All of the facilities at the Barra Olympic Park except for the aquatics center are scheduled to be completed by year’s end and everything in the smaller Deodoro cluster is expected to be up and running by next spring. “We are literally making a miracle happen here,” Paes declared, stating that all construction was on schedule.

From the moment Rio was tapped as the first South American city to host the Games, which was both an historic and risky choice, skeptics predicted that it would take just such a miracle for the organizers to meet the fast-forward timeline given the lengthy list of infrastructure requirements that amounted to an ambitious urban renewal.

Although Rio did have 18 existing sports facilities, including massive Maracana Stadium for the opening ceremonies and Joao Havelange Stadium for track and field, the city had to build the better part of two Olympic complexes from scratch. It also had to upgrade its airport, renovate its waterfront, construct multiple roads, modernize and expand its public transit system, and purify a polluted bay that 12 million cariocas use as their toilet.

That malodorous but vital cleanup task, which government officials acknowledge won’t be done in time for the Games, attracted worldwide attention on the eve of this month’s venue dress rehearsals in rowing, sailing, and open-water swimming.

An independent study of water quality at the Olympic venues commissioned by the Associated Press recently concluded that athletes would be exposed to dangerously high levels of viruses and bacteria from raw human sewage that in some cases were 1.7 million times higher than what would be considered hazardous on a Southern California beach. “Everybody runs the risk of infection in these polluted waters,” local hepatologist Carlos Terra said.

Rio has been trying to do in seven years what in most cities takes decades at a time when the cash-strapped country also was spending nearly $4 billion building and renovating stadiums for last summer’s soccer World Cup. When the International Olympic Committee evaluation commission did its technical study before the 2016 vote, Rio was listed fifth behind Doha, the Qatar capital that didn’t make the cut.

“Rio did not win because it had the best infrastructure,” Paes observed. “It had bad conditions, bad public transportation, inequality, traffic jams, landslides, and all the other problems cities in Brazil have. Tokyo, Chicago, London, and others were all on another level of development and you cannot compare them with Rio. Rio is Rio. You must compare Rio now and Rio post-Games with Rio in 2009.”

The gold standard for an Olympic host city renaissance is Barcelona. Rio’s cultural seaside cousin remade itself for the 1992 Olympics, which catapulted the Catalan capital into Europe’s hot new city.

Similar five-ringed fever was seen as a potential catalyst to transform not only Rio but the entire nation. “Today Brazil was upgraded from a second-class country to a first-class country,” then-president Luiz Inacio Lula da Silva proclaimed after Rio had prevailed by a 66-32 count on the third ballot.

The IOC, which had gone to England and China for its previous two summer sites, wasn’t expecting a rapid urban makeover in a tropical country. A more realistic model was Greece, which took years to get preparations for the Athens Games into high gear.

Like the country’s traditional syrtaki dance, preparations began slowly and climaxed in a whirlwind, and Athens ended up producing marvelous Games after all. “Dear Greek friends, you have won,” then-IOC president Jacques Rogge declared at the Closing Ceremony.

So the IOC was prepared for a samba rhythm with Rio, which had staged a successful Pan American Games in 2007. But the two host countries are magnitudes apart in terms of size and complexity. Brazil is more than 60 times larger than Greece with nearly 20 times as many people and an unwieldy political structure.

After nearly five years of debating and dithering, the federal, state, and local governments still weren’t on the same page with the Rio organizers. “At some point . . . they have to decide who is doing what,” said Gilbert Felli, the IOC’s executive director for the Games.

The international sports federations were so alarmed by the somnolent pace of progress on the venues that some began pushing the IOC last year for a Plan B that either would use leftover Pan Am facilities and shift some events to Sao Paulo or move the Olympics to a different country.

“They have many words but not money, and words are not enough,” said Francesco Ricci Bitti, president of the International Tennis Federation. “They are delaying, delaying, and delaying.”

Once the IOC took a more direct hand, the venues began taking shape more rapidly. As of last week, the Barra park that will be at the center of the Games was said by officials to be 82 percent complete, and the man-made golf course, the most controversial of all the venues, is expected to be ready in time for November’s test event.

“We are continuing to work because there is still much to do,” said Brazilian president Dilma Rousseff, who proclaimed that Rio will be “doubtlessly the most beautiful natural landscape for the Olympic Games since ancient Greece.”

With a year to go it’s Brazil or bust, and with the countdown clock ticking and the jackhammers still pounding, the Lords of the Rings are hoping that flying down to Rio won’t be a febrile folly. “We were confident then and we are confident now,” said IOC president Thomas Bach, who played beach soccer when he arrived last week, “that these Olympic Games will amaze the world.”

Material from wire services was used in this report. John Powers can be reached at jpowers@globe.com.