

First, who gets to decide that the pope is incapacitated? And second, church law states that for any resignation to be valid the pope must have freely chosen to make the decision. Some legal experts wonder whether a letter written more than a decade ago would pass muster as a “freely chosen” decision.

In recent history, the question was most acute during the long decline of Pope John Paul II, who began suffering from Parkinson’s disease years before his death in 2005. Although he could still answer “Yes” or “No,” his evident decline fed questions in the Vatican rumor mill about who was really making decisions, said Father Reese.

A far more dire consequence that some of John Paul’s defenders attribute to his slowed facilities was the lack of a response from the church to widespread accusations of child sexual abuse by Catholic clergy.

“The church has paid a significant price for that in the 20 years since then,” said Dr. Miles Pattenden, a history professor and the author of “The Cambridge History of the Papacy.”

The problem of ailing popes is not exactly a new one, but is far more complicated in the age of social media, instant communication and greater longevity. “In the bad old days, the doctors would kill him quickly through bleeding and terrible medicine,” said Father Reese, or “just lock him in a back room and run the church.”

Before 1700, there was an established precedent of a “cardinal nephew,” a close relative of an ailing pope, running matters in his name. “An old or elderly pope who was infirm would stay in bed and his nephew would run the government,” said Dr. Pattenden, and while the practice was still extant the nephew would often become pope himself when his older relative passed away.

Church leaders have long been reluctant to address the issue of defining when a pope might be considered incapacitated because it impedes the theory that a pope must have complete freedom to act, said Mr. Allen. “Nobody can tell a pope when it is over,” he said. “It is extraordinarily delicate to navigate how you get to the end of a papacy without the pope’s express will.”

The hurdles are evident in specific provisions like Canon 335 of Vatican law, which states that if the Holy See is “vacant or entirely impeded” then nothing can be altered in the governance of the church. But “entirely impeded” is not defined.

Francis’s predecessor, Benedict XVI, resigned at 85, citing age and infirmity, but he was an extreme outlier — no pope had stepped down in almost 600 years.

Given the leaps in modern medicine to prolong life, an international group of Catholic legal experts has proposed norms to flesh out what constituted “entirely impeded,” and had called for the transfer of governing authority to the entire College of Cardinals or a commission, depending on how long the pope was incapacitated. If a committee of experts deemed the pope’s incapacity irreversible, cardinals in the Curia would have to declare his governing days over and call a conclave to name a new pope.

But those were only proposals and to date no established law has seen the light of day.

With the pope’s condition slowly improving, there appears to be less urgency, but depending on how long his hospital stay lasts, it still could have an effect.

The streets of Vatican City and the churches of Rome are full of pilgrims here for the church’s Jubilee Year, but their hopes of seeing Pope Francis have been dashed while he remains in the hospital.

“The big reason that most people would come to Rome during a Jubilee Year is to see the pope,” Mr. Allen said. “If you cannot see the pope, that puts a damper on things.”

While the Trump administration in Washington was cutting environmental programs, delegates at U.N. biodiversity talks in Rome made modest progress Thursday on a series of measures to support nature.

Governments gathered to tackle global biodiversity losses that are unprecedented in human history, driven by the ways people have transformed the world.

The seismic geopolitical changes of recent weeks loomed over the talks as countries negotiated in a large conference room, fighting for small steps toward consensus. Delegates painstakingly negotiated the language of diplomatic texts even as Britain announced reductions to its overseas development aid and as the United States continued cutting its international aid programs.

“We have sent a light of hope,” said Susana Muhamad, Colombia’s departing environment minister, who presided over the meeting. “The common good — the environment, the protection of life and the capacity to come together for something bigger than each national interest — is possible.”

Many developing countries are rich in biodiversity but poor economically, and three days of tense negotiations centered on whether a new fund would be part of a plan to mobilize $200 billion a year for nature funding by 2030.

African and Latin American countries have demanded a new fund, arguing that the way they currently gain access to multilateral money is unfair and inefficient. But many donor countries have fought their proposed fund, saying it would be expensive to set up and manage, diverting money that could otherwise be spent on conservation itself. In the end, delegates agreed on a process to decide whether a new fund would be created. Still, it was a hard-won compromise and the room erupted in applause.

Delegates also approved a framework for monitoring nations’ progress on biodiversity commitments made in Montreal in 2022, which included an agreement to conserve 30 percent of the world’s land and water.

“We now have a road map to secure the finances required to avert the biodiversity crisis, and the means to monitor and review progress,” said Martin Harper, chief executive of BirdLife International, a science and advocacy group. “These crucial steps must now be backed up with real money from developed nations.”

Countries have recognized a biodiversity financing gap of $700 billion a year. In a landmark agreement in 2022, they agreed to mobilize at least $200 billion a year by 2030 from public and private sources and to find an additional $500 billion a year by 2030 by phasing out or reforming subsidies that harm nature. It’s a huge amount of money to find in five years, even in the most favorable political climate.

The talks unfolded with one country conspicuously absent: the United States.

“I can’t remember the last time the U.S. didn’t show up, but it’s been a very, very long time,” said David Ainsworth, a spokesman for the secretariat that manages the United Nations biodiversity treaty that underlies the talks, the Convention on Biological Diversity.

The United States has long held an awkward but nonetheless influential role in global biodiversity negotiations. It is the only country in the world, with the exception of the Vatican, that has not ratified the treaty. Still, the United States has long wielded considerable influence from the sidelines of the talks.

Now, all that has been thrown into question. In recent years, at least $385 million of U.S. biodiversity funding was funneled through the Agency for International Development, which is being dismantled by the Trump administration. Other streams of U.S. biodiversity funding are also at risk.

The White House did not answer questions on its plans for biodiversity funding or why it did not send delegates to the talks.

Monica Medina, who served as biodiversity envoy in the Biden administration, called the American absence in Rome “a deafening silence” and said cuts to biodiversity funding could result in devastating extinctions.

“U.S. funding has been a very important part of how we have kept some of the biodiversity we all love — elephants, whales, rhinos and polar bears — from going extinct,” Ms. Medina said. “We might not be able to keep some of these amazing animals around for our kids and grandkids without some of this funding.”

The meeting in Rome was a resumption of talks held last fall in Cali, Colombia, officially called the 16th Conference of the Parties to the Convention on Biological Diversity, or COP16. After reaching a groundbreaking agreement in overtime on a new way for companies to compensate countries for their use of genetic material, the talks lost quorum and were suspended.

One promising outcome of the Rome talks, according to stakeholders, was a move to begin an international dialogue of environment and finance ministers from developed and developing countries.

Over the course of the negotiations, some delegates made impassioned pleas for nature.

“Biodiversity cannot wait for a bureaucratic process that lasts forever while the environmental crisis continues to get worse,” a government delegate from Bolivia told the gathered nations on Wednesday. “Forests are burning, rivers are in agony and animals are disappearing.”