WASHINGTON (AP) — For some time now, President Donald Trump has been encouraging people to think of Mexico as a portal for international terrorists who "pour" into the U.S. Except, he says, for 10 who were recently caught by the U.S.: "These are very serious people."

These 10 do not exist, except as a federal statistic that Trump and his vice president put through a rhetorical grinder in service of describing emigrants from Mexico as a menace.

There is, in fact, genuine concern about the potential for Islamic extremists to make their way across the border into the U.S. But that concern applies more to the Canadian border, where Trump is not planning to put up a wall.

Here is how the myth of the "very serious" 10 developed over the months, culminating this week in Trump's assertion that "we caught 10 terrorists. These are over the last very short period of time — 10."

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In July 2017, a State Department report on terrorism comes out, breaking down perceived threats by country, and it does not fit into Trump's story about danger from the south.

It says: "There are no known international terrorist organizations operating in Mexico, no evidence that any terrorist group has targeted U.S. citizens in Mexican territory, and no credible information that any member of a terrorist group has traveled through Mexico to gain access to the United States."

At most, it notes that "Mexican government officials observed on social media an increase in terrorist group sympathizers in its territory over the previous year."

That report dwells much more on the northern neighbor, home to "Canada-based violent extremists inspired by terrorist groups such as ISIS and al-Qaida and their affiliates and adherents." Moreover, Canada has experienced episodes of terrorism at the hands of sympathizers of those organizations. And, the report says, Canada faces a "significant challenge in prosecuting individuals who have traveled abroad to engage in terrorism, due to the difficulty in proving association with terrorist organizations or having committed specific terrorist acts."

Yet the State Department credits both Canada and Mexico with cooperating with the U.S. on terrorism and strengthening protections. And no evidence has emerged that terrorists are pouring in from Canada, either.

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In January, a joint report by the Homeland Security and Justice departments states that Homeland Security had 2,554 "encounters" worldwide with people on a terrorist watch list who were trying to travel to the U.S. That breaks down to an average of seven per day, and is the seed of what becomes Trump's claim about the "very serious" 10.

The vast majority, 2,170, were trying to come by air, with 335 by land and the rest by sea. Nothing ties them specifically to Mexico.

Vice President Mike Pence misrepresents the finding the next month, attributing all of those encounters to the Mexico border: "I learned yesterday at the Hidalgo border center that along the southern border of the United States, we actually still apprehend 1,100 individuals a day, who are attempting to enter this country illegally, including seven individuals a day who are either known or suspected terrorists."

There's more wrong with his statement than that.

The people encountered by Homeland Security were denied entry to the U.S., not apprehended en masse as Pence says. In some cases law enforcement authorities were notified for "appropriate action," says the report, giving no detail on arrests or any charges.

As well, it may be a stretch to characterize everyone on the watch list as a "known or suspected" terrorist. The list is an initial trip-wire identifying people who, rightly or wrongly, raise some concern. The standard for placing someone on the list is reasonable suspicion, a lower bar than the probable cause needed to arrest someone for an alleged crime.

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In June, the average of seven per day grows to 10, without new statistics being announced. "On average, my department now blocks 10 known or suspected terrorists a day from traveling to or attempting to enter the United States," says Homeland Security Secretary Kirstjen Nielsen.

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In September, the next annual State Department terrorism report is released, finding "no credible evidence indicating that international terrorist groups have established bases in Mexico, worked with Mexican drug cartels or sent operatives via Mexico into the United States. The U.S. southern border remains vulnerable to potential terrorist transit, although terrorist groups likely seek other means of trying to enter the United States." More episodes of terrorism are noted in Canada.

Late in the month, with the Trump administration signaling concern about Central American migrants making their way north, and Trump continually prodding for money to build his wall, the statistic is brought out, and misshapen, again.

"In the last fiscal year, we apprehended more than 10 terrorists or suspected terrorists per day at our southern border from countries that are referred to in the lexicon as other than Mexico," Pence says. "That means from the Middle East region."

The average of 10 a day has grown again, to more than 10. And Pence again says this all happened at the Mexico border.

This time, when Pence's office is asked about the claim, his people acknowledge the error. Later the same day, the vice president correctly attributes the number to all points of entry into the U.S.

Trump, though, continues to drum up fear about Muslim extremism coming via Mexico and migrants, tweeting Oct. 22: "Sadly, it looks like Mexico's Police and Military are unable to stop the Caravan heading to the Southern Border of the United States. Criminals and unknown Middle Easterners are mixed in. I have alerted Border Patrol and Military that this is a National Emergy. Must change laws!"

The next day he tells reporters in the Oval Office: "I spoke with Border Patrol this morning. ... They say that over the course of the year, over the course of many years, they have intercepted good and bad people, including people from the Middle East."

Yet he acknowledges, "There's no proof of anything."

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This month, Trump's proposed southern wall becomes the prickly centerpiece of a budget struggle that threatens to idle parts of the government starting Dec. 21 barring a deal.

In his contentious public meeting with the Democratic leaders, Rep. Nancy Pelosi and Sen. Chuck Schumer, Trump makes a case for varied menaces from Mexico, describing those trying to get into the U.S. as carriers of disease — an assertion he did not support — and criminals.

He renders the average of 10 watch-list travelers per day encountered by Homeland Security last year as "10 terrorists," never mind suspects, recently "caught" trying to get in.

The White House's support for that assertion? The same report that got the ball weirdly rolling almost a year ago: a daily average of encounters in 2017, not 10 terrorists in the flesh, very serious or not.

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Associated Press writers Jill Colvin and Colleen Long contributed to this report.