I cannot believe that I am about to write the following sentence, but here goes: Donald Trump, the frontrunner for the Republican nomination for the presidency, is accusing Ted Cruz's father, Rafael Cruz, of being connected to John F. Kennedy's assassination.

"[Ted Cruz's] father was with Lee Harvey Oswald prior to Oswald's being — you know, shot," Trump told Fox News on Tuesday. "What is this, right prior to his being shot, and nobody even brings it up. They don't even talk about that. That was reported, and nobody talks about it."

"Reported" is a strong word for the sourcing behind this information. It appears to come from a National Enquirer article that's almost comically shoddy in its methods. The entirety of the evidence is that there's a picture of a guy standing near Lee Harvey Oswald handing out pro-Castro pamphlets and that guy looks kind of like Rafael Cruz.

"There's more similarity than dissimilarity," Mitch Goldstone, president and CEO of ScanMyPhotos, told the National Enquirer. "It looks to be the same person and I can say as much with a high degree of confidence."

This is, needless to say, a rather thin foundation for this charge. As the Miami Herald archly noted, "The explosive suggestion that Cruz's father would have had any affiliation with Oswald is not corroborated in any other way."

The hoax watchers at Snopes agree. "The sum total of the evidence presented in support of this claim is that some grainy pictures of a man purportedly photographed handing out 'Hands off Cuba!' pamphlets with Lee Harvey Oswald in New Orleans during the summer of 1963 bear a passing resemblance to a picture of Rafael Cruz, Sr. taken nine years earlier," they wrote. That didn't stop the allegation from being picked up in the odder corners of the internet, like InfoWars.com.

To understand this bizarre vignette in a bizarre campaign, it's worth going back to the March 13 edition of Meet the Press, where moderator Chuck Todd pressed Donald Trump on falling for a different internet hoax.

In this case, the hoax was that an Ohio protestor who rushed Trump's stage over the weekend was tied to ISIS. Trump wasn't just making this up — he was sent video of the protestor, Thomas DiMassimo, standing in front of the ISIS flag while Arabic music plays in the background. But the clip, which Trump retweeted, was doctored footage.

Trump was unrepentant. "All I know is what's on the internet," he shrugged.

The comment launched much mockery. But it's actually both more damning and more important than it seems. There's plenty of good information on the internet. Trump has a repeated habit of choosing bad information, both on and offline.

His tendency to solicit, repeat, and retweet self-serving falsehoods served up by sycophants and hangers-on should be taken seriously. Among the most important tasks the president has is knowing what to believe, whom to listen to, which facts to trust, and which theories to explore. Trump's terrible judgment in this regard is one of the many reasons he's not qualified for the office.

Trump's record here also undermines the strongest argument for his candidacy: that his showman's persona is just a front, and at heart he's a calm, thoughtful, coolheaded businessman who will surround himself with the best people and govern in a pragmatic, results-oriented fashion.

If you want to see just how obviously false that argument for Trump is, start by looking at his doctor's note.

"His physical strength and stamina are extraordinary"

It's tradition for presidential candidates to release a note from their physician testifying to their fitness to fulfill the duties of the presidency. On December 14, Donald Trump submitted his entry to this quadrennial custom. It is, I think, one of the most revealing documents of the campaign.

The letter, supposedly written by Dr. Harold Bornstein of Lenox Hill Hospital in Manhattan, is four paragraphs long and defies parody.

"If elected, Mr. Trump, I can state unequivocally, will be the healthiest individual ever elected to the presidency," Bornstein writes. "His blood pressure, 110/65, and laboratory test results were astonishingly excellent. ... His physical strength and stamina are extraordinary."

This is ... not how most doctor notes read. "Reached for comment regarding this, a spokesperson at the American Medical Association just giggled," reported the Daily Beast.

The letter sounds as if it were written by Trump himself, and perhaps it was. No matter who wrote the final draft, though, the underlying message is clear: Trump has entrusted his health to a sycophant who will say whatever Trump tells him to say however Trump tells him to say it.

There are many positions where one might accept a pliable crony. But "personal physician" should not be one of them. The fact that Trump would entrust his health to a doctor who would sign off on a note like this should terrify his family and friends. But more than that, it should disqualify him from the presidency.

One of the dangers of the presidency is that it's easy for anyone who controls nuclear weapons to insulate himself from hard truths and unpleasant critiques. But good decisions require difficult conversations. Presidents often have to hear things they don't want to hear — that an idea isn't good, that a strategy has become unworkable, that a policy doesn't add up, that a trusted subordinate is underperforming, that a strategy won't survive public or judicial scrutiny.

A good president needs to surround himself with people willing to stand up to him, people who aren't cowed by the trappings of the office. But Trump didn't even choose a personal physician unintimidated by the trappings of his wealth. A good president needs people around him who will say things that might make him angry. But Trump has managed to surround himself with people who will say the exact things he wants said, in the exact way he would say them.

Of course, as is often the case with Trump, it gets worse.

"I retweeted somebody that was supposedly an expert. Am I gonna check every statistic?"

On January 22, Donald Trump retweeted a dumb Jeb Bush joke from an account called WhiteGenocideTM.

As you might expect, WhiteGenocideTM's racial politics are not the sort of thing that major presidential candidates typically associate themselves with. The account had, among other trenchant observations, tweeted that "Hitler SAVED Europe."

Yes. Except for the fact that Hitler SAVED Europe. https://t.co/AvtEvJ3OFC — Donald Trumpovitz (@WhiteGenocideTM) January 14, 2016

The resulting furor led Marshall Kirkpatrick to analyze the Twitter accounts Trump had retweeted that week. He found that 62 percent of the accounts Trump boosted that week followed multiple white supremacist Twitter accounts.

Trump's rather nonjudgmental attitude toward retweeting sycophants has led to more than a few missteps.

Sometimes those mistakes are farcical: Gawker created a Twitter bot under the name "Il Duce" that tweeted Benito Mussolini quotes at Donald Trump; sure enough, Trump retweeted it.

Sometimes those missteps are more dangerous. Trump promoted, for instance, a graphic showing that 81 percent of white people who are murdered are murdered by black people; the real number, according to the FBI, is 14 percent. Furthermore, the graphic cited the "Crime Statistics Bureau," an organization that doesn't even exist.

"I retweeted somebody that was supposedly an expert," Trump said when confronted by Bill O'Reilly. "Am I gonna check every statistic?"

Well, yeah, when you're a major presidential candidate with almost 8 million Twitter followers, you are supposed to check every statistic.

Trump was an irresponsible businessman, he's an irresponsible candidate, and he'd be a terrible president

When you ask Donald Trump how he'll achieve any of the absurd things he's promised, his answer is always the same: He'll get the best minds in business — they're good friends of his, you know, great people, really great people — and they'll help him do what these dim-bulb politicians can't.

For instance, when Scott Pelley asked Trump on 60 Minutes how he would make good on his trade policies, Trump said, "I have the smartest people on Wall Street lined up already," as if that served as an answer.

But Trump doesn't have those people lined up. His campaign is not thick with endorsements from the smartest minds of the business world, much less the political or policy worlds.

Nor does Trump's business background suggest he would be able to line those people up. Trump is better at being a celebrity than he is at being a businessman. A number of his enterprises have gone bankrupt, and he would be richer today if he had simply invested his inheritance in the stock market rather than in his various business efforts.

What we've learned about Trump throughout this campaign, though, goes further than that: He has a lousy bullshit detector, he doesn't gravitate toward the smartest people on any given topic, and he doesn't much care about finding the best information.

Worse, verifying the information he receives just isn't a great passion for Trump — he believes the information he wants to believe, and he's not particularly interested in learning that he was wrong. His knowledge of policy has remained thin, and, despite being burned again and again, his credulousness toward online sycophants has persisted.

These are bad traits in a candidate, but they would be disastrous traits in a president. America can't entrust its future to someone who thinks "All I know is what's on the internet" is a sufficient response to repeating falsehoods for the umpteenth time. America can't hand over its nuclear arsenal to someone who will believe any conspiracy theory he's presented with as long as its confirms his priors.

We need to do better.

Watch: Donald Trump's rise is a scary moment in America