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Image copyright AFP Image caption Chinese President Xi Jinping had dinner with the Obamas during their state visit to the US "The United States welcomes the rise of a China that is peaceful, stable, prosperous and a responsible player in global affairs."
So said President Obama on Friday, standing alongside President Xi at the White House.
This line didn't make the news. After all, Mr Obama repeats it whenever he meets his Chinese counterpart. But the qualified American welcome to the rise of China is still the principle which underpins the most important bilateral relationship in the world, and it is the principle which has guided American policy on China through eight presidencies and four decades.
Never before in history has a great power risen so fast, and in so many different spheres, as the China we see today. In three short decades, the backwards farming nation of both presidents' childhoods has become the world's largest manufacturer and largest trader. It may overtake the US during President Xi's time in office to become the world's largest economy.
China's trade with the US alone has risen from $2bn in 1979, when relations were established, to nearly $600bn last year.
The relationship is broad and deep, marked by intense co-operation but also intense competition. Both presidents are conscious of the dangers of the former giving way to the latter. The Thucydides trap
In his speech in Seattle, President Xi Jinping tackled the threat of the two nations' strategic rivalry head on:
"There is no such thing as the so-called Thucydides trap in the world. But should major countries time and again make the mistakes of strategic miscalculation, they might create such traps for themselves."
Thucydides trap? What was a 2,000-year-old Greek historian doing in a speech otherwise peppered with references to American writers and American culture?
The trap named after him describes the structural stress caused when a rapid rise - like China's - changes the balance of power with an established rival that escalates to war. The rising power wants a bigger say in how its world is managed and the existing power holds onto the status quo.
Thucydides identified it as the trend which led an up-and-coming Athens to war with Sparta in the 5th Century BC. It appeared in Mr Xi's speech nearly 2,500 years later because some strategic thinkers believe that what was true of Athens and Sparta is true of China and the US today.
In some ways the dangers are arguably greater today. Sparta and Athens were at least close in language and culture. Britain and Germany before the First World War also shared many values, and contemporaries could see that strategic competition was in the interests of neither and that war would be catastrophic. But that didn't mean they could prevent it. Common ground
So as President Obama and President Xi performed the rituals of the visit, complete with 21-gun salute and state dinner, what both sides can agree on is the urgency of avoiding the Thucydides trap. Sustaining the US welcome for a rising China requires them to find and strengthen areas of co-operation.
Hence announcements on climate change, with President Obama welcoming President Xi's commitment to a carbon trading programme. They also celebrated co-operation on Iran's nuclear programme and said they would continue to work together to tackle the North Korean nuclear problem. China signed off on a multi-billion dollar deal to buy Boeing aircraft, making the point that there are concrete benefits in American jobs and profit from good relations with China.
President Xi worked hard to keep the tone positive. Chinese good manners dictate that on great ceremonial occasions, the visitor should avoid unpleasantness and find nice things to say about the host. So he reeled off a list of his favourite American authors and movies. But when President Xi goes to Russia he loves Pushkin and Tolstoy, when he goes to France it's Moliere and Dumas.
Beneath the feel-good flattery, the unspoken message was one of inexorable Chinese power. President Obama did manage to extract an agreement that both governments would refrain from cyber-theft of intellectual property for commercial gain, and that they would work on "international rules of the road for appropriate conduct in cyberspace".
But he also wondered aloud whether words would be followed by actions, and warned that he might still impose sanctions if Chinese hacking persisted. President Xi denies that China has perpetrated any hacking, insisting instead that it is the victim of cyber hacking. On tensions in the South China Sea, they made no progress. Hardnosed diplomacy
Overall, the Obama administration tried to put a brave face on the usefulness of this summit, but it is the third major encounter in three years between the two presidents and the relationship has not lived up to early American hopes. President Obama has invested heavily in trying to build a personal rapport, and on this occasion too they had a two-hour conversation over a private dinner. But one is seven years into his presidency with just a year to go, the other is three years in with seven still ahead.
President Xi's foreign policy is more confident, articulate and focused than that of any of his predecessors. His assertiveness has caught the Obama administration off guard, whether it's the ambitious island building in the South China Sea, a new development bank to challenge US dominance of global financial institutions, or the push to reshape the physical and diplomatic architecture of Asia through China's "One Belt One Road" strategy.
The inescapable conclusion is that President Xi has made a hardnosed calculation that his counterpart is too preoccupied with American problems in the Middle East and Russia to push back against his own muscular version of the way China should rise.
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Image copyright AFP